Linear Time-varying Discrete-time Systems Research Articles

The poles and zeros of a linear time-varying system

For linear time-varying discrete-time and continuous-time systems, a notion of poles and zeros is developed in terms of factorizations of operator polynomials with time-varying coefficients. In the discrete-time case, it is shown that the poles can be computed by solving a nonlinear recursion with time-varying coefficients. In the continuous-time case, the poles can be calculated by solving a nonlinear differential equation with time-varying coefficients. The theory is applied to the study of the zero-input response and asymptotic stability. It is shown that if a time-varying analogue of the Vandermonde matrix is invertible, the zero-input response can be decomposed into a sum of modes associated with the poles. Stability is then studied in terms of the components of the modal decomposition.

Stabilizability and Stable-Proper Factorizations for Linear Time-Varying Systems

The objective of this paper is to study in detail stabilizability and stable-proper factorizations for linear time-varying discrete-time systems. Our main results are: (i) If a linear-time-varying system can be stabilized by dynamic state feedback, then it can also be stabilized by memoryless state feedback. (ii) A complete characterization of the existence of stable-proper factorizations for linear time-varying input/output operators. This characterization is nontrivial; there exist input/output operators that do not admit stable-proper factorizations.

Polynomial matrix-fraction representations for linear time-varying systems

This paper deals with the existence and associated realization theory of skew polynomial fraction representations for linear time-varying discrete-time systems. It is shown that the time-varying analog of the polynomial model always has a free state-space. Necessary and sufficient conditions for the existence of Bezout factorizations are obtained in terms system theoretic properties of state-space realizatios.

離散時間形式線形時変系へのモデル規範形学習制御

離散時間形式線形時変系へのモデル規範形学習制御

Uniformly optimal control of linear feedback systems

The problem of uniformly optimal control is posed in the context of linear time-varying discrete-time systems; the controller is optimal uniformly over a pre-specified set of exogenous signals. Existence of an optimal controller is proved and a formula for the minimum cost is derived. The time-invariant case is treated in the frequency domain. It is shown that for time-invariant systems an optimal time-varying controller is no better than an optimal time-invariant controller.

A Transfer-Function Approach to Linear Time-Varying Discrete-Time Systems

In the first part of the paper a transfer-function approach is developed for the class of linear time-varying discrete-time systems. The theory is specified in terms of skew (noncommutative) rings of polynomials and formal power series, both with coefficients in a ring of time functions. The transfer-function matrix is defined to be a matrix whose entries belong to a skew ring of formal power series. It is shown that various system properties, such as asymptotic stability, can be characterized in terms of the skew-ring framework. In the last part of the paper, the transfer-function framework is applied to the study of feedback control. New results are obtained on assignability of system dynamics by using dynamic output feedback and dynamic state feedback. The results are applied to the control of an armature-controlled do motor with a variable loading.

Invertibility of linear time-varying discrete-time systems

For linear time-varying discrete-time systems, left, right, weak left, and weak right ivnertibility are defined. It is shown that left and weak left invertibility are dual D right and weak right invertibility respectively.

Detectability and Stabilizability of Time-Varying Discrete-Time Linear Systems

The concepts of detectability and stabilizability are explored for time-varying systems. We study duality, invariance under feedback, an extended version of the lemma of Lyapunov, existence of stabilizing feedback laws, linear quadratic filtering and control, and the existence of approximate canonical forms.

Regelung zeitvarianter zeitdiskreter linearer Mehrfachsysteme/ Control of linear time-varying discrete-time multiple-input multiple-output systems

Article Regelung zeitvarianter zeitdiskreter linearer Mehrfachsysteme/ Control of linear time-varying discrete-time multiple-input multiple-output systems was published on December 1, 1979 in the journal at - Automatisierungstechnik (volume 27, issue 1-12).

Regelung zeitvarianter zeitdiskreter linearer Mehrfachsysteme /Control of linear time-varying discrete-time multiple-input multiple-output systems

Article Regelung zeitvarianter zeitdiskreter linearer Mehrfachsysteme /Control of linear time-varying discrete-time multiple-input multiple-output systems was published on December 1, 1979 in the journal at - Automatisierungstechnik (volume 27, issue 1-12).

A direct way to stabilize continuous-time and discrete-time linear time-varying systems

We derive, in a parallel fashion, state-feedback control laws for continuous-time and discrete-time linear time-varying systems; these laws assure that the zero solution of the closed-loop system is (globally) uniformly exponentially stable at a prescribed rate.

Algebraic Theory of Linear Time-Varying Systems

An algebraic theory of linear time-varying discrete-time systems is developed in terms of a module structure defined over a noncommutative polynomial ring. The module setup is induced from a semilinear transformation that is derived from the given system. Various structural properties of the module framework are explored including the concepts of cyclicity and n-cyclicity. The module theory is then applied to the study of reachability and state feedback. Results on the construction of feedback controllers are obtained that resemble pole or coefficient assignability in the theory of time-invariant systems.

On the estimation of noise covariances in linear discrete-time systems

This paper is concerned with an approach for estimating or tracking the time-varying input and measurement noise covariances in time-varying discrete-time linear systems. The approach is firstly to introduce the estimators for the case where the noise co-variances are unknown constants. (The estimators are defined as the mean squares of the estimators of noises based on all the available measurement data.) They arc then transformed in sequential form, and are subsequently modified by incorporating a fading memory to yield estimates for time-varying noise covariances. The time-varying noise covariance estimates are evaluated as the fading mean squares of the estimates of noises based on all the measurement data up to present time. A numerical example for a simple system indicates acceptable performance of the proposed method.

On explicit solution, stability, and reduction of a class of linear time-varying discrete-time systems

Explicit results on solution, stability and reduction of a class of discrete-time linear time-varying systems are presented. It is shown that the state transition matrix can be computed explicitly by φ(k,0)=A1 kA2 k ,where A 1and A 2are constant matrices. The stability is completely determined by eigenvalues of A 1 and A 2. It is also shown that the system can be reduced to a time-invariant one by an algebraic transformation. Examples are given to show that stability criteria of linear time-invariant systems cannot, in general, be applied to time-varying cases.