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where Az = P-IA P, Bz = P-IB, and Cz = CP. The observability matrix ofthe
original system is given by
QO.x -.
and that of the transformed system in phase-variable form is given by
so that
Let
Then,
QO.z -
P -. oo1Qo.z
ez -z-z
ez = (Az - Lz Cz)ez
(5.302)
(5.303)
(5.304)
(5.305)
(5.306)
y - y - Czez (5.307)
where z and y are the estimated state and Lhe output vectors in the transformed
system. Thus, the design procedure is as follows: 1) design the observer in the
transformed space and obtain the gain matrix Lz and 2) transform back to the
original system to get the corresponding gain matrix Lx as follows.
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 525
We have z = P. -1X,z = P-Ix,.so that ez = z - z = P-l(x - x) = P-lex and
ez = P-Ie,.. Then,
ex = (A - Lx)ex . (5.308)
y - y = Cex
where
Lx -. PLz
(5.309)
(5.310)
Here, Lx is the gain matrix of the full-state observer corresponding to the given
system.
Now one question that remains to be answered is how to construct the trans-
formation matrix P. We will discuss one method that is based on knowing the
characteristic equation or the eigenvalues of the given system. This approach is
feasible because we have software tools like MAIIAB ti:at are available. We will
illustrate this method in a later example.
Example 5.14
Given the plant,
l+[:l. '
Convert it to the phase;variable form.
Solution. For the given system, the contro
Q.=,B AB A2B,-O'
The rank of this matrix is three because all the columns are linearly independent.
The inverse of this matrix exists and is given by
-1.35 0.4 0.3
Q;i= 1.30 -0.7 0.35 .
-0.15 0.10 -0.05
Then,
Pi-[0 O 1l[B AB A2Bl-'
=[0 0 1] 1.30 -0.7
-0.15 0.10
=[-0.15 0.10 -0,05]
526 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Similarly,
so that
P2 - Pi A
=[-0.05 0.20 -0.10
P3 = Pl A2
=[0.15 0.90 0.05]
-0.15 0.10
P - -0.05 0.20
0.15 0.90
PAP-I =
PB=[{]
Then, the phase-variable form of the given system is given by
where z - Px.
[ili: = [-q :,3 :] [z;:l+[ll,
Example 5.15
Fortheplant, .
25(s + 2)
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