Modern Control Engineering

714 Chapter 9 / Control Systems Analysis in State Space

Substitutingl 1 =2, and l 3 =1into these three equations gives

Solving for a 0 (t),a 1 (t), and a 2 (t), we obtain

Hence,

A–9–15. Show that the system described by

(9–114)

(9–115)

where

is completely output controllable if and only if the composite m*nrmatrixP, where

is of rank m. (Notice that complete state controllability is neither necessary nor sufficient for

complete output controllability.)

Solution.Suppose that the system is output controllable and the output y(t)starting from any y(0),

the initial output, can be transferred to the origin of the output space in a finite time interval

0 tT. That is,

y(T)=Cx(T)= 0 (9–116)

P=CCB  CAB  CA^2 B  p  CAn-^1 BD

C=m*n matrix

B=n*r matrix

A=n*n matrix

y=output vector (m-vector) (mn)

u=control vector (r-vector)

x=state vector (n-vector)

y=Cx

x#=Ax+Bu

= C

e2t

0

0

12et-12e2t+13te2t

e2t

• 3et+3e2t
  
  
  • 4et+4e2t
    0
    et
  
  
  
  

S

+Aet-e2t+te2tBC

4

0

0

16

4

9

12

0

1

S

eAt=A4et-3e2t+2te2tBC

1

0

0

0

1

0

0

0

1

S +A-4et+4e2t-3te2tBC

2

0

0

1

2

3

4

0

1

S

a 2 (t)=et-e2t+te2t

a 1 (t)=- 4 et+ 4 e^2 t- 3 te^2 t

a 0 (t)=4et-3e2t+2te2t

a 0 (t)+a 1 (t)+a 2 (t)=et

a 0 (t)+ 2 a 1 (t)+ 4 a 2 (t)=e2t

a 1 (t)+ 4 a 2 (t)=te2t

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