A Survey of the Probability Density Function Control for Stochastic Dynamic Systems
---
paper_title: Recent advances on stochastic distribution control: Probability density function control

paper_content:

This semi-plenary paper presents a brief and selected survey on the advances on stochastic distribution control, where the purpose of the controller design is to control the shape of output probability density functions (pdf) of non-Gaussian and general stochastic systems. This research was motivated through the requirement of distribution shape control of a number of practical systems in 1996. Following the developments since 1996, much research has been performed internationally and journal special issues and invited session at major conferences have been seen since 2001. It is expected that this survey will provide readers with some up-to-date information on this new area.
---
paper_title: A Fokker-Planck control framework for multidimensional stochastic processes

paper_content:

An efficient framework for the optimal control of probability density functions (PDFs) of multidimensional stochastic processes is presented. This framework is based on the Fokker-Planck equation that governs the time evolution of the PDF of stochastic processes and on tracking objectives of terminal configuration of the desired PDF. The corresponding optimization problems are formulated as a sequence of open-loop optimality systems in a receding-horizon control strategy. Many theoretical results concerning the forward and the optimal control problem are provided. In particular, it is shown that under appropriate assumptions the open-loop bilinear control function is unique. The resulting optimality system is discretized by the Chang-Cooper scheme that guarantees positivity of the forward solution. The effectiveness of the proposed computational framework is validated with a stochastic Lotka-Volterra model and a noised limit cycle model.
---
paper_title: A Stochastic Averaging Approach for Feedback Control Design of Nonlinear Systems Under Random Excitations

paper_content:

This paper presents a method for designing and quantifying the performance of feedback stochastic controls for nonlinear systems. The design makes use of the method of stochastic averaging to reduce the dimension of the state space and to derive the Ito stochastic differential equation for the response amplitude process. The moment equation of the amplitude process closed by the Rayleigh approximation is used as a means to characterize the transient performance of the feedback control. The steady state and transient response of the amplitude process are used as the design criteria for choosing the feedback control gains. Numerical examples are studied to demonstrate the performance of the control.
---
paper_title: Shaping probability density functions using a switching linear controller

paper_content:

Abstract Linear controllers for dynamic systems have been used in a stochastic context to decrease the variance of certain key variables. An extension is to use nonlinear controllers for shaping the entire probability density function (PDF). Typically, polynomial controllers are preferred due to their flexibility and also because their continuity makes them mathematically attractive. However, due to restrictions on stability, these controllers tend to be quasi-linear around the desired operating point. So, in this paper, a switching linear controller is presented as an alternative for PDF shaping. It is shown that, despite the discontinuity, an analytical solution of the PDF can be obtained using the Fokker–Planck–Kolmogorov equation. Also, for a simple case, even the Gram-Charlier coefficients can be computed analytically. Thus, it is demonstrated that the PDF shaping problem with a switching controller does not cause any further complications than using polynomial controllers. It is then shown in simulation that switching controllers are more robust and could create more skewness due to their discontinuity.
---
paper_title: Nonlinear filter design using Fokker-Planck-Kolmogorov probability density evolutions

paper_content:

The Fokker-Planck-Kolmogorov equation (FPKE) in conjunction with Bayes conditional density update formula provides optimal estimates for a general continuous-discrete nonlinear filtering problem. It is well known that the analytical solution of FPKE and Bayes formula are extremely difficult to obtain except in a few special cases. Hence, we address this problem using numerical approaches. The efficient numerical solution of FPKE presented relies on the key issue of adaptively calculating the domain over which the state probability density function is to be evaluated, which is done using Chebyshev's inequality. Application to a passive tracking example shows that this approach can provide consistent estimators when measurement nonlinearities and noise levels are high.
---
paper_title: Particle Model Predictive Control for Probability Density Functions

paper_content:

In industrial processes such as steel making, it is often important to suppress the dispersion of end-product quality, which motivates a control problem of probability density functions (PDFs) representing the dispersion of the state of dynamical systems. Although we consider a system whose state equation is represented by a nonlinear ordinary differential equation, the time evolution of a PDF of the state is governed by a partial differential equation (PDE) with a finite dimensional control input. Model predictive control (MPC), also known as receding horizon control, is applied for the optimal control of the dispersion. An MPC method of the PDF is formulated exactly using the time evolution of the PDF, which we call exact formulation. We propose another control method for the PDF, which is formulated by Monte Carlo (MC) approximation, and call this control method particle MPC (PMPC). We apply the proposed control method to a one-dimensional nonlinear system and compare the results of the two control methods, which show that the proposed control method is valid and has a high potential for real-time implementation.
---
paper_title: Control of stationary probability density of nonlinear systems subject to Poisson white noise excitation

paper_content:

An innovative design procedure for controlling a nonlinear system subject to Poisson white noise excitation to target a specified stationary probability density function is proposed based on the ap...
---
paper_title: Stochastic averaging of n-dimensional non-linear dynamical systems subject to non-Gaussian wide-band random excitations

paper_content:

The averaged generalized Fokker–Planck–Kolmogorov (GFPK) equation for response of n-dimensional (n-d) non-linear dynamical systems to non-Gaussian wide-band stationary random excitation is derived from the standard form of equation of motion. The explicit expressions for coefficients of the fourth-order approximation of the averaged GFPK equation are given in series form. Conditions for convergences of these series are pointed out. The averaged GFPK equation is then reduced to that for 1-d dynamical systems derived by Stratonovich and compared with the closed form of GFPK equation for n-d dynamical systems subject to Poisson white noise derived by Di Paola and Falsone. Finally, this averaged GFPK equation is further reduced to that for quasi linear system subject to non-Gaussian wide-band stationary random excitation. Stationary probability density for quasi linear system subject to filtered Poisson white noise is obtained. Theoretical results for an example are confirmed by using Monte-Carlo simulation for different parameter values.
---
paper_title: Feedback control of nonlinear stochastic systems for targeting a specified stationary probability density

paper_content:

In the present paper, an innovative procedure for designing the feedback control of multi-degree-of-freedom (MDOF) nonlinear stochastic systems to target a specified stationary probability density function (SPDF) is proposed based on the technique for obtaining the exact stationary solutions of the dissipated Hamiltonian systems. First, the control problem is formulated as a controlled, dissipated Hamiltonian system together with a target SPDF. Then the controlled forces are split into a conservative part and a dissipative part. The conservative control forces are designed to make the controlled system and the target SPDF have the same Hamiltonian structure (mainly the integrability and resonance). The dissipative control forces are determined so that the target SPDF is the exact stationary solution of the controlled system. Five cases, i.e., non-integrable Hamiltonian systems, integrable and non-resonant Hamiltonian systems, integrable and resonant Hamiltonian systems, partially integrable and non-resonant Hamiltonian systems, and partially integrable and resonant Hamiltonian systems, are treated respectively. A method for proving that the transient solution of the controlled system approaches the target SPDF as t->~ is introduced. Finally, an example is given to illustrate the efficacy of the proposed design procedure.
---
paper_title: Nonlinear stochastic control via stationary probability density functions

paper_content:

Presents a control strategy for externally excited stochastic systems with parametric uncertainties. The objective is to drive the system to match a target probability density function (PDF) in steady state. The control consists of a nonlinear feedback and a switching term for handling parameter uncertainty. The fast rate of convergence to the stationary PDF is observed in the time evolution of the mean and variance of the closed loop system through Monte Carlo simulations. The dependence of the control performance on the number of terms in the polynomial control series and on the rate of convergence to the steady state distribution are studied numerically. This control design based on the stationary PDF is also applied to the tracking of a time-varying probability density function.
---
paper_title: Optimal control of probability density functions of stochastic processes*

paper_content:

