Observability Of Linear Systems Research Articles

Approximation with the output of linear control systems

In this paper we give a new proof that for controllable and observable linear systems every L 2 [ 0 , T ] function can be approximated in the L 2 [ 0 , T ] sense with an output function generated by an L 2 [ 0 , T ] input function. We also give a new characterization of how continuous functions on [ 0 , T ] are uniformly approximated by an output generated by a continuous input function. The relative degree of the transfer function of the system determines those functions that can be approximated. We further show that if the initial data is allowed to vary then every continuous function is uniformly approximated by outputs generated by continuous functions.

Observability in linear systems of equations and inequalities: Applications

This paper deals with the problem of observability of linear systems of equations and inequalities in which some of the unknowns and some of the right-hand side constants are known and one seeks the subset of the remaining unknowns and constants that can be uniquely determined. A general methodology for solving this problem, first in the case of linear equations and finally the case of inequalities, is presented. The proposed methods are illustrated by their application to the telephone network problem.

ADAPTIVE LEARNING CONTROL OF LINEAR SYSTEMS BY OUTPUT ERROR FEEDBACK

This paper addresses the problem of designing an output error feedback tracking control for single-input, single-output, minimum phase, observable linear systems. The reference output signal is assumed to be smooth and periodic with known period. By developing in Fourier series expansion a suitable periodic input reference signal, an output error feedback adaptive learning control is designed which ‘learns’ the input reference signal by identifying its Fourier coefficients: exponential tracking of both the input and the output reference signals is achieved if the Fourier series expansion is finite, while arbitrary small tracking errors are guaranteed otherwise.

FUNCTIONAL OBSERVERS FOR LINEAR SYSTEMS WITH UNKNOWN INPUTS

ABSTRACTIn this paper, a straightforward approach is presented to design functional observers for linear systems with unknown inputs. Necessary and sufficient conditions are provided for the existence of the observers. Some illustrated examples are included.

On stochastic optimal control of partially observable nonlinear quasi Hamiltonian systems

A stochastic optimal control strategy for partially observable nonlinear quasi Hamiltonian systems is proposed. The optimal control forces consist of two parts. The first part is determined by the conditions under which the stochastic optimal control problem of a partially observable nonlinear system is converted into that of a completely observable linear system. The second part is determined by solving the dynamical programming equation derived by applying the stochastic averaging method and stochastic dynamical programming principle to the completely observable linear control system. The response of the optimally controlled quasi Hamiltonian system is predicted by solving the averaged Fokker-Planck-Kolmogorov equation associated with the optimally controlled completely observable linear system and solving the Riccati equation for the estimated error of system states. An example is given to illustrate the procedure and effectiveness of the proposed control strategy.

Optimality for the linear quadratic non‐Gaussian problem via the asymmetric Kalman filter

AbstractIn the linear non‐Gaussian case, the classical solution of the linear quadratic Gaussian (LQG) control problem is known to provide the best solution in the class of linear transformations of the plant output if optimality refers to classical least‐squares minimization criteria. In this paper, the adaptive linear quadratic control problem is solved with optimality based on asymmetric least‐squares approach, which includes least‐squares criteria as a special case. Our main result gives explicit solutions for this optimal quadratic control problem for partially observable dynamic linear systems with asymmetric observation errors. The main difficulty is to find the optimal state estimate. For this purpose, an asymmetric version of the Kalman filter based on asymmetric least‐squares estimation is used. We illustrate the applicability of our approach with numerical results. Copyright © 2004 John Wiley & Sons, Ltd.

