Modern Control Engineering

836 Chapter 10 / Control Systems Design in State Space

we find that the rank of

isn+1.Thus, the system defined by Equation (10–168) is completely state controllable.

A–10–13. Consider the system shown in Figure 10–49. Using the pole-placement-with-observer approach,

design a regulator system such that the system will maintain the zero position Ay 1 =0andy 2 =0B

in the presence of disturbances. Choose the desired closed-loop poles for the pole-placement part

to be

and the desired poles for the minimum-order observer to be

First, determine the state feedback gain matrix Kand observer gain matrix Ke.Then, obtain

the response of the system to an arbitrary initial condition—for example,

wheree 1 ande 2 are defined by

Assume that m 1 =1kg,m 2 =2kg,k=36Nm, and b=0.6N-sm.

Solution.The equations for the system are

By substituting the given numerical values for m 1 , m 2 , k,andband simplifying, we obtain

Let us choose the state variables as follows:

x 4 =y# 2

x 3 =y# 1

x 2 =y 2

x 1 =y 1

y

$

2 =18y 1 - 18y 2 +0.3y

1 - 0.3y

2

y

$

1 =-36y 1 +36y 2 - 0.6y

1 +0.6y

2 +u

m 2 y

$

2 =kAy 1 - y 2 B+bAy

1 - y

2 B

m 1 y

$

1 =kAy 2 - y 1 B+bAy

2 - y

1 B+u

e 2 =y 2 - y 2

e 1 =y 1 - y 1

e 1 ( 0 )=0.1, e 2 ( 0 )=0.05

y 1 ( 0 )=0.1, y 2 ( 0 )=0, y# 1 ( 0 )=0, y# 2 ( 0 )= 0

s=-15, s=- 16

s=- 2 +j2 13 , s=- 2 - j2 13 , s=-10, s=- 10

CBˆAˆ BˆAˆ^2 BˆpAˆn BˆD

m 1 m 2

y 1 y 2

u

k

b

Regulator

Figure 10–49

Mechanical system.

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