Modern Control Engineering

744 Chapter 10 / Control Systems Design in State Space

We assume that the plant given by Equation (10–31) is completely state controllable. The

transfer function of the plant can be given by

To avoid the possibility of the inserted integrator being canceled by the zero at the origin

of the plant, we assume that Gp(s)has no zero at the origin.

Assume that the reference input (step function) is applied at t=0.Then, for t>0,

the system dynamics can be described by an equation that is a combination of Equations

(10–31) and (10–34):

(10–35)

We shall design an asymptotically stable system such that x(q),j(q), and u(q) approach

constant values, respectively. Then, at steady state, and we get y(q)=r.

Notice that at steady state we have

(10–36)

Noting that r(t)is a step input, we have r(q)=r(t)=r(constant) for t>0.By

subtracting Equation (10–36) from Equation (10–35), we obtain

B (10–37)

x#(t)-x#(q)

j

(t)-j

(q)

R = B

A

- C

0

0

RB

x(t)-x(q)

j(t)-j(q)

R + B

B

0

RCu(t)-u(q)D

B

x#(q)

j

(q)

R = B

A

- C

0

0

RB

x(q)

j(q)

R +B

B

0

Ru(q)+ B

0

1

Rr(q)

j

(t)=0,

B

x#(t)

j

(t)

R = B

A

- C

0

0

RB

x(t)

j(t)

R +B

B

0

Ru(t)+ B

0

1

Rr(t)

Gp(s)=C(s I-A)-^1 B

C= 1 *n constant matrix

B=n* 1 constant matrix

A=n*n constant matrix

r =reference input signal (step function, scalar)

j=output of the integrator (state variable of the system, scalar)

y =output signal (scalar)

u =control signal (scalar)

y

K

A

 kI B  C

r j x

.

j u

+– +– +

+

Figure 10–6

Type 1 servo system.

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