# Linear Systems: Discrete-Time, Time-Varying, State Variable Descriptions

**DOI:**https://doi.org/10.1007/978-1-4471-5102-9_191-1

## Abstract

Discrete-time processes that can be modeled by linear difference equations with time-varying coefficients can be written in terms of state variable descriptions of the form \(x(k + 1) = A(k)x(k) + B(k)u(k),\ y(k) = C(k)x(k) + D(k)u(k)\). The response of such systems due to a given input and initial conditions is derived. Equivalence of state variable descriptions is also discussed.

## Keywords

Linear systems Discrete-time Time-varying State variable descriptions## Introduction

Discrete-time systems arise in a variety of ways in the modeling process. There are systems that are inherently defined only at discrete points in time; examples include digital devices, inventory systems, and economic systems such as banking where interest is calculated and added to savings accounts at discrete time interval. There are also systems that describe continuous-time systems at discrete points in time; examples include simulations of continuous processes using digital computers and feedback control systems that employ digital controllers and give rise to sampled-data systems.

*x*(

*k*) (\(k \in \mathbb{Z}\), the set of integers) is a column vector of dimension

*n*(\(x(k) \in {\mathbb{R}}^{n}\)); the output is \(y(k) \in {\mathbb{R}}^{m}\) and the input is \(u(k) \in {\mathbb{R}}^{m}\).

*A*(

*k*),

*B*(

*k*),

*C*(

*k*), and

*D*(

*k*) are matrices with entries functions of time

*k*,

*A*(

*k*) = [

*a*

_{ ij }(

*k*)], \(a_{ij}(k) : \mathbb{Z} \rightarrow \mathbb{R}\) (\(A(k) \in {\mathbb{R}}^{n\times n}\), \(B(k) \in {\mathbb{R}}^{n\times m}\), \(C(k) \in {\mathbb{R}}^{p\times n}\), \(D(k) \in {\mathbb{R}}^{p\times m}\)). The vector difference equation in (1) is the state equation, while the algebraic equation is the output equation. Note that in the time-invariant case,

*A*(

*k*) =

*A*,

*B*(

*k*) =

*B*,

*C*(

*k*) =

*C*, and

*D*(

*k*) =

*D*.

The advantage of the state variable description (1) is that given an input *u*(*k*), *k* ≥ *k* _{0} and an initial condition *x*(*k* _{0}) = *x* _{0}, the state trajectories or motions for *k* ≥ *k* _{0} can be conveniently characterized. To determine the expressions, we first consider the homogeneous state equation and the corresponding initial value problem.

## Solving *x*(*k* + 1) = *A*(*k*)*x*(*k*); *x*(*k* _{0}) = *x* _{0}

*state transition matrix*of (2) given by

## System Response

*y*(

*k*) of (1) is

*state response*(when

*u*(

*k*) = 0 and the system is driven only by the initial state conditions) and the

*input response*(when

*x*(

*k*

_{0}) = 0 and the system is driven only by the input

*u*(

*k*)); this illustrates the linear systems principle of superposition.

## Equivalence of State Variable Descriptions

*P*

^{− 1}(

*k*) exists. Then

## Summary

State variable descriptions for linear discrete-time time-varying systems were introduced and the state and output responses to inputs and initial conditions were derived. The equivalence of state variable representations was also discussed.

## Cross-References

“Linear Systems: Discrete-Time, Time-Invariant, State Variable Descriptions,” Panos J. Antsaklis

“Linear Systems: Discrete-Time, Impulse Response Descriptions,” Panos J. Antsaklis

“Linear Systems: Continuous-Time, Time-Varying, State Variable Descriptions,” Panos J. Antsaklis

“Sampled Data Systems,” Panos J. Antsaklis and H. L. Trentelman

## Recommended Reading

The state variable descriptions received wide acceptance in systems theory beginning in the late 1950s. This was primarily due to the work of R.E. Kalman. For historical comments and extensive references, see Kailath (1980). The use of state variable descriptions in systems and control opened the way for the systematic study of systems with multi-inputs and multi-outputs.

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