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

**DOI:**https://doi.org/10.1007/978-1-4471-5102-9_190-1

## Abstract

Continuous-time processes that can be modeled by linear differential equations with time-varying coefficients can be written in terms of state variable descriptions of the form \(\dot{x}(t) = A(t)x(t) + B(t)u(t),\ y(t) = C(t)x(t) + D(t)u(t)\). The response of such systems due to a given input and initial conditions is derived using the Peano-Baker series. Equivalence of state variable descriptions is also discussed.

## Keywords

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

*x*(

*t*) (\(t \in \mathbb{R}\), the set of reals) is a column vector of dimension

*n*(\(x(t) \in {\mathbb{R}}^{n}\)) and

*A*(

*t*),

*B*(

*t*),

*C*(

*t*),

*D*(

*t*) are matrices with entries functions of time

*t*. \(A(t) = [a_{ij}(t)],\ a_{ij}(t) : \mathbb{R} \rightarrow \mathbb{R}\). \(A(t) \in {\mathbb{R}}^{n\times n},B(t) \in {\mathbb{R}}^{n\times m},C(t) \in {\mathbb{R}}^{p\times n},D(t) \in {\mathbb{R}}^{p\times m}\). The input vector is \(u(t) \in {\mathbb{R}}^{m}\) and the output vector is \(y(t) \in {\mathbb{R}}^{p}\). The vector differential equation in (1) is the

*state equation*, while the algebraic equation is the

*output equation*.

The advantage of the state variable description (1) is that given an input \(u(t),\ t \geq 0\) and an initial condition *x*(*t* _{0}) = *x* _{0}, the state trajectory or motion for \(t \geq t_{0}\) can be conveniently characterized. To derive the expressions, we first consider the homogenous state equation and the corresponding initial value problem.

## Solving \(\dot{x}(t) = A(t)x(t);\ x(t_{0}) = x_{0}\)

*x*(

*t*) = [

*x*

_{1}(

*t*),

*…*,

*x*

_{ n }(

*t*)]

^{ T }is the state (column) vector of dimension

*n*and

*A*(

*t*) is an

*n*×

*n*matrix with entries functions of time that take on values from the field of reals (\(A \in {\mathbb{R}}^{n\times n}\)).

Under certain assumptions on the entries of *A*(*t*), a solution of (2) exists and it is unique. These assumptions are satisfied, and a solution exists and is unique in the case, for example, when the entries of *A*(*t*) are continuous functions of time. In the following we make this assumption.

*method of successive approximations*which when applied to

*m*→

*∞*,

*ϕ*

_{ m }converges to the unique solution of (3), assuming the

*f*satisfies certain conditions.

*m*→

*∞*, and under the above continuity assumptions on

*A*(

*t*),

*ϕ*

_{ m }(

*t*) converges to the unique solution of (2), i.e.,

*n*×

*n*matrix \(\Phi (t,t_{0})\) is called the

*state transition matrix*of (2). The defining series (6) is called the Peano-Baker series.

*A*(

*t*) =

*A*, a constant matrix, then (6) becomes

## System Response

*variation of constants formula*. This result can be shown via direct substitution of (10) into (9); note that \(\phi (t) = \Phi (t_{0},t_{0})x_{0} = x_{0}\). That (10) is a solution can also be shown using a change of variables in (9), namely,

*state response*(when

*u*(

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

*input response*(when

*x*

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

*u*(

*t*)); this illustrates the linear system principle of superposition.

*y*(

*t*) \((= C(t)x(t) + D(t)u(t))\) is

*δ*(

*t*) and is derived based on the basic property for

*δ*(

*t*), namely,

*δ*(

*t*−

*τ*) denotes an impulse applied at time

*τ*=

*t*.

## Properties of the State Transition Matrix \(\Phi (t,t_{0})\)

In general it is difficult to determine \(\Phi (t,t_{0})\) explicitly; however, \(\Phi (t,t_{0})\) may be readily determined in a number of special cases including the cases in which *A*(*t*) = *A*, *A*(*t*) is diagonal, *A*(*t*)*A*(*τ*) = *A*(*τ*)*A*(*t*).

*n*linearly independent initial conditions

*x*

_{0i }, the corresponding

*n*solutions

*ϕ*

_{ i }(

*t*) are also linearly independent. Let a

*fundamental matrix*\(\Psi (t)\) of \(\dot{x} = A(t)x\) be an

*n*×

*n*matrix, the columns of which are a set of linearly independent solutions \(\phi _{1}(t),\ldots ,\phi _{n}(t)\). The state transition matrix \(\Phi \) is the fundamental matrix determined from solutions that correspond to the initial conditions \({[1,0,0,\ldots ]}^{T},\ {[0,1,0,\ldots ,0]}^{T},\ldots {[0,0,\ldots ,1]}^{T}\) (recall that \(\Phi (t_{0},t_{0}) = I\)). The following are properties of \(\Phi (t,t_{0})\):

- (i)
\(\Phi (t,t_{0}) = \Psi (t){\Psi }^{-1}(t_{0})\) with \(\Psi (t)\) any fundamental matrix.

- (ii)
\(\Phi (t,t_{0})\) is nonsingular for all

*t*and*t*_{0}. - (iii)
\(\Phi (t,\tau ) = \Phi (t,\sigma )\Phi (\sigma ,\tau )\) (semigroup property).

- (iv)
\({[\Phi (t,t_{0})]}^{-1} = \Phi (t_{0},t)\).

## Equivalence of State Variable Descriptions

*P*

^{− 1}(

*t*) exists and

*P*and

*P*

^{− 1}are continuous. Then the system

## Summary

State variable descriptions for continuous-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: Continuous-Time, Time-Invariant, State Variable Descriptions,” Panos J. Antsaklis

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

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

## Recommended Reading

## Bibliography

- Antsaklis PJ, Michel AN (2006) Linear systems. Birkhauser, BostonGoogle Scholar
- Brockett RW (1970) Finite dimensional linear systems. Wiley, New YorkGoogle Scholar
- Kailath T (1980) Linear systems. Prentice-Hall, Englewood CliffsGoogle Scholar
- Miller RK, Michel AN (1982) Ordinary differential equations. Academic, New YorkGoogle Scholar
- Rugh WJ (1996) Linear system theory, 2nd edn. Prentice-Hall, Englewood CliffsGoogle Scholar
- Sontag ED (1990) Mathematical control theory: deterministic finite dimensional systems. Texts in applied mathematics, vol 6. Springer, New YorkGoogle Scholar