Design of reduced-order state estimators for linear time-varying multivariable systems

The design of reduced-order state estimators for linear time-varying multivariable systems is considered. Employing the concepts of matrix operators and the method of canonical transformations, this paper shows that there exists a reduced-order state estimator for linear time-varying systems that are ‘lexicography-fixedly observable’. In addition, the eigenvalues of the estimator can be arbitrarily assigned. A simple algorithm is proposed for the design of the state estimator.     

Canonical transformation for a class of time-varying multivariable systems

This paper concerns the transformation of time-varying multivariable systems into canonical structures. The study employs differential matrix operators. It leads to a systematic and straightforward technique for developing canonical forms for time-varying multivariable systems that are uniformly observable and «lexicography-fixed’. It shows that canonical forms derived by other authors are special cases of the canonical form of this paper. The derived canonical forms are not unique. However, their structures are controlled by the designer     

Fault estimator design for a class of switched systems with time-varying delay

This article studies the problem of fault estimator design for switched time-delay systems with impulsive control. The delay signal is assumed to be uniformly bounded, differentiable and has a bounded derivative. The problem is first formulated in the framework of H _∞ filtering by incorporating a priori knowledge of the fault into the design procedure. A hybrid controller, composed of a fault estimator and an impulsive controller, is constructed. Some delay-dependent sufficient conditions are derived on the existence of the hybrid controller by using the multiple Lyapunov functional approach. In addition, based on the cone complementarity algorithm, the solutions to the parameter matrices are obtained by solving a set of linear matrix inequalities. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.     

Design of single-functional observers for linear time-varying multivariable systems

A method is presented for the design of an observer capable of reconstructing a single linear functional of the states of a linear time-varying system. The method is based on a canonical transformation of uniformly observable time-varying multi-variable system. It is shown that this canonical form, along with certain pre-specified structures of observer coefficient matrices, leads to a systematic and straightforward procedure for designing a functional observer of minimal order.     

Design of state feedback for large-scale multivariable systems

This note presents a general method which reduces the computational requirements in the state feedback design of large-scale multivariable systems. The given system is first transformed into a general block canonical form by using simple equivalent transformation. The state feedback problem is then reformulated in terms of a Sylvester equation. Finally, the transformed system matrices along with certain assumed block forms for unknown matrices enable the Sylvester equation to be decomposed and solved effectively.     

Design of multivariable linear time-varying tracking systems with stochastic disturbances

A new designing method of multivariable linear time-varying tracking systems with stochastic disturbances for polynomial command inputs is proposed. It has been shown that it is possible to choose the matrices of gain elements placed in feedbacks and parallel paths of the closed-loop system so that the output y(t) tracks the polynomial command input v(t) with (E expectation operator).     

Self-tuning regulators for a class of multivariable systems

Control of a class of multivariable systems described by linear vector difference equations with constant but unknown parameters is discussed. A multivariable minimum variance strategy is first presented. This gives a generalization of the minimum variance strategy for single-input single-output systems. A multivariable self-tuning regulator based on the minimum variance strategy is then proposed. It uses a recursive least squares estimator and a linear controller obtained directly from the current estimates. The asymptotic properties of the algorithm are discussed. If the estimated parameters converge, the resulting controller will under certain conditions give the minimum variance strategy. The analysis also gives insight into the case when several single-input single-output self-tuning regulators are operating in cascade mode.     

Sensor diagnosis and state estimation for a class of skew symmetric time-varying systems

In this contribution we investigate the problem of simultaneous observer based sensor fault reconstruction and state estimation of a class of linear time-varying (LTV) systems that are skew-symmetric models. The main features concern the use of a bank of observers to detect and isolate faulty sensors and in the same time provide unbiased state estimation. On the other hand, we introduced a switching gain technique to deal with singular points. Stability analysis is achieved thanks to the Barbalat’s lemma and without solving the well-known time-varying Sylvester equation. The proposed approach is extended to more general LTV systems of any order.     

Finite-time stabilization by state feedback control for a class of time-varying nonlinear systems

In this paper, finite-time stabilization is considered for a class of nonlinear systems dominated by a lower-triangular model with a time-varying gain. Based on the finite-time Lyapunov stability theorem and dynamic gain control design approach, state feedback finite-time stabilization controllers are proposed with gains being tuned online by two dynamic equations. Different from many existing finite-time control designs for lower-triangular nonlinear systems, the celebrated backstepping method is not utilized here. It is observed that our design procedure is much simpler, and the resulting control gains are in general not as high as those provided by the backstepping method. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.     

Minimal-order observer designs for linear time-varying multivariable systems

This note presents two simple approaches for the design of minimal-order observer for linear time-varying systems. Both techniques rely on canonical transformations of the given system. As a consequence, simple expressions are obtained which permit the observer constraint equation to be solved effectively. The observers possess the desirable property of having a time-invariant observer coefficient matrix with arbitrary eigenvalues. A comparison is made between the proposed methods from which conclusions are drawn. An illustrative example is also included.     

Recursive solutions to deadbeat observers for time-varying multivariable systems

This paper describes a new construction of the deadbeat observer for the state estimation of linear time-varying multivariable systems. The proposed design methods are computationally attractive because of their simplicity and recursive nature, and are recommendable even for time-invariant multivariable systems.     

Servotracking for a class of multivariable systems with parasitics

In this paper we show that the reduced-order controllers obtained by Porter (1976) and Porter and Tsingas (1978) for asymptotic tracking of constant reference inputs and polynomial-type reference inputs respectively for a class of singularly perturbed multivariable linear lime-invariant plants are special cases of a more general controller derived from multivariable servomechanism theory. These controllers essentially consist of a servocompensator, a stabilizing control law and a reduced-order observer.     

Arbitrary eigenvalue assignments for linear time-varying multivariable control systems

The problem of eigenvalue assignments for a class of linear time-varying multi-variable systems is considered. Using matrix operators and canonical transformations, it is shown that a time-varying system that is ‘lexicography-fixedly controllable’ can be made via state feedback to be equivalent to a time-invariant system whose eigenvalues are arbitrarily assignable. A simple algorithm for the design of the state feedback is provided.     

Feedback stabilization of a class of distributed systems and construction of a state estimator

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.     

Robust real-time algorithms for identification of linear multivariable time-varying systems

The paper presents a discussion on the problem of the robust real-time identification of linear multivariable time-varying dynamic systems working in a noisy environment. Two methodologically different approaches to the design of such algorithms are presented. The first is based on the one-step estimation, optimal in the sense of the minimal conditional mean-square error, combined with convenient approximations of the underlying error covariance matrix. The second is based on the general formulation of robustified stochastic approximation algorithms, characterized by a suitable non-linear transformation of normalized residuals. Particular algorithms are derived on the basis of step-by-step optimization with respect to the weighting matrix of the algorithm. Monte Carlo simulation results illustrate the discussion, and show the efficiency of the proposed robust algorithms in the presence of large disturbance realizations, the so-called outliers.     

Adaptive control for a class of nonlinear systems with time-varying delays in state and input

This paper is concerned with the adaptive stabilization problem of uncertain input delayed systems.A solution to this problem is given for a class of uncertain nonlinear systems with time-varying delays in both state and input.An adaptive asymptotically stabilizing controller,which can guarantee the stability of the closed-loop system and the convergence of the original system state,is designed by means of the Lyapunov-Krasovskii functional stability theory combined with linear matrix inequalities (LMIs) an...     

Robust predictive extended state observer for a class of nonlinear systems with time-varying input delay

ABSTRACT

This paper deals with asymptotic stabilisation of a class of nonlinear input-delayed systems via dynamic output feedback in the presence of disturbances. The proposed strategy has the structure of an observer-based control law, in which the observer estimates and predicts both the plant state and the external disturbance. A nominal delay value is assumed to be known and stability conditions in terms of linear matrix inequalities are derived for fast-varying delay uncertainties. Asymptotic stability is achieved if the disturbance or the time delay is constant. The controller design problem is also addressed and a numerical example with an unstable system is provided to illustrate the usefulness of the proposed strategy.     

Observable state space realizations for multivariable systems

This paper derives two canonical state space forms (i.e., the observer canonical form and the observability canonical form) from multiple-input multiple-output systems described by difference equations. The state space model is expressed by the first-order difference equation and is equivalent to the input–output representation. More specifically, by setting the different state variables, the difference equations or the input–output representations can be transformed into two observable canonical forms and the canonical state space model can be also transformed into the difference equations. Finally, two examples are given.     

Global robust output regulation for a class of multivariable systems

In this paper, we study the global robust output regulation problem for a class of multivariable nonlinear systems. The problem is first converted into a stabilization problem of an augmented system composed of the original plant and an internal model. The augmented system is a multi‐input system containing both dynamic uncertainty and time‐varying static uncertainty. By decomposing the multi‐input control problem into several single‐input control problems, we will solve the problem by solving several single‐input control problems via a recursive approach utilizing the changing supply function technique. The theoretical result is applied to the speed tracking control and load torque disturbance rejection problem of a surface‐mounted permanent magnet synchronous motor. Copyright © 2011 John Wiley & Sons, Ltd.     

Canonical state space representations for multivariable systems

The problem of state space representations for multiple-input, multiple-output linear time-invariant systems is considered. The known definitions of the first and second canonical forma for scalar systems are used to facilitate the derivation of explicit canonical expressions for multivariable systems.