Lecture12 pole-placement-multi-inputs-2020

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6. rD
Example: Consider the system described by
x(k+',=
[+ I 1,il
*.[; 1]
",-,
[r 2
vrr):[1 3lt] oo,
Find a feedback gain matrix K so as to locate the closed loop of the system at zr,z: -0.3 + 0.4 j ,
u,q= 0. Further, find the steady state outputs due to step inputs.
Solution:
lo LL 13 0 I
CoN=[HG*HG2*HG3*Hf =11 1 3 0 12 -1 I
"nr 'tl-lo
1 o 1 o 1 |
11 -1 5 -1 22 -3 I
coNR:lh1 GttlttzGhzl:[: 1 i ll
Lr 3 :, -"1
P
w,1:l+1 1l]
*':[;,
l] 1]
coNRr:inverse(coNR):[j
1 + +]
The closed loop characteristic equation is

7. r,
d:P GP=
[i i i i]
and H^:
[i il
z4 + 0.6 z3 + 0.25 *
And hence K" is given by
*" =
[]n i -'.ru ;:-l and K = K^ p'r : [01. s.zss _?.u o.tus]
The closed loop system is given by
x(k+l) : (G-H*K) x(k) + Hr(k)
Note, if we change the locations of the closed loop poles we will see that the steady state values
of the outputs will change also which is not what we want. What we want is that the outputs will
follow the inputs at steady state if the number of inputs equal the number of outputs. To achieve
that we'need to add integrators between the errors ( (k) - y(k) and the corresponding inputs u(k).
[-o.s -2.3s L.6 -.65 I I'0 1 I
x(k+,;=:
I_3:3 :i:31 2:2
jrrjrloo. lo, I l,ror
L o.a 3.3s 0.4 z.6s I [r -11
and x,, = ( r - c, ) H
fif
:
f-;l;ilt,aul
and y,, : c x,, :
[-rr.ilrrf]
L L9.3077 l