Abstract A Fokker‐Planck framework for the formulation of an optimal control strategy of stochastic processes is presented. Within this strategy, the control objectives are defined based on the probability density functions of the stochastic processes. The optimal control is obtained as the minimizer of the objective under the constraint given by the Fokker‐Planck model. Representative stochastic processes are considered with different control laws and with the purpose of attaining a final target configuration or tracking a desired trajectory. In this latter case, a receding‐horizon algorithm over a sequence of time windows is implemented.
---
paper_title: Infinite Dimensional Optimization and Control Theory

paper_content:

Part I. Finite Dimensional Control Problems: 1. Calculus of variations and control theory 2. Optimal control problems without target conditions 3. Abstract minimization problems: the minimum principle for the time optimal problem 4. Abstract minimization problems: the minimum principle for general optimal control problems Part II. Infinite Dimensional Control Problems: 5. Differential equations in Banach spaces and semigroup theory 6. Abstract minimization problems in Hilbert spaces: applications to hyperbolic control systems 7. Abstract minimization problems in Banach spaces: abstract parabolic linear and semilinear equations 8. Interpolation and domains of fractional powers 9. Linear control systems 10. Optimal control problems with state constraints 11. Optimal control problems with state constraints: The abstract parabolic case Part III. Relaxed Controls: 12. Spaces of relaxed controls: topology and measure theory 13. Relaxed controls in finite dimensional systems: existence theory 14. Relaxed controls in infinite dimensional spaces: existence theory.
---
paper_title: Probability Via Expectation

paper_content:

1 Uncertainty, Intuition, and Expectation.- 1 Ideas and Examples.- 2 The Empirical Basis.- 3 Averages over a Finite Population.- 4 Repeated Sampling: Expectation.- 5 More on Sample Spaces and Variables.- 6 Ideal and Actual Experiments: Observables.- 2 Expectation.- 1 Random Variables.- 2 Axioms for the Expectation Operator.- 3 Events: Probability.- 4 Some Examples of an Expectation.- 5 Moments.- 6 Applications: Optimization Problems.- 7 Equiprobable Outcomes: Sample Surveys.- 8 Applications: Least Square Estimation of Random Variables.- 9 Some Implications of the Axioms.- 3 Probability.- 1 Events, Sets and Indicators.- 2 Probability Measure.- 3 Expectation as a Probability Integral.- 4 Some History.- 5 Subjective Probability.- 4 Some Basic Models.- 1 A Model of Spatial Distribution.- 2 The Multinomial, Binomial, Poisson and Geometric Distributions.- 3 Independence.- 4 Probability Generating Functions.- 5 The St. Petersburg Paradox.- 6 Matching, and Other Combinatorial Problems.- 7 Conditioning.- 8 Variables on the Continuum: The Exponential and Gamma Distributions.- 5 Conditioning.- 1 Conditional Expectation.- 2 Conditional Probability.- 3 A Conditional Expectation as a Random Variable.- 4 Conditioning on a ? Field.- 5 Independence.- 6 Statistical Decision Theory.- 7 Information Transmission.- 8 Acceptance Sampling.- 6 Applications of the Independence Concept.- 1 Renewal Processes.- 2 Recurrent Events: Regeneration Points.- 3 A Result in Statistical Mechanics: The Gibbs Distribution.- 4 Branching Processes.- 7 The Two Basic Limit Theorems.- 1 Convergence in Distribution (Weak Convergence).- 2 Properties of the Characteristic Function.- 3 The Law of Large Numbers.- 4 Normal Convergence (the Central Limit Theorem).- 5 The Normal Distribution.- 6 The Law of Large Numbers and the Evaluation of Channel Capacity.- 8 Continuous Random Variables and Their Transformations.- 1 Distributions with a Density.- 2 Functions of Random Variables.- 3 Conditional Densities.- 9 Markov Processes in Discrete Time.- 1 Stochastic Processes and the Markov Property.- 2 The Case of a Discrete State Space: The Kolmogorov Equations.- 3 Some Examples: Ruin, Survival and Runs.- 4 Birth and Death Processes: Detailed Balance.- 5 Some Examples We Should Like to Defer.- 6 Random Walks, Random Stopping and Ruin.- 7 Auguries of Martingales.- 8 Recurrence and Equilibrium.- 9 Recurrence and Dimension.- 10 Markov Processes in Continuous Time.- 1 The Markov Property in Continuous Time.- 2 The Case of a Discrete State Space.- 3 The Poisson Process.- 4 Birth and Death Processes.- 5 Processes on Nondiscrete State Spaces.- 6 The Filing Problem.- 7 Some Continuous-Time Martingales.- 8 Stationarity and Reversibility.- 9 The Ehrenfest Model.- 10 Processes of Independent Increments.- 11 Brownian Motion: Diffusion Processes.- 12 First Passage and Recurrence for Brownian Motion.- 11 Action Optimisation Dynamic Programming.- 1 Action Optimisation.- 2 Optimisation over Time: the Dynamic Programming Equation.- 3 State Structure.- 4 Optimal Control Under LQG Assumptions.- 5 Minimal-Length Coding.- 6 Discounting.- 7 Continuous-Time Versions and Infinite-Horizon Limits.- 8 Policy Improvement.- 12 Optimal Resource Allocation.- 1 Portfolio Selection in Discrete Time.- 2 Portfolio Selection in Continuous Time.- 3 Multi-Armed Bandits and the Gittins Index.- 4 Open Processes.- 5 Tax Problems.- 13 Finance: 'Risk-Free' Trading and Option Pricing.- 1 Options and Hedging Strategies.- 2 Optimal Targeting of the Contract.- 3 An Example.- 4 A Continuous-Time Model.- 5 How Should it Be Done?.- 14 Second-Order Theory.- 1 Back to L2.- 2 Linear Least Square Approximation.- 3 Projection: Innovation.- 4 The Gauss-Markov Theorem.- 5 The Convergence of Linear Least Square Estimates.- 6 Direct and Mutual Mean Square Convergence.- 7 Conditional Expectations as Least Square Estimates: Martingale Convergence.- 15 Consistency and Extension: The Finite-Dimensional Case.- 1 The Issues.- 2 Convex Sets.- 3 The Consistency Condition for Expectation Values.- 4 The Extension of Expectation Values.- 5 Examples of Extension.- 6 Dependence Information: Chernoff Bounds.- 16 Stochastic Convergence.- 1 The Characterization of Convergence.- 2 Types of Convergence.- 3 Some Consequences.- 4 Convergence in rth Mean.- 17 Martingales.- 1 The Martingale Property.- 2 Kolmogorov's Inequality: the Law of Large Numbers.- 3 Martingale Convergence: Applications.- 4 The Optional Stopping Theorem.- 5 Examples of Stopped Martingales.- 18 Large-Deviation Theory.- 1 The Large-Deviation Property.- 2 Some Preliminaries.- 3 Cramer's Theorem.- 4 Some Special Cases.- 5 Circuit-Switched Networks and Boltzmarm Statistics.- 6 Multi-Class Traffic and Effective Bandwidth.- 7 Birth and Death Processes.- 19 Extension: Examples of the Infinite-Dimensional Case.- 1 Generalities on the Infinite-Dimensional Case.- 2 Fields and ?-Fields of Events.- 3 Extension on a Linear Lattice.- 4 Integrable Functions of a Scalar Random Variable.- 5 Expectations Derivable from the Characteristic Function: Weak Convergence324.- 20 Quantum Mechanics.- 1 The Static Case.- 2 The Dynamic Case.- References.
---
paper_title: A Hamiltonian approach using partial differential equations for open-loop stochastic optimal control

paper_content:

This paper utilizes a minimum principle for infinite dimensional systems for the optimal control of systems constrained by the Fokker-Plank equation governing the evolution of a state probability density function. From the backwards evolution of the corresponding adjoint system, we define a Hamiltonian and use its gradient to construct a numerical optimal control. The basic nature of the adjoint system allows for all of the necessary terms defining the control to be inferred from stochastic process samples which is exploited in provided examples. Solving stochastic optimal control problems utilizing stochastic processes is a promising approach for solving open-loop stochastic optimal control problems of non-linear dynamic systems with a multi-dimensional state vector.
---
paper_title: Application of linearization methods with probability density criteria in control problems

paper_content:

The application of statistical and equivalent linearization methods with criteria in probability density space to the determination of quasi-optimal control for the nonlinear dynamic system with the quadratic criterion is discussed in the paper. To determine the quasi-optimal control two modified versions of standard iterative procedure are proposed, where two versions of statistical and equivalent linearization methods were combined with the optimal control method for linear systems with the mean-square criterion. The detailed considerations and numerical results are given for the Duffing oscillator.
---
paper_title: Output PDFs control for linear stochastic systems with arbitrarily bounded random parameters: a new application of the Laplace transform

paper_content:

This paper presents a controller design for the shape control of output probability density functions for non-Gaussian dynamic stochastic systems represented by an ARMAX model. The coefficients in the ARMAX model are random and are represented by their known probability density functions and the system is also subjected to a random noise. All these random parameters and noise are assumed bounded and non-Gaussian. To formulate a simple relationship among the probability density function of the system output and those of random parameters and the noise term, the Laplace transform is applied to all the probability density functions. As a result, a simple mathematical relationship amongst all the transferred probability density functions of the system output and random parameters has been established. Since controlling the shape of the output probability density function is equivalent to controlling the shape of its Laplace transform, a new performance function is introduced, whose minimization is performed so as to design an optimal control input sequence that makes the shape of the output probability density function follow a target distribution. A gradient search technique is applied in the optimization.
---
paper_title: Optimal probability density function control for NARMAX stochastic systems

paper_content:

This paper presents a new control strategy for a class of non-Gaussian stochastic systems so that the output probability density function (PDF) of the system can be made to follow a desired PDF. The system considered is represented by an Nonlinear AutoRegressive and Moving Average with eXogenous (NARMAX) inputs with input channel time-delay and non-Gaussian noise. A multi-step-ahead nonlinear cumulative cost function is used to improve tracking performance. For this purpose, a relationship between the PDFs of all the inputs and the PDFs of multiple-step-ahead output is formulated by constructing an auxiliary multivariate mapping. By minimizing this performance function, a new explicit predictive controller design algorithm is established with less conservatism than some previous results. Furthermore, an improved approach is developed to guarantee the local stability of the closed-loop system by tuning the weighting parameters recursively. Simulations are given to demonstrate the effectiveness of the proposed control algorithm and desired results have been obtained.
---
paper_title: Nonlinear stochastic control via stationary probability density functions

paper_content:

Presents a control strategy for externally excited stochastic systems with parametric uncertainties. The objective is to drive the system to match a target probability density function (PDF) in steady state. The control consists of a nonlinear feedback and a switching term for handling parameter uncertainty. The fast rate of convergence to the stationary PDF is observed in the time evolution of the mean and variance of the closed loop system through Monte Carlo simulations. The dependence of the control performance on the number of terms in the polynomial control series and on the rate of convergence to the steady state distribution are studied numerically. This control design based on the stationary PDF is also applied to the tracking of a time-varying probability density function.
---
paper_title: Joint stochastic distribution tracking control for multivariate descriptor systems with non-gaussian variables

paper_content:

This article considers a new tracking control problem for a class of nonlinear stochastic descriptor systems where the tracked target is a given joint probability density function (JPDF). The controlled plants can be represented by multivariate discrete-time descriptor systems with non-Gaussian disturbances and nonlinear output equations. The control objective is to find crisp algorithms such that the conditional output JPDFs can follow the given target JPDF. A direct relationship is established between the JPDFs of the transformed tracking error and the stochastic input. An optimisation approach is presented such that the distances between the output JPDF and the desired one are minimised. Furthermore, a stabilisation suboptimal control strategy is proposed using the linear matrix inequality-based Lyapunov theory. Finally, simulations are provided to demonstrate the effectiveness of the stochastic tracking control algorithms.
---
paper_title: On the Dynamic Response of Continuous Systems Including Model Uncertainty

paper_content:

This paper presents a technique for obtaining the response of linear continuous systems with parameter uncertainties subjected to deterministic excitation
---
paper_title: Shaping probability density functions using a switching linear controller

paper_content:

Abstract Linear controllers for dynamic systems have been used in a stochastic context to decrease the variance of certain key variables. An extension is to use nonlinear controllers for shaping the entire probability density function (PDF). Typically, polynomial controllers are preferred due to their flexibility and also because their continuity makes them mathematically attractive. However, due to restrictions on stability, these controllers tend to be quasi-linear around the desired operating point. So, in this paper, a switching linear controller is presented as an alternative for PDF shaping. It is shown that, despite the discontinuity, an analytical solution of the PDF can be obtained using the Fokker–Planck–Kolmogorov equation. Also, for a simple case, even the Gram-Charlier coefficients can be computed analytically. Thus, it is demonstrated that the PDF shaping problem with a switching controller does not cause any further complications than using polynomial controllers. It is then shown in simulation that switching controllers are more robust and could create more skewness due to their discontinuity.
---
paper_title: Probability density function for stochastic response of non-linear oscillation system under random excitation

paper_content:

Abstract Probability density function (PDF) of stochastic responses is a critical topic in uncertainty analysis. In this paper, orthogonal decomposition technique was extended to discuss non-stationary response of non-linear oscillator under random excitation. The PDF of stochastic reponses is represented by a set of standardized multivariable orthogonal polynomials. According to the Galerkin scheme, the original problem, which has to solve the Fokker–Planck–Kolmogorov (FPK) equation, was converted to a first-order linear ordinary differential equation, in terms of unknown time-dependent coefficients. Then, stationary and non-stationary PDFs of uncertainty responses were obtained. In numerical examples, first-order and second-order non-linear systems exposed to the Gaussian white noise were considered. Finally, the accuracy of the proposed method was demonstrated through appropriate comparisons to Monte-Carlo simulation and analytical results.
---
paper_title: Control design for first-order processes: shaping the probability density of the process state

paper_content:

Abstract While the long term behaviour of a stationary stochastic process is concisely summarized by the probability density function (PDF), many well-established regulatory control techniques focus only on two quantities derived from the PDF, the mean and variance, as key design targets. This paper presents a technique for the design of control laws that shape all aspects of the PDF of a first-order process. By developing relationships between the moments of the PDF and the coefficients in a control law parameterization, an approximate control law is developed to shape the process PDF as desired.
---
paper_title: Response variability in structural dynamics

paper_content:

This paper presents a technique for treating uncertainties in the dynamic models of a structural system. The formulation of the method is presented for a simple case of a single-degree-of-freedom linear oscillator. The uncertainties are modelled as random variables and are assumed to be time-independent. The solution is expanded as a series involving the random terms, and a system of linear ordinary differential equations for the unknowns of the problem is derived using the weighted residual method. The system of equations is then integrated in time and the response variability is computed. Validation calculations show that the results from the method agree well with those obtained by other solution techniques. Finally, some possible applications and extensions of the present work are discussed.
---
paper_title: Bounded Dynamic Stochastic Systems: Modelling and Control

paper_content:

1 Preliminaries.- 1.1 Introduction.- 1.2 An example: flocculation model.- 1.3 The aim of the new development.- 1.4 The structure of the book.- 1.5 Random variables and stochastic processes.- 1.5.1 Random variables and their distribution functions.- 1.5.2 Mean and variance.- 1.5.3 Random vector.- 1.5.4 Conditional mean.- 1.6 Stochastic processes.- 1.7 Some typical distributions.- 1.7.1 Gaussian distribution.- 1.7.2 Uniform distribution.- 1.7.3 ? distribution.- 1.8 Conclusions.- 2 Control of SISO Stochastic Systems: A Fundamental Control Law.- 2.1 Introduction.- 2.2 Preliminaries on B-splines artificial neural networks.- 2.3 Model representation.- 2.3.1 Static models.- 2.3.2 Dynamic models.- 2.4 System modelling and parameter estimation.- 2.4.1 Modelling of static systems.- 2.4.2 Modelling of linear dynamic systems.- 2.5 Control algorithm design.- 2.5.1 Control algorithm for static systems.- 2.5.2 Control algorithm for linear dynamic systems.- 2.5.3 Constraints on input energy for dynamic systems.- 2.6 Discussions.- 2.6.1 Adaptive control.- 2.6.2 Modelling and control of time delay systems.- 2.6.3 On-line measurement of Vk.- 2.6.4 Controllability, observability and stability.- 2.7 Examples.- 2.7.1 Static system modelling.- 2.7.2 A design example for dynamic systems.- 2.8 Conclusions.- 3 Control of MIMO Stochastic Systems: Robustness and Stability.- 3.1 Introductionx.- 3.2 Model representation.- 3.2.1 State space form.- 3.2.2 The input-output form.- 3.3 The controller using V(k).- 3.3.1 Measurement of V(k).- 3.3.2 Feedback control using V(k).- 3.3.3 Stability issues.- 3.4 The controller using f(y, U(k)).- 3.4.1 The formulation of control algorithm.- 3.4.2 Stability issues.- 3.5 An illustrative example.- 3.5.1 Control algorithm design.- 3.5.2 Simulation results.- 3.6 Conclusions and discussions.- 4 Realization of Perfect Tracking.- 4.1 Introduction.- 4.2 Preliminaries and model representation.- 4.3 Main result.- 4.4 Simulation results.- 4.4.1 Controller design.- 4.4.2 Simulation results.- 4.5 An LQR based algorithm.- 4.6 Conclusions.- 5 Stable Adaptive Control of Stochastic Distributions.- 5.1 Introduction.- 5.2 Model representation.- 5.3 On-line estimation and its convergence.- 5.4 Adaptive control algorithm design.- 5.5 Stability analysis.- 5.6 A simulated example.- 5.7 Conclusions.- 6 Model Reference Adaptive Control.- 6.1 Introduction.- 6.2 Model representation.- 6.3 An adaptive controller design.- 6.3.1 Construction of the reference model.- 6.3.2 Construction of error dynamics.- 6.4 Adaptive tuning rules for K(t) and Q(t).- 6.5 Robust adaptive control scheme.- 6.5.1 Control scheme when ?(t) ? 0.- 6.5.2 Control scheme when both e0 and ? are present.- 6.6 A case study.- 6.7 Conclusions and discussions.- 7 Control of Nonlinear Stochastic Systems.- 7.1 Introduction.- 7.2 Model representation.- 7.3 Control algorithm design.- 7.4 Stability issues.- 7.5 A neural network approach.- 7.5.1 Training of the neural networks.- 7.5.2 A linearised control algorithm.- 7.6 Two examples.- 7.7 Calculation of ?.- 7.8 Conclusions.- 8 Application to Fault Detection.- 8.1 Introduction.- 8.2 Model representation.- 8.3 Fault detection.- 8.3.1 Fault detection for static systems.- 8.3.2 Dynamic systems.- 8.3.3 Fault detection signal.- 8.4 An adaptive diagnostic observer.- 8.5 Discussions.- 8.6 An identification based FDD.- 8.7 Fault diagnosis.- 8.7.1 The algorithm.- 8.7.2 An applicability study.- 8.8 Discussions and conclusions.- 9 Advanced Topics.- 9.1 Introduction.- 9.2 Square root models.- 9.3 Control algorithm design.- 9.3.1 Finding weights from ?(y, u(k)).- 9.3.2 The control algorithm.- 9.4 Simulations.- 9.5 Continuous-time models.- 9.6 The control algorithm.- 9.7 Control of the mean and variance.- 9.7.1 The control of output mean value.- 9.7.2 The control of output variance.- 9.8 Singular stochastic systems.- 9.8.1 Model representation.- 9.8.2 Control algorithm design.- 9.9 Pseudo ARMAX systems.- 9.10 Filtering issues.- 9.11 Conclusions.- References.
---
paper_title: Regulatory control design for stochastic processes: shaping the probability density function

paper_content:

This paper presents a regulatory controller synthesis technique that makes a pre-selected probability density function (PDF) for the closed-loop (CL) process the target of the design. The proposed design technique, referred to as PDF-shaping, approximately solves the integral equation giving the PDF of the CL process dynamics. The parameterization of the closed-loop process dynamics results in the parameterization of a controller that yields the required closed-loop PDF. Controller synthesis is then a matter of selecting, based on engineering or economic concerns, a target distribution, and then applying the proposed technique to find the approximate closed-loop dynamics. The special case of PDF-shaping using Gram-Charlier PDFs is presented, as well as the general case. Results of this novel approach to controller synthesis are demonstrated with numerical simulations for an example process.
---
paper_title: Gradient descent optimisation for ILC-based stochastic distribution control

paper_content:

Stochastic Distribution Control (SDC) for non-Gaussian system is a mathematically complicated yet practical problem to solve. The most recent solution involves a Radial Basis Function Neural Network (RBFNN) framework to approximate non-Gaussian output Probability Density Function (PDF). The dynamic weights of such neural network are controlled within each batch of ILC, using a dedicated adaptive controller. Then, between adjacent batches, an Iterative Learning Control (ILC) algorithm is applied to tune RBFNN centres and widths. The most practical ILC-based SDC applications use sum of squared PDF tracking errors (calculated within each batch of ILC) to tune RBFNN parameters through so called P-type ILC laws. This paper first analyses the practical disadvantages of P-type ILC laws to use in SDC applications which include robustness, sensitivity, and operational problems. Then, based on the gradients of squared PDF tracking errors (calculated within batches), it develops a new ILC-based non-Gaussian stochastic control. Simulation results demonstrate the effectiveness of proposed gradient descent optimisation method for non-Gaussian stochastic distribution control.
---
paper_title: MIMO Probability Density Function Control Using Simple LOG-MLP Neural Networks

paper_content:

This paper presents a new model for the control of multivariable output probability density function (PDF). Firstly, a Multi-Layer Perceptron (MLP) neural network is adopted to approximate the static output PDF of the MIMO systems. Nonlinear principal component analysis (NLPCA) is then used to reduce the order of the obtained static neural network model and the dynamics of the system is considered based on the lower-order model. After this, an integrated solution is provided to set up the system with lower-order dynamics for the purpose of stochastic distribution control. The controller design is then presented in detail. Finally, a simulation example is given to demonstrate the effectiveness of the method and encouraging results have been obtained.
---
paper_title: Periodic learning of B-spline models for output PDF control: application to MWD control

paper_content:

Periodic learning of B-spline basis functions model for the output probability density function (PDF) control of non-Gaussian systems is studied in this paper using the recursive least square algorithm. Within each control interval, the basis functions are fixed and the control input design is performed that controls the shape of the output PDFs. However, between each control interval, periodic learning techniques are used to tune the shape of the basis functions. This has been shown to be able to improve the accuracy of the B-spline approximation model. As such, the overall B-spline model of the output PDFs becomes a dual-model related to both time and space variables. The algorithm has been applied to a simulation study of the molecular weight distribution (MWD) control of a styrene polymerization process, leading to some interesting results.
---
paper_title: Fault tolerant control based on stochastic distributions via MLP neural networks

paper_content:

An optimal fault tolerant control (FTC) scheme using output probability density functions (PDFs) is studied for the general stochastic continuous time systems. Being different from the classical FTC problems, the measured information is the stochastic distribution of the system output rather than its value. The control objective is to use the output PDFs to design control schemes that can compensate the fault and attenuate the disturbance. A multi-layer perceptron (MLP) neural network is applied to approximate the output PDFs, with which nonlinear principal component analysis (NLPCA) can be used to reduce the model order. For the established continuous-time weighting system with disturbances and uncertainties which is used to link the input and the weights, an LMI-based feasible FTC method is presented to assure that the fault can be well measured and compensated, where the H"~ performance index for the uncertain error systems is optimized.
---
paper_title: PDF Control of Stochastic Parameter System Using Linear Matrix Inequalities

paper_content:

This paper presents a probability density function (PDF)tracking control strategy for stochastic parameter system based on a square root B-spline model for the output probabil- ity density functions.The objective is to control the PDF of system output to follow a desired PDF.Using the B-spline ap- proximation the tracking problem of PDFs is transferred to the tracking of given weights values which correspond to the given PDF.At the same time,the system is transferred to a MIMO system,whose output is the weight value vector.As a result, the tracking of given weights values is transformed to a tracking problem of a MIMO system.Furthermore,a sufficient condition of the PDF of system output to follow a desired PDF is given and the control strategy is obtained by solving several linear ma- trix inequalities.A simulated example is used to demonstrate the efficiency of the proposed approach and encouraging results have been gained.
---
paper_title: Lyapunov-based controller design for bounded dynamic stochastic distribution control

paper_content:

Following developments in the modelling and control of the output probability density function of dynamic stochastic systems, an approach for the controller design is presented using square root approximation, where a set of B-spline functions are used to approximate the square root of the measured output probability density function to guarantee its positiveness. A performance function is defined which measures the tracking error of the output probability density function with respect to a given distribution. Instead of finding an optimal control which minimises this performance function and then analysing the stability of the closed loop system, the new approach directly uses the performance function as a Lyapunov function to design the required controller. As a result, the controller obtained not only guarantees the decreasing of the performance function with respect to time, but also stabilises the closed-loop system, realising an asymptotically tracking performance of the output probability density function with respect to its target distribution. The algorithm described has been tested on a simulated example and desired results have been achieved.
---
paper_title: An adaptive PI controller for non-Gaussian stochastic systems

paper_content:

In this paper, a new algorithm for an adaptive Proportional-Integrator (PI)controller for nonlinear systems subjected to stochastic non-Gaussian disturbance is studied. The minimum entropy control is applied to decrease the closed-loop tracking error under an iterative learning control (ILC) basis. The key issue here is to divide the control horizon into a number of equally time-domain intervals called batches. Within each interval there are a fixed number of sample points. The design procedure is divided into two main algorithms, within each batch and between any two adjacent batches. D-type ILC laws are employed to tune the PI controller coefficients between two adjacent batches. However. within each batch, the PI coefficients are fixed. A sufficient condition has been established to guarantee the stability of the closed-loop system. An illustrated example of one-link manipulator with revolute joints actuated by a DC motor is included to demonstrate the use of control algorithm, and satisfactory results have been obtained.
---
paper_title: Modelling and control of the flame temperature distribution using probability density function shaping

paper_content:

This paper presents three control algorithms for the output probability density function (PDF) control of the 2D and 3D flame distribution systems. For the 2D flame distribution systems, control methods for both static and dynamic flame systems are presented, where at first the temperature distribution of the gas jet flames along the cross-section is approximated. Then the flame energy distribution (FED) is obtained as the output to be controlled by using a B-spline expansion technique. The general static output PDF control algorithm is used in the 2D static flame system, where the dynamic system consists of a static temperature model of gas jet flames and a second-order actuator. This leads to a second-order closed-loop system, where a state space model is used to describe the dynamics with the weights of the B-spline functions as the state variables. Finally, a predictive control algorithm is designed for such an output PDF system. For the 3D flame distribution systems, all the temperature values of the flames are firstly mapped into one temperature plane, and the shape of the temperature distribution on this plane can then be controlled by the 3D flame control method proposed in this paper. Three cases are studied for the proposed control methods and desired simulation results have been obtained.
---
paper_title: Recent advances on stochastic distribution control: Probability density function control

paper_content:

This semi-plenary paper presents a brief and selected survey on the advances on stochastic distribution control, where the purpose of the controller design is to control the shape of output probability density functions (pdf) of non-Gaussian and general stochastic systems. This research was motivated through the requirement of distribution shape control of a number of practical systems in 1996. Following the developments since 1996, much research has been performed internationally and journal special issues and invited session at major conferences have been seen since 2001. It is expected that this survey will provide readers with some up-to-date information on this new area.
---
paper_title: Probability density function predictive control based on the rational square-root B-spline model

paper_content:

A predictive control algorithm for probability density function (PDF) is presented based on the rational square-root B-spline based PDF model. The rational square-root PDF model is studied first, based on the analysis of the relationship between the pseudo weight and the actual weight, the rational square-root B-spline model can be easily set up; then the predictive model is deduced; a performance index is chosen next to design the predictive controller; and simulation study is performed for the molecular weight distribution (MWD) control of a styrene polymerization process to test the validity of the predictive control algorithm.
---
paper_title: Bounded Dynamic Stochastic Systems: Modelling and Control

paper_content:

1 Preliminaries.- 1.1 Introduction.- 1.2 An example: flocculation model.- 1.3 The aim of the new development.- 1.4 The structure of the book.- 1.5 Random variables and stochastic processes.- 1.5.1 Random variables and their distribution functions.- 1.5.2 Mean and variance.- 1.5.3 Random vector.- 1.5.4 Conditional mean.- 1.6 Stochastic processes.- 1.7 Some typical distributions.- 1.7.1 Gaussian distribution.- 1.7.2 Uniform distribution.- 1.7.3 ? distribution.- 1.8 Conclusions.- 2 Control of SISO Stochastic Systems: A Fundamental Control Law.- 2.1 Introduction.- 2.2 Preliminaries on B-splines artificial neural networks.- 2.3 Model representation.- 2.3.1 Static models.- 2.3.2 Dynamic models.- 2.4 System modelling and parameter estimation.- 2.4.1 Modelling of static systems.- 2.4.2 Modelling of linear dynamic systems.- 2.5 Control algorithm design.- 2.5.1 Control algorithm for static systems.- 2.5.2 Control algorithm for linear dynamic systems.- 2.5.3 Constraints on input energy for dynamic systems.- 2.6 Discussions.- 2.6.1 Adaptive control.- 2.6.2 Modelling and control of time delay systems.- 2.6.3 On-line measurement of Vk.- 2.6.4 Controllability, observability and stability.- 2.7 Examples.- 2.7.1 Static system modelling.- 2.7.2 A design example for dynamic systems.- 2.8 Conclusions.- 3 Control of MIMO Stochastic Systems: Robustness and Stability.- 3.1 Introductionx.- 3.2 Model representation.- 3.2.1 State space form.- 3.2.2 The input-output form.- 3.3 The controller using V(k).- 3.3.1 Measurement of V(k).- 3.3.2 Feedback control using V(k).- 3.3.3 Stability issues.- 3.4 The controller using f(y, U(k)).- 3.4.1 The formulation of control algorithm.- 3.4.2 Stability issues.- 3.5 An illustrative example.- 3.5.1 Control algorithm design.- 3.5.2 Simulation results.- 3.6 Conclusions and discussions.- 4 Realization of Perfect Tracking.- 4.1 Introduction.- 4.2 Preliminaries and model representation.- 4.3 Main result.- 4.4 Simulation results.- 4.4.1 Controller design.- 4.4.2 Simulation results.- 4.5 An LQR based algorithm.- 4.6 Conclusions.- 5 Stable Adaptive Control of Stochastic Distributions.- 5.1 Introduction.- 5.2 Model representation.- 5.3 On-line estimation and its convergence.- 5.4 Adaptive control algorithm design.- 5.5 Stability analysis.- 5.6 A simulated example.- 5.7 Conclusions.- 6 Model Reference Adaptive Control.- 6.1 Introduction.- 6.2 Model representation.- 6.3 An adaptive controller design.- 6.3.1 Construction of the reference model.- 6.3.2 Construction of error dynamics.- 6.4 Adaptive tuning rules for K(t) and Q(t).- 6.5 Robust adaptive control scheme.- 6.5.1 Control scheme when ?(t) ? 0.- 6.5.2 Control scheme when both e0 and ? are present.- 6.6 A case study.- 6.7 Conclusions and discussions.- 7 Control of Nonlinear Stochastic Systems.- 7.1 Introduction.- 7.2 Model representation.- 7.3 Control algorithm design.- 7.4 Stability issues.- 7.5 A neural network approach.- 7.5.1 Training of the neural networks.- 7.5.2 A linearised control algorithm.- 7.6 Two examples.- 7.7 Calculation of ?.- 7.8 Conclusions.- 8 Application to Fault Detection.- 8.1 Introduction.- 8.2 Model representation.- 8.3 Fault detection.- 8.3.1 Fault detection for static systems.- 8.3.2 Dynamic systems.- 8.3.3 Fault detection signal.- 8.4 An adaptive diagnostic observer.- 8.5 Discussions.- 8.6 An identification based FDD.- 8.7 Fault diagnosis.- 8.7.1 The algorithm.- 8.7.2 An applicability study.- 8.8 Discussions and conclusions.- 9 Advanced Topics.- 9.1 Introduction.- 9.2 Square root models.- 9.3 Control algorithm design.- 9.3.1 Finding weights from ?(y, u(k)).- 9.3.2 The control algorithm.- 9.4 Simulations.- 9.5 Continuous-time models.- 9.6 The control algorithm.- 9.7 Control of the mean and variance.- 9.7.1 The control of output mean value.- 9.7.2 The control of output variance.- 9.8 Singular stochastic systems.- 9.8.1 Model representation.- 9.8.2 Control algorithm design.- 9.9 Pseudo ARMAX systems.- 9.10 Filtering issues.- 9.11 Conclusions.- References.
---
paper_title: Constrained PI Tracking Control for Output Probability Distributions Based on Two-Step Neural Networks

