Observers for linear distributed-parameter systems: A survey
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paper_title: Toward a practical theory for distributed parameter systems

paper_content:

The control of dist,ributed parameter (DP) systems represents a real challenge, both from a theoretical and a practical point of view, to the systems engineer. Distribut.ed parameter systems arise in various application areas, such as chemical proms systems, aerospace systems, magneto-hydrodynamic systems, and communicat. ions systems, to ment.ion just a few. Thus, there is sufficient motivation for research directed t,oward the analysis, synt.hesis, and design techniques for DP systems. On t.he surface, it. may appear that t.he available theory for distributed parameter systems is almost at the same level as that associated with lumped systems. However, there exists a much wider gap between the theory and its applications. In the remainder of this correspondence, we shall briefly discus the reasons for this gap and suggest, certain tentative approaches which may contribute to the development of a theory and computat,ional algorithms which take into account. some of the practical problems associated with the design of controllers for DP systems. In order to make these concepts clear it. becomes necessary to briefly review, in an informal manner, what a DP system is and in what sense it differs, from both a mat,hemat.ical and a practical point of view, from a conventional lumped system.
---
paper_title: Nonlinear Observers—A State-of-the-Art Survey

paper_content:

The state-of-the-art of nonlinear state estimators or “observers” is reviewed. The use of these observers in real time nonlinear compensators is evaluated in terms of their on-line computational requirements. Their robustness properties are evaluated in terms of the extent to which the design requires a “perfect” model.
---
paper_title: Sliding mode observers: a survey

paper_content:

Sliding mode observers have unique properties, in that the ability to generate a sliding motion on the error between the measured plant output and the output of the observer ensures that a sliding mode observer produces a set of state estimates that are precisely commensurate with the actual output of the plant. It is also the case that analysis of the average value of the applied observer injection signal, the so-called equivalent injection signal, contains useful information about the mismatch between the model used to define the observer and the actual plant. These unique properties, coupled with the fact that the discontinuous injection signals which were perceived as problematic for many control applications have no disadvantages for software-based observer frameworks, have generated a ground swell of interest in sliding mode observer methods in recent years. This article presents an overview of both linear and non-linear sliding mode observer paradigms. The use of the equivalent injection signal in problems relating to fault detection and condition monitoring is demonstrated. A number of application specific results are also described. The literature in the area is presented and qualified in the context of continuing developments in the broad areas of the theory and application of sliding mode observers.
---
paper_title: Observing the State of a Linear System

paper_content:

In much of modern control theory designs are based on the assumption that the state vector of the system to be controlled is available for measurement. In many practical situations only a few output quantities are available. Application of theories which assume that the state vector is known is severely limited in these cases. In this paper it is shown that the state vector of a linear system can be reconstructed from observations of the system inputs and outputs. It is shown that the observer, which reconstructs the state vector, is itself a linear system whose complexity decreases as the number of output quantities available increases. The observer may be incorporated in the control of a system which does not have its state vector available for measurement. The observer supplies the state vector, but at the expense of adding poles to the over-all system.
---
paper_title: Some recent applications of distributed parameter systems theory - A survey

paper_content:

A survey of some recent applications of distributed parameter systems theory is presented. The practical areas discussed range from process control problems in an industrial plant to the identification, monitoring and control of air and water quality in our environment. Some new, promising areas of application are discussed along with suggestions for future research emphasis.
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paper_title: A finite dimensional sliding mode observer for a spatially continuous process

paper_content:

Modeling and control of systems represented by partial differential equations (PDEs) is an interesting research field as the process under investigation is infinite dimensional and most commonly used techniques are for finite dimensional systems. This paper considers the development of a finite dimensional observer obtained after an appropriate model reduction stage. A sliding mode observer is considered as it utilizes the sign of a quantity and ensures good reconstruction performance. The paper compares the fictitious state variables that are obtained after projecting the instantaneous snapshots on an eigenbasis and the state variables predicted by the observer. The results emphasize that the designed observer functions well on some set of operating conditions, which are elaborated in the paper.
---
paper_title: Some recent applications of distributed parameter systems theory - A survey

paper_content:

A survey of some recent applications of distributed parameter systems theory is presented. The practical areas discussed range from process control problems in an industrial plant to the identification, monitoring and control of air and water quality in our environment. Some new, promising areas of application are discussed along with suggestions for future research emphasis.
---
paper_title: Observation and Control for Operator Semigroups

paper_content:

The evolution of the state of many systems modeled by linear partial diﬁerentialequations (PDEs) or linear delay-diﬁerential equations can be described by operatorsemigroups. The state of such a system is an element in an inﬂnite-dimensionalnormed space, whence the name \inﬂnite-dimensional linear system".The study of operator semigroups is a mature area of functional analysis, which isstill very active. The study of observation and control operators for such semigroupsis relatively more recent. These operators are needed to model the interactionof a system with the surrounding world via outputs or inputs. The main topicsof interest about observation and control operators are admissibility, observability,controllability, stabilizability and detectability. Observation and control operatorsare an essential ingredient of well-posed linear systems (or more generally systemnodes). Inthisbookwedealonlywithadmissibility, observabilityandcontrollability.We deal only with operator semigroups acting on Hilbert spaces.This book is meant to be an elementary introduction into the topics mentionedabove. By \elementary" we mean that we assume no prior knowledge of ﬂnite-dimensional control theory, and no prior knowledge of operator semigroups or ofunbounded operators. We introduce everything needed from these areas. We doassume that the reader has a basic understanding of bounded operators on Hilbertspaces, diﬁerential equations, Fourier and Laplace transforms, distributions andSobolev spaces on
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paper_title: Design of optimal controllers for distributed systems using finite dimensional state observers

paper_content:

The problem of constructing an "observer" to enable us to implement an approximate optimal control for a distributed parameter system is examined where the state is measured at a few pre-specified points. The observer is formulated as the output of a dynamical system described by a set of ordinary differential equations. Both distributed and boundary control problems are studied and the observer-formulation is set up for both cases. Some reasonable assumptions have been made in order that the approximation introduced by the eigenfunction expansion technique be satisfactory. For the case of the boundary control problem, a simple example is solved to illustrate the method.
---
paper_title: An Introduction to Infinite-Dimensional Linear Systems Theory

paper_content:

1 Introduction.- 1.1 Motivation.- 1.2 Systems theory concepts in finite dimensions.- 1.3 Aims of this book.- 2 Semigroup Theory.- 2.1 Strongly continuous semigroups.- 2.2 Contraction and dual semigroups.- 2.3 Riesz-spectral operators.- 2.4 Delay equations.- 2.5 Invariant subspaces.- 2.6 Exercises.- 2.7 Notes and references.- 3 The Cauchy Problem.- 3.1 The abstract Cauchy problem.- 3.2 Perturbations and composite systems.- 3.3 Boundary control systems.- 3.4 Exercises.- 3.5 Notes and references.- 4 Inputs and Outputs.- 4.1 Controllability and observability.- 4.2 Tests for approximate controllability and observability.- 4.3 Input-output maps.- 4.4 Exercises.- 4.5 Notes and references.- 5 Stability, Stabilizability, and Detectability.- 5.1 Exponential stability.- 5.2 Exponential stabilizability and detectability.- 5.3 Compensator design.- 5.4 Exercises.- 5.5 Notes and references.- 6 Linear Quadratic Optimal Control.- 6.1 The problem on a finite-time interval.- 6.2 The problem on the infinite-time interval.- 6.3 Exercises.- 6.4 Notes and references.- 7 Frequency-Domain Descriptions.- 7.1 The Callier-Desoer class of scalar transfer functions.- 7.2 The multivariable extension.- 7.3 State-space interpretations.- 7.4 Exercises.- 7.5 Notes and references.- 8 Hankel Operators and the Nehari Problem.- 8.1 Frequency-domain formulation.- 8.2 Hankel operators in the time domain.- 8.3The Nehari extension problem for state linear systems.- 8.4 Exercises.- 8.5 Notes and references.- 9 Robust Finite-Dimensional Controller Synthesis.- 9.1 Closed-loop stability and coprime factorizations.- 9.2 Robust stabilization of uncertain systems.- 9.3 Robust stabilization under additive uncertainty.- 9.4 Robust stabilization under normalized left-coprime-factor uncertainty.- 9.5 Robustness in the presence of small delays.- 9.6 Exercises.- 9.7 Notes and references.- A. Mathematical Background.- A.1 Complex analysis.- A.2 Normed linear spaces.- A.2.1 General theory.- A.2.2 Hilbert spaces.- A.3 Operators on normed linear spaces.- A.3.1 General theory.- A.3.2 Operators on Hilbert spaces.- A.4 Spectral theory.- A.4.1 General spectral theory.- A.4.2 Spectral theory for compact normal operators.- A.5 Integration and differentiation theory.- A.5.1 Integration theory.- A.5.2 Differentiation theory.- A.6 Frequency-domain spaces.- A.6.1 Laplace and Fourier transforms.- A.6.2 Frequency-domain spaces.- A.6.3 The Hardy spaces.- A.7 Algebraic concepts.- A.7.1 General definitions.- A.7.2 Coprime factorizations over principal ideal domains.- A.7.3 Coprime factorizations over commutative integral domains.- References.- Notation.
---
paper_title: Nonlinear Observers—A State-of-the-Art Survey

paper_content:

The state-of-the-art of nonlinear state estimators or “observers” is reviewed. The use of these observers in real time nonlinear compensators is evaluated in terms of their on-line computational requirements. Their robustness properties are evaluated in terms of the extent to which the design requires a “perfect” model.
---
paper_title: Sliding mode observers: a survey

paper_content:

Sliding mode observers have unique properties, in that the ability to generate a sliding motion on the error between the measured plant output and the output of the observer ensures that a sliding mode observer produces a set of state estimates that are precisely commensurate with the actual output of the plant. It is also the case that analysis of the average value of the applied observer injection signal, the so-called equivalent injection signal, contains useful information about the mismatch between the model used to define the observer and the actual plant. These unique properties, coupled with the fact that the discontinuous injection signals which were perceived as problematic for many control applications have no disadvantages for software-based observer frameworks, have generated a ground swell of interest in sliding mode observer methods in recent years. This article presents an overview of both linear and non-linear sliding mode observer paradigms. The use of the equivalent injection signal in problems relating to fault detection and condition monitoring is demonstrated. A number of application specific results are also described. The literature in the area is presented and qualified in the context of continuing developments in the broad areas of the theory and application of sliding mode observers.
---
paper_title: An introduction to observers

paper_content:

Observers which approximately reconstruct missing state-variable information necessary for control are presented in an introductory manner. The special topics of the identity observer, a reduced-order observer, linear functional observers, stability properties, and dual observers are discussed.
---
paper_title: Observing the State of a Linear System

paper_content:

In much of modern control theory designs are based on the assumption that the state vector of the system to be controlled is available for measurement. In many practical situations only a few output quantities are available. Application of theories which assume that the state vector is known is severely limited in these cases. In this paper it is shown that the state vector of a linear system can be reconstructed from observations of the system inputs and outputs. It is shown that the observer, which reconstructs the state vector, is itself a linear system whose complexity decreases as the number of output quantities available increases. The observer may be incorporated in the control of a system which does not have its state vector available for measurement. The observer supplies the state vector, but at the expense of adding poles to the over-all system.
---
paper_title: A finite dimensional sliding mode observer for a spatially continuous process

paper_content:

Modeling and control of systems represented by partial differential equations (PDEs) is an interesting research field as the process under investigation is infinite dimensional and most commonly used techniques are for finite dimensional systems. This paper considers the development of a finite dimensional observer obtained after an appropriate model reduction stage. A sliding mode observer is considered as it utilizes the sign of a quantity and ensures good reconstruction performance. The paper compares the fictitious state variables that are obtained after projecting the instantaneous snapshots on an eigenbasis and the state variables predicted by the observer. The results emphasize that the designed observer functions well on some set of operating conditions, which are elaborated in the paper.
---
paper_title: Observers and parameter determination for distributed parameter systems

paper_content:

The aim of the paper is to investigate the estimation of unknown states and unknown functions for distributed parameter systems. First we construct finite dimensional state observers to estimate the unknown states and give the error estimates for the distributed parameter systems of parabolic type with unknown input sources. Next we consider the determination of unknown input distribution functions using these estimated states. The problems of unknown function determination are not necessarily well-posed even if the distributed parameter systems are identifiable. A well-posed approximation method by regularization is applied to obtain the approximate functions which depend continuously on the measurement data. We also construct finite dimensional state observers for the distributed parameter systems of hyperbolic type with unknown input sources. Simple numerical examples are presented.
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paper_title: Observers for linear multivariable systems with applications

paper_content:

This paper presents an algorithm for the design of asymptotic state estimators (observers) for index-invariant uniformly observable time-varying linear finite-dimensional multivariable systems. The results obtained indicate that asymptotic estimators can be employed in optimally designed regulators provided an increase from the optimal cost is tolerable. It is also shown that any uniformly observable and uniformly controllable plant with index-invariant observability and controllability matrices can be stabilized with an observer.
---
paper_title: Design of optimal controllers for distributed systems using finite dimensional state observers

paper_content:

The problem of constructing an "observer" to enable us to implement an approximate optimal control for a distributed parameter system is examined where the state is measured at a few pre-specified points. The observer is formulated as the output of a dynamical system described by a set of ordinary differential equations. Both distributed and boundary control problems are studied and the observer-formulation is set up for both cases. Some reasonable assumptions have been made in order that the approximation introduced by the eigenfunction expansion technique be satisfactory. For the case of the boundary control problem, a simple example is solved to illustrate the method.
---
paper_title: Discrete-Time Observers and Parameter Determination for Distributed Parameter Systems with Discrete-Time Input–Output Data

paper_content:

The aim of this paper is to study the estimation of unknown states and unknown input distribution functions for distributed parameter systems with discrete-time input–output data. First we construct finite dimensional discrete-time state observers to estimate unknown states and give the error estimates for distributed parameter systems with unknown inputs. Next we consider the determination of unknown input distribution functions using these estimated states. We discuss the relationship between observability and identifiability. The problem of determination of unknown functions is not well posed in general even if the distributed parameter systems are identifiable. We present and discuss a feasible approximation method by regularization which gives a constructive procedure to obtain approximately a true input distribution function. We also investigate limit properties of approximate solutions as the number of sampling periods tends to $ + \infty $.
---
paper_title: Toward a practical theory for distributed parameter systems

paper_content:

The control of dist,ributed parameter (DP) systems represents a real challenge, both from a theoretical and a practical point of view, to the systems engineer. Distribut.ed parameter systems arise in various application areas, such as chemical proms systems, aerospace systems, magneto-hydrodynamic systems, and communicat. ions systems, to ment.ion just a few. Thus, there is sufficient motivation for research directed t,oward the analysis, synt.hesis, and design techniques for DP systems. On t.he surface, it. may appear that t.he available theory for distributed parameter systems is almost at the same level as that associated with lumped systems. However, there exists a much wider gap between the theory and its applications. In the remainder of this correspondence, we shall briefly discus the reasons for this gap and suggest, certain tentative approaches which may contribute to the development of a theory and computat,ional algorithms which take into account. some of the practical problems associated with the design of controllers for DP systems. In order to make these concepts clear it. becomes necessary to briefly review, in an informal manner, what a DP system is and in what sense it differs, from both a mat,hemat.ical and a practical point of view, from a conventional lumped system.
---
paper_title: Sliding mode observers: a survey

paper_content:

Sliding mode observers have unique properties, in that the ability to generate a sliding motion on the error between the measured plant output and the output of the observer ensures that a sliding mode observer produces a set of state estimates that are precisely commensurate with the actual output of the plant. It is also the case that analysis of the average value of the applied observer injection signal, the so-called equivalent injection signal, contains useful information about the mismatch between the model used to define the observer and the actual plant. These unique properties, coupled with the fact that the discontinuous injection signals which were perceived as problematic for many control applications have no disadvantages for software-based observer frameworks, have generated a ground swell of interest in sliding mode observer methods in recent years. This article presents an overview of both linear and non-linear sliding mode observer paradigms. The use of the equivalent injection signal in problems relating to fault detection and condition monitoring is demonstrated. A number of application specific results are also described. The literature in the area is presented and qualified in the context of continuing developments in the broad areas of the theory and application of sliding mode observers.
---
paper_title: Backstepping observers for a class of parabolic PDEs

paper_content:

In this paper we design exponentially convergent observers for a class of parabolic partial integro-differential equations (P(I)DEs) with only boundary sensing available. The problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system. Observer gain (output injection function) is shown to satisfy a well-posed hyperbolic PDE that is closely related to the hyperbolic PDE governing backstepping control gain for the state-feedback problem. For several physically relevant problems the observer gains are obtained in closed form. The observer gains are then used for an output-feedback design in both collocated and anti-collocated setting of sensor and actuator. The order of the resulting compensator can be substantially lowered without affecting stability. Explicit solutions of a closed loop system are found in particular cases.
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paper_title: Observers for systems characterized by semigroups

paper_content:

The theory of observers is generalized from finite dimensional linear systems to abstract linear systems characterized by semigroups on Banach spaces. Sufficient conditions are given for both identity and reduced-order observers to exist for the abstract system. It is shown that the spectrum of a closed-loop control system using an observer is the union of the spectrum of the observer and the spectrum of the closed-loop system with state feedback. The observer theory for the abstract system is used to show that observability is a sufficient condition for the existence of an observer for a system modeled by a linear functional differential equation.
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paper_title: Implementation of distributed parameter state observers

paper_content:

With the aid of simplifying assumptions a one-dimensional mathematical model for a three-dimensional aluminium slab (100 cm long, 25 cm wide, 2 cm thick) has been developed and the modelling parameter of the apparatus have been determined experimentally |14|. The observer problem considered here is the real time state reconstruction of the slab temperature profile using only a limited number of thermocouple measurements.
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paper_title: Sensors and observers in distributed parameter systems

paper_content:

Luenberger observer theory is extended to distributed parameter systems. This extension is based on the consideration of sensors. For systems with infinite dimensional state spaces, it is possible to construct the state vector asymptotically (or a part of the state vector) by a ‘good’ choice of sensors. We show that the link between detectability and sensor structure may be of some interest in the construction of observers.
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paper_title: Receding window observer and dynamic feedback control of discrete infinite dimensional systems

paper_content:

A discrete-time infinite-dimensional system is considered. The receding window filtering idea is introduced to reconstruct the state of the system asymptotically from incomplete measurements. This observer is used in the stabilizing dynamic feedback control of the system. Robustness properties of such dynamic controllers are derived. >
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paper_title: Observer theory for distributed-parameter systems

paper_content:

This paper examines the problem of the approximate reconstruction of the unknown state variables in distributed-parameter systems. New results on the observer theory for important classes of linear and non-linear operator, partial differential, and partial differential-integral equations in describing distributed-parameter systems are presented. The specific developments employ the recent results on Lyapunov stability theory, along with the theory of linear and non-linear semigroup operators, and their infinitesimal generators. The questions of observability, stability of the state reconstruction error dynamics associated with the proposed observer structure are discussed. The theoretical results are illustrated with some applications to problems of the kinetic lumping of complex distributed-parameter chemical reaction systems, as well as the observer design for linear and non-linear distributed-parameter diffusion systems.
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paper_title: Feedback stabilization of a class of distributed systems and construction of a state estimator

paper_content:

In this paper, we study feedback stabilization of a class of distributed systems governed by partial differential equations of parabolic type and its application to constructing a state estimator for asymptotic state identification. It is proved that, when a controller (an observation) can be arbitrarily constructed, observability (controllability) of the system is necessary and sufficient for stabilizing the system so that it has an arbitrarily large damping constant. As an application of this result, it is shown that a state estimator can be constructed, the output of which approaches asymptotically the real state of the system with an arbitrary convergence rate.
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paper_title: Output-feedback boundary control of an uncertain heat equation with noncollocated observation: A sliding-mode approach

paper_content:

The boundary stabilization problem of a one-dimensional unstable heat conduction system with boundary disturbance is investigated using a sliding-mode approach. This infinite-dimensional system, mathematical modeled by a parabolic partial differential equation (PDE), is powered with a Dirichlet type boundary actuator and only sensing at opposite end. By applying the Volterra integral transformation, a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties. The associated Lyapunov function is used for designing an infinite-dimensional sliding manifold, on which the system exhibits the same type of stability and robustness against of bounded exogenous boundary disturbance. By utilizing the similar transformation, an infinite-dimensional sliding-mode observer is proposed to reconstruct the system' states, which is with robustness to boundary disturbance. Moreover, the relative degree of the chosen sliding function with respect to the output-feedback boundary control input is zero. A continuous control law satisfying the reaching condition is obtained by passing a discontinuous (signum) signal through an integrator.
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paper_title: Feedback stabilization of distributed parameter systems by a functional observer

paper_content:

Feedback stabilization of unstable parabolic equations is of great interest. The fact that it is not necessarily possible to stabilize the equations by means of static feedback schemes when both observation and control can be realized only through the boundary is illustratively shown by a simple example. In view of this, a functional observer of Luenberger type is derived and then utilized in order to stabilize unstable parabolic equations for which observation of the state and control can be carried out only through the boundary.
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paper_title: Adaptive observers for structurally perturbed infinite dimensional systems

paper_content:

The aim of this investigation is to construct an adaptive observer for a class of infinite-dimensional plants having a structured perturbation with an unknown constant parameter, such as the case of static output feedback with an unknown gain. The adaptive observer uses the nominal dynamics of the unperturbed plant and an adaptation law based on the Lyapunov redesign method. We obtain conditions on the system to ensure uniform boundedness of the estimator dynamics and the parameter estimates, and convergence of the estimator error. Examples illustrating the approach are presented along with some numerical results.
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paper_title: Adaptive observers for slowly time varying infinite dimensional systems

paper_content:

We consider a class of infinite dimensional systems with an unknown time varying perturbation in the input term. The goal here is twofold, namely to estimate the state and identify the unknown parameter in the input term using only input and output measurements. An adaptive observer along with a parameter adaptive law that is based on Lyapunov redesign is presented and, under certain conditions imposed on the plant, is shown to achieve state error convergence. Parameter convergence can be established by imposing the additional condition of persistence of excitation. Examples that illustrate the applicability of this approach to a parabolic partial differential equation and a delay system are included along with some numerical results.
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paper_title: The generation of adaptive law structures for globally convergent adaptive observers

paper_content:

Observing the state of an unknown linear system by means of a parametrized representation of the standard Luenberger observer with an additional adaptive loop (adaptive observer) is considered. A general technique on how to construct suitable adaptive laws for this additional loop is presented and it is proven that these adaptive laws always result in global and arbitrarily fast convergence of the adaptive process. Because the adaptive laws can assume a variety of different structures, both structural and parametric degrees of freedom in the adaptive law (rather than the so far available parametric degrees alone) are obtained, which can be used in a future optimization of the adaptive observer performance.
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paper_title: Finite-dimensional adaptive observers applied to distributed parameter systems

paper_content:

The application of adaptive observers to distributed parameter systems is considered. The effects of the infinite-dimensional unmodeled dynamics (residuals) on the observer state and parameter estimates are analyzed. For the purpose of the analysis, the state and parameter estimates of the nth-order observer are interpreted as estimates of an nth-order reduced-order model (ROM) of the infinite-dimensional plant. These estimates can then be compared to the states and parameters of a true ROM and bounds for the errors can be derived. It is proven that if the input provides definite excitation and the residual energy is bounded over a finite time interval, the estimation errors are ultimately bounded, with no assumption regarding stability of the plant or boundedness of the plant input. >
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paper_title: Adaptive monitoring and accommodation of nonlinear actuator faults in positive real infinite dimensional systems

paper_content:

We consider a class of positive real infinite dimensional systems which are subjected to incipient actuator faults. The actuator fault is modelled as a time varying transition from an initial (linear or even hysteretic) map into a(another) hysteretic map at the onset of the fault occurrence. An infinite dimensional adaptive detection observer is utilized to generate a residual signal in order to detect the fault occurrence and to assist in the fault accommodation. This is done via an automated control reconfiguration which utilizes information on the new hysteretic map and adjusts the controller via a static right inverse of the new actuator map. A robust modification is utilized in order to avoid false alarms caused by unmodelled dynamics. An example is included to illustrate the applicability of the proposed detection scheme.
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paper_title: An adaptive observer for single-input single-output linear systems

paper_content:

A full order adaptive observer is described for observing the state of a single-input single-output observable continuous differential system with unknown parameters. Convergence of the observer states to those of the system is accomplished by directly changing the parameters of the observer using an adaptive law based upon Lyapunov stability theory. Observer eigenvalues may be freely chosen. Some restriction is placed upon the system input in that it must be sufficiently rich in frequencies in order to insure convergence.
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paper_title: Adaptive Observers for a Class of Infinite Dimensional Systems

paper_content:

Abstract An adaptive observer for an infinite dimensional plant with a structured perturbation is developed to estimate the state of the plant. The class under consideration includes a collocated (passive) feedback with uncertain feedback gain. This observer is based on the nominal part of the plant and the adaptation law for estimating the uncertain gain. The proposed adaptation law developed is based on the Lyapunov stability method. It is proved that the resulting estimator dynamics are stable in the sense that the tracking error converges to zero and the parameter estimate is bounded.
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paper_title: Control of Distributed Parameter Systems with Spillover using an Augmented Observer

paper_content:

Modern modal control methods for flexible structures, based on a truncated model of the structure's dynamics, have control and observation spillover which can reduce the stability margin of the controlled structure. The sensor output is often filtered to reduce observation spillover, however, it introduces signal distortion which perturbs the closed-loop system pole locations. This can reduce stability margins and jeopardize convergence of the observer. When the filter equation is included in the standard observer equations the separation principle between the controller and the observer will no longer holds, and the closed-loop poles cannot be located using standard pole placement methods. A new method is presented, where a first order filter is included in an augmented observer. The separation principle, and controllability and observability properties of the original system are shown to remain when using the augmented observer. Filter and spillover produced pole perturbations in the different methods are illustrated with a numerical example.
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paper_title: Dynamics And Control Of Structures

paper_content:

Newtonian Mechanics. Principles of Analytical Mechanics. Concepts from Linear System Theory. Lumped--Parameter Structures. Control of Lumped--Parameter Systems: Classical Approach. Control of Lumped--Parameter Systems: Modern Approach. Distributed--Parameter Structures: Exact and Approximate Methods. Control of Distributed Structures. A Review of Literature on Structural Control. References. Author Index. Subject Index.
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paper_title: Feedback control of flexible systems

paper_content:

Feedback control is developed for the class of flexible systents described by the generalized wave equation with damping. The control force distribution is provided by a number of point force actuators and the system displacements and/or their velocities are measured at various points. A feedback controller is developed for a finite number of modes of the flexible system and the controllability and observability conditions necessary for successful operation are displayed. The control and observation spillover due to the residual (uncontrolled) modes is examined and the combined effect of control and observation spillover is shown to lead to potential instabilities in the closed-loop system. Some remedies for spillover, including a straightforward phase-locked loop prefilter, are suggested to remove the instability mechanism. The concepts of this paper are illustrated by some numerical studies on the feedback control of a simply-supported Euler-Bernoulli beam with a single actuator and sensor.
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paper_title: Second-order observers for second-order distributed parameter systems in R2

paper_content:

Abstract This note addresses observer design for second-order distributed parameter systems in R 2 . Particularly, second-order distributed parameter systems without distributed damping are studied. Based on finite number of measurements, exponentially stable observer is designed. The existence, uniqueness and stability of solutions of the observers are based on semigroup theory.
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paper_title: Robust Observer-Based Control of a Vibrating Beam

paper_content:

A new multiple-input multiple-output time domain stability criterion for large flexible structures is illustrated by application to observer-based control of a Bernoulli-Euler beam. Upper norm bounds of the state transition matrix of the residual dynamics and total spillover matrix (impulse response matrix of the residual model) are investigated. The approach can be easily extended to a practical consideration of the unavoidable saturation of the actuators and gives an insight into the stabilization analysis of saturating control of flexible structures.
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paper_title: Natural consensus filters for second order infinite dimensional systems

paper_content:

Abstract This work proposes consensus filters for a class of second order infinite dimensional systems. The proposed structure of the consensus filters is that of a local filter written in the natural setting of a second order formulation with the additional coupling that enforces consensus by minimizing the disagreement between all local filters. An advantage of the second order formulation imposed on the local filters is the natural interpretation that they retain, namely that the time derivative of the estimated position is equal to the estimated velocity. Stability analysis of the collective dynamics is achieved via the use of a parameter-dependent Lyapunov functional, and which guarantees that asymptotically, all filters agree with each other and that they all converge to the true state of the second order system.
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paper_title: Natural second-order observers for second-order distributed parameter systems

paper_content:

The aim of this manuscript is to present an alternative method for designing state observers for second-order distributed parameter systems without resorting to a first-order formulation. This method has the advantage of utilizing the algebraic structure that second-order systems enjoy with the obvious computational savings in observer gain calculations. The proposed scheme ensures that the derivative of the estimated position is indeed the estimate of the velocity component and to achieve such a result, a parameter-dependent Lyapunov function was utilized to ensure the asymptotic convergence of the state estimation error.
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paper_title: Design of consensus and adaptive consensus filters for distributed parameter systems

paper_content:

This work establishes an abstract framework that considers the distributed filtering of spatially varying processes using a sensor network. It is assumed that the sensor network consists of groups of sensors, each of which provides a number of state measurements from sensing devices that are not necessarily identical and which only transmit their information to their own sensor group. A modification to the local spatially distributed filters provides the non-adaptive case of spatially distributed consensus filters which penalize the disagreement amongst themselves in a dynamic manner. A subsequent modification to this scheme incorporates the adaptation of the consensus gains in the disagreement terms of all local filters. Both the well-posedness of these two consensus spatially distributed filters and the convergence of the associated observation errors to zero in appropriate norms are presented. Their performance is demonstrated on three different examples of a diffusion partial differential equation with point measurements.
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paper_title: Natural consensus filters for second order infinite dimensional systems

paper_content:

Abstract This work proposes consensus filters for a class of second order infinite dimensional systems. The proposed structure of the consensus filters is that of a local filter written in the natural setting of a second order formulation with the additional coupling that enforces consensus by minimizing the disagreement between all local filters. An advantage of the second order formulation imposed on the local filters is the natural interpretation that they retain, namely that the time derivative of the estimated position is equal to the estimated velocity. Stability analysis of the collective dynamics is achieved via the use of a parameter-dependent Lyapunov functional, and which guarantees that asymptotically, all filters agree with each other and that they all converge to the true state of the second order system.
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paper_title: Consensus and Cooperation in Networked Multi-Agent Systems

paper_content:

This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis frame- work is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small- world networks, Markov processes and gossip-based algo- rithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions. Simu- lation results are presented that demonstrate the role of small- world effects on the speed of consensus algorithms and cooperative control of multivehicle formations.
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paper_title: End-point sensing and state observation of a flexible-link robot

paper_content:

This paper presents an end-point sensor system and the development of an observer to reconstruct the states of a flexible-link robot. The sensor system includes a tip displacement sensor and an accelerometer. Based on the assumed-models method, an observer is developed using the Kalman filtering algorithm. Experimental results are given to demonstrate the effectiveness of the observer.
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paper_title: Robust discrete‐time nonlinear sliding mode state estimation of uncertain nonlinear systems

paper_content:

In this paper, we propose a discrete-time nonlinear sliding mode observer for state and unknown input estimations of a class of single-input/single-output nonlinear uncertain systems. The uncertainties are characterized by a state-dependent vector and a scalar disturbance/unknown input. The discrete-time model is derived through Taylor series expansion together with nonlinear state transformation. A design methodology that combines the discrete-time sliding mode (DSM) and a nonlinear observer design is adopted, and a strategy is developed to guarantee the convergence of the estimation error to a bound within the specified boundary layer. A relation between sliding mode gain and boundary layer is established for the existence of DSM, and the estimation is made robust to external disturbances and uncertainties. The unknown input or disturbance can also be estimated through the sliding mode. The conditions for the asymptotical stability of the estimation error are analysed. Application to a bioreactor is given and the simulation results demonstrate the effectiveness of the proposed scheme. Copyright © 2006 John Wiley & Sons, Ltd.
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paper_title: A robust observer for discrete time nonlinear systems

paper_content:

Abstract This paper extends the results developed in (Ciccarella et al., 1993) and presents a robust observer for discrete time nonlinear systems. A simple, robust and easy to implement algorithm is given whose convergence properties are guaranteed for autonomous and forced systems. Combined parameter and state estimation is made for a numerical example, which compares the robust observer to the observer given in (Ciccarella et al., 1993).
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Title: Observers for Linear Distributed-Parameter Systems: A Survey

Section 1: INTRODUCTION
Description 1: Provide an overview of Distributed-Parameter Systems (DPSs) and the need for observers. Discuss the motivation for the survey and outline the paper's structure.

Section 2: LUMPED-PARAMETER OBSERVER DESIGN
Description 2: Briefly recall the observer design problem for lumped-parameter systems, including the mathematical background and key concepts.

Section 3: DISTRIBUTED-PARAMETER SYSTEMS
Description 3: Introduce models of DPSs, focusing on their representation via partial differential equations and abstract state space equations. Discuss the concept of lumping and order reduction methods.

Section 4: DISTRIBUTED-PARAMETER OBSERVERS
Description 4: Present the various approaches to observer design for DPSs, including finite-dimensional and infinite-dimensional observers. Discuss generalization of observer theory to infinite-dimensional systems and provide an overview of specific observer design methods.

Section 5: SECOND-ORDER DISTRIBUTED-PARAMETER SYSTEMS
Description 5: Discuss observer design for second-order DPSs such as flexible structures and vibration systems. Include descriptions of natural observers and other specialized methods for second-order systems.

Section 6: DISTRIBUTED ESTIMATION
Description 6: Cover recent developments in distributed estimation using sensor networks. Explain how local observers and consensus methods are applied in this context.

Section 7: SUMMARY AND CHALLENGES
Description 7: Summarize the survey's findings and identify future research challenges, including robustness against model uncertainty, discrete-time observer analysis, and applications to real-world systems.