Robotic Manipulators State Observation Via One-Time Gain Switching

High-gain Luenberger-like state observation would provide the best possible tracking error of nonlinear systems at the expense of higher transient variance and slower convergence to the real state. Conversely, low-gain Luenberger-like observation usually provides lower transient variance of the error system, but might lead to nonzero or nonconvergent steady tracking error. This paper presents new switching-gain observer designs that capitalize on the benefits of low-gain, high-gain, and sliding mode observations. Switching and non-switching observer designs are both considered. Since adaptive observers are often of complex structures and might be difficult to implement, a simpler adaptation form is presented using one-time switching between two predetermined solutions. A nonlinear coordinate transformation is applied to obtain a linear observable system with nonlinear perturbation terms characterized by a Lipschitz constant and/or a finite bound on the norm. Sufficient conditions are derived for the existence of sliding mode, asymptotic stability of the error system, and the independence of the reconstruction error system from the perturbation inputs during the sliding mode. The new observer designs are applied to flexible-joint manipulators in order to explore their performance capabilities. The switching-gain observer was shown to be a reasonable compromise that is easy to be implemented.

Robust adaptive compensation of biased sinusoidal disturbances with unknown frequency

Asymptotically stable, observable linear systems of order n which are not required to be minimum phase and are affected by an additive noisy biased sinusoidal disturbance with unknown bias, magnitude, phase and frequency are considered. The problem of designing an output feedback compensator which regulates the output to zero for any initial condition and for any biased sinusoidal disturbance with no noise is addressed, under the assumption that the system parameters are known. This problem is solved by a (2 n+6)-order compensator which generates asymptotically convergent estimates of the biased sinusoidal disturbance and of its parameters, including frequency. The robustness of the closed loop system with respect to sufficiently small additive unmodelled noise is characterized in terms of input-to-state stability.

Linear operator equations with applications in control and signal processing

The author gives several definitions, including the definition of linear vector spaces, inner products, and Hilbert spaces. He defines linear operators and the Hilbert adjoint operator, and gives several illustrative examples. He presents a diagram which he says is key to understanding linear operator equations. It is a pedagogically important tool for understanding linear operators. Its details are discussed. When attention is restricted to linear matrix equations, the singular-value decomposition completely characterizes the fundamental subspaces of the operator, as is also discussed. The author presents several applications of the theory, including least squares, minimum-norm solutions, controllability and observability of linear systems, optimal control, optimal estimation, and modeling mechanical systems. The examples were chosen to illustrate the wide variety of problems that can be solved using the theory presented in the previous sections.

INPUT RECONSTRUCTION BY MEANS OF SYSTEM INVERSION: APPLICATION TO FAULT DETECTION AND ISOLATION

This paper attempts to look at the fundamental problem of fault detection and isolation (FDI) in nonlinear systems. Using the idea of input reconstruction by means of dynamic inversion the authors first discuss the properties of input (or fault) observability in linear systems. The extension of the results to nonlinear systems as well as the mathematical conditions of the calculation of the inverse system, which provides the inverse in finite algorithmic steps, are given. The applicability of the inversion process to fault reconstruction in nonlinear systems is demonstrated.

The Class of All Reduced-order State Observers

The class of all state observers for linear systems is characterized. A free parameter in observers is restricted within the ring of strictly proper stable rational functions. The order of observers is lower than the known ones by the number of output. When input disturbances exist in the plant, the resultant parameterization of estimation error dynamics is also given, which provides a tool to solve reduced-order robust estimation problem.

Robust Regulation for Nonlinear Systems via an Observed-Based Generalized Immersion

The problem of output regulation of the system affected by unknown constant parameteres is considered here. Under certain assumptions, such a problem is known to be solvable using error feedback. The corresponding necessary and sufficient conditions basically include the solvability of the so-called regulator equations and the existence of a finite dimensional immersion of the exogenous system with outputs into the one having suitable observability and controllability properties. Its model is then directly used for dynamic compensator construction. Usually, such an immersion may be selected as the one to an observable linear system with outputs, but for many interesting cases, this kind of finite dimensional immersion is difficult or even impossible to find. In order to achieve constructive procedures for wider classes, this paper investigates a more general type of immersion, namely to nonlinear system containing a copy of exosystem or its part. Such a generalized immersion enables to solve robust output regulation problem via dynamic feedback compensator using error and exosystem state measurement. When the exosystem states are not completely measurable, a modified observed-based generalized immersion is then presented. Examples together with computer simulation are included to clarify the suggested approach.

