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Fig. 5.44 Unit-step response of full-state feedback design ofExample 5.15.
Comparing the coefficients of the above characteristic equation with that of the
desired characteristic equation, we get ki = 110.7543, k2 - 53.6, and k3 = 1-10.
Thus, the given system with full-state feedback is given by
[{:i']
0 1
0 O ,:,Ol[x[:l+[{lr,t,
-125.7543 -76.6 -
Now, we verify the design by simulating a response to a unit-step input using
MATLAB.4 The results ofthe simulation are shown in Fig.5.44.It can be observed
that the design requirements have been met.
Given the system
Example 5.16
/ji:] = [:5 -3 -,] [x::] +[{].,t,
y=[l 0 0] C::]
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 529
1) Express the given plant in dual phase-variable form. 2) Design a full-state
observer so that the closed-loop characteristic equation is given by S3 + lOOS2 +
1500s + 40,000 - 0. 3) Verify the design for u(t) = 25t assunung xi(0) = 5,
X2(0) : 1.5, and X3(0) : 0.25.
Solution. We have the given system
x - Ax + Bu
where
-5 .2 0
A- 1 -3 1
o 1-
y = Cx
B=[{] C=[l 0 0]
It is converuent to use MATLAB4 to compute the observability matrix, which for
this system is found to be
Qo.l=[-,~s -,6 0;]
Using MATLAB,4 we find that this system has full rank of three; hence the system
is observable. Furthermore,
Q-,l = [i:.i
:011 00:5]
Next step is to express the given plantin the dualphase-variable form. We know that
the elements of the first column of the matrix in dual phase-variable form are the
coefficients of the characteristic equation. Conversely, given the coefficients of the
charactenstic equation, we can directly write down the dual phase-variable form
of matrix A.
As we know, the characteristic equation is given by
A(s) -. lsl - Al = 0
S3+9S2+20s+8 = 0
0.5
0.45
0.4
0,35
o 0.3
g
E 0.25
<
0.2
0.15
0.1
0.05
o
530 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Then, the dual phase-variable fc if the giver
Furthermore, we assume that
Cz=[l 0 0]
Then,
QO.z = (_g;A2):]
P - Qo IQo*z
= [-2
-9 1]
Now let us design the full-state observer for the system transformed in dual
phase-variable form. We have
-(9+11) 1
Az-LzCz= -(20+12) 'g ,]
-(8+13) 0
The characteristic equation of the observer is given by
S3 + (9 +II)SZ + (20 +I2)S +8 +l3 = 0
The desired characteristic equation is
S3 + lOOS2 + 150Cs + 40,000 -. O
Comparing the coefficients oflike powers of s, we get li - 91,12 = 1480, and
l3 = 39,992. Then, the gain matrix corresponding to the given system is obtained
by Lransforming back as
~.r = P,z = [,j:'84,]
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 531
:j
4
ZJ
a
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