## 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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