Control Theory and Applications 10.1 System Theory 235
Part A | 10.1
In the above, jj D det./ stands for the determinant of
a matrix and adj./ is the conjugate transpose of the matrix cofactors. In effect, the following conclusions are
made:
The characteristic polynomial of an LTI system with
description in state-space as in (10.34) is the determinant of matrix .sI n A/, p c .s/ D jsI n Aj.
The poles of a system with the general form (10.34),
the roots of p c .s/ D jsI n Aj, are the eigenvalues of
matrix A.
The above general remarks are now demonstrated in
the special case of an LTI–SISO system, with a transfer
function of the following format
H.s/ D
Y.s/
U.s/
D
b 0
s n C a n1 s n1 C C C C C a 1 s C a 0
: (10.43)
It is noted here that although the system is SISO the
state vector is of n dimension, that is, equal to the order of the system. However, there are more than one
choices of the state vector’s components. The problem
of choosing the components of the state vector in order
to formulate a state-space description of an LTI system,
which is otherwise known only in the form of either
a scalar transfer function or a transfer function matrix,
is referred to as the realization problem [10.1, 2]. In
general case of MIMO–LTI systems is not solvable.
However, in the case of LTI–SISO systems with transfer function of the form (10.43), a minimal description,
a description that includes no redundant components
can be deduced as
H.s/ D
Y.s/
U.s/
D
b 0
s n C a n1 s n1 C C C C C a 1 s C a 0
D
b 0
s n C a n1 s n1 C C C C C a 1 s C a 0
X 1 .s/
X 1 .s/
:
After introducing the auxiliary signal as above, the following differential equation is obtained in the time domain. Appropriate initial conditions of the form (10.9)
for solving the differential equation must also be provided.
 d
n
dt n C a n1
d
n1
dt n1 C C C C C a 1
d
dt
C a 0
Ã
x 1 .t/
D u.t/ , x
.n/
1 .t/ D u.t/
n1
X
v D0
a v x
.v /
1 .t/
and y.t/ D b 0 x 1 .t/ :
(10.44)
In the above, x
.v /
1 .t/ stands for the v th order time
derivative of the auxiliary signal x 1 .t/. An appropriate
state vector x is
x D
h
x 1 .t/ x
.1/
1 .t/ ::: x
.n1/
1
.t/
i T :
(10.45)
For the components of x the following differential equations hold,
P
x v D x v C1 ; v D 1; : : : ; n 1
and P
x n D D
n
X
v D1
a .v 1/ x v C u :
(10.46)
If the above are arranged in the matrix format the following equations are obtained
2
6
6
6
6
6
4
P
x 1
P
x 2
P
x 3
: : :
P
x n
3
7
7
7
7
7
5
D
2
6
6
6
6
6
4
0
1
0
: : :
0
0
0
1
: : :
0
0
0
0
: : :
0
: : :
: : :
: : :
: : :
: : :
a 0 a 1 a 2 : : : a n1
3
7
7
7
