Modern Control Engineering

652 Chapter 9 / Control Systems Analysis in State Space

EXAMPLE 9–1 Consider the system given by

Obtain state-space representations in the controllable canonical form, observable canonical form,

and diagonal canonical form.

Controllable Canonical Form:

Observable Canonical Form:

Diagonal Canonical Form:

Eigenvalues of an nnMatrix A. The eigenvalues of an n*nmatrixAare the

roots of the characteristic equation

|lI-A|=0

The eigenvalues are also called the characteristic roots.

Consider, for example, the following matrix A:

The characteristic equation is

|lI-A|=

=l^3 +6l^2 +11l+6

=(l+1)(l+2)(l+3)=0

The eigenvalues of Aare the roots of the characteristic equation, or –1, –2,and–3.

Diagonalization of nnMatrix. Note that if an n*nmatrixAwith distinct

eigenvalues is given by

3

l

0

6

- 1

l

11

0

- 1

l+ 6

3

A= C

0

0

- 6

1

0

- 11

0

1

- 6

S

y(t)=[2 - 1]B

x 1 (t)

x 2 (t)

R

B

x

1 (t)

x# 2 (t)

R = B

- 1

0

0

- 2

RB

x 1 (t)

x 2 (t)

R + B

1

1

Ru(t)

y(t)=[0 1]B

x 1 (t)

x 2 (t)

R

B

x# 1 (t)

x

2 (t)

R = B

0

1

- 2

- 3

RB

x 1 (t)

x 2 (t)

R+ B

3

1

Ru(t)

y(t)=[3 1]B

x 1 (t)

x 2 (t)

R

B

x# 1 (t)

x# 2 (t)

R = B

0

- 2

1

- 3

RB

x 1 (t)

x 2 (t)

R + B

0

1

Ru(t)

Y(s)

U(s)

=

s+ 3

s^2 +3s+ 2

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