paper_content:

In this paper, a new method for the control of the shape of the conditional output probability density function (pdf) for general nonlinear dynamic stochastic systems is presented using two-step neural networks (NNs). Following the square-root B-spline NN approximation to the measured output pdf, the problem is transferred into the tracking of dynamic weights. Different from the previous related works, time-delay dynamic NNs with undetermined parameters are employed to identify the nonlinear relationships between the control input and the weighting vectors. In order to achieve the required control objective and satisfy the state constraints due to the property of output pdfs, a constrained PI tracking controller is designed by solving a class of linear matrix inequalities and algebraic equations. With the proposed tracking controller and adaptive projection algorithms, both identification and tracking errors can be made to converge to zero, and the state constraints can also be simultaneously guaranteed. Finally, two simulated examples are given, which effectively demonstrate the use of the proposed control algorithm.
---
paper_title: Data driven output joint probability density function control for multivariate non-linear non-Gaussian systems

paper_content:

This study presents a novel data-based joint probability density function (JPDF) control strategy for multivariate non-linear non-Gaussian stochastic systems so that the output JPDF of the system can be made to follow a desired JPDF. The output JPDF, which is usually immeasurable, is estimated according to the output sequence of the system. The multi-step-ahead cumulative performance index is constructed with respect to the control objective and is minimised based on an intelligent optimisation algorithm. By minimising this performance function, a new predictive controller design algorithm is established with more simple formulation and less computation load than existed results. Furthermore, a new approach is developed to guarantee ‘convergence in distribution’. Finally, simulations are given to demonstrate the effectiveness of the proposed control algorithm and some desired results have been obtained.
---
paper_title: Statistic information tracking of Non-Gaussian systems: A data-driven control framework based on adaptive NN modeling

paper_content:

A new type of data-driven control framework for Non-Gaussian stochastic systems is established in this paper. Different from the traditional feedback style, the driven information for tracking problem is the statistic information set (SIS) of the output rather than the output value. The set of statistical information (including the moments and the entropy) or probability density functions (PDFs) of the output are the measured information and the controlled objective. Under this framework, a mixed two-step adaptive neural network (NN) modeling is established with combining a static NN for description of the statistic information or PDFs and a dynamic one for identification of the relationship between input and output weight vectors. An adaptive PI tracking controller based on the proposed dynamic NNs is designed so as to track a target stochastic distribution. Finally, simulation results on a model in paper-making processes are given to demonstrate the effectiveness.
---
paper_title: An ILC-Based Adaptive Control for General Stochastic Systems With Strictly Decreasing Entropy

paper_content:

In this paper, a new method for adaptive control of general nonlinear and non-Gaussian unknown stochastic systems has been proposed. The method applies the minimum entropy control scheme to decrease the closed-loop randomness of the output under an iterative learning control (ILC) basis. Both modeling and control of the plant are performed using dynamic neural networks. For this purpose, the whole control horizon is divided into a certain number of time domain subintervals called batches and a pseudo-D-type ILC law is employed to train the plant model and controller parameters so that the entropy of the closed-loop tracking error is made to decrease batch by batch. The method has the advantage of decreasing the output uncertainty versus the advances of batches along the time horizon. The analysis on the proposed ILC convergence is made and a set of demonstrable experiment results is also provided to show the effectiveness of the obtained control algorithm, where encouraging results have been obtained.
---
paper_title: Neural PID control strategy for superheated steam temperature based on minimum entropy

paper_content:

A novel control algorithm is applied to control superheated steam temperature in a power plant. Since the disturbances existing in practical processes are probably non-Gaussian, the performance index of system is constructed by minimizing the entropy. The optimal parameters of controller are obtained and applied to control superheated steam temperature in a power plant. The simulation results verify its effectiveness
---
paper_title: Cascade control of superheated steam temperature with neuro-PID controller.

paper_content:

In this paper, an improved cascade control methodology for superheated processes is developed, in which the primary PID controller is implemented by neural networks trained by minimizing error entropy criterion. The entropy of the tracking error can be estimated recursively by utilizing receding horizon window technique. The measurable disturbances in superheated processes are input to the neuro-PID controller besides the sequences of tracking error in outer loop control system, hence, feedback control is combined with feedforward control in the proposed neuro-PID controller. The convergent condition of the neural networks is analyzed. The implementation procedures of the proposed cascade control approach are summarized. Compared with the neuro-PID controller using minimizing squared error criterion, the proposed neuro-PID controller using minimizing error entropy criterion may decrease fluctuations of the superheated steam temperature. A simulation example shows the advantages of the proposed method.
---
paper_title: Statistical Information Based Single Neuron Adaptive Control for Non-Gaussian Stochastic Systems

paper_content:

Based on information theory, the single neuron adaptive control problem for stochastic systems with non-Gaussian noises is investigated in this paper. Here, the statistic information of the output within a receding window rather than the output value is used for the tracking problem. Firstly, the single neuron controller structure, which has the ability of self-learning and self-adaptation, is established. Then, an improved performance criterion is given to train the weights of the single neuron. Furthermore, the mean-square convergent condition of the proposed control algorithm is formulated. Finally, comparative simulation results are presented to show that the proposed algorithm is superior to the PID controller. The contributions of this work are twofold: (1) the optimal control algorithm is formulated in the data-driven framework, which needn’t the precise system model that is usually difficult to obtain; (2) the control problem of non-Gaussian systems can be effectively dealt with by the simple single neuron controller under improved minimum entropy criterion.
---
paper_title: A data-driven based adaptive fault diagnosis scheme for nonlinear stochastic distribution systems via 2-step neural networks and descriptor model

paper_content:

A data-driven based adaptive sensor fault diagnosis (FD) and compensation scheme for stochastic distribution control (SDC) systems is studied in this paper, where an augmented descriptor model is employed. Unlike traditional SDC systems, the driven information is the output probability density function (OPDF), which is a kind of image mapping information to the true output values. A mixed 2-step adaptive neural network (NN) framework is studied, where the static NN is to describe the OPDF while the dynamic NN is to identify nonlinearity, uncertainty of system and to refine the OPDF model based on data of the input and statistic information of the output. To identify the sensor fault, an augmented descriptor system is employed, where the augmented state includes the plant state and the sensor fault. As a result, an adaptive strategy is given for nonlinear parameter estimation and sensor fault identification simultaneously. A sensor compensation rule is given to restore the plant by adding it to output feedback controller. The simulation examples are given to verify the effectiveness of the presented algorithm.
---
paper_title: Minimum entropy based run-to-run control for semiconductor processes with uncertain metrology delay

paper_content:

A novel run-to-run control methodology for semiconductor processes with uncertain metrology delay which is developed by combining the minimum error entropy and the optimal control strategy is presented. In most semiconductor processes, the product quality data from the previous run are not often available before the start of the next run. Thus, the corrective step is often delayed by one batch or more, and the duration of the delay is uncertain with stochastic characteristics. Coupled with inaccurate process models, the delay may lead to significant variations of the process outputs even with the use of exponentially weighted moving average (EWMA) controllers. This paper proposes a new method of handling the uncertain metrology delay from a probability viewpoint. The fundamentals of the run-to-run control systems are first reexamined, and then an innovative performance index is given by incorporating the entropy (or information potential) and the mean value of tracking error with constraints on control input energy. The probability density function (PDF) based optimal control algorithm is proposed for processes where the disturbance and delay are non-Gaussian and the stability of the algorithm is analyzed. In addition, the methodology of the proposed control strategy is extended to include recursive PDF estimation and on-line real time implementation. The paper also includes a simulation example of minimum entropy control of a tungsten chemical-vapor deposition process to illustrate the methodology. Furthermore, comparisons between the conventional EWMA method and the proposed method are done to show the advantages of our newly proposed method.
---
paper_title: Multi-objective minimum entropy controller design for stochastic processes

paper_content:

Minimum variance control is an established method in control of systems corrupted by noise. In these cases, as it is not possible to directly control the actual value of the system variables, one aims to reduce the variations instead. However, when the system noises are non-Gaussian, this approach fails because non-Gaussian noise cannot be characterised by simple measures such as variance. In these cases, the Entropy is proposed as a generalisation of the variance measure and the control objective becomes that of minimising the Entropy. Previously a limited form of this problem has been solved using first order Newtonian methods. In this paper, the control objective is first expanded to also include an error term related to the closed loop tracking performance, and the combined problem is then solved using a fast global optimisation search algorithm. The effectiveness of the approach is demonstrated through a case study based on a first principle model of a nonlinear heat exchanger.
---
paper_title: Minimum entropy control of closed-loop tracking errors for dynamic stochastic systems

paper_content:

The entropy has been used to characterize the uncertainty of the tracking error for general nonlinear and non-Gaussian stochastic systems. A recursive optimization solution has been developed and the local stability condition of the closed-loop system has been established. The generality of this algorithm has been proved by the special case study of the minimum variance control for linear Gaussian systems.
---
paper_title: Observer-based parametric decoupling controller design for a class of multi-variable non-linear uncertain systems

paper_content:

ABSTRACTThis paper presents a novel decoupling control strategy for Lipschitz multi-variable non-linear uncertain systems. Using the explicit parametric design, an observer-based output feedback controller has been developed with free parameters while the closed-loop system can be further described by transfer function matrix with these free parameters. The coupling effects of the systems would be attenuated if the free parameters are optimised where the performance criterion is given based on the norm of the transfer functions. Moreover, the sufficient conditions of stabilization have been obtained for observer, controller and closed-loop system, respectively. Following the procedure of the presented control strategy, an illustrative numerical example is given to demonstrate the effectiveness of the presented control strategy. In addition, the similar design approach has been discussed for filtering problem which is a potential extension of the presented control strategy.
---
paper_title: Decoupling control in statistical sense: minimised mutual information algorithm

paper_content:

This paper presents a novel concept to describe the couplings among the outputs of the stochastic systems which are represented by NARMA models. Compared with the traditional coupling description, the presented concept can be considered as an extension using statistical independence theory. Based on this concept, the decoupling control in statistical sense is established with the necessary and sufficient conditions for complete decoupling. Since the complete decoupling is difficult to achieve, a control algorithm has been developed using the Cauchy-Schwarz mutual information criterion. Without modifying the existing control loop, this algorithm supplies a compensative controller to minimise the statistical couplings of the system outputs and the local stability has been analysed. In addition, a further discussion illustrates the combination of the presented control algorithm and data-based mutual information estimation. Finally, a numerical example is given to show the feasibility and efficiency of the proposed algorithm.
---
paper_title: Minimum Error Entropy Filter for Fault Detection of Networked Control Systems

paper_content:

In this paper, fault detection of networked control systems with random delays, packet dropout and noises is studied. The filter is designed using a minimum error entropy criterion. The residual generated by the filter is then evaluated to detect faults in networked control systems. An illustrative networked control system is used to verify the effectiveness of the proposed approach.
---
paper_title: Particle Model Predictive Control for Probability Density Functions

paper_content:

In industrial processes such as steel making, it is often important to suppress the dispersion of end-product quality, which motivates a control problem of probability density functions (PDFs) representing the dispersion of the state of dynamical systems. Although we consider a system whose state equation is represented by a nonlinear ordinary differential equation, the time evolution of a PDF of the state is governed by a partial differential equation (PDE) with a finite dimensional control input. Model predictive control (MPC), also known as receding horizon control, is applied for the optimal control of the dispersion. An MPC method of the PDF is formulated exactly using the time evolution of the PDF, which we call exact formulation. We propose another control method for the PDF, which is formulated by Monte Carlo (MC) approximation, and call this control method particle MPC (PMPC). We apply the proposed control method to a one-dimensional nonlinear system and compare the results of the two control methods, which show that the proposed control method is valid and has a high potential for real-time implementation.
---
paper_title: Article Function Based Fault Detection for Uncertain Multivariate Nonlinear Non-Gaussian Stochastic Systems Using Entropy Optimization Principle

paper_content:

In this paper, the fault detection in uncertain multivariate nonlinear non-Gaussian stochastic systems is further investigated. Entropy is introduced to characterize the stochastic behavior of the detection errors, and the entropy optimization principle is established for the fault detection problem. The principle is to maximize the entropies of the stochastic detection errors in the presence of faults and to minimize the entropies of the detection errors in the presence of disturbances. In order to calculate the entropies, the formulations of the joint probability density functions (JPDFs) of the stochastic errors are presented in terms of the known JPDFs of both the disturbances and the faults. By using the novel performance indexes and the formulations for the entropies of the detection errors, new fault detection design methods are provided for the considered multivariate nonlinear non-Gaussian plants. Finally, a simulation example is given to illustrate the efficiency of the proposed fault detection algorithm.
---
paper_title: Design of new fault diagnosis and fault tolerant control scheme for non-Gaussian singular stochastic distribution systems

paper_content:

New fault diagnosis (FD) and fault tolerant control (FTC) algorithms for non-Gaussian singular stochastic distribution control (SDC) systems are presented in this paper. Different from general SDC systems, in singular SDC systems, the relationship between the weights and the control input is expressed by a singular state space model, which increases the difficulty in the FD and FTC design. The proposed approach relies on an iterative learning observer (ILO) for fault estimation. The fault may be constant, fast-varying or slow-varying. Based on the estimated fault information, the fault tolerant controller can be designed to make the post-fault probability density function (PDF) still track the given distribution. Simulations are given to show the effectiveness of the proposed FD and FTC algorithms.
---
paper_title: Minimum entropy control for non-linear and non-Gaussian two-input and two-output dynamic stochastic systems

paper_content:

In this study, the problem of control algorithm design for a class of nonlinear two-input and two-output systems with non-Gaussian disturbances is investigated, where a general non-linear auto-regressive moving average with exogenous model is used to describe the system. Based on the deduced probability density functions of tracking errors, a new performance index is established using the entropy and joint entropy so as to characterise the uncertainty of the tracking errors of the closed-loop system. This performance also includes the expectations of tracking errors and the constrains of control energy. A recursive optimisation control algorithm is obtained by minimising the performance index. Moreover, the local stability condition of the closed-loop systems is established after some formulations. Finally, the comparative simulation results are presented to show that the performance of the proposed algorithm is superior to that of proportional–integral–derivative controller.
---
paper_title: Minimized coupling in probability sense for a class of multivariate dynamic stochastic control systems

paper_content:

This paper presents a novel concept which is firstly established to describe the probabilistic property of the couplings among system states. Based on this concept, a new algorithm is presented to minimize the elements of the states covariance matrix for a class of multivariate dynamic stochastic nonlinear systems, which are represented by a set of Itō stochastic differential equations. Since the measurable covariance matrix is dynamically related to the control inputs, this controller combines feedback linearization, covariance control and LQR can thus attenuate the pairwise dependence of the states. Moreover, decoupling in probability sense can be realized. The mean square stability is proved for the closed loop systems. To evaluate the performance of the closed loop systems with different controllers, the assessment criterion is proposed. An illustrative example is utilized to demonstrate the use of the control algorithm, and desired results have been obtained.
---
paper_title: Statistic Tracking Control for Non-Gaussian Systems Using T-S Fuzzy Model

paper_content:

Abstract This paper studies a new type of control framework for dynamical stochastic systems, called statistic tracking control. Non-Gaussian systems are considered and the tracked objective is the statistical information of a given target probability density function (PDF). Following neural network approximation to the performance function, the concerned problem is transferred into the tracking of given weights. Different from the previous related works, the time delay T-S fuzzy models with the exogenous disturbances are applied to represent the nonlinear weighting dynamics. Meanwhile, the generalized PI controller structure and the improved convex LMI algorithms are proposed to fulfil the tracking problem. Furthermore, in order to enhance the robust performance, the peak-to-peak measure is applied to optimize the tracking performance.
---
paper_title: EKF-Based Enhanced Performance Controller Design for Nonlinear Stochastic Systems

paper_content:

In this paper, a novel control algorithm is presented to enhance the performance of the tracking property for a class of nonlinear and dynamic stochastic systems subjected to non-Gaussian noises. Although the existing standard PI controller can be used to obtain the basic tracking of the systems, the desired tracking performance of the stochastic systems is difficult to achieve due to the random noises. To improve the tracking performance, an enhanced performance loop is constructed using the EKF-based state estimates without changing the existing closed loop with a PI controller. Meanwhile, the gain of the enhanced performance loop can be obtained based upon the entropy optimization of the tracking error. In addition, the stability of the closed loop system is analyzed in the mean-square sense. The simulation results are given to illustrate the effectiveness of the proposed control algorithm.
---
paper_title: A data-driven based adaptive fault diagnosis scheme for nonlinear stochastic distribution systems via 2-step neural networks and descriptor model

paper_content:

A data-driven based adaptive sensor fault diagnosis (FD) and compensation scheme for stochastic distribution control (SDC) systems is studied in this paper, where an augmented descriptor model is employed. Unlike traditional SDC systems, the driven information is the output probability density function (OPDF), which is a kind of image mapping information to the true output values. A mixed 2-step adaptive neural network (NN) framework is studied, where the static NN is to describe the OPDF while the dynamic NN is to identify nonlinearity, uncertainty of system and to refine the OPDF model based on data of the input and statistic information of the output. To identify the sensor fault, an augmented descriptor system is employed, where the augmented state includes the plant state and the sensor fault. As a result, an adaptive strategy is given for nonlinear parameter estimation and sensor fault identification simultaneously. A sensor compensation rule is given to restore the plant by adding it to output feedback controller. The simulation examples are given to verify the effectiveness of the presented algorithm.
---
paper_title: Minimum entropy incremental control for nonlinear stochastic systems with non-Gaussian disturbances and uncertain parameters

paper_content:

In this paper, the problem of controller design for a general nonlinear stochastic system is investigated. The randomness of the considered stochastic system comes from external excitations and system parameters. To achieve minimum entropy control, the probability density function (PDF) of the tracking error is given by using recursive method firstly. Then, a new performance index is formulated, which contain the entropy of the tracking error, the mathematical expectation of the squared error and the control energy. A recursive optimization control algorithm is obtained by minimizing the performance index. Stability analysis is established and encouraging simulation results are included to illustrate the efficiency of the proposed method.
---
paper_title: Fault detection and diagnosis for non-Gaussian stochastic distribution systems with time delays via RBF neural networks.

paper_content:

A new fault detection and diagnosis (FDD) problem via the output probability density functions (PDFs) for non-gausian stochastic distribution systems (SDSs) is investigated. The PDFs can be approximated by radial basis functions (RBFs) neural networks. Different from conventional FDD problems, the measured information for FDD is the output stochastic distributions and the stochastic variables involved are not confined to Gaussian ones. A (RBFs) neural network technique is proposed so that the output PDFs can be formulated in terms of the dynamic weighings of the RBFs neural network. In this work, a nonlinear adaptive observer-based fault detection and diagnosis algorithm is presented by introducing the tuning parameter so that the residual is as sensitive as possible to the fault. Stability and Convergency analysis is performed in fault detection and fault diagnosis analysis for the error dynamic system. At last, an illustrated example is given to demonstrate the efficiency of the proposed algorithm, and satisfactory results have been obtained.
---
paper_title: Shaping probability density functions using a switching linear controller

paper_content:

Abstract Linear controllers for dynamic systems have been used in a stochastic context to decrease the variance of certain key variables. An extension is to use nonlinear controllers for shaping the entire probability density function (PDF). Typically, polynomial controllers are preferred due to their flexibility and also because their continuity makes them mathematically attractive. However, due to restrictions on stability, these controllers tend to be quasi-linear around the desired operating point. So, in this paper, a switching linear controller is presented as an alternative for PDF shaping. It is shown that, despite the discontinuity, an analytical solution of the PDF can be obtained using the Fokker–Planck–Kolmogorov equation. Also, for a simple case, even the Gram-Charlier coefficients can be computed analytically. Thus, it is demonstrated that the PDF shaping problem with a switching controller does not cause any further complications than using polynomial controllers. It is then shown in simulation that switching controllers are more robust and could create more skewness due to their discontinuity.
---

Title: A Survey of the Probability Density Function Control for Stochastic Dynamic Systems

Section 1: INTRODUCTION
Description 1: Introduce the importance of PDF control in stochastic dynamic systems, discuss the assumptions and limitations of Gaussian models, and outline different PDF control strategies.

Section 2: MODEL-BASED DIRECT EVOLUTION PDF CONTROL
Description 2: Discuss PDF control methods focused on directly evolving the PDF, primarily using partial differential equations and inverse formulas.

Section 3: MODEL-BASED TRANSFORMED EVOLUTION PDF CONTROL
Description 3: Explain how PDFs of stochastic systems can be transformed to simplify control design, including approaches like orthogonal decomposition and weight-based neural networks.

Section 4: DATA-BASED PDF CONTROL
Description 4: Detail model-free PDF control strategies using data-driven approaches such as kernel density estimation and dynamic neural networks.

Section 5: EXTENSIONS: MINIMUM ENTROPY CONTROL, FILTERING, FAULT DIAGNOSIS AND PROBABILISTIC DECOUPLING
Description 5: Explore advanced topics in PDF control like minimum entropy control, filtering, fault diagnosis, and probabilistic decoupling of stochastic dynamic systems.

Section 6: CONCLUSION
Description 6: Summarize the developments and applications of PDF control in stochastic dynamic systems, highlight the benefits and limitations of different methods, and suggest future research directions.