From Modern Control Systems: An Introduction

1.9 SOLUTION OF STATE EQUATIONS FOR LINEAR TIME-INVARIANT SYSTEMS

1.9.1 State Transition Matrix (STM)

The state equation of a linear time-invariant system is given by


For a homogeneous (unforced) system


we have


Take the Laplace Transform on both sides


Equation (1.20) may also be written as


Take the inverse Laplace


Equation (1.21) gives the solution of the LTI homogeneous state Equation (1.19). From Equation (1.21) it is observed that the initial state x(0) at t = 0, is driven to a state x( t) at time t. This transition in state is carried out by the matrix exponential e A t. Because of this property, e A t is termed as the State Transition Matrix and is denoted by ( t).

Thus,


where, is called the Resolvent Matrix.

As e A t represents a power series of the matrix A t, thus,


1.9.1.1 Properties of the State Transition Matrix

  1. ?(0) = I.

    Proof. ?(t) = e At

    Put t = 0

    We have ?(0) = I.

  2. ? -1(t) = ?(-t)

    Proof. ?(t) = e At

    Postmultiply both sides by e -At

    We have ?(t)e -At = e At.e -At

    or, ?(t) ?(-t) = I

    Premultiply both sides by ? -1( t).

    We have ? -1( t) ?(t) ?(-t) = ? -1( t)

    or, ?


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