Modern Control Engineering

668 Chapter 9 / Control Systems Analysis in State Space

EXAMPLE 9–6 Obtain the time response of the following system:

whereu(t)is the unit-step function occurring at t=0,or

u(t)=1(t)

For this system,

The state-transition matrix was obtained in Example 9–5 as

The response to the unit-step input is then obtained as

or

If the initial state is zero, or x(0)= 0 , then x(t)can be simplified to

9–5 Some Useful Results in Vector-Matrix Analysis

In this section we present some useful results in vector-matrix analysis that we use in

Section 9–6. Specifically, we present the Cayley–Hamilton theorem, the minimal poly-

nomial, Sylvester’s interpolation method for calculating and the linear independence

of vectors.

Cayley–Hamilton Theorem. The Cayley–Hamilton theorem is very useful in

proving theorems involving matrix equations or solving problems involving matrix

equations.

Consider an n*nmatrixAand its characteristic equation:

|lI-A|=ln+a 1 ln–1+p+an–1l+an=0

The Cayley–Hamilton theorem states that the matrix Asatisfies its own characteristic

equation, or that

An+a 1 An–1+p+an–1A+anI= 0 (9–44)

To prove this theorem, note that is a polynomial in lof degree n-1.

That is,

adj(l I-A)=B 1 ln-^1 +B 2 ln-^2 +p+Bn- 1 l+Bn

adj(l I-A)

eAt,

B

x 1 (t)

x 2 (t)

R =C

1

2

• e-t+

1

2

e-2t

e-t-e-2t

S

B

x 1 (t)

x 2 (t)

R= B

2e-t-e-2t

• 2e-t+2e-2t

e-t-e-2t

• e-t+2e-2t

RB

x 1 (0)

x 2 (0)

R+ B

1

2 - e

• t+ 1
  2 e
  
  
  • 2t
    e-t-e-2t
  
  
  
  

R

x(t)=eAt x(0)+

3

t

0

B

2e-(t-t)-e-2(t-t)

• 2e-(t-t)+2e-2(t-t)

e-(t-t)-e-2(t-t)

• e-(t-t)+2e-2(t-t)

RB

0

1

R[1]dt

(t)=eAt= B

2e-t-e-2t

• 2e-t+2e-2t

e-t-e-2t

• e-t+2e-2t

R

(t)=eAt

A= B

0

- 2

1

- 3

R, B= B

0

1

R

B

x

1

x# 2

R= B

0

- 2

1

- 3

RB

x 1

x 2

R + B

0

1

Ru

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