Control Allocation- A Survey
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paper_title: Constrained nonlinear control allocation with singularity avoidance using sequential quadratic programming

paper_content:

Control allocation problems can be formulated as optimization problems, where the objective is typically to minimize the use of control effort (or power) subject to actuator rate and position constraints, and other operational constraints. Here we consider the additional objective of singularity avoidance, which is essential to avoid loss of controllability in some applications, leading to a nonconvex nonlinear program. We suggest a sequential quadratic programming approach, solving at each sample a convex quadratic program approximating the nonlinear program. The method is illustrated by simulated maneuvers for a marine vessel equipped with azimuth thrusters. The example indicates reduced power consumption and increased maneuverability as a consequence of the singularity-avoidance.
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paper_title: Constrained quadratic programming techniques for control allocation

paper_content:

The paper considers the objective of optimally specifying redundant actuators under constraints, a problem commonly referred to as control allocation. The problem is posed as a mixed /spl lscr//sub 2/-norm optimization objective and converted to a quadratic programming formulation. The implementation of an interior-point algorithm is presented. Alternative methods including fixed-point and active set methods are used to evaluate the reliability, accuracy and efficiency of the primal-dual interior-point method. While the computational load of the interior-point method is found to be greater for problems of small size, convergence to the optimal solution is also more uniform and predictable. In addition, the properties of the algorithm scale favorably with problem size.
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paper_title: Efficient Optimal Constrained Control Allocation via Multiparametric Programming

paper_content:

Constrained control allocation is studied, and it is shown how an explicit piecewise linear representation of the optimal solution can be computed numerically using multiparametric quadratic programming. Practical benefits of the approach include simple and efficient real-time implementation that permits software verifiability. Furthermore, it is shown how to handle control deficiency, reconfigurability, and flexibility to incorporate, for example, rate constraints. The algorithm is demonstrated on several overactuated aircraft control configurations, and the computational complexity is compared to other explicit approaches from the literature. The applicability of the method is further demonstrated using overactuated marine vessel dynamic position experiments on a scale model in a basin.
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paper_title: Efficient active set algorithms for solving constrained least squares problems in aircraft control allocation

paper_content:

In aircraft control, control allocation can be used to distribute the total control effort among the actuators when the number of actuators exceeds the number of controlled variables. The control allocation problem is often posed as a constrained least squares problem to incorporate the actuator position and rate limits. Most proposed methods for real-time implementation, like the redistributed pseudoinverse method, only deliver approximate, and sometimes unreliable solutions. We investigate the use of classical active set methods for control allocation. We develop active set algorithms that always find the optimal control distribution, and show by simulation that the timing requirements are in the same range as for two previously proposed solvers.
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paper_title: Evaluation of Optimization Methods for Control Allocation

paper_content:

The performanceand computational requirements ofoptimization methodsfor control allocation areevaluated. Two control allocation problems are formulated: a direct allocation method that preserves the directionality of the moment and a mixed optimization method that minimizes the error between the desired and the achieved momentsaswellasthecontroleffort.Theconstrainedoptimizationproblemsaretransformedinto linearprograms so that they can be solved using well-tried linear programming techniques such as the simplex algorithm. A variety of techniques that can be applied for the solution of the control allocation problem in order to accelerate computations are discussed. Performance and computational requirements are evaluated using aircraft models with different numbers of actuators and with different properties. In addition to the two optimization methods, three algorithms with low computational requirements are also implemented for comparison: a redistributed pseudoinverse technique, a quadratic programming algorithm, and a e xed-point method. The major conclusion is that constrained optimization can be performed with computational requirements that fall within an order of magnitude of those of simpler methods. The performance gains of optimization methods, measured in terms of the error between the desired and achieved moments, are found to be small on the average but sometimes signie cant.Avariety ofissuesthataffecttheimplementation ofthevariousalgorithmsin ae ight-controlsystem are discussed.
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paper_title: A Survey of Control Allocation Methods for Ships and Underwater Vehicles

paper_content:

Control allocation problems for marine vessels can be formulated as optimization problems, where the objective typically is to minimize the use of control effort (or power) subject to actuator rate and position constraints, power constraints as well as other operational constraints. In addition, singularity avoidance for vessels with azimuthing thrusters represent a challenging problem since a non-convex nonlinear program must be solved. This is useful to avoid temporarily loss of controllability in some cases. In this paper, a survey of control allocation methods for overactuated vessels are presented.
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paper_title: Lyapunov stability analysis of daisy chain control allocation

paper_content:

A demonstration that feedback control of systems with redundant controls can be reduced to feedback control of systems without redundant controls and control allocation is presented. It is shown that control allocation can introduce unstable zero dynamics into the system, which is important if input/output inversion control techniques are utilized. The daisy chain control allocation technique for systems with redundant groups of controls is also presented. Sufficient conditions are given to ensure that the daisy chain control allocation does not introduce unstable zero dynamics into the system. Aircraft flight control examples are given to demonstrate the derived results.
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paper_title: Resolving actuator redundancy – Optimal control vs . control allocation

paper_content:

This paper considers actuator redundancy management for a class of overactuated nonlinear systems. Two tools for distributing the control effort among a redundant set of actuators are optimal control design and control allocation. In this paper, we investigate the relationship between these two design tools when the performance indexes are quadratic in the control input. We show that for a particular class of nonlinear systems, they give exactly the same design freedom in distributing the control effort among the actuators. Linear quadratic optimal control is contained as a special case. A benefit of using a separate control allocator is that actuator constraints can be considered, which is illustrated with a flight control example.
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paper_title: Control Allocation and Zero Dynamics

paper_content:

Closed-loop stability for dynamic inversion controllers depends on the stability of the zero dynamics. The zero dynamics, however, depend on a generally nonlinear control allocation function that optimally distributes redundant controls. Therefore, closed-loop stability depends on the control allocation function. A sufe cient condition is provided for globally asymptotically stable zero dynamics with a class of admissible nonlinear control allocation functions. It is shown that many common control allocation functions belong to the class of functions that are covered by the aforementioned zero dynamics stability condition. Aircraft e ight control examples are given to demonstrate the utility of the results.
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paper_title: Hierarchical actuator systems

paper_content:

Mechanical actuators are integral components of many engineered systems. Many of the presently available actuator systems lack the desired stroke, power, controllability and reliability. The hierarchical actuator is a natural extension of the trend toward improving the performance of actuators through increments in geometric complexity and control. The hierarchical concept is to build integrated actuators out of a combination of smaller actuators. The smaller actuators are arranged geometrically and controlled so as to extend the performance of the total actuator into ranges that are not possible with actuators that are based on a few active elements and levels of control. Precision, speed increase, force output, load sharing, efficiency under smooth load/displacement control, smooth motion, stroke amplification/reduction and redundancy are all possible. Mechanics and mechanisms of hierarchical actuators are examined, along with a few experiments to demonstrate the operating principles.
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paper_title: DESIGN AND CONTROL OF UNDER-ACTUATED AND OVER-ACTUATED MECHANICAL SYSTEMS - CHALLENGES OF MECHANICS AND MECHATRONICS

paper_content:

The majority of current mechanical systems used in machinery and especially those which are controlled by microprocessors can be described as equal-actuated. This means that the number of actuators (drives, controls) is equal to the number of degrees-of-freedom. The mechanical systems can have directly such property or it can be as such treated during the design and operation. Classical rigid mechanisms can have such property naturally. Generally all flexible mechanisms violate this property as not all flexible degrees-of-freedom can be actuated and thus directly controlled. However, there are important mechanical systems which do not fulfil this criterion at equality of actuators and degrees-of-freedom. The examples of under-actuated systems are bio-mechanical systems during dynamic phase of motion, technical systems of cranes, vehicles, underwater robots, missiles with failed engines, inverted pendulum, and ball on the beam. The examples of over-actuated systems are again biomechanical systems during the contact with ground and recently introduced redundantly actuated parallel robots. The paper first deals with the introduction and definition of the property of under-actuation, equal-actuation and over-actuation. Then the paper deals with the control of under-actuated or over-actuated systems and the challenge of how to design such systems. For the covering abstract see ITRD E125059.
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paper_title: Model predictive control: Theory and practice - A survey

paper_content:

Abstract We refer to Model Predictive Control (MPC) as that family of controllers in which there is a direct use of an explicit and separately identifiable model. Control design methods based on the MPC concept have found wide acceptance in industrial applications and have been studied by academia. The reason for such popularity is the ability of MPC designs to yield high performance control systems capable of operating without expert intervention for long periods of time. In this paper the issues of importance that any control system should address are stated. MPC techniques are then reviewed in the light of these issues in order to point out their advantages in design and implementation. A number of design techniques emanating from MPC, namely Dynamic Matrix Control, Model Algorithmic Control, Inferential Control and Internal Model Control, are put in perspective with respect to each other and the relation to more traditional methods like Linear Quadratic Control is examined. The flexible constraint handling capabilities of MPC are shown to be a significant advantage in the context of the overall operating objectives of the process industries and the 1-, 2-, and ∞-norm formulations of the performance objective are discussed. The application of MPC to non-linear systems is examined and it is shown that its main attractions carry over. Finally, it is explained that though MPC is not inherently more or less robust than classical feedback, it can be adjusted more easily for robustness.
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paper_title: Gas-Lift Allocation under Precedence Constraints: Piecewise-Linear Formulation and K-Covers

paper_content:

Artificial lifting is a costly, but indispensable means to recover oil from high depth reservoirs. Continuous gas-lift works by injecting high pressure gas at the bottom of the production tubing to gasify the oil column, thereby forcing the flow of fluid to surface facilities. The problem consists in deciding which wells should produce and allocating a limited lift-gas rate to the active ones, subject to lower and upper bounds on gas injection, activation precedence constraints, and nonlinearities and discontinuities of the well performance curves. To this end, this paper develops a piecewise linear formulation of the lift-gas allocation problem that allows the application of powerful integer-programming algorithms. More specifically, it analyzes the constraint polyhedron of the piecewise linear formulation and extends cover inequalities of the knapsack polytope to the problem at hand.
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paper_title: A survey of industrial model predictive control technology

paper_content:

This paper provides an overview of commercially available model predictive control (MPC) technology, both linear and nonlinear, based primarily on data provided by MPC vendors. A brief history of industrial MPC technology is presented first, followed by results of our vendor survey of MPC control and identification technology. A general MPC control algorithm is presented, and approaches taken by each vendor for the different aspects of the calculation are described. Identification technology is reviewed to determine similarities and differences between the various approaches. MPC applications performed by each vendor are summarized by application area. The final section presents a vision of the next generation of MPC technology, with an emphasis on potential business and research opportunities. r 2002 Elsevier Science Ltd. All rights reserved.
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paper_title: Predictive Control Allocation for a Thermal Management System Based on an Inner Loop Reference Model—Design, Analysis, and Experimental Results

paper_content:

This paper addresses the challenge of controlling an overactuated engine thermal management system where two actuators, with different dynamic authorities and saturation limits, are used to obtain tight temperature regulation. A modular control strategy is proposed that combines model predictive control allocation (MPCA) with the use of an inner loop reference model. This results in an inner loop controller that closely matches a dynamic specification for input-output performance while addressing actuator dynamics and saturation constraints. This paper presents the design and implementation strategy and illustrates the effectiveness of the proposed solution through real-time simulation and experimental results.
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paper_title: Real-Time Production Optimization of Oil and Gas Production Systems: A Technology Survey

paper_content:

The information flow used for optimization of an offshore oil production plant is described. The elements in this description include data acquisition, data storage, processing facility model updating, well model updating, reservoir model updating, production planning, reservoir planning, and strategic planning. Methods for well allocation, gas lift and gas/water injection optimization and updating of the models are reviewed in relationship with the information flow described. Challenges of real time optimization are discussed.
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paper_title: Optimal thrust allocation for marine vessels

paper_content:

Abstract This paper presents a new thrust-allocation scheme which significantly reduces the fuel consumption for dynamic positioning of ships when rotatable azimuth thrusters are used. Thrust allocation is the problem of determining the thrust and the direction of each of the n thruster devices of a ship, given the desired forces and moment from the control law. The problem of singular configurations is pointed out and solved by using a modification of the singular-value decomposition. A filtering scheme is proposed to control the azimuth directions to reduce the wear and tear, and to minimize the required thrust and power in a general manner, allowing the enabling and disabling of thrusters. Both new theory and full-scale sea trials are presented.
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paper_title: Matrix analysis

paper_content:

Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints.
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paper_title: Nonlinear inversion flight control for a supermaneuverable aircraft

paper_content:

Nonlinear dynamic inversion affords the control system designer a straightforward means of deriving control laws for nonlinear systems. The control inputs are used to cancel unwanted terms in the equations of motion using negative feedback of these terms. In this paper, we discuss the use of nonlinear dynamic inversion in the design of a flight control system for a Supermaneuvera ble aircraft. First, the dynamics to be controlled are separated into fast and slow variables. The fast variables are the three angular rates and the slow variables are the angle of attack, sideslip angle, and bank angle. A dynamic inversion control law is designed for the fast variables using the aerodynamic control surfaces and thrust vectoring control as inputs. Next, dynamic inversion is applied to the control of the slow states using commands for the fast states as inputs. The dynamic inversion system was compared with a more conventional, gain-scheduled system and was shown to yield better performance in terms of lateral acceleration, sideslip, and control deflections.
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paper_title: Closed-form solutions to constrained control allocation problem

paper_content:

This paper describes the results of recent research into the problem of allocating several flight control effectors to generate moments acting on a flight vehicle. The results focus on the use of various generalized inverse solutions and a hybrid solution utilizing daisy chaining. In this analysis, the number of controls is greater than the number of moments being controlled, and the ranges of the controls are constrained to certain limits. The control effectors are assumed to be individually linear in their effects throughout their ranges of motion and independent of one another in their effects. A standard of comparison is developed based on the volume of moments or moment coefficients a given method can yield using admissible control deflections. Details of the calculation of the various volumes are presented. Results are presented for a sample problem involving 10 flight control effectors. The effectivenesses of the various allocation schemes are contrasted during an aggressive roll about the velocity vector at low dynamic pressure. The performance of three specially derived generalized inverses, a daisy-chaining solution, and direct control allocation are compared.
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paper_title: Adaptive control of nonlinear systems: A case study of underwater robotic systems

paper_content:

The problem of controlling underwater mobile robots in 6 degrees of freedom (DOF) is addressed. Underwater mobile robots where the number of thrusters and control surfaces exceeds the number of controllable DOF are considered in detail. Unlike robotic manipulators underwater mobile robots should include a velocity dependent thruster configuration matrix B(q), which modifies the standard manipulator equation to: Mq + C(q)q + g(x) = B(q)u where x = J(x)q. Uncertainties in the thruster configuration matrix due to unmodeled nonlinearities and partly known thruster characteristics are modeled as multiplicative input uncertainty. This article proposes two methods to compensate for the model uncertainties: (1) an adaptive passivity-based control scheme and (2) deriving a hybrid (adaptive and sliding) controller. The hybrid controller combines the adaptive scheme where M, C, and g are estimated on-line with a switching term added to the controller to compensate for uncertainties in the input matrix B. Global stability is ensured by applying Barbalat's Lyapunov-like lemma. The hybrid controller is simulated for the horizontal motion of the Norwegian Experimental Remotely Operated Vehicle (NEROV).
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paper_title: CONTROL ALLOCATION APPROACHES

paper_content:

The control allocation problem is defined in terms of solving or approximately solving a system of linear equations subject to constraints for redundant actuator control variable commands. There is one equation for each controlled axis. The constraints arise from actuation rate and position limits. An axis priority weighting is introduced when the equations cannot be solved exactly because of the constraints. A actuator command preference weighting and preferred values are introduced to uniquely solve the equations when there are more unknowns than equations. Several approaches to this problem are discussed. The approaches are broadly grouped into linear and quadratic programming approaches. While the linear programming approach is well suited to solving the general problem a computationally faster but approximate solution was found with the quadratic programming approach. Example results based on tailless aircraft flight control are presented to illustrate the key aspects of the approaches to control allocation. Conclusions and recommendations are stated relative to the various approaches.
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paper_title: Constrained control allocation

paper_content:

This paper addresses the problem of the allocation of several airplane flight controls to the generation of specified body-axis moments. The number of controls is greater than the number of moments being controlled, and the ranges of the controls are constrained to certain limits. They are assumed to be individually linear in their effect throughout their ranges of motion and independent of one another in their effects. The geometries of the subset of the constrained controls and of its image in moment space are examined. A direct method of allocating these several controls is presented that guarantees the maximum possible moment can be generated within the constraints of the controls. It is shown that no single generalized inverse can yield these maximum moments everywhere without violating some control constraint. A method is presented for the determination of a generalized inverse that satisfies given specifications which are arbitrary but restricted in number. We then pose and solve a minimization problem that yields the generalized inverse that best approximates the exact solutions. The results are illustrated at each step by an example problem involving three controls and two moments.
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paper_title: Constrained control allocation

paper_content:

This paper addresses the problem of the allocation of several airplane flight controls to the generation of specified body-axis moments. The number of controls is greater than the number of moments being controlled, and the ranges of the controls are constrained to certain limits. They are assumed to be individually linear in their effect throughout their ranges of motion and independent of one another in their effects. The geometries of the subset of the constrained controls and of its image in moment space are examined. A direct method of allocating these several controls is presented that guarantees the maximum possible moment can be generated within the constraints of the controls. It is shown that no single generalized inverse can yield these maximum moments everywhere without violating some control constraint. A method is presented for the determination of a generalized inverse that satisfies given specifications which are arbitrary but restricted in number. We then pose and solve a minimization problem that yields the generalized inverse that best approximates the exact solutions. The results are illustrated at each step by an example problem involving three controls and two moments.
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paper_title: Lyapunov stability analysis of daisy chain control allocation

paper_content:

A demonstration that feedback control of systems with redundant controls can be reduced to feedback control of systems without redundant controls and control allocation is presented. It is shown that control allocation can introduce unstable zero dynamics into the system, which is important if input/output inversion control techniques are utilized. The daisy chain control allocation technique for systems with redundant groups of controls is also presented. Sufficient conditions are given to ensure that the daisy chain control allocation does not introduce unstable zero dynamics into the system. Aircraft flight control examples are given to demonstrate the derived results.
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paper_title: Research on allocation efficiency of the redistributed pseudo inverse algorithm

paper_content:

Allocation efficiency is an important performance index to measure the quality of the allocation algorithm. In order to compute the efficiency, the volume of the subset of attainable moments must be solved. The efficiency of the redistributed pseudo inverse (RPI) algorithm depends on the choice of the pseudo-inverse matrix. The subset of attainable moments of RPI is a complex non-convex polyhedron. By analyzing two-dimensional and three-dimensional allocation problems with a “micro-element” method, here we propose an approximate calculation algorithm to compute the volume of the non-convex polyhedron. In order to improve the allocation efficiency of RPI, genetic algorithm is used to find the best pseudo-inverse matrix. The simulation results show that the best pseudo-inverse matrix can be easily chosen by the proposed method and the high allocation efficiency is achieved.
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paper_title: Evaluation of Optimization Methods for Control Allocation

paper_content:

The performanceand computational requirements ofoptimization methodsfor control allocation areevaluated. Two control allocation problems are formulated: a direct allocation method that preserves the directionality of the moment and a mixed optimization method that minimizes the error between the desired and the achieved momentsaswellasthecontroleffort.Theconstrainedoptimizationproblemsaretransformedinto linearprograms so that they can be solved using well-tried linear programming techniques such as the simplex algorithm. A variety of techniques that can be applied for the solution of the control allocation problem in order to accelerate computations are discussed. Performance and computational requirements are evaluated using aircraft models with different numbers of actuators and with different properties. In addition to the two optimization methods, three algorithms with low computational requirements are also implemented for comparison: a redistributed pseudoinverse technique, a quadratic programming algorithm, and a e xed-point method. The major conclusion is that constrained optimization can be performed with computational requirements that fall within an order of magnitude of those of simpler methods. The performance gains of optimization methods, measured in terms of the error between the desired and achieved moments, are found to be small on the average but sometimes signie cant.Avariety ofissuesthataffecttheimplementation ofthevariousalgorithmsin ae ight-controlsystem are discussed.
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paper_title: Fast Implementation of Direct Allocation with Extension to Coplanar Controls

paper_content:

The direct allocation method is considered for the control allocation problem. The original method assumed that every three columns of the controls effectiveness matrix were linearly independent. Here, the condition is relaxed, so that systems with coplanar controls can be considered. For fast online execution, an approach using spherical coordinates is also presented, and results of the implementation demonstrate improved performance over a sequential search. Linearized state-space models of a C-17 aircraft and of a tailless aircraft are used in the evaluation. Oincreasethe reliability ofaircraft,cone gurations with a large numberofactuatorsandcontrolsurfacesareadvantageous.Recone gurable control laws may be used to exploit all of the available control power despite failures and damages. 1;2 Control allocation is the problem of distributing the control requirements among multiple actuators to satisfy the desired objectives while accounting for the limited range of the actuators. Although solutions exist for the control allocation problem, an issue of current interest is that of the feasibilityoftheirimplementationonexistingcomputersforaircraft with a large number of actuators. 3 The direct allocation approach 4i6 is based on the concept of the attainable moment set (AMS), which is the set of all of the moment vectors that are achievable within the control constraints. The methodofdirectallocation allowsoneto achieve100%oftheAMS, whereas some other approaches such as daisy chaining, pseudoinverse,and generalized inversesolutionshavebeen shown to achieve
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paper_title: Evaluation of Optimization Methods for Control Allocation

paper_content:

The performanceand computational requirements ofoptimization methodsfor control allocation areevaluated. Two control allocation problems are formulated: a direct allocation method that preserves the directionality of the moment and a mixed optimization method that minimizes the error between the desired and the achieved momentsaswellasthecontroleffort.Theconstrainedoptimizationproblemsaretransformedinto linearprograms so that they can be solved using well-tried linear programming techniques such as the simplex algorithm. A variety of techniques that can be applied for the solution of the control allocation problem in order to accelerate computations are discussed. Performance and computational requirements are evaluated using aircraft models with different numbers of actuators and with different properties. In addition to the two optimization methods, three algorithms with low computational requirements are also implemented for comparison: a redistributed pseudoinverse technique, a quadratic programming algorithm, and a e xed-point method. The major conclusion is that constrained optimization can be performed with computational requirements that fall within an order of magnitude of those of simpler methods. The performance gains of optimization methods, measured in terms of the error between the desired and achieved moments, are found to be small on the average but sometimes signie cant.Avariety ofissuesthataffecttheimplementation ofthevariousalgorithmsin ae ight-controlsystem are discussed.
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paper_title: Closed-form solutions to constrained control allocation problem

paper_content:

This paper describes the results of recent research into the problem of allocating several flight control effectors to generate moments acting on a flight vehicle. The results focus on the use of various generalized inverse solutions and a hybrid solution utilizing daisy chaining. In this analysis, the number of controls is greater than the number of moments being controlled, and the ranges of the controls are constrained to certain limits. The control effectors are assumed to be individually linear in their effects throughout their ranges of motion and independent of one another in their effects. A standard of comparison is developed based on the volume of moments or moment coefficients a given method can yield using admissible control deflections. Details of the calculation of the various volumes are presented. Results are presented for a sample problem involving 10 flight control effectors. The effectivenesses of the various allocation schemes are contrasted during an aggressive roll about the velocity vector at low dynamic pressure. The performance of three specially derived generalized inverses, a daisy-chaining solution, and direct control allocation are compared.
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paper_title: Constrained control allocation - Three-moment problem

paper_content:

This paper presents a method for the solution of the constrained control allocation problem for the case of three moments. The control allocation problem is to find the "best" combination of several flight control effectors for the generation of specified body-axis moments. The number of controls is greater than the number of moments being controlled, and the ranges of the controls are constrained to certain limits. The controls are assumed to be individually linear in their effect throughout their ranges of motion and complete in the sense that they generate moments in arbitrary combinations. The best combination of controls is taken to be an apportioning of the controls that yields the greatest total moment in a specified ratio of moments without exceeding any control constraint. The method of solving the allocation problem is presented as an algorithm and is demonstrated for a problem of seven aerodynamic controls on an F-18 airplane.
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paper_title: Constrained control allocation

paper_content:

This paper addresses the problem of the allocation of several airplane flight controls to the generation of specified body-axis moments. The number of controls is greater than the number of moments being controlled, and the ranges of the controls are constrained to certain limits. They are assumed to be individually linear in their effect throughout their ranges of motion and independent of one another in their effects. The geometries of the subset of the constrained controls and of its image in moment space are examined. A direct method of allocating these several controls is presented that guarantees the maximum possible moment can be generated within the constraints of the controls. It is shown that no single generalized inverse can yield these maximum moments everywhere without violating some control constraint. A method is presented for the determination of a generalized inverse that satisfies given specifications which are arbitrary but restricted in number. We then pose and solve a minimization problem that yields the generalized inverse that best approximates the exact solutions. The results are illustrated at each step by an example problem involving three controls and two moments.
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paper_title: Sensitivity Analysis of Linear Programming and Quadratic Programming Algorithms for Control Allocation

paper_content:

The next generation (NextGen) transport aircraft configurations being investigated as part of the NASA Aeronautics Subsonic Fixed Wing Project have more control surfaces, or control effectors, than existing transport aircraft configurations. Conventional flight control is achieved through two symmetric elevators, two antisymmetric ailerons, and a rudder. The five control surfaces, reduced to three command variables, produce moments along the three main axes of the aircraft and enable the pilot to control the attitude of the aircraft. Next generation aircraft will have additional redundant control effectors to control the three moments, creating a situation where the aircraft is over-actuated and where a simple relationship no longer exists between the required control surface deflections and the desired moments. NextGen flight controllers will incorporate control allocation algorithms to determine the optimal effector commands to attain the desired moments, taking into account the effector limits. Approaches to solving the problem using linear programming and quadratic programming algorithms have been proposed and tested. It is of great interest to understand their relative advantages and disadvantages and how design parameters may affect their properties. In this paper, we investigate the sensitivity of the effector commands with respect to the desired moments and show on some examples the sensitivity of the solutions provided by the linear programming and quadratic programming methods.
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paper_title: Control Allocation with Load Balancing

paper_content:

Next generation aircraft with a large number of actuators will require advanced control allocation methods to compute the actuator commands needed to follow desired trajectories while respecting system constraints. Previously, algorithms were proposed to minimize the l1 or l2 norms of the tracking error and of the actuator deflections. The paper discusses the alternative choice of the l∞ ∞ ∞ ∞ norm, or sup norm. Minimization of the control effort translates into the minimization of the maximum actuator deflection (min-max optimization). The paper shows how the problem can be solved effectively by converting it into a linear program and solving it using a simplex algorithm. Properties of the algorithm are also investigated through examples. In particular, the min-max criterion results in a type of load balancing, where the load is the desired command and the algorithm balances this load among various actuators. The solution using the l∞ ∞ ∞ ∞ norm also results in better robustness to failures and to lower sensitivity to nonlinearities in illustrative examples.
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paper_title: Adaptable method of managing jets and aerosurfaces for aerospace vehicle control

paper_content:

An actuator selection procedure is presented that uses linear programming to optimally specify bounded aerosurface deflections and jet firings in response to differential torque and/or force commands. This method creates a highly adaptable interface to vehicle control logic by automatically providing intrinsic actuator decoupling, dynamic response to actuator reconfiguration, dynamic upper bound and objective specification, and the capability of coordinating hybrid operation with dissimilar actuators. The objective function minimized by the linear programming algorithm is adapted to realize several goals, i.e., discourage large aerosurface deflections, encourage the use of certain aerosurfaces (speedbrake, body flap) as a function of vehicle state, minimize drag, contribute to translational control, and adjust the balance between jet firings and aerosurface activity during hybrid operation. A vehicle model adapted from Space Shuttle aerodynamic data is employed in simulation examples that drive the actuator selection with a six-axis vehicle controller tracking a scheduled re-entry trajectory.
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paper_title: Numerical recipes in C

paper_content:

Note: Includes bibliographical references, 3 appendixes and 2 indexes.- Diskette v 2.06, 3.5''[1.44M] for IBM PC, PS/2 and compatibles [DOS] Reference Record created on 2004-09-07, modified on 2016-08-08
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paper_title: Evaluation of Optimization Methods for Control Allocation

paper_content:

The performanceand computational requirements ofoptimization methodsfor control allocation areevaluated. Two control allocation problems are formulated: a direct allocation method that preserves the directionality of the moment and a mixed optimization method that minimizes the error between the desired and the achieved momentsaswellasthecontroleffort.Theconstrainedoptimizationproblemsaretransformedinto linearprograms so that they can be solved using well-tried linear programming techniques such as the simplex algorithm. A variety of techniques that can be applied for the solution of the control allocation problem in order to accelerate computations are discussed. Performance and computational requirements are evaluated using aircraft models with different numbers of actuators and with different properties. In addition to the two optimization methods, three algorithms with low computational requirements are also implemented for comparison: a redistributed pseudoinverse technique, a quadratic programming algorithm, and a e xed-point method. The major conclusion is that constrained optimization can be performed with computational requirements that fall within an order of magnitude of those of simpler methods. The performance gains of optimization methods, measured in terms of the error between the desired and achieved moments, are found to be small on the average but sometimes signie cant.Avariety ofissuesthataffecttheimplementation ofthevariousalgorithmsin ae ight-controlsystem are discussed.
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paper_title: Resource balancing control allocation

paper_content:

Next generation aircraft with a large number of actuators will require advanced control allocation methods to compute the actuator commands needed to follow desired trajectories while respecting system constraints. Previously, algorithms were proposed to minimize the l 1 or l 2 norms of the tracking error and of the control effort. The paper discusses the alternative choice of using the l 1 norm for minimization of the tracking error and a normalized l 221E; norm, or sup norm, for minimization of the control effort. The algorithm computes the norm of the actuator deflections scaled by the actuator limits. Minimization of the control effort then translates into the minimization of the maximum actuator deflection as a percentage of its range of motion. The paper shows how the problem can be solved effectively by converting it into a linear program and solving it using a simplex algorithm. Properties of the algorithm are investigated through examples. In particular, the min-max criterion results in a type of resource balancing, where the resources are the control surfaces and the algorithm balances these resources to achieve the desired command. A study of the sensitivity of the algorithms to the data is presented, which shows that the normalized l 221E; algorithm has the lowest sensitivity, although high sensitivities are observed whenever the limits of performance are reached.
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paper_title: Interior-Point Algorithms for Control Allocation

paper_content:

Linear-programming formulations of control allocation problems are considered, including those associated with direct allocation and mixed 1 -norm objectives. Primal-dual and predictor-corrector path-following interior-point algorithms, that are shown to be well suited for the control-allocation problems, are described in some detail with an emphasis on preferred implementations. The performance of each algorithm is evaluated for computational efficiency and for accuracy using linear models of a C-17 transport and a tailless fighter aircraft. Appropriate choices of stopping tolerances and other algorithm parameters are studied. Comparisons of speed and accuracy are made to the simplex method. Results show that real-time implementation of the algorithms is feasible, without requiring excessive number of computations.
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paper_title: Constrained quadratic programming techniques for control allocation

paper_content:

The paper considers the objective of optimally specifying redundant actuators under constraints, a problem commonly referred to as control allocation. The problem is posed as a mixed /spl lscr//sub 2/-norm optimization objective and converted to a quadratic programming formulation. The implementation of an interior-point algorithm is presented. Alternative methods including fixed-point and active set methods are used to evaluate the reliability, accuracy and efficiency of the primal-dual interior-point method. While the computational load of the interior-point method is found to be greater for problems of small size, convergence to the optimal solution is also more uniform and predictable. In addition, the properties of the algorithm scale favorably with problem size.
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paper_title: Efficient active set algorithms for solving constrained least squares problems in aircraft control allocation

paper_content:

In aircraft control, control allocation can be used to distribute the total control effort among the actuators when the number of actuators exceeds the number of controlled variables. The control allocation problem is often posed as a constrained least squares problem to incorporate the actuator position and rate limits. Most proposed methods for real-time implementation, like the redistributed pseudoinverse method, only deliver approximate, and sometimes unreliable solutions. We investigate the use of classical active set methods for control allocation. We develop active set algorithms that always find the optimal control distribution, and show by simulation that the timing requirements are in the same range as for two previously proposed solvers.
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paper_title: Two Reconfigurable Flight-Control Design Methods: Robust Servomechanism and Control Allocation

paper_content:

Two methods for control system reconfiguration have been investigated. The first method is a robust servomechanism control approach (optimal tracking problem) that is a generalization of the classical proportional-plus-integral control to multiple input-multiple output systems. The second method is a control-allocation approach based on a quadratic programming formulation. A globally convergent fixed-point iteration algorithm has been developed to make onboard implementation of this method feasible. These methods have been applied to reconfigurable entry flight control design for the X-33 vehicle. Examples presented demonstrate simultaneous tracking of angle-of-attack and roll angle commands during failures of the fight body flap actuator. Although simulations demonstrate success of the first method in most cases, the control-allocation method appears to provide uniformly better performance in all cases.
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paper_title: Practical Methods of Optimization

paper_content:

Preface Table of Notation Part 1: Unconstrained Optimization Introduction Structure of Methods Newton-like Methods Conjugate Direction Methods Restricted Step Methods Sums of Squares and Nonlinear Equations Part 2: Constrained Optimization Introduction Linear Programming The Theory of Constrained Optimization Quadratic Programming General Linearly Constrained Optimization Nonlinear Programming Other Optimization Problems Non-Smooth Optimization References Subject Index.
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paper_title: Interior-Point Algorithms for Control Allocation

paper_content:

Linear-programming formulations of control allocation problems are considered, including those associated with direct allocation and mixed 1 -norm objectives. Primal-dual and predictor-corrector path-following interior-point algorithms, that are shown to be well suited for the control-allocation problems, are described in some detail with an emphasis on preferred implementations. The performance of each algorithm is evaluated for computational efficiency and for accuracy using linear models of a C-17 transport and a tailless fighter aircraft. Appropriate choices of stopping tolerances and other algorithm parameters are studied. Comparisons of speed and accuracy are made to the simplex method. Results show that real-time implementation of the algorithms is feasible, without requiring excessive number of computations.
---
paper_title: A nonlinear programming approach for control allocation

paper_content:

A novel nonlinear programming based control allocation scheme is developed. The performance of this nonlinear control allocation algorithm is compared with that of other control allocation approaches, including a mixed optimization scheme, a redistributed pseudo-inverse approach, and a direct allocation (geometric) method. The control allocation methods are first compared using open-loop measures such as the ability to attain commanded moments for a prescribed maneuver. The methods are then compared in closed-loop with a dynamic inversion-based control law. Next, the performance of the different algorithms is compared for different reference trajectories under a variety of failure conditions. Finally, we perform some preliminary studies employing "split actuators" that increase available control authority under failure conditions. All studies are conducted on a re-entry vehicle simulation.
---
paper_title: Efficient active set algorithms for solving constrained least squares problems in aircraft control allocation

paper_content:

In aircraft control, control allocation can be used to distribute the total control effort among the actuators when the number of actuators exceeds the number of controlled variables. The control allocation problem is often posed as a constrained least squares problem to incorporate the actuator position and rate limits. Most proposed methods for real-time implementation, like the redistributed pseudoinverse method, only deliver approximate, and sometimes unreliable solutions. We investigate the use of classical active set methods for control allocation. We develop active set algorithms that always find the optimal control distribution, and show by simulation that the timing requirements are in the same range as for two previously proposed solvers.
---
paper_title: Numerical recipes in C

paper_content:

Note: Includes bibliographical references, 3 appendixes and 2 indexes.- Diskette v 2.06, 3.5''[1.44M] for IBM PC, PS/2 and compatibles [DOS] Reference Record created on 2004-09-07, modified on 2016-08-08
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paper_title: Certification aspects of the fast gradient method for solving the dual of parametric convex programs

paper_content:

This paper examines the computational complexity certification of the fast gradient method for the solution of the dual of a parametric convex program. To this end, a lower iteration bound is derived such that for all parameters from a compact set a solution with a specified level of suboptimality will be obtained. For its practical importance, the derivation of the smallest lower iteration bound is considered. In order to determine it, we investigate both the computation of the worst case minimal Euclidean distance between an initial iterate and a Lagrange multiplier and the issue of finding the largest step size for the fast gradient method. In addition, we argue that optimal preconditioning of the dual problem cannot be proven to decrease the smallest lower iteration bound. The findings of this paper are of importance in embedded optimization, for instance, in model predictive control.
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paper_title: Flexible Piecewise Function Evaluation Methods Based on Truncated Binary Search Trees and Lattice Representation in Explicit MPC

paper_content:

Efficient methods for evaluation of piecewise functions defined over convex polyhedral partitions are proposed. As an important application, the explicit model predictive control problem is considered which requires a piecewise affine (PWA) control law to be evaluated online. The widely used binary search tree (BST) method is modified to be able to deal with a wider class of problems for which the BST method becomes prohibitive in terms of preprocessing time or memory requirements. The proposed method combines an orthogonal truncated binary search tree (OTBST) and lattice representation for PWA functions in a unified structure enjoying the advantages of both approaches. Both OTBST and Lattice-based OTBST (LOTBST) methods enable the designer to tradeoff between preprocessing time, storage requirement, and online computation time. The OTBST approach can be applied to more general partitions, e.g., discontinues and overlapping, while the LOTBST is directed towards more efficient evaluation of PWA functions, associated to the explicit MPC solutions. The key feature of the proposed methods is that the exact solution can be computed with predefined worst case online computation time guarantees. The computations are readily implementable using fixed-point arithmetic on a low cost microcontroller since there is no recursive accumulation of round-off errors, and the online algorithm is simple with a small footprint suitable for formal verification of correctness of implementation. Using several examples it is shown that the proposed LOTBST leads to a considerably less preprocessing time and memory requirement comparing to the pure BST and less online computation time comparing to the pure lattice representation.
---
paper_title: Integrated Adaptive Guidance and Control for Re-Entry Vehicles with Flight Test Results

paper_content:

To enable autonomous operation of future reusable launch vehicles, reconfiguration technologies will be needed to facilitate mission recovery following a major anomalous event. The Air Force’s Integrated Adaptive Guidance and Control program developed such a system for Boeing’s X-40A, and the total in-flight simulator research aircraft was employed to flight test the algorithms during approach and landing. The inner loop employs a modelfollowing/dynamic-inversion approach with optimal control allocation to account for control-surface failures. Further, the reference-model bandwidth is reduced if the control authority in any one axis is depleted as a result of control effector saturation. A backstepping approach is utilized for the guidance law, with proportional feedback gains that adapt to changes in the reference model bandwidth. The trajectory-reshaping algorithm is known as the optimum-path-to-go methodology. Here, a trajectory database is precomputed off line to cover all variations under consideration. An efficient representation of this database is then interrogated in flight to rapidly find the “best” reshaped trajectory, based on the current state of the vehicle’s control capabilities. The main goal of the flight-test program was to demonstrate the benefits of integrating trajectory reshaping with the essential elements of control reconfiguration and guidance adaptation. The results indicate that for more severe, multiple control failures, control reconfiguration, guidance adaptation, and trajectory reshaping are all needed to recover the mission.
---
paper_title: Dynamic control allocation with asymptotic tracking of time-varying control input commands

paper_content:

In this paper, the problem of control allocation - distribution of control power among redundant control effectors, under a set of constraints - for the inner loop of a re-entry vehicle guidance and control system is studied. Our control allocation scheme extends a previously developed model-predictive algorithm by providing asymptotic tracking of time-varying control input commands. The approach accounts for non-negligible dynamics of the actuators with hard constraints, setting it apart from most existing control allocation schemes, where a static relationship between control surface deflections (actuator outputs) and moments about a three-body axis (plant inputs) is assumed. The approach is readily extended to encompass a variety of linear actuator dynamics without the need for redesign of the overall control allocation scheme, allowing for increased effectiveness of the inner loop in terms of speed of maneuverability. Simulation results, with consideration given toward implementation, are provided for an experimental reusable launch vehicle, and are compared to those of static control allocation schemes.
---
paper_title: Fault tolerant control using sliding modes with on-line control allocation

paper_content:

This paper proposes an on-line sliding mode control allocation scheme for fault tolerant control. The effectiveness level of the actuators is used by the control allocation scheme to redistribute the control signals to the remaining actuators when a fault or failure occurs. The paper provides an analysis of the sliding mode control allocation scheme and determines the nonlinear gain required to maintain sliding. The on-line sliding mode control allocation scheme shows that faults and even certain total actuator failures can be handled directly without reconfiguring the controller. The simulation results show good performance when tested on different fault and failure scenarios.
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paper_title: Fault tolerant adaptive control allocation schemes for overactuated systems

paper_content:

This paper presents a fault-tolerant adaptive control allocation scheme for overactuated systems subject to loss of effectiveness actuator faults. The main idea is to use an ‘ad hoc’ online parameters estimator, coupled with a control allocation algorithm, in order to perform online control reconfiguration whenever necessary. Time-windowed and recursive versions of the algorithm are proposed for nonlinear discrete-time systems and their properties analyzed. Two final examples have been considered to show the effectiveness of the proposed scheme. The first considers a simple linear system with redundant actuators and it is mainly used to exemplify the main properties and potentialities of the scheme. In the second, a realistic marine vessel scenario under propeller and thruster faults is treated in full details. Copyright © 2010 John Wiley & Sons, Ltd.
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paper_title: Model predictive control allocation for overactuated systems - stability and performance

paper_content:

Overactuated systems often arise in automotive, aerospace, and robotics applications, where for reasons of redundancy or performance constraints, it is beneficial to equip a system with more control inputs than outputs. This necessitates control allocation methods that distribute control effort amongst many actuators to achieve a desired effect. Until recently, most methods have treated the control allocation as static in the sense that different dynamic authorities of the actuators were not taken into account. Recent advances have used model predictive control allocation (MPCA) to consider the dynamic authorities of the actuators over a receding horizon. In this paper, we consider the dynamic control allocation problem for overactuated systems where each actuator has different dynamic control authority and hard saturation limits. A modular control design approach is proposed, where the controller consists of an outer loop controller that synthesizes a desired virtual control input signal and an inner loop controller that uses MPCA to achieve the desired virtual control signal. We derive sufficient stability conditions for the composite feedback system and show how these conditions may be realized by imposing an additional constraint on the MPCA design. An automotive example is provided to illustrate the effectiveness of the proposed algorithm.
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paper_title: A new control allocation method that accounts for effector dynamics

paper_content:

Given a time history of desired moments, the control allocation problem is to solve for the effector inputs so that some norm of the error between the achieved and desired moments is minimized. Existing methods solve for the actuator deflections, while accounting for magnitude and rate limitations of the effectors. In this paper, we propose the dynamic control allocation (DCA) Method, that also accounts for effector dynamics, in addition to magnitude and rate limits. We show through numerical experiments that the DCA method allocates the desired moments according to effector bandwidths - that is the slow effectors are allocated the lower frequencies in the desired moments. The numerical simulations also show that the DCA outperforms the existing simplex algorithm based LP method, that does not account for actuator dynamics.
---
paper_title: Model predictive dynamic control allocation with actuator dynamics

paper_content:

A model predictive, dynamic control allocation algorithm is developed in this paper for the inner loop of a re-entry vehicle guidance and control system. The purpose of the control allocation portion of the guidance and control architecture is to distribute control power among redundant control effectors to meet the desired control objectives under a set of constraints. Most existing algorithms neglect the actuator dynamics or deal with the actuator dynamics separately, thereby assuming a static relationship between actuator outputs (in our case, control surface deflections) and plant inputs (i.e., moments about the three body axis). We propose a dynamic control allocation scheme based on model-based predictive control (MPC) that directly takes into account actuators with noneligible dynamics and hard constraints. Model-based predictive control schemes compute the control inputs by optimizing an open-loop control objective over a future time interval at each control step. In our setup, the model-predictive control allocation problem is posed as a sequential quadratic programming problem with dynamic constraints, which can be cast into a linear complementary problem (LCP) and therefore solved by linear programming approaches in a finite number of iterations. The time-varying affine internal model used in the MPC design enhances the ability of the control loop to deal with unmodeled system nonlinearities. The approach can be easily extended to encompass a variety of linear actuator dynamics without the need to redesign the overall scheme. Results are based on the model of an experimental reusable launch vehicle, and compared with that of existing static control allocation schemes.
---
paper_title: Dynamic allocation for input redundant control systems

paper_content:

In this paper we address control systems with redundant actuators and characterize the concepts of weak and strong input redundancy. Based on this characterization, we propose a dynamic augmentation to a control scheme which performs the plant input allocation with the goal of employing each actuator in a suitable way, based on its magnitude and rate limits. The proposed theory is first developed for redundant plants without saturation and then extended to the case of magnitude saturation first and of magnitude and rate saturation next. Several simulation examples illustrate the proposed technique and show its advantages for practical application.
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paper_title: Dynamic model predictive control allocation using CVXGEN

paper_content:

Control allocation deals with the allocation of control among a redundant set of effectors, while taking into account the individual constraints. The use of model predictive control (MPC) for control allocation allows the response times of the actuators to be accounted for, and to take advantage of predictions of the virtual control input as well as differences in dynamic control authority and cost of use among the actuators. Quadratic programming (QP) is essential for implementation of the optimal constrained control allocation strategies. The main contributions of the present paper are the investigation of using the software system CVXGEN and the MPC-based control allocation method. CVXGEN synthesizes a customized portable and library-free C-source code QP interior-point solver for the specific QP problem resulting from the MPC formulation, exploiting structural properties of the specific QP and optimizing the source code for execution speed. Two case studies, one being a missile auto-pilot, illustrates the benefits of using the MPC formulation, and the efficiency of CVXGEN.
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paper_title: Stabilization of Automotive Vehicles Using Active Steering and Adaptive Brake Control Allocation

paper_content:

In this work a dynamic control allocation approach is presented for an automotive vehicle yaw stabilization scheme. The stabilization strategy consists of a high level module that deals with the vehicle motion control objective (yaw rate reference generation and tracking), a low level module that handles the braking control for each wheel (longitudinal slip control and maximal tire-road friction parameter estimation), and an intermediate level dynamic control allocation module that generates the longitudinal slip reference for the low level brake control module and commands front wheel steering angle corrections. The control allocation design is such that the actual torque about the yaw axis tends to the desired torque calculated form the high level module, with desirable distribution of control forces satisfying actuator constraints and minimal control effort objectives. Conditions for uniform asymptotic stability are given for the case when the control allocation includes adaptation of the tire-road maximal friction coefficients, and the scheme has been implemented in a realistic non linear multi body vehicle simulation environment. The simulation cases show that the yaw control allocation strategy stabilizes the vehicle in extreme maneuvers where the non linear vehicle yaw dynamics otherwise (without active braking or active steering) becomes unstable in the sense of over- or under steering. The control allocation implementation is efficient and suitable for low cost automotive electronic control units.
---
paper_title: A nonlinear programming approach for control allocation

paper_content:

A novel nonlinear programming based control allocation scheme is developed. The performance of this nonlinear control allocation algorithm is compared with that of other control allocation approaches, including a mixed optimization scheme, a redistributed pseudo-inverse approach, and a direct allocation (geometric) method. The control allocation methods are first compared using open-loop measures such as the ability to attain commanded moments for a prescribed maneuver. The methods are then compared in closed-loop with a dynamic inversion-based control law. Next, the performance of the different algorithms is compared for different reference trajectories under a variety of failure conditions. Finally, we perform some preliminary studies employing "split actuators" that increase available control authority under failure conditions. All studies are conducted on a re-entry vehicle simulation.
---
paper_title: Equivalence of hybrid dynamical models

paper_content:

This paper establishes equivalences among five classes of hybrid systems: mixed logical dynamical (MLD) systems, linear complementarity (LC) systems, extended linear complementarity (ELC) systems, piecewise affine (PWA) systems, and max-min-plus-scaling (MMPS) systems. Some of the equivalences are established under (rather mild) additional assumptions. These results are of paramount importance for transferring theoretical properties and tools from one class to another, with the consequence that for the study of a particular hybrid system that belongs to any of these classes, one can choose the most convenient hybrid modeling framework.
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paper_title: Control of Systems Integrating Logic, Dynamics, and Constraints

paper_content:

This paper proposes a framework for modeling and controlling systems described by interdependent physical laws, logic rules, and operating constraints, denoted as mixed logical dynamical (MLD) systems. These are described by linear dynamic equations subject to linear inequalities involving real and integer variables. MLD systems include linear hybrid systems, finite state machines, some classes of discrete event systems, constrained linear systems, and nonlinear systems which can be approximated by piecewise linear functions. A predictive control scheme is proposed which is able to stabilize MLD systems on desired reference trajectories while fulfilling operating constraints, and possibly take into account previous qualitative knowledge in the form of heuristic rules. Due to the presence of integer variables, the resulting on-line optimization procedures are solved through mixed integer quadratic programming (MIQP), for which efficient solvers have been recently developed. Some examples and a simulation case study on a complex gas supply system are reported.
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paper_title: Optimizing nonlinear control allocation

paper_content:

Control allocation is commonly utilized in overactuated mechanical systems in order to optimally generate a requested generalized force using a redundant set of actuators. Using a control-Lyapunov approach, we develop an optimizing control allocation algorithm in the form of a dynamic update law, for a general class of nonlinear systems. The asymptotically optimal control allocation in interaction with an exponentially stable trajectory-tracking controller guarantees uniform boundedness and uniform global exponential convergence.
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paper_title: Adaptive Control Allocation

paper_content:

In this work we address the control allocation problem for a nonlinear over-actuated time-varying system where parameters affine in the effector model may be assumed unknown. Instead of optimizing the control allocation at each time instant, a dynamic approach is considered by constructing update-laws that represent asymptotically optimal allocation search and adaptation. Using Lyapunov analysis for cascaded set-stable systems, uniform global/local asymptotic stability is guaranteed for the sets described by the system, the optimal allocation update-law and the adaptive update-law.
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paper_title: Adaptive control allocation for non-linear systems with internal dynamics

paper_content:

An adaptive control allocation method is presented for a general class of non-linear systems with internal dynamics and unknown parameters. A certainty equivalence indirect adaptive approach is used to estimate the unknown parameters. Based on the estimated parameters, model reference control and control allocation techniques are used to control the non-linear system subject to control constraints and internal dynamics stabilisation. A Lyapunov design approach with the property of convergence to a positively invariant set is proposed. The derived adaptive control allocation is in the form of a dynamic update law, which, together with a stable model reference control, guarantees that the closed-loop non-linear system be input-to-state stable.
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paper_title: Constrained Nonlinear Finite-Time Control Allocation

paper_content:

This paper proposed an optimal control allocation method for a general class of overactuated nonlinear systems with internal dynamics. Dynamic inversion technique is used for the commanded subsystem to track a stable model reference control law. The corresponding control allocation law has to guarantee the stability of the internal dynamics and satisfy control constraints. The proposed method is based on a Lyapunov design approach with finite-time convergence to optimality. The derived control allocation is in the form of a dynamic update law, which, together with the stable model reference control law, guarantees the stability of the closed-loop nonlinear system.
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paper_title: Adaptive nonlinear control allocation of non-minimum phase uncertain systems

paper_content:

This paper presents an adaptive nonlinear control allocation method for a general class of non-minimum phase uncertain systems. Indirect adaptive approach and Lyapunov design approach are applied to the design of adaptive control allocation. The derived adaptive control allocation law, together with a stable model reference control, guarantees that the closed-loop nonlinear system is input-to-state stable.
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paper_title: ON OPTIMIZING NONLINEAR ADAPTIVE CONTROL ALLOCATION WITH ACTUATOR DYNAMICS

paper_content:

Abstract In this work we address the optimizing control allocation problem for a nonlinear over-actuated time-varying system where parameters affine in the dynamic actuator and effector model may be assumed unknown. In-stead of finding the optimal control allocation at each time instant, a dynamic approach is considered by constructing update-laws that represent asymptotically optimal allocation search and adaptation. Using Lyapunov analysis for cascaded set-stable systems, uniform global/local asymptotic stability is guaranteed for the optimal set described by the system, the optimal allocation update-law and the adaptive update-law. Simulations of a scaled-model ship, manoeuvred at low-speed, demonstrate the performance of the proposed allocation scheme.
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paper_title: Nonlinear Control Allocation for Non-Minimum Phase Systems

paper_content:

In this brief, we propose a control allocation method for a particular class of uncertain over-actuated affine nonlinear systems, with unstable internal dynamics. Dynamic inversion technique is used for the commanded output to track a smooth output reference trajectory. The corresponding control allocation law has to guarantee the boundedness of the states, including the internal dynamics, and satisfy control constraints. The proposed method is based on a Lyapunov design approach with finite-time convergence to a given invariant set. The derived control allocation is in the form of a dynamic update law which, together with a sliding mode control law, guarantees boundedness of the output tracking error as well as of the internal dynamics. The effectiveness of the control law is tested on a numerical model of the non-minimum phase planar vertical take-off and landing (PVTOL) system.
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paper_title: Comparison of Static and Dynamic Control Allocation Techniques for Integrated Vehicle Control

paper_content:

Abstract Comparison of static and dynamic control allocation techniques for nonlinear constrained optimal distribution of tire forces in a vehicle control system is presented. The total body forces and moments, obtained from a high level controller, are distributed among tire forces, which are constrained to nonlinear constraint of saturation, through two approaches. For the static control allocation technique the interior-point method is employed to perform the nonlinear constrained optimization problem. Also, by the dynamic control allocation technique a dynamic update law is derived for desired forces of each tire. Through simulation results efficiency of two approaches in enhancing vehicle handling and stability is evaluated and the results are compared.
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paper_title: Concepts for constrained control allocation of mixed quadratic and linear effectors

paper_content:

Concepts for new constrained control allocation strategies are developed that deal with systems where moments are nonlinearly related to effector deflections such as those encountered in the case of yawing moment contributions from left-right effector pairs on aircraft.. These concepts are illustrated by considering single and multiple left-right pair effector mixing problems for moments that lie in the roll-yaw moment plane. Methods for generating the boundary of an attainable moment set for a class of multiple non-linear effectors and for clipping unattainable moment commands with axis prioritization are presented.
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paper_title: Control allocation and compensation for over-actuated systems with non-linear effectors

paper_content:

Control allocation techniques for inversion-based control laws have evolved that ensure that commands to the control effectors do not violate rate or position limits of the individual effectors. These allocation techniques typically assume that all effector dynamics are fast relative to the system being controlled and are therefore neglected. Unfortunately there are practical cases where this assumption breaks down and it becomes desirable to compensate for lags between the commands to the effectors and the effector response. In this paper, a technique for compensating for individual effector dynamics while respecting actuator constraints is proposed.
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paper_title: Method for Determination of Nonlinear Attainable Moment Sets

paper_content:

A method for generating attainable moment sets for left-right pairs of nonlinear control effectors on aircraft is presented. It is shown that the determination of the attainable moment set boundary can be posed as a constrained, nonlinear optimization problem. The first-order necessary conditions for a point to be on the boundary are simply the Kuhn-Tucker first-order necessary conditions. An algorithm is given where the Kuhn-Tucker points for a given point in the pitch-roll plane are constructed for each control effector configuration that forms a candidate boundary. The Kuhn-Tucker points are then checked for feasibility, and the points on the boundary are those that produce extremal values of the objective function. The nonlinear attainable moment set boundary is used to determine feasibility of moment commands generated by the flight control system. Infeasible commands are clipped to the boundary of the attainable moment set. If a commanded moment is feasible, a constrained optimization problem is solved for the control surface deflections.
---
paper_title: Integrated Adaptive Guidance and Control for Re-Entry Vehicles with Flight Test Results

paper_content:

To enable autonomous operation of future reusable launch vehicles, reconfiguration technologies will be needed to facilitate mission recovery following a major anomalous event. The Air Force’s Integrated Adaptive Guidance and Control program developed such a system for Boeing’s X-40A, and the total in-flight simulator research aircraft was employed to flight test the algorithms during approach and landing. The inner loop employs a modelfollowing/dynamic-inversion approach with optimal control allocation to account for control-surface failures. Further, the reference-model bandwidth is reduced if the control authority in any one axis is depleted as a result of control effector saturation. A backstepping approach is utilized for the guidance law, with proportional feedback gains that adapt to changes in the reference model bandwidth. The trajectory-reshaping algorithm is known as the optimum-path-to-go methodology. Here, a trajectory database is precomputed off line to cover all variations under consideration. An efficient representation of this database is then interrogated in flight to rapidly find the “best” reshaped trajectory, based on the current state of the vehicle’s control capabilities. The main goal of the flight-test program was to demonstrate the benefits of integrating trajectory reshaping with the essential elements of control reconfiguration and guidance adaptation. The results indicate that for more severe, multiple control failures, control reconfiguration, guidance adaptation, and trajectory reshaping are all needed to recover the mission.
---
paper_title: Reconfigurable Control Allocation against Aircraft Control Effector Failures

paper_content:

Due to increased requirements on the reliability, maneuverability and survivability of modern and future manned and unmanned aerial vehicles, more control effectors/surfaces are being introduced. This introduces redundant or overactuated control effectors and requires the control allocation function, together with baseline flight control law, to be implemented in the overall flight control systems. In particular, in the case of control effector (actuator) failures or control surface damages, an effective re-distribution (or reallocation) of the control surface deflections with the remaining healthy control effectors is needed in order to achieve acceptable performance even in the presence of control effector failures. In this paper, as a preliminary study of reconfigurable control allocation (control re-allocation) applied to a realistic and nonlinear aircraft model, a pseudo-inverse method and a fixed-point algorithm are implemented and tested in an ADMIRE (aero-data model in research environment) benchmark aircraft model. Partial loss of control effectiveness faults have been implemented in the ADMIRE benchmark and used for evaluating the control re-allocation schemes. Simulation results show satisfactory results for accommodating the control effector failures.
---
paper_title: Dynamic control allocation with asymptotic tracking of time-varying control input commands

paper_content:

In this paper, the problem of control allocation - distribution of control power among redundant control effectors, under a set of constraints - for the inner loop of a re-entry vehicle guidance and control system is studied. Our control allocation scheme extends a previously developed model-predictive algorithm by providing asymptotic tracking of time-varying control input commands. The approach accounts for non-negligible dynamics of the actuators with hard constraints, setting it apart from most existing control allocation schemes, where a static relationship between control surface deflections (actuator outputs) and moments about a three-body axis (plant inputs) is assumed. The approach is readily extended to encompass a variety of linear actuator dynamics without the need for redesign of the overall control allocation scheme, allowing for increased effectiveness of the inner loop in terms of speed of maneuverability. Simulation results, with consideration given toward implementation, are provided for an experimental reusable launch vehicle, and are compared to those of static control allocation schemes.
---
paper_title: Real-Time Adaptive Control Allocation applied to a High Performance Aircraft

paper_content:

This paper presents the development and application of one approach to the control of aircraft with large numbers of control effectors. This approach, referred to as real-time adaptive control allocation, combines a nonlinear method for control allocation with actuator failure detection and isolation. The control allocator maps moment (or angular acceleration) commands into physical control effector commands as functions of individual control effectiveness and availability. The actuator failure detection and isolation algorithm is a model-based approach that uses models of the actuators to predict actuator behavior and an adaptive decision threshold to achieve acceptable false alarm/missed detection rates. This integrated approach provides control reconfiguration when an aircraft is subjected to actuator failure, thereby improving maneuverability and survivability of the degraded aircraft. This method is demonstrated on a next generation military aircraft (Lockheed-Martin Innovative Control Effector) simulation that has been modified to include a novel nonlinear fluid flow control control effector based on passive porosity. Desktop and real-time piloted simulation results demonstrate the performance of this integrated adaptive control allocation approach. Introduction Future aircraft are being proposed with many more control effectors than the traditional elevator, aileron, and rudder. Innovative control effectors under study range from thrust vectoring to all-moving wing tips and actuated forebody surfaces to more exotic effectors such as shape-change materials and fluidflow devices (figure 1). For example, hundreds or even thousands of micro fluid-flow devices could be distributed in arrays across the upper and lower surface of a wing to allow direct control of the wing’s boundary layer. Optimal use of this large number of effectors will be challenging, but the potential control power and redundancy offers the flight controls designer freedom to maximize mission performance and enhance survivability. * Research Engineer † Senior Research Engineer No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. In particular, reconfigurable control approaches seek to take advantage of this control redundancy to mitigate the deleterious effects of control effector failure or battle damage [Buffington 1998, Brinker 1999, Tallant 1999]. A key element in these approaches is reconfigurable control allocation. The reconfigurable control allocator [Lallman 1985, Durham 1992, Buffington 1997, Enns 1998] maps moment (or angular acceleration) commands into physical control effector commands as a function of individual control effectiveness and availability. This paper presents the development of an integrated reconfigurable control allocation method which combines a nonlinear method for control allocation with actuator failure detection and isolation (FDI) (figure 2). This integrated approach provides control reconfiguration when an aircraft is subjected to actuator failure, thereby improving maneuverability and survivability of the degraded aircraft. Desktop and realtime piloted simulation are used to demonstrate the performance of this integrated adaptive control allocation approach. Real-Time Adaptive Control Allocation The adaptive control allocation is an integrated approach that combines two main elements: a nonlinear control allocation approach and actuator failure detection and isolation. These elements are discussed in more detail in the following sections. Control Allocation In general, the forces and moments generated by the control effectors are a function of vehicle flight condition and effector commands. In the case where, at a given flight condition, the control effectiveness is relatively linear with effector command and there is minimal control coupling, the system can be considered as a linear control allocation problem. The linear control allocation problem can be stated as follows. Given the system
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paper_title: Constrained control allocation based on nonlinear compensation with application to flight control

paper_content:

In flight control, control allocation is used to distribute the total control command to each actuator when the number of actuators exceeds the number of controlled variables. A new optimal control allocation method based on nonlinear compensation was proposed in this paper. The quadratic term and mutual-interference effects were introduced into control effectiveness matrix. The linear combination of control objective and error objective was selected as optimal target. Then the optimal control allocation problem of redundancy system was solved by transformed to nonlinear programming formulation. Comparison with other allocation method and control system simulation results show the proposed method has timing properties similar to the redistributed pseudo-inverse method and achieves the objective accurately and optimally.
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paper_title: Lyapunov stability analysis of daisy chain control allocation

paper_content:

A demonstration that feedback control of systems with redundant controls can be reduced to feedback control of systems without redundant controls and control allocation is presented. It is shown that control allocation can introduce unstable zero dynamics into the system, which is important if input/output inversion control techniques are utilized. The daisy chain control allocation technique for systems with redundant groups of controls is also presented. Sufficient conditions are given to ensure that the daisy chain control allocation does not introduce unstable zero dynamics into the system. Aircraft flight control examples are given to demonstrate the derived results.
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paper_title: Sensitivity Analysis of Linear Programming and Quadratic Programming Algorithms for Control Allocation

paper_content:

The next generation (NextGen) transport aircraft configurations being investigated as part of the NASA Aeronautics Subsonic Fixed Wing Project have more control surfaces, or control effectors, than existing transport aircraft configurations. Conventional flight control is achieved through two symmetric elevators, two antisymmetric ailerons, and a rudder. The five control surfaces, reduced to three command variables, produce moments along the three main axes of the aircraft and enable the pilot to control the attitude of the aircraft. Next generation aircraft will have additional redundant control effectors to control the three moments, creating a situation where the aircraft is over-actuated and where a simple relationship no longer exists between the required control surface deflections and the desired moments. NextGen flight controllers will incorporate control allocation algorithms to determine the optimal effector commands to attain the desired moments, taking into account the effector limits. Approaches to solving the problem using linear programming and quadratic programming algorithms have been proposed and tested. It is of great interest to understand their relative advantages and disadvantages and how design parameters may affect their properties. In this paper, we investigate the sensitivity of the effector commands with respect to the desired moments and show on some examples the sensitivity of the solutions provided by the linear programming and quadratic programming methods.
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paper_title: CONTROL ALLOCATION WITH ADAPTIVE FAILURE CONTROL

paper_content:

In the age of modern aircraft and fly-by-wire control systems, the inclusion of mechanical backup systems for handling instances of control failures is becoming more uncommon. As a result, pilots are forced to fully rely on the failure immunities designed into these systems and be assured that, should such failures occur, the aircraft maintain adequate flying qualities long enough for a safe ejection or an emergency landing. This paper demonstrates the control reconfiguration aspects of a control allocation with rate limiting algorithm to adapt to various control failures while still achieving desired moments, thus providing a safe environment for the pilot.
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paper_title: Reconfigurable Control Allocation Technology Using Weighted Least Squares for Nonlinear System in Unmanned Aerial Vehicle

paper_content:

Reconfigurable control allocation research is important to multidisciplinary science and engineering applications. In particular, the proposed research plays a significant role in enhancing the safety, reliability and fault tolerance capability of Unmanned Aerial Vehicle (UAV), which is one of the most active research and development areas. The main objective of this paper is to introduce and evaluate UAV reconfigurable control system design against control surfaces faults without modifying the baseline controller. The faults introduced are in the form of partial loss and stuck at unknown position on the UAV control surfaces. Weighted Least Squares (WLS) control reallocation algorithm with application to UAV was investigated. The paper is undertaken in a nonlinear UAV model ALTAV (Almost-Light-Than-Air-Vehicles), developed by Quanser incorporation. Different faults have been introduced in control surfaces with different trajectory commanded inputs. Gaussian noise was introduced in the model. Comparisons were made under normal situation, the case without control reallocation, and the case with control reallocation method. Simulation results show the satisfactory reconfigurable flight control system performance using the WLS control reallocation method for ALTAV nonlinear UAV benchmark.
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paper_title: Adaptive control allocation in the presence of actuator failures

paper_content:

This paper proposes a control allocation framework where a feedback adaptive signal is designed for a group of redundant actuators and then it is adaptively allocated among all group members. In the adaptive control allocation structure, cooperative actuators are grouped and treated as an equivalent control effector. A state feedback adaptive control signal is designed for the equivalent effector and adaptively allocated to the member actuators. Two adaptive control allocation algorithms, guaranteeing closed-loop stability and asymptotic state tracking when partial and total loss of control effectiveness occur, are developed. Proper grouping of the actuators reduces the controller complexity without reducing their efficacy. The implementation and effectiveness of the strategies proposed is demonstrated in detail using several examples.
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paper_title: Control allocation and management for aircraft with multiple effectors

paper_content:

This paper focuses on the control redundancy problem due to the increase of control effectors of advanced aircrafts. Based on the current control allocation research, this paper further puts forward the concept of control allocation and management system. The bases sequenced control allocation method is proposed firstly, and then some management strategies are introduced to the control allocation process of bases sequenced method. The candidate effectors and optimal indexes are managed according to the information of flight conditions, mission requirements and effectors working conditions, and engineering experience is also considered in this system. The key aspects of the proposed system are illustrated by a 6DOF nonlinear aircraft model. The simulation results show that the functions of control allocation system are extended and the system adaptability to the flight status, mission requirements and effectors failure conditions are improved.
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paper_title: Constrained quadratic programming techniques for control allocation

paper_content:

The paper considers the objective of optimally specifying redundant actuators under constraints, a problem commonly referred to as control allocation. The problem is posed as a mixed /spl lscr//sub 2/-norm optimization objective and converted to a quadratic programming formulation. The implementation of an interior-point algorithm is presented. Alternative methods including fixed-point and active set methods are used to evaluate the reliability, accuracy and efficiency of the primal-dual interior-point method. While the computational load of the interior-point method is found to be greater for problems of small size, convergence to the optimal solution is also more uniform and predictable. In addition, the properties of the algorithm scale favorably with problem size.
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paper_title: Control Allocation with Load Balancing

paper_content:

Next generation aircraft with a large number of actuators will require advanced control allocation methods to compute the actuator commands needed to follow desired trajectories while respecting system constraints. Previously, algorithms were proposed to minimize the l1 or l2 norms of the tracking error and of the actuator deflections. The paper discusses the alternative choice of the l∞ ∞ ∞ ∞ norm, or sup norm. Minimization of the control effort translates into the minimization of the maximum actuator deflection (min-max optimization). The paper shows how the problem can be solved effectively by converting it into a linear program and solving it using a simplex algorithm. Properties of the algorithm are also investigated through examples. In particular, the min-max criterion results in a type of load balancing, where the load is the desired command and the algorithm balances this load among various actuators. The solution using the l∞ ∞ ∞ ∞ norm also results in better robustness to failures and to lower sensitivity to nonlinearities in illustrative examples.
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paper_title: Control Allocation and Zero Dynamics

paper_content:

Closed-loop stability for dynamic inversion controllers depends on the stability of the zero dynamics. The zero dynamics, however, depend on a generally nonlinear control allocation function that optimally distributes redundant controls. Therefore, closed-loop stability depends on the control allocation function. A sufe cient condition is provided for globally asymptotically stable zero dynamics with a class of admissible nonlinear control allocation functions. It is shown that many common control allocation functions belong to the class of functions that are covered by the aforementioned zero dynamics stability condition. Aircraft e ight control examples are given to demonstrate the utility of the results.
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paper_title: Model predictive dynamic control allocation with actuator dynamics

paper_content:

A model predictive, dynamic control allocation algorithm is developed in this paper for the inner loop of a re-entry vehicle guidance and control system. The purpose of the control allocation portion of the guidance and control architecture is to distribute control power among redundant control effectors to meet the desired control objectives under a set of constraints. Most existing algorithms neglect the actuator dynamics or deal with the actuator dynamics separately, thereby assuming a static relationship between actuator outputs (in our case, control surface deflections) and plant inputs (i.e., moments about the three body axis). We propose a dynamic control allocation scheme based on model-based predictive control (MPC) that directly takes into account actuators with noneligible dynamics and hard constraints. Model-based predictive control schemes compute the control inputs by optimizing an open-loop control objective over a future time interval at each control step. In our setup, the model-predictive control allocation problem is posed as a sequential quadratic programming problem with dynamic constraints, which can be cast into a linear complementary problem (LCP) and therefore solved by linear programming approaches in a finite number of iterations. The time-varying affine internal model used in the MPC design enhances the ability of the control loop to deal with unmodeled system nonlinearities. The approach can be easily extended to encompass a variety of linear actuator dynamics without the need to redesign the overall scheme. Results are based on the model of an experimental reusable launch vehicle, and compared with that of existing static control allocation schemes.
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paper_title: Computationally Efficient use of MPC and Dynamic Inversion for Reconfigurable Flight Control

paper_content:

†This paper features the combination of model-based predictive control and dynamic inversion into a constrained and globally valid control method for fault-tolerant flight-control purposes. The fact that the approach is both constrained and model-based creates the possibility to incorporate additional constraints, or even a new model, in case of a failure. Both of these properties lead to the fault-tolerant qualities of the method. Efficient distribution of the desired control moves over the control effectors creates the possibility to separate the input allocation problem from the inversion loop when redundant actuators are available. An important aspect that is considered here is the computational complexity of the presented methods. This complexity is reduced through application of an intelligent constraint mapping algorithm that allows for a strict separation of the applied model-predictive controller and the nonlinear dynamic inversion method. Furthermore, the complexity is reduced through application of a method that only uses the full set of available inputs, or actuators, when absolutely necessary, e.g. in a failure situation. Part of this paper consists of the application of the proposed theory to an aerospace benchmark of moderately high complexity. It is shown through an example that the theory is well-suited to the task, provided that fault-detection and isolation information is available continuously.
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paper_title: Evaluation of Optimization Methods for Control Allocation

paper_content:

The performanceand computational requirements ofoptimization methodsfor control allocation areevaluated. Two control allocation problems are formulated: a direct allocation method that preserves the directionality of the moment and a mixed optimization method that minimizes the error between the desired and the achieved momentsaswellasthecontroleffort.Theconstrainedoptimizationproblemsaretransformedinto linearprograms so that they can be solved using well-tried linear programming techniques such as the simplex algorithm. A variety of techniques that can be applied for the solution of the control allocation problem in order to accelerate computations are discussed. Performance and computational requirements are evaluated using aircraft models with different numbers of actuators and with different properties. In addition to the two optimization methods, three algorithms with low computational requirements are also implemented for comparison: a redistributed pseudoinverse technique, a quadratic programming algorithm, and a e xed-point method. The major conclusion is that constrained optimization can be performed with computational requirements that fall within an order of magnitude of those of simpler methods. The performance gains of optimization methods, measured in terms of the error between the desired and achieved moments, are found to be small on the average but sometimes signie cant.Avariety ofissuesthataffecttheimplementation ofthevariousalgorithmsin ae ight-controlsystem are discussed.
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paper_title: IMPROVING CONTROL ALLOCATION ACCURACY FOR NONLINEAR AIRCRAFT DYNAMICS

paper_content:

The forces and moments produced by a vehicle’s aerodynamic control surfaces are often nonlinear functions of control surface deflection. This phenomenon limits the accuracy of state-of-the-art control allocation algorithms since all of the approaches are based on the assumption that the control variable rates are linear functions of the surface deflections and that control variable rate increments are not produced for zero deflections. The errors introduced by this assumption are currently mitigated by the robustness resulting from feedback control laws. A method for improving the performance of the feedback control/control allocation system is presented that directly attacks the inaccuracies introduced by these linear assumptions. The approach is demonstrated using a dynamic inversion-based control law developed for a lifting body with four control surfaces. The commanded body rates constitute the input to a dynamic inversion control law that forms the inner-loop control structure. The outputs of the dynamic inversion algorithm serve as the inputs to a linear programming based control allocator. The control allocation algorithm is a mixed optimization scheme that minimizes the norm of the error and the deviation of the control input from a preferred value. Whether the underlying dynamic system is linear or nonlinear in the way that the controls affect the system state, the control allocator requires a linear approximation to the system inputs. In this work, instead of using a linear approximation to describe the control variable rate due to a control surface deflection (a control subspace, that contains the zero element), an affine relationship is utilized. This allows one to modify the input to the control allocator by providing an additional intercept term that corrects for the errors introduced by the control allocation algorithm’s assumption of linearity. Since on-line control allocators perform computations at flight control system update rates, the decision about how far to move the control surface in the next time interval is critical. In order to accurately compute the next set of surface deflection commands, the local slope at the current operating point is used and the intercept is accounted for at the input to the control allocator. With relatively little computational and design overhead, the accuracy of current control allocation algorithms can be improved utilizing an affine relationship for a vehicle’s control dependent accelerations. Simulation experiments show a significant improvement in tracking commanded body rates when the control allocation correction term is used.
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paper_title: Stability Properties and Cross-Coupling Performance of the Control Allocation Scheme CAPIO

paper_content:

This paper presents a stability analysis and an application of a recently developed algorithm to recover from Pilot Induced Oscillations (CAPIO). When actuators are ratesaturated due to either an aggressive pilot command, high gain of the ight control system or some anomaly in the system, the eective delay in the control loop may increase. This eective delay increase manifests itself as a phase shift between the commanded and actual system signals and can instigate Pilot Induced Oscillations (PIO). CAPIO reduces the eective time delay by minimizing the phase shift between the commanded and the actual attitude accelerations. To establish theoretical results pertinent to CAPIO stability analysis, a scalar case is presented. In addition, we present simulation results for aircraft with cross-coupling which demonstrates the potential of CAPIO serving as an eective PIO handler in adverse conditions.
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paper_title: Resource balancing control allocation

paper_content:

Next generation aircraft with a large number of actuators will require advanced control allocation methods to compute the actuator commands needed to follow desired trajectories while respecting system constraints. Previously, algorithms were proposed to minimize the l 1 or l 2 norms of the tracking error and of the control effort. The paper discusses the alternative choice of using the l 1 norm for minimization of the tracking error and a normalized l 221E; norm, or sup norm, for minimization of the control effort. The algorithm computes the norm of the actuator deflections scaled by the actuator limits. Minimization of the control effort then translates into the minimization of the maximum actuator deflection as a percentage of its range of motion. The paper shows how the problem can be solved effectively by converting it into a linear program and solving it using a simplex algorithm. Properties of the algorithm are investigated through examples. In particular, the min-max criterion results in a type of resource balancing, where the resources are the control surfaces and the algorithm balances these resources to achieve the desired command. A study of the sensitivity of the algorithms to the data is presented, which shows that the normalized l 221E; algorithm has the lowest sensitivity, although high sensitivities are observed whenever the limits of performance are reached.
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paper_title: Guidance, Navigation, and Control System Design for Tripropeller Vertical-Takeoff and-Landing Unmanned Air Vehicle

paper_content:

In this paper, we present the design and development of the guidance, navigation, and control system of a small vertical-takeoff-and-landing unmanned air vehicle based on a 6 degrees-of-freedom nonlinear dynamic model. The vertical-takeoff-and-landing unmanned air vehicle is equipped with three propellers for vertical thrust, and thrust differential together with a set of yaw trim flaps are used for 3 degrees-of-freedom attitude and thrust control actuation. The focus is on the 6 degrees-of-freedom flight control algorithm design using the trajectory linearization control method, along with simulation verification and robustness tests. Hardware and software implementation of the flight controller and onboard navigation sensors are also briefly discussed.
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paper_title: Model-Predictive Dynamic Control Allocation Scheme for Reentry Vehicles

paper_content:

Allocation of control authority among redundant control effectors, under hard constraints, is an important component of the inner loop of a reentry vehicle guidance and control system. Whereas existing control allocation schemes generally neglect actuator dynamics, thereby assuming a static relationship between control surface deflections and moments about a three-body axis, in this work a dynamic control allocation scheme is developed that implements a form of model-predictive control. In the approach proposed here, control allocation is posed as a sequential quadratic programming problem with constraints, which can also be cast into a linear complementarity problem and therefore solved in a finite number of iterations. Accounting directly for nonnegligible dynamics of the actuators with hard constraints, the scheme extends existing algorithms by providing asymptotic tracking of time-varying input commands for this class of applications. To illustrate the effectiveness of the proposed scheme, a high-fidelity simulation for an experimental reusable launch vehicle is used, in which results are compared with those of static control allocation schemes in situations of actuator failures.
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paper_title: Control allocation in overactuated aircraft under position and rate limiting

paper_content:

Existing approaches to control allocation in overactuated aircraft under control input saturation are conservative. This is due to the fact that the solution to the corresponding constrained optimization problem may not be attainable with a chosen control input weighting matrix. On the other hand, there may exist a weighting matrix for which a solution exists under control input constraints. In this paper we propose two control allocation algorithms (CAAs) that can be used either online or off-line, and that, under an assumption that at least one solution to the tracking control design exists, finds that solution. The properties of the proposed CAAs are illustrated through simulations of the F/A-18C/D aircraft.
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paper_title: Two Reconfigurable Flight-Control Design Methods: Robust Servomechanism and Control Allocation

paper_content:

Two methods for control system reconfiguration have been investigated. The first method is a robust servomechanism control approach (optimal tracking problem) that is a generalization of the classical proportional-plus-integral control to multiple input-multiple output systems. The second method is a control-allocation approach based on a quadratic programming formulation. A globally convergent fixed-point iteration algorithm has been developed to make onboard implementation of this method feasible. These methods have been applied to reconfigurable entry flight control design for the X-33 vehicle. Examples presented demonstrate simultaneous tracking of angle-of-attack and roll angle commands during failures of the fight body flap actuator. Although simulations demonstrate success of the first method in most cases, the control-allocation method appears to provide uniformly better performance in all cases.
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paper_title: Constrained control allocation

paper_content:

This paper addresses the problem of the allocation of several airplane flight controls to the generation of specified body-axis moments. The number of controls is greater than the number of moments being controlled, and the ranges of the controls are constrained to certain limits. They are assumed to be individually linear in their effect throughout their ranges of motion and independent of one another in their effects. The geometries of the subset of the constrained controls and of its image in moment space are examined. A direct method of allocating these several controls is presented that guarantees the maximum possible moment can be generated within the constraints of the controls. It is shown that no single generalized inverse can yield these maximum moments everywhere without violating some control constraint. A method is presented for the determination of a generalized inverse that satisfies given specifications which are arbitrary but restricted in number. We then pose and solve a minimization problem that yields the generalized inverse that best approximates the exact solutions. The results are illustrated at each step by an example problem involving three controls and two moments.
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paper_title: Integrated Adaptive Guidance and Control for Re-Entry Vehicles with Flight Test Results

paper_content:

To enable autonomous operation of future reusable launch vehicles, reconfiguration technologies will be needed to facilitate mission recovery following a major anomalous event. The Air Force’s Integrated Adaptive Guidance and Control program developed such a system for Boeing’s X-40A, and the total in-flight simulator research aircraft was employed to flight test the algorithms during approach and landing. The inner loop employs a modelfollowing/dynamic-inversion approach with optimal control allocation to account for control-surface failures. Further, the reference-model bandwidth is reduced if the control authority in any one axis is depleted as a result of control effector saturation. A backstepping approach is utilized for the guidance law, with proportional feedback gains that adapt to changes in the reference model bandwidth. The trajectory-reshaping algorithm is known as the optimum-path-to-go methodology. Here, a trajectory database is precomputed off line to cover all variations under consideration. An efficient representation of this database is then interrogated in flight to rapidly find the “best” reshaped trajectory, based on the current state of the vehicle’s control capabilities. The main goal of the flight-test program was to demonstrate the benefits of integrating trajectory reshaping with the essential elements of control reconfiguration and guidance adaptation. The results indicate that for more severe, multiple control failures, control reconfiguration, guidance adaptation, and trajectory reshaping are all needed to recover the mission.
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paper_title: Attitude control of a satellite with redundant thrusters

paper_content:

Redundant thrusters are generally used for a reliable attitude control system. Also, redundant thrusters yield a better performance if they are used appropriately. In this paper, the authors propose an efficient redundancy management algorithm to reduce the fuel consumption. The algorithm is based on a linear programming problem which is a constrained optimization problem. For the algorithm, a cost function is defined as a quantity related to the fuel consumption for a maneuver. The independent variables are the thrusters' on-times which are control input variables of a satellite dynamic model. The advantage of the proposed method is verified by numerical examples. The examples show that the proposed method consumes less fuel than an existing method for a given maneuvering command. A sub-optimal algorithm is also discussed for an onboard computation. The proposed algorithm is applied to two maneuvers: move-to-rest and rest-to-rest. This is verified by a numerical simulation.
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paper_title: Quantized Control Allocation of Reaction Control Jets and Aerodynamic Control Surfaces

paper_content:

Amixed-integer linear programming approach tomixing continuous andpulsed control effectors is proposed. The method is aimed at applications involving reentry vehicles that are transitioning from exoatmospheric flight to endoatmospheric flight. In this flight phase, aerodynamic surfaces are weak and easily saturated, and vehicles typically rely on pulsed reaction control jets for attitude control. Control laws for these jets have historically been designed using single-axis phase-plane analysis, which has proven to be sufficient formany applications wheremultiaxis coupling is insignificant and when failures have not been encountered. Here, we propose using a mixed-integer linear programming technique to blend continuous control effectors and pulsed jets to generate moments commanded by linear or nonlinear control laws. When coupled with fault detection and isolation logic, the control effectors can be reconfigured to minimize the impact of control effector failures or damage. When the continuous effectors can provide the desired moments, standard linear programming methods can be used to mix the effectors; however, when the pulsed effectors must be used to augment the aerodynamic surfaces, mixed-integer linear programming techniques are used to determine the optimal combination of jets to fire. The reaction jet control allocator acts as a nonuniformquantizer that applies amoment vector to the vehicle, which approximates the desired moment generated by a continuous control law. Lyapunov theory is applied to develop amethod for determining the region of attraction associated with a quantized vehicle attitude control system.
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paper_title: Multi-saturation anti-windup structure for satellite control

paper_content:

Future space missions demand high precision control in both angular and linear axes. Therefore propulsion systems providing effort in both axes with a high precision are needed. Microthrusters can satisfy these requirements. These actuators are usually subject to present an allocation problem. Moreover, because of this demand on high precision, the maximal propulsion capacity appears to be critically low, leading to a possible saturation of the actuators. A multi-saturation based model for a highly non-linear allocation function is presented. An anti-windup strategy dealing with the actuator saturation is proposed. Simulations consider a two satellites flight formation scenario. They show the improvement on performance and stability tolerance to off-nominal initial conditions. Simulations are based on a nominal model provided by Thales Alenia Space (TAS).
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paper_title: Fault-tolerant control allocation for Mars entry vehicle using adaptive control

paper_content:

Accurate and reliable control of planetary entry is a major challenge for planetary exploration vehicles. For Mars entry, uncertainties in atmospheric properties such as winds aloft and density pose a major problem for meeting precision landing requirements. Anticipated manned missions to Mars will also require levels of safety and fault tolerance not required during earlier robotic missions. This paper develops a nonlinear fault-tolerant controller specifically tailored for addressing the unique environmental and mission demands of future Mars entry vehicles. The controller tracks a desired trajectory from entry interface to parachute deployment, and has an adaptation mechanism that reduces tracking errors in the presence of uncertain parameters such as atmospheric density, and vehicle properties such as aerodynamic coefficients and inertias. This nonlinear control law generates the commanded moments for a discrete control allocation algorithm, which then generates the optimal controls required to follow the desired trajectory. The reaction control system acts as a non-uniform quantizer, which generates applied moments that approximate the desired moments generated by a continuous adaptive control law. If a fault is detected in the control jets, it reconfigures the controls and minimizes the impact of control failures or damage on trajectory tracking. It is assumed that a fault identification and isolation scheme already exists to identify failures. A stability analysis is presented, and fault tolerance performance is evaluated with non real-time simulation for a complete Mars entry trajectory tracking scenario using various scenarios of control effector failures. The results presented in the paper demonstrate that the control algorithm has a satisfactory performance for tracking a pre-defined trajectory in the presence of control failures, in addition to plant and environment uncertainties. Copyright © 2010 John Wiley & Sons, Ltd.
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paper_title: Thrust allocation with power management functionality on dynamically positioned vessels

paper_content:

A thrust allocation method with capabilities to assist the power management system on dynamically positioned ships is proposed in this paper. Its main benefits are reduction in frequency and/or load variations on the electric network, and a formulation of thruster bias which can be released when required by the power management system. To reduce load variations without increasing overall power consumption it is necessary to deviate from the thrust command given by the dynamic positioning system or joystick. The resulting deviation in position and velocity of the vessel is tightly controlled, and results show that small deviations are sufficient to fulfill the objective. For simplicity, the study has been limited to thrusters with fixed direction, having in mind that generalizations are fairly straightforward.
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paper_title: Thrust allocation with linear constrained quadratic cost function

paper_content:

In this paper, a method of thrust allocation based on a linearly constrained quadratic cost function capable of handling rotating azimuths is presented. The problem formulation accounts for magnitude and rate constraints on both thruster forces and azimuth angles. The advantage of this formulation is that the solution can be found with a finite number of iterations for each time step. Experiments with a model ship are used to validate the thrust allocation system.
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paper_title: Optimum allocation for multiple thrusters

paper_content:

In one concept for a Mobile Offshore Base (MOB), the several modules are kept in alignment and properly positioned with respect to one another by the use of many steerable thruster units. The research reported here develops an algorithm using linear programming to rationally allocate commanded thrust and thruster azimuth to multiple thrusters (more than two) in order to best achieve the desired force system on the MOB module. The linear programming approach allows explicit consideration of many of the real-world characteristics of the thrusters, including maximum thrust capability, maximum slew rate of the thrusters, etc. The algorithm was programmed and the results of the optimum allocation for two different scenarios are presented.
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paper_title: Antispin Thrust Allocation for Marine Vessels

paper_content:

In extreme weather conditions, thrusters on ships and rigs may be subject to severe thrust losses caused by ventilation and in-and-out-of-water events. When a thruster ventilates, air is sucked down from the surface and into the propeller. In more severe cases, parts of or even the whole propeller can be out of the water. These losses vary rapidly with time and cause increased wear and tear in addition to reduced thruster performance. In this paper, a thrust allocation strategy is proposed to reduce the effects of thrust losses and to reduce the possibility of multiple ventilation events. This thrust allocation strategy is named antispin thrust allocation, based on the analogous behavior of antispin wheel control of cars. The proposed thrust allocation strategy is important for improving the life span of the propulsion system and the accuracy of positioning for vessels conducting station keeping in terms of dynamic positioning or thruster-assisted position mooring. Application of this strategy can result in an increase of operational time and, thus, increased profitability. The performance of the proposed allocation strategy is demonstrated with experiments on a model ship.
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paper_title: Optimal thrust allocation for marine vessels

paper_content:

Abstract This paper presents a new thrust-allocation scheme which significantly reduces the fuel consumption for dynamic positioning of ships when rotatable azimuth thrusters are used. Thrust allocation is the problem of determining the thrust and the direction of each of the n thruster devices of a ship, given the desired forces and moment from the control law. The problem of singular configurations is pointed out and solved by using a modification of the singular-value decomposition. A filtering scheme is proposed to control the azimuth directions to reduce the wear and tear, and to minimize the required thrust and power in a general manner, allowing the enabling and disabling of thrusters. Both new theory and full-scale sea trials are presented.
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paper_title: Power Optimal Thruster Allocation

paper_content:

The thruster allocation is a complex mapping from a demanded force and turning moment to a set of thruster pitch/ rpm and azimuth setpoints. Since the number of degrees of freedom of thruster controls is normally very high, there exist many such mappings.
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paper_title: Optimal Speed and Heading Control for Stabilization of Parametric Oscillations in Ships

paper_content:

In this chapter, two strategies to actively control ships experiencing parametric roll resonance are proposed. Both approaches aim at changing the frequency of the parametric excitation by controlling the Doppler shift, which, in recent results, has been shown to be effective to reduce the roll angle significantly. However, exactly how to change the frequency of the parametric excitation to stabilize the parametric oscillations has remained an open research topic. Thus, two optimal control philosophies that alter the ship’s forward speed and heading angle, which in turn changes the encounter frequency and consequently the frequency of the parametric excitation, are presented. This is referred to as frequency detuning. As a first approach, the methodology of extremum seeking (ES) control is adapted for ships in parametric roll resonance to iteratively determine the optimal setpoint of the encounter frequency in order to avoid one of the conditions for parametric roll. The encounter frequency is consequently mapped to the ship’s forward speed and heading angle by a control allocation. This is formulated as a constrained nonlinear optimization problem. Second, the use of a model predictive control (MPC) is proposed. By addressing both states and input constraints explicitly, the MPC formulation is used to change the ship’s forward speed and heading angle in an optimal manner to reduce parametric roll oscillations. The effectiveness of the proposed control approaches to stabilize parametric oscillations, by simultaneously changing the ship’s forward speed and heading angle optimally, is illustrated by computer simulations. This clearly verifies the concept of frequency detuning.
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paper_title: Frequency Detuning Control by Doppler Shift

paper_content:

In this work, a control strategy for ship parametric roll resonance is developed by changing the frequency of the parametric excitation. This is achieved by varying the ship’s forward speed. This changes the perceived frequency of the waves, known as the encounter frequency (and thus the frequency of parametric excitation), via Doppler shift. A novel control philosophy based on a 1-DOF roll model, capable of describing the roll motion when the encounter frequency is non-constant, is proposed. The stability properties of the closed-loop system are mathematically proven and the effectiveness of the control system is demonstrated by simulating the closed-loop system using both a 6-DOF ship model and a simplified 1-DOF model describing the roll motion. The simulations are in agreement with the theoretical analysis.
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paper_title: Cooperative manipulation on the water using a swarm of autonomous tugboats

paper_content:

In this paper we present a strategy that allows a swarm of autonomous tugboats to cooperatively move a large object on the water. The two main challenges are: (1) the actuators are unidirectional and experience saturation; (2) the hydrodynamics of the system are difficult to characterize. The primary theoretical contribution of the paper addresses the first challenge. We present a tracking controller and force allocation strategy that, despite actuator limitations, result in asymptotically convergent tracking for a certain class of reference trajectories. The primary practical contribution is the introduction of a set of adaptive control laws that address the second challenge by compensating for unknown, and difficult to measure, hydrodynamic parameters. Experimental verification of the controllers is presented using a 1:36 scale model of a U.S. Navy ship, inside the Naval Academy's unique 380 ft testing tank.
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paper_title: Comprehensive framework for tracking control and thrust allocation for a highly overactuated autonomous surface vessel

paper_content:

In this paper, we present a comprehensive trajectory tracking framework for cooperative manipulation scenarios involving marine surface ships. Our experimental platform is a small boat equipped with six thrusters, but the technique presented here can be applied to a multiship manipulation scenario such as a group of autonomous tugboats transporting a disabled ship or unactuated barge. The primary challenges of this undertaking are as follows: (1) the actuators are unidirectional and experience saturation; (2) the hydrodynamics of the system are difficult to characterize; and (3) obtaining acceptable performance under field conditions (i.e., global positioning system errors, wind, waves, etc.) is arduous. To address these issues, we present a framework that includes trajectory generation, tracking control, and force allocation that, despite actuator limitations, results in asymptotically convergent trajectory tracking. In addition, the controller employs an adaptive feedback law to compensate for unknown—difficult to measure—hydrodynamic parameters. Field trials are conducted utilizing a 3-m vessel in a nearby estuary. © 2010 Wiley Periodicals, Inc. **image** © 2011 Wiley Periodicals, Inc.
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paper_title: Fault-accommodating thruster force allocation of an AUV considering thruster redundancy and saturation

paper_content:

A new approach to the fault-accommodating allocation of thruster forces of an autonomous underwater vehicle (AUV) is investigated in this paper. This paper presents a framework that exploits the excess number of thrusters to accommodate thruster faults during operation. First, a redundancy resolution scheme is presented that considers the presence of an excess number of thrusters along with any thruster faults and determines the reference thruster forces to produce the desired motion. This framework is then extended to incorporate a dynamic state feedback technique to generate reference thruster forces that are within the saturation limit of each thruster. Results from both computer simulations and experiments are provided to demonstrate the viability of the proposed scheme.
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paper_title: Adaptive control of nonlinear systems: A case study of underwater robotic systems

paper_content:

The problem of controlling underwater mobile robots in 6 degrees of freedom (DOF) is addressed. Underwater mobile robots where the number of thrusters and control surfaces exceeds the number of controllable DOF are considered in detail. Unlike robotic manipulators underwater mobile robots should include a velocity dependent thruster configuration matrix B(q), which modifies the standard manipulator equation to: Mq + C(q)q + g(x) = B(q)u where x = J(x)q. Uncertainties in the thruster configuration matrix due to unmodeled nonlinearities and partly known thruster characteristics are modeled as multiplicative input uncertainty. This article proposes two methods to compensate for the model uncertainties: (1) an adaptive passivity-based control scheme and (2) deriving a hybrid (adaptive and sliding) controller. The hybrid controller combines the adaptive scheme where M, C, and g are estimated on-line with a switching term added to the controller to compensate for uncertainties in the input matrix B. Global stability is ensured by applying Barbalat's Lyapunov-like lemma. The hybrid controller is simulated for the horizontal motion of the Norwegian Experimental Remotely Operated Vehicle (NEROV).
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paper_title: How the Four Wheels should Share Forces in an Optimum Cooperative Chassis Control

paper_content:

This paper investigates the effectiveness of weighting coefficients adaptation in simultaneous optimum distribution of lateral and longitudinal tire forces for improvement of vehicle handling and stability. Three different cases of weighting coefficients adaptation are considered in this study. Similar weighting coefficients for rear and front wheels are adopted in case (1). In case (2), the weighting coefficients of front wheels are greater than the corresponding value of rear wheels. Finally for case (3), the weighting coefficients of rear wheels are adopted to be greater than the values of front wheels. It is concluded that weighting coefficients adaptation can exert a large influence on the vehicle handling performance.
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paper_title: A Global Approach to Vehicle Control: Coordination of Four Wheel Steering and Wheel Torques

paper_content:

Currently, advanced control systems implemented on production ground vehicles have the goal of promoting maneuverability and stability. With proper coordination of steering and braking action, these goals may be achieved even when road conditions are severe. This paper considers the effect of steering and wheel torques on the dynamics of vehicular systems. Through the input-output linearization technique, the advantages of four-wheel steering (4WS) system and independent torques control are clear from a mathematical point of view. A sliding mode controller is also designed to modify driver’s steering and braking commands to enhance maneuverability and safety. Simulation results show the maneuverability and safety are improved. Although the controller design is based on a four-wheel steering vehicle, the algorithm can also be applied to vehicles of different configurations with slight changes.
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paper_title: Simultaneous Optimal Distribution of Lateral and Longitudinal Tire Forces for the Model Following Control

paper_content:

This paper presents a proposed optimum tire force distribution method in order to optimize tire usage and find out how the tires should share longitudinal and lateral forces to achieve a target vehicle response under the assumption that all four wheels can be independently steered, driven, and braked. The inputs to the optimization process are the driver's commands (steering wheel angle, accelerator pedal pressure, and foot brake pressure), while the outputs are lateral and longitudinal forces on all four wheels. Lateral and longitudinal tire forces cannot be chosen arbitrarily, they have to satisfy certain specified equality constraints. The equality constraints are related to the required total longitudinal force, total lateral force, and total yaw moment. The total lateral force and total moment required are introduced using the model responses of side-slip angle and yaw rate while the total longitudinal force is computed according to driver's command (traction or braking). A computer simulation of a closed-loop driver-vehicle system subjected to evasive lane change with braking is used to prove the significant effects of the proposed optimal tire force distribution method on improving the limit handling performance. The robustness of the vehicle motion with the proposed control against the coefficient of friction variation as well as the effect of steering wheel angle amplitude is discussed.
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paper_title: Real-time performance of control allocation for actuator coordination in heavy vehicles

paper_content:

This paper shows how real-time optimisation for actuator coordination, known as control allocation, can be a viable choice for heavy vehicle motion control systems. For this purpose, a basic stability control system implementing the method is presented. The real-time performance of two different control allocation solvers is evaluated and the use of dynamic weighting is analysed. Results show that sufficient vehicle stability can be achieved when using control allocation for actuator coordination in heavy vehicle stability control. Furthermore, real-time simulations indicate that the optimisation can be performed with the computational capacity of today's standard electronic control units.
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paper_title: MPC-based approach to active steering for autonomous vehicle systems

paper_content:

In this paper a novel approach to autonomous steering systems is presented. A model predictive control (MPC) scheme is designed in order to stabilize a vehicle along a desired path while fulfilling its physical constraints. Simulation results show the benefits of the systematic control methodology used. In particular we show how very effective steering manoeuvres are obtained as a result of the MPC feedback policy. Moreover, we highlight the trade off between the vehicle speed and the required preview on the desired path in order to stabilize the vehicle. The paper concludes with highlights on future research and on the necessary steps for experimental validation of the approach.
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paper_title: Stabilization of Automotive Vehicles Using Active Steering and Adaptive Brake Control Allocation

paper_content:

In this work a dynamic control allocation approach is presented for an automotive vehicle yaw stabilization scheme. The stabilization strategy consists of a high level module that deals with the vehicle motion control objective (yaw rate reference generation and tracking), a low level module that handles the braking control for each wheel (longitudinal slip control and maximal tire-road friction parameter estimation), and an intermediate level dynamic control allocation module that generates the longitudinal slip reference for the low level brake control module and commands front wheel steering angle corrections. The control allocation design is such that the actual torque about the yaw axis tends to the desired torque calculated form the high level module, with desirable distribution of control forces satisfying actuator constraints and minimal control effort objectives. Conditions for uniform asymptotic stability are given for the case when the control allocation includes adaptation of the tire-road maximal friction coefficients, and the scheme has been implemented in a realistic non linear multi body vehicle simulation environment. The simulation cases show that the yaw control allocation strategy stabilizes the vehicle in extreme maneuvers where the non linear vehicle yaw dynamics otherwise (without active braking or active steering) becomes unstable in the sense of over- or under steering. The control allocation implementation is efficient and suitable for low cost automotive electronic control units.
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paper_title: Yaw disturbance attenuation by robust decoupling of car steering

paper_content:

Abstract Robust decoupling of the lateral and yaw motions of a car has been achieved by feedback of the integrated yaw rate into front wheel steering. In the present paper the yaw disturbance attenuation is analyzed for a generic single-track vehicle model. The frequency limit, up to which yaw disturbances are attenuated, is calculated. For specific vehicle data, it is shown that this control law significantly reduces the influence of yaw disturbances on yaw rate and side-slip angle for low frequencies. This safety advantage is experimentally verified for μ-split braking.
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paper_title: Coordination of steering and individual wheel braking actuated vehicle yaw stability control

paper_content:

Active safety of road transport requires, among other things, the improvement of road vehicle yaw stability by active control. One approach for yaw dynamics improvement is to use differential braking, thereby creating the moment that is necessary to counteract the undesired yaw motion. An alternative approach is to command additional steering angles to create the counteracting moment. The maximum benefit, of course, can be gained through coordinated and combined use of both methods of corrective yaw motion generation in a control strategy. This problem has been approached by using a revised model regulator here as the main controller that utilizes coordinated steering and individual wheel braking actuation, with the aim of achieving better vehicle yaw stability control. Independent use of the individual means of actuation are treated first. Possible strategies for combined and coordinated use of steering and individual wheel braking action in a vehicle yaw dynamics controller are then presented. Simulation results on a nonlinear two track vehicle model are used to illustrate the effectiveness of the coordinated approach.
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paper_title: Coordinated vehicle dynamics control with control distribution

paper_content:

This paper investigates a hierarchically coordinated vehicle dynamics control approach with individual wheel torque and steering actuation. A high-level robust nonlinear sliding mode controller is designed to determine the generalized forces/moments required to achieve vehicle motion objectives. A weighted pseudo-inverse based control allocation method is employed for computationally efficient distribution of control effort to the slip and slip angle of each wheel. To avoid saturation, tire-road friction estimation is an essential part of the control distribution scheme. Two adverse driving scenario simulations are used to evaluate the effectiveness of this control system.
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paper_title: Vehicle dynamics integrated control for four-wheel-distributed steering and four-wheel-distributed traction/braking systems

paper_content:

In this article, vehicle dynamics integrated control algorithm using an on-line non-linear optimization method is proposed for 4-wheel-distributed steering and 4-wheel-distributed traction/braking systems. The proposed distribution algorithm minimizes work load of each tire, which is controlled to become the same value. The global optimality of the convergent solution of the recursive algorithm can be proved by extension to convex problems. This implies that theoretical limited performance of vehicle dynamics integrated control is clarified. Furthermore, the effect of this vehicle dynamics control for the 4-wheel-distributed steering and 4-wheel-distributed traction/braking systems is demonstrated by simulation to compare with the combination of the various actuators.
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paper_title: Comparison of Static and Dynamic Control Allocation Techniques for Integrated Vehicle Control

paper_content:

Abstract Comparison of static and dynamic control allocation techniques for nonlinear constrained optimal distribution of tire forces in a vehicle control system is presented. The total body forces and moments, obtained from a high level controller, are distributed among tire forces, which are constrained to nonlinear constraint of saturation, through two approaches. For the static control allocation technique the interior-point method is employed to perform the nonlinear constrained optimization problem. Also, by the dynamic control allocation technique a dynamic update law is derived for desired forces of each tire. Through simulation results efficiency of two approaches in enhancing vehicle handling and stability is evaluated and the results are compared.
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paper_title: The Effectiveness of ESC (Electronic Stability Control) in Reducing Real Life Crashes and Injuries

paper_content:

ESC (Electronic Stability Control) was introduced on the mass market in 1998. Since then, several studies showing the positive effects of ESC have been presented. In this study, data from crashes occurring in Sweden during 1998 to 2004 were used to evaluate the effectiveness of ESC on real life crashes. To control for exposure, induced exposure methods were used, where ESC-sensitive to ESC-insensitive crashes and road conditions were matched in relation to cars equipped with and without ESC. Cars of similar or in some cases identical make and model were used to isolate the role of ESC. The study shows the positive and consistent effects of ESC overall and in circumstances where the road has low friction. The overall effectiveness on all injury crash types except rear end crashes was 16.7 +/- 9.3 %, while for serious and fatal crashes the effectiveness was 21.6 +/- 12.8 %. The corresponding estimates for crashes with injured car occupants were 23.0+/-9.2% and 26.9+/-13.9%. For serious and fatal loss-of control type crashes on wet roads the effectiveness was 56.2 +/- 23.5 % and for roads covered with ice or snow the effectiveness was 49.2+/-30.2%. It was estimated that for Sweden, with a total of 500 vehicle related deaths annually, that 80-100 fatalities could be saved annually if all cars had ESC. On the basis of the results, it is recommended that all new cars sold should have ESC as standard equipment.
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paper_title: Global and local chassis control based on load sensing

paper_content:

Modern cars are equipped with an increasing number of active systems in order to help the driver, e.g. to better deal with unexpected changes in the vehicle dynamics. To improve the global vehicle dynamics, comfort and handling, coordination between traditionally stand-alone active systems is required. This demands for a Global Chassis Controller.
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paper_title: On the Control Allocation for Coordinated Ground Vehicle Dynamics Control Systems

paper_content:

This paper presents a new control allocation scheme for advanced coordinated vehicle dynamics control (CVDC) that takes into account vehicle operating condition and tire-surface friction coefficient. Individual tire slip and slip angles are selected as the control variables to resolve the inherent tire force nonlinear constraints. A real-time adaptable accelerated fixed-point (AFP) method is proposed to solve the magnitude and rate constrained quadratic programming control allocation (CA) problem. It could achieve faster convergence rates when control variable saturations occur. The performance of this control allocation approach is evaluated for adverse driving conditions simulated using the CarSimreg vehicle simulation package.
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paper_title: Combined Lateral Force and Yaw Moment Control to Maximize Stability as well as Vehicle Responsiveness during Evasive Maneuvering for Active Vehicle Handling Safety

paper_content:

Aiming to maximize not only stability limit but also vehicle responsiveness combined yaw moment and lateral force control as a cooperative control is presented. The total lateral force for front and rear wheels as well as direct yaw moment are introduced using the models response of side-slip angle and yaw rate. The side-slip response is the 2DOF vehicle response while the yaw rate response is a suggested first order lag to prevent oscillation in relation to steering input. Three different cases of combining lateral force and direct yaw moment control have been investigated using computer simulations. In order to show the effect of the cooperative control on vehicle stability and responsiveness, a direct yaw moment control to follow the yaw rate response is taken as comparison case. The effect of cooperative control is proved by computer simulations of the vehicle response to a single sine wave steer input with braking.
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paper_title: Control of a ground vehicle using quadratic programming based control allocation techniques

paper_content:

This paper examines the use of control allocation techniques for the control of multiple inputs to a ground vehicle to track a desired yaw rate trajectory while minimizing vehicle sideslip. The proposed controller uses quadratic programming accompanied by linear quadratic regulator gains designed around a linear vehicle model to arrive at a combination of vehicle commands. Several failure scenarios are examined and the results for two different quadratic programming approaches are presented along with a discussion of the advantages each method has to offer.
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paper_title: Energy-efficient control allocation with applications on planar motion control of electric ground vehicles

paper_content:

This paper presents a control method for tracking electric ground vehicle planar motions while achieving the optimal energy consumption. Sliding mode control and an energy-efficient control allocation (CA) scheme are synthesized to track the desired vehicle longitudinal, lateral, and yaw motions. By explicitly incorporating actuator efficiencies and actuator operating modes into the coordination of redundant in-wheel motors equipped on electric ground vehicles, vehicle planar motion control and operating energy optimization are achieved simultaneously. Different maneuvers are tested for comparisons between the standard and the energy-efficient CA schemes. Based on experimental data and some reasonable assumptions on the efficiencies of in-wheel motors, the energy-efficient CA dictates different torque distributions on all the wheels under consideration of different efficiencies. Simulation results indicate that, in comparison to the results by the standard CA scheme, less energy is consumed when the energy-efficient CA scheme is applied for controlling the electric ground vehicle planar motions.
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paper_title: Utilization of Actuators to Improve Vehicle Stability at the Limit: From Hydraulic Brakes Toward Electric Propulsion

paper_content:

The capability of over-actuated vehicles to maintain stability during limit handling is studied in this paper. A number of important differently actuated vehicles, equipped with hydraulic brakes toward more advanced chassis solutions, are presented. A virtual evaluation environment has specifically been developed to cover the complex interaction between the driver and the vehicle under control. In order to fully exploit the different actuators setup, and the hard nonconvex constraints they possess, the principle of control allocation by nonlinear optimization is successfully employed. The final evaluation is made by exposing the driver and the over-actuated vehicles to a safety-critical double lane change. Thereby, the differently actuated vehicles are ranked by a quantitative indicator of stability.
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paper_title: DESIGN AND CONTROL OF UNDER-ACTUATED AND OVER-ACTUATED MECHANICAL SYSTEMS - CHALLENGES OF MECHANICS AND MECHATRONICS

paper_content:

The majority of current mechanical systems used in machinery and especially those which are controlled by microprocessors can be described as equal-actuated. This means that the number of actuators (drives, controls) is equal to the number of degrees-of-freedom. The mechanical systems can have directly such property or it can be as such treated during the design and operation. Classical rigid mechanisms can have such property naturally. Generally all flexible mechanisms violate this property as not all flexible degrees-of-freedom can be actuated and thus directly controlled. However, there are important mechanical systems which do not fulfil this criterion at equality of actuators and degrees-of-freedom. The examples of under-actuated systems are bio-mechanical systems during dynamic phase of motion, technical systems of cranes, vehicles, underwater robots, missiles with failed engines, inverted pendulum, and ball on the beam. The examples of over-actuated systems are again biomechanical systems during the contact with ground and recently introduced redundantly actuated parallel robots. The paper first deals with the introduction and definition of the property of under-actuation, equal-actuation and over-actuation. Then the paper deals with the control of under-actuated or over-actuated systems and the challenge of how to design such systems. For the covering abstract see ITRD E125059.
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paper_title: Nonlinear hierarchical control allocation for vehicle yaw stabilization and rollover prevention

paper_content:

In ambition to minimize potential interferences between yaw stabilization and rollover prevention of an automotive vehicle, this work presents a new approach to integrate both objectives. It introduces rollover prevention in form of a nonlinear constraint on the control allocation of a yaw stabilizing controller, yielding a hierarchical allocation problem. A suitable algorithm in form of a dynamic update law addressing this problem is derived. Its implementation is computationally efficient and suitable for low cost automotive electronic control units. The proposed rollover constraint design does not require any sensory equipment in addition to the yaw stabilizing algorithm. Actuation is conducted by differential braking, while an extension to further actuators is possible. The method is validated using an industrial multi-body vehicle simulator.
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paper_title: Active Suspension Control to Improve Vehicl0e Ride and Steady-State Handling

paper_content:

In this paper, a full-vehicle active suspension system is designed to simultaneously improve vehicle ride comfort and steady-state handling performance. First, a linear suspension model of a vehicle and a nonlinear handling model are described. Next, the link between the suspension model and vehicle steady-state handling characteristics is analysed. Then, an H-infinity controller for the suspension is designed to achieve integrated ride-comfort and handling control. Finally, the controller is verified by computer simulations.
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paper_title: Stabilization of Automotive Vehicles Using Active Steering and Adaptive Brake Control Allocation

paper_content:

In this work a dynamic control allocation approach is presented for an automotive vehicle yaw stabilization scheme. The stabilization strategy consists of a high level module that deals with the vehicle motion control objective (yaw rate reference generation and tracking), a low level module that handles the braking control for each wheel (longitudinal slip control and maximal tire-road friction parameter estimation), and an intermediate level dynamic control allocation module that generates the longitudinal slip reference for the low level brake control module and commands front wheel steering angle corrections. The control allocation design is such that the actual torque about the yaw axis tends to the desired torque calculated form the high level module, with desirable distribution of control forces satisfying actuator constraints and minimal control effort objectives. Conditions for uniform asymptotic stability are given for the case when the control allocation includes adaptation of the tire-road maximal friction coefficients, and the scheme has been implemented in a realistic non linear multi body vehicle simulation environment. The simulation cases show that the yaw control allocation strategy stabilizes the vehicle in extreme maneuvers where the non linear vehicle yaw dynamics otherwise (without active braking or active steering) becomes unstable in the sense of over- or under steering. The control allocation implementation is efficient and suitable for low cost automotive electronic control units.
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paper_title: Optimal Rollover Prevention With Steer by Wire and Differential Braking

paper_content:

This paper uses Model Predictive Control theory to develop a framework for automobile stability control. The framework is then demonstrated with a roll mode controller which seeks to actively limit the peak roll angle of the vehicle while simultaneously tracking the driver’s yaw rate command. Initially, control law presented assumes knowledge of the complete input trajectory and acts as a benchmark for the best performance any controller could have on this system. This assumption is then relaxed by only assuming that the current driver steering command is available. Numerical simulations on a nonlinear vehicle model show that both control structures effectively track the driver intended yaw rate during extreme maneuvers while also limiting the peak roll angle. During ordinary driving, the controlled vehicle behaves identically to an ordinary vehicle. These preliminary results shows that for double lane change maneuvers, it is possible to limit roll angle while still closely tracking the driver’s intent.Copyright © 2003 by ASME
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paper_title: An Enhancement to an Electronic Stability Control System to Include a Rollover Control Function

paper_content:

This paper proposes a method to enhance existing electronic stability control systems such that a certain level of rollover mitigation performance is achieved. Such an enhancement is conducted through a control algorithm using only the standard ESC sensors. The analysis presented here reveals that a rollover mitigation system such as this will face a trade-off between the vehicle’s responsiveness and the control robustness due to error in the roll dynamics model and state estimation.
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paper_title: An Optimal Traction Control Scheme for Off-Road Operation of Robotic Vehicles

paper_content:

Active degrees of freedom provide a robotic vehicle the ability to enhance its performance in all terrain conditions. While active suspension systems are now commonplace in on-road vehicles, their application to off-road terrains has been little investigated. A fundamental component of such an application is a need to translate desired body motion commands into actuator values through the use of proprioceptive algorithms. The diverse nature of the terrains that might be encountered places variable demands upon the operation of the vehicle. This entails the potential use of a diverse set of algorithms designed to optimize mobility and performance. This paper presents a cohesive control scheme designed for the operation of an autonomous vehicle under all conditions. The ideas presented have been tested in simulation, and some have been used extensively in the field
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paper_title: Research on traction force allocation for omni-directional mobile robots

paper_content:

Eccentric centroid is one of the most serious problems of omni-directional mobile robot design. The reason is the limited precision in producing causes. Nowadays, most control approaches are studied basing on ideal models with regular centroid. Thus extra energy is needed to adjust difference between ideal centroid and actual centroid. In this paper, we propose an approach with eccentric centroid to analyze the distribution of traction on each wheel in this condition and get an optimization solution set of this traction with linear programming algorithm. We conducted simulation experiments to demonstrate the algorithm is of positive effect in reducing cost of extra energy.
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paper_title: Robust control allocation of redundantly actuated variable structure systems

paper_content:

This paper proposes new robust control allocation method of redundantly actuated variable structure systems. The original system is divided into two subsystems by factorizing the input matrix. Then all system uncertainties can be treated with the virtual control input. The control strategy has two steps. At the first step the virtual control input is found with sliding mode control with perturbation estimation method (SMCPE) considering system uncertainties then at the second step the real control input is calculated by using the Quadratic Optimization method. The simulation is conducted with the redundantly actuated leg-wheel hybrid structures which have 4 actuators and 3DOFs. The state-space model is derived from the dynamic equation with kinematic and dynamic constraints. The effect of kinematic error is also studied.
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paper_title: Development of a two-joint robotic fish for real-world exploration

paper_content:

Research on biomimetic robotic fish has been undertaken for more than a decade. Various robotic fish prototypes have been developed around the world. Although considerable research efforts have been devoted to understanding the underlying mechanism of fish swimming and construction of fish-like swimming machines, robotic fish have largely remained laboratory curiosities. This paper presents a robotic fish that is designed for application in real-world scenarios. The robotic fish adopts a rigid torpedo-shaped body for the housing of power, electronics, and payload. A compact parallel four-bar mechanism is designed for propulsion and maneuvering. Based on the kinematic analysis of the tail mechanism, the motion control algorithm of joints is presented. The swimming performance of the robotic fish is investigated experimentally. The swimming speed of the robotic fish can reach 1.36 m/s. The turning radius is 1.75 m. Powered by the onboard battery, the robotic fish can operate for up to 20 h. Moreover, the advantages of the biomimetic propulsion approach are shown by comparing the power efficiency and turning performance of the robotic fish with that of a screw-propelled underwater vehicle. The application of the robotic fish in a real-world probe experiment is also presented. © 2010 Wiley Periodicals, Inc. © 2011 Wiley Periodicals, Inc.
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paper_title: Globally Optimal Force Allocation in Active Mechanisms With Four Frictional Contacts

paper_content:

Mechanisms interacting with their environments that possess complete contact force controllability such as multifingered hands and walking vehicles 2 are considered in this article. In these systems, the redundancy in actuation can be used to optimize the force distribution characteristics. The resulting optimization problems can be highly nonlinear. Here, the redundancy in actuation is characterized using geometric reasoning which leads to simplifications in the formulation of the optimization problems. Next, advanced polynomial continuation techniques are adapted to solve for the global optimum of an important nonlinear optimization problem for the case of four frictional contacts. The algorithms developed here are not suited for real-time implementation. However, these algorithms can be used in off-line force planning, and they can be used to develop look-up tables for certain applications. The outputs of these algorithms can also be used as a baseline to evaluate the effectiveness of sub-optimal schemes.
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paper_title: Force distribution of a six-legged walking robot with high constant speed

paper_content:

For a walking robot with high constant body speed, the dynamic effects of the legs on the transfer phase are dominant compared with other factors. This paper presents a new force distribution algorithm to maximize walkable terrain without slipping considering the dynamic effects of the legs on the transfer phase. Maximizing the walkable terrain means having the capability of walking on more slippery ground under the same constraint, namely constant body speed. A simple force distribution algorithm applied to the proposed walking model with a pantograph leg shows an improvement in the capability of preventing foot-slippage compared with one using a pseudo-inverse method.
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paper_title: A Hierarchical Control Distribution Approach for Large Scale Over Actuated Systems

paper_content:

A general hierarchical methodology for control distribution in highly redundant system is presented. The new method makes use of distribution functions to approximate the feasible solution set and to keep in check the "curse of dimensionality". To improve the performance of the distribution function approach a hierarchical approach is proposed which decomposes a large scale control distribution problem in to many small scale control distribution problems to compromise the need for real-time computation against optimality. The main advantage of the proposed hierarchical approach is the de-coupling of many small scale problems from each other. The convergence and accuracy of the proposed method are demonstrated by numerical studies.
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paper_title: A control allocation approach for energetic swarm control

paper_content:

In this paper, an energetic swarm controller is developed that controls the swarm temperature, swarm centre position, and swarm potential energy. A sliding control approach is combined with a control allocation process to solve the overactuated control problem. The control allocation problem is solved using nonlinear programming software which allows the optimization problem to be solved with input saturation constraints. Furthermore, a low level trajectory controller based on dynamic feedback linearization is developed in order to improve the trajectory tracking performance of the individual swarm members. Application to a group of wheeled mobile robots is used to demonstrate the approach. Together, these results allow energetic swarm controllers to be implemented on wheeled mobile robot (WMR) systems with uncertainty and input saturation constraints.
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paper_title: Force allocation in a large-scale distributed active surface

paper_content:

This paper investigates the problem of closed-loop control of a small number of parameters by allocating actuation in a system with many binary degrees of freedom, using an actual large-scale air-jet table as an example. In this system, the desired force and torque are produced by a large number of spatially distributed binary air jets directing individual forces on an object. Various algorithms for solving the force allocation problem-determining the appropriate valve states in this hyper-redundant system-are investigated. The algorithms range from discrete optimal search to continuous constrained optimization to a hybrid hierarchical approach that can be distributed. The latter consists of using the continuous optimal solutions to recursively break the large optimization problems into smaller problems that can be solved using optimal search methods or precomputed lookup tables. A tradeoff between computation time and allocation error was found. The optimal algorithms yield low errors but the time is exponential in the number of actuators, while the continuous solutions execute quickly but yield larger errors. The hybrid hierarchical optimal algorithms give the best compromise between these conflicting goals, and their applicability spans the full range of degrees of freedom from a few to many thousands. These hierarchical algorithms are useful in many such highly redundant systems.
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paper_title: Two new design concepts for snake robot locomotion in unstructured environments

paper_content:

This communication presents and justifies ideas related to motion control of snake robots that are currently the subject of ongoing investigations by the authors. In particular, we highlight requirements for intelligent and efficient snake robot locomotion in unstructured environments, and subsequently we present two new design concepts for snake robots that comply with these requirements. The first design concept is an approach for sensing environment contact forces, which is based on measuring the joint constraint forces at the connection between the links of the snake robot. The second design concept involves allowing the cylindrical surface of each link of a snake robot to rotate by a motor inside the link in order to induce propulsive forces on the robot from its environments. The paper details the advantages of the proposed design concepts over previous snake robot designs.
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paper_title: Constrained optimization based control of real time large-scale systems: airjet object movement system

paper_content:

The control of real time, nonlinear, large-scale systems - systems with large aggregations of sensors and actuators - is seldom explored in actual operating physical systems. In such many-element systems, control issues such as actuation allocation, fusion of sensor data, and system identification emerge as challenging problems for large-scale system control. In this work, constrained optimization is used to solve these problems as applied to the control of an object moving system with 1,152 actuators and 32,000 sensors with a 2 ms control loop time. Solutions for allocating actuation among large numbers of actuators using hierarchical constrained optimization and fusing the output of many sensors into a small number of final measurements under tight real time constraints have been developed. This paper demonstrate that hyper-redundant systems are capable of system self-identification, and that constrained optimization can effectively solve problems associated with control of many-element systems.
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Title: Control Allocation – A Survey
Section 1: Introduction
Description 1: Summarize the objective of the survey and provide an overview of control allocation methods, with emphasis on recent developments across various application domains.
  
Section 2: Over-actuated mechanical systems
Description 2: Explain over-actuated mechanical systems, including the use of actuators and effectors, and the design of control algorithms.

Section 3: Control allocation introduction
Description 3: Define the primary objective of control allocation algorithms and discuss the general approach to control input computation.

Section 4: Control allocation for linear effector models
Description 4: Describe the assumption of linear effector models in control allocation and their applicability in various systems.

Section 5: Unconstrained linear control allocation
Description 5: Discuss methods to solve control allocation problems when there are no constraints on the control inputs, including the use of generalized inverses.

Section 6: Constrained linear control allocation
Description 6: Explain various strategies for control allocation problems where constraints on the control inputs must be satisfied, such as redistributed pseudo-inverse and linear programming.

Section 7: Error minimization using linear programming
Description 7: Detail approaches to minimize errors in achieving requested control inputs using linear programming techniques.

Section 8: Error minimization using quadratic programming
Description 8: Discuss the quadratic programming approach to minimize control allocation errors and its computational methods.

Section 9: Explicit approaches to constrained error minimization
Description 9: Explore explicit solutions for constrained error minimization in control allocation problems using methods like multi-parametric quadratic programming.

Section 10: Dynamics and fault tolerance
Description 10: Address dynamic control allocation methods that consider actuator dynamics and fault tolerance in the allocation process.

Section 11: Control allocation for nonlinear effector models
Description 11: Present the challenges and methods used in control allocation for systems with nonlinear effector models.

Section 12: Nonlinear programming methods
Description 12: Discuss the use of nonlinear programming techniques in control allocation and the associated computational challenges.

Section 13: Mixed-integer programming methods
Description 13: Describe the application of mixed-integer programming methods for control allocation problems involving piecewise linear models.

Section 14: Dynamic optimum-seeking methods
Description 14: Explain Lyapunov-function based dynamic optimization methods for control allocation.

Section 15: Direct nonlinear allocation
Description 15: Explore the method of direct allocation for nonlinear effector models and its historical development.

Section 16: Aerospace
Description 16: Review control allocation methods specific to aerospace applications, including aircraft and spacecraft control systems.

Section 17: Ships, underwater vehicles and offshore vessels
Description 17: Discuss various control allocation methods used in maritime applications for both surface and underwater vehicles.

Section 18: Automotive and ground vehicles
Description 18: Survey control allocation methods applied in automotive industry for yaw stability, electric propulsion, rollover prevention, and mobile robots.

Section 19: Other application areas
Description 19: Highlight control allocation techniques used in other fields such as legged robots, flapping wing vehicles, multi-agent systems, and distributed actuator arrays.

Section 20: Conclusions
Description 20: Summarize the state of understanding of control allocation, highlighting the well-established methods for linear models, and ongoing research challenges for nonlinear models.