Page 987 - The Mechatronics Handbook

0066_frame_Ch33.fm Page 11 Wednesday, January 9, 2002 8:00 PM

LMM in the State of Space
The used variables are: x 1 , the rod position; x 2 , the rod velocity; x 3 , the rod acceleration; x 4 , the spool
position; and x 5 , the spool velocity.

x ˙ 1 = x 2 t(), x ˙ 2 = x 3 t(), x ˙ 4 = x 5 t()

x ˙ 3 = – w Z x 2 t() 2D Z w Z x 3 t() + k Z w Z x 4 t() (33.16)
2
2
–
x ˙ 5 = – w V x 4 t() 2D V w V x 5 t() + k V w V ut()
2
2
–
Thus the MM of the axis in state-space form becomes
x ˙ t() = Ax t() + bu t()
(33.17)
T
y t() = c x t()
where
 
 0 1 0 0 0   0 
 0 0 1 0 0   0 

A =   0 – 2 – 0   , b =   0  , c = ( 1 0000) (33.18)
T
 w Z 2D Z w Z k Z   
 0 0 0 0 1   0 
 2   
 0 0 0 – w V – 2D V w V  k V

Controller Design
T
The characteristic polynomial is obtained from det[sI − (A C − b C r )] = 0, where A C and b C are the
controllable forms of the matrices A and b. If A ≠ A C , the use of transformation matrix T is advisable,
−1
T
in order to obtain A C and b C . Thus A C = TAT , and b C = Tb. The matrix F = A C − b C r has the form
 0 1 .. 0 
 0 0 .. 0 
F =   . . .. .   (33.19)
 
 0 0 .. 1 
 
−a 0 – r 1 −a 1 – r 2 .. −a n−1 – r n
The characteristic polynomial of the matrix F is

s + ( a n−1 + r n )s n−1 + … + ( a 1 + r 2 )s + ( a 0 + r 1 ) (33.20)
n
The poles chosen for the closed-loop determine the polynomial

s + p n−1 s n−1 + p n−2 s n−2 + … + p 1 s + p 0 (33.21)
n
T
The polynomials (33.20) and (33.21) are identical; therefore, the coefficients of matrix r R are
r v = p v−1 – a v−1 , v = 1,…, n

If

T
A = Ac, r = r R T

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