Modern Control Engineering

656 Chapter 9 / Control Systems Analysis in State Space

Then we may take as another set of state variables any set of functions

provided that, for every set of values there corresponds a unique set of

valuesx 1 ,x 2 ,p,xn,and vice versa. Thus, if xis a state vector, then where

is also a state vector, provided the matrix Pis nonsingular. Different state vectors convey

the same information about the system behavior.

9–3 Transformation of System Models with MATLAB

In this section we shall consider the transformation of the system model from transfer

function to state space, and vice versa. We shall begin our discussion with the

transformation from transfer function to state space.

Let us write the closed-loop transfer function as

Once we have this transfer-function expression, the MATLAB command

[A, B, C, D] = tf2ss(num,den)

will give a state-space representation. It is important to note that the state-space repre-

sentation for any system is not unique. There are many (indeed, infinitely many) state-

space representations for the same system. The MATLAB command gives one possible

such state-space representation.

State-Space Formulation of Transfer-Function Systems. Consider the

transfer-function system

(9–22)

There are many (again, infinitely many) possible state-space representations for this

system. One possible state-space representation is

y =[1 0 0]C

x 1

x 2

x 3

S +[0] u

C

x

1

x

2

x

3

S = C

0

0

- 10

1

0

- 5

0

1

- 6

SC

x 1

x 2

x 3

S +C

0

10

- 50

Su

Y(s)

U(s)

=

10s+ 10

s^3 +6s^2 +5s+ 10

Y(s)

U(s)

=

numerator polynomial in s

denominator polynomial in s

=

num

den

xˆ =Px

xˆ,

xˆ 1 ,xˆ 2 ,p,xˆn ,

xˆn=XnAx 1 , x 2 ,p, xnB







xˆ 2 =X 2 Ax 1 , x 2 ,p, xnB

xˆ 1 =X 1 Ax 1 , x 2 ,p, xnB

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