Observation control for discrete-continuous stochastic systems

Presents a theoretical framework for the optimization of observations in partially observable linear discrete-continuous stochastic systems. The problem of achievement of the best quality of estimation with discrete and continuous observations is reduced to a deterministic one with impulse or generalized control. An approach based on the discontinuous time transformation is presented and used to reduce the original optimization problem with impulsive control to the standard one with bounded controls. The existence and description of the optimal generalized observation process are also discussed. The illustrative example of the observations optimal control is presented.

Design of linear functional observers for linear systems with unknown inputs

This paper presents a reduced-order linear functional state observer for linear systems with unknown inputs. A simple observer construction procedure is provided. A numerical example is given to illustrate the properties of the observer. The example deals with a linear system comprising of 20 states, 2 inputs, 10 outputs and 5 unknown inputs for which a fourth-order observer is designed to estimate two linear functions of the states.

Uniformly observable linear nonstationary systems with many outputs and their canonical forms

Uniformly observable linear nonstationary systems with many outputs and their canonical forms

On Concepts of Controllability for Deterministic and Stochastic Systems

The new necessary and sufficient conditions, which are formulated in terms of convergence of a certain sequence of operators involving the resolvent of the negative of the controllability operator, are found for deterministic linear stationary control systems to be completely and approximately controllable, respectively. These conditions are applied to study the S-controllability (a property of attaining an arbitrarily small neighborhood of each point in the state space with a probability arbitrarily close to one) and C--controllability (the S--controllability fortified with some uniformity) of stochastic systems. It is shown that the S--controllability (the C--controllability) of a partially observable linear stationary control system with an additive Gaussian white noise disturbance on all the intervals [0,T] for T>0 is equivalent to the approximate (complete) controllability of its deterministic part on all the intervals [0,T] for T>0.

On the characteristic polynomial of matrices with prescribed columns and the stabilization and observability of linear systems

Let A 2 F , B 2 F , where F is an arbitrary eld. In this paper, the possible characteristic polynomials of [A B ], when some of its columns are prescribed and the other columns vary, are described. The characteristic polynomial of [A B ] is de ned as the largest determinantal divisor (or the product of the invariant factors) of [xIn A B ]. This result generalizes a previous theorem by H. Wimmer which studies the same problem when t = 0. As a consequence, it is extended to arbitrary elds a result, already proved for in nite elds, that describes all the possible characteristic polynomials of a square matrix when an arbitrary submatrix is xed and the other entries vary. Finally, applications to the stabilization and observability of linear systems by state feedback are studied. AMS subject classi cations. 15A18, 93B60

Observers for Linear Systems with Unknown Inputs

This paper presents a constructive solution to the design problem of observers to estimate both system state and unknown inputs. We propose an equivalent system, which is free of unknown inputs to derive the observer. Based on the equivalent system description, one can design an observer to estimate both the system state and the unknown inputs simultaneously. Experimental results for a tank system demonstrate the applicability of the proposed methodologies.

Moving horizon observers and observer-based control

In this paper two topics are explored. A new approach to the problem of obtaining an estimate of the state of a nonlinear system is proposed. The moving horizon observer produces an estimate of the state of the nonlinear system at time t either by minimizing, or approximately minimizing, a cost function over the preceding interval (horizon) [t-T,t]; as t advances, so does the horizon. Convergence of the estimator is established under the assumption that the corresponding global optimization problem can be (approximately) solved and a uniform reconstructability condition is satisfied; the latter condition is automatically satisfied for linear observable systems. The utility of the estimator for receding horizon control is explored. In particular, stability of a composite moving horizon system, comprising a moving horizon regulator and a moving horizon observer, is established. >

Remarks on linearization of discrete-time autonomous systems and nonlinear observer design

This paper presents necessary and sufficient conditions under which a discrete-time autonomous system with outputs is locally state equivalent to an observable linear system or a system in the nonlinear observer form (Krener and Isidori, 1983). In particular, an open problem raised in Lee and Nam (1991), namely the observer linearization problem, is solved for a nonlinear system which may not be invertible (i.e., the mapping f may not be a local diffeomorphism ). As a consequence, the nonlinear observer design problem is solved by means of exact linearization techniques.