Modern Control Engineering

706 Chapter 9 / Control Systems Analysis in State Space

By identifying the coefficients aiof the minimal polynomial (which is the same as the characteristic

polynomial in this case), we have

The inverse of Acan then be obtained from Equation (9–100) as follows:

A–9–11. Show that if matrix Acan be diagonalized, then

wherePis a diagonalizing transformation matrix that transforms Ainto a diagonal matrix, or

P–1AP=D, where Dis a diagonal matrix.

Show also that if matrix Acan be transformed into a Jordan canonical form, then

whereSis a transformation matrix that transforms Ainto a Jordan canonical form J, or S–1AS=J.

Solution.Consider the state equation

If a square matrix can be diagonalized, then a diagonalizing matrix (transformation matrix) exists

and it can be obtained by a standard method. Let Pbe a diagonalizing matrix for A. Let us define

Then

whereDis a diagonal matrix. The solution of this last equation is

Hence,

x(t)=Pxˆ(t)=PeDt P-^1 x(0)

xˆ(t)=eDt xˆ(0)

xˆ

=P-^1 APxˆ=Dxˆ

x=Pxˆ

x

=Ax

eAt=SeJt S-^1

eAt=PeDt P-^1

= C

3

17

7

17

1

17

6

17

• 173
  2
  17

- 174

2

17

• 177

S

=

1

17

C

3

7

1

6

- 3

2

- 4

2

- 7

S

=

1

17

cC

7

- 2

- 2

0

7

2

- 4

8

9

S+ 3 C

1

3

1

2

- 1

0

0

- 2

- 3

S - 7 C

1

0

0

0

1

0

0

0

1

Ss

A-^1 =-

1

a 3

AA^2 +a 1 A+a 2 IB=

1

17

AA^2 + 3 A- 7 IB

a 1 =3, a 2 =-7, a 3 =- 17

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