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At. Similarly, we can get X2(At), X3(At),. - . , x" (At) by successively solving the
other equations.
The phase-variab/e form of representation has another advantage that the ele-
ments of the last row constitute the coefficients of the characteristic equation as
follows:
A(s) = s +an _lSn-l.+...+alS +00 = 0 (5.221)
This property is useful in the design of compensators using the pole-placement
method, which we will discuss a little later.
5.10.6 Conversion of Differential Equations to
Phase-Variable Form
Let the given dynamical system be represented by the following linear differ-
ential equation:
dy
diy d" y+..-+aid aoy=u(t) (5.222)
dty + an_i dt l1 d~ + ao
Let us select a set of state vanables such that each subsequent state variable is
defined as the derivative of the previous state variable. That is,
Then,
XI =y
XI - X2
X2 -. X3
dn-iy
X2 - y - Xl, - -.,Xn = d~y = k" _1 (5.223)
(5.224)
(5.225)
Xn - -aoxi - alx2 - a2X3 - - - - - an -lXn + u(t) (5.226)
Or, in matrix form,
x -. Ax + Bu
(5.227)
where
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 515
The output is given by
B-
Xl
X2
y=Cx=[l O ... O] X3
xl
(5.228)
(5.229)
(5.230)
(5.231)
Thus, by choosing each successive state variable to be the derivative ofthe previous
one, we are able to express the given differential equation in the state-space, phase-
variable form.
5.10.7 Conversion of General State-Space Representation to
Phase-Variable Form
A system given in a general state-space form can be expressed iri the phase-
variable form if the system is controllable. Let
x - Ax + Bu
y = Cx
(5.232)
(5.233)
oioo.o
ooio.o
A= 0 0 O 1 O
..,,
....-.
-ao -ai -a2 ' ' -an_i
516 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
be the given plant, which is notin phase-variable form. We assume that this system
is controllable. Furthermore, let us assume that there exists a matrix P, which is
defined as
z - Px (5.234)
which transforms the given system into phase-variable form,
1
0
O
00
10
01
0
O
0
0
0
-ao -ai ' ' ' -an_i
The transformation matrix P has the form
p\\ P12 ' ' '
p21p22"
p31 P32 . ' -
P -.
.-...
.....
Pnl pn2 ' ' -
We have
Zi - Pix
so that
u (5.235)
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