Modern Control Engineering

712 Chapter 9 / Control Systems Analysis in State Space

In the present case we have

(9–106)

By substituting l 1 ,l 4 ,p,lmforlin Equation (9–106), we obtain the following m-2equations:

(9–107)

By differentiating Equation (9–106) with respect to l, we obtain

(9–108)

where

Substitution of l 1 forlin Equation (9–108) gives

Referring to Equation (9–105), this last equation becomes

(9–109)

Similarly, differentiating Equation (9–106) twice with respect to land substituting l 1 forl,we

obtain

This last equation can be written as

(9–110)

Rewriting Equations (9–110), (9–109), and (9–107), we get

(9–111)

a 0 +a 1 lm+a 2 l^2 m+p+am- 1 lmm-^1 =fAlmB







a 0 +a 1 l 4 +a 2 l^24 +p+am- 1 lm 4 -^1 =fAl 4 B

a 0 +a 1 l 1 +a 2 l^21 +p+am- 1 lm 1 -^1 =fAl 1 B

a 1 + 2 a 2 l 1 +p+(m-1)am- 1 lm 1 -^2 =f¿Al 1 B

a 2 + 3 a 3 l 1 +p+

(m-1)(m-2)

2

am- 1 lm 1 -^3 =

f–Al 1 B

2

f–Al 1 B= 2 a 2 + 6 a 3 l 1 +p+(m-1)(m-2)am- 1 lm 1 -^3

d^2

d^2 l

f(l)^2

l=l 1

=f–Al 1 B=

d^2

dl^2

a(l)^2

l=l 1

f¿Al 1 B=a 1 + 2 a 2 l 1 +p+(m-1)am- 1 lm 1 -^2

d

dl

f(l)^2

l=l 1

=f¿Al 1 B=

d

dl

a(l)^2

l=l 1

Al-l 1 B^2 h(l)=

d

dl

Cg(l)Al-l 1 B^3 Al-l 4 BpAl-lmBD

d

dl

f(l)=Al-l 1 B^2 h(l)+

d

dl

a(l)

fAlmB=aAlmB







fAl 4 B=aAl 4 B

fAl 1 B=aAl 1 B

=g(l)CAl-l 1 B^3 Al-l 4 BpAl-lmBD+a(l)

f(l)=g(l)f(l)+a(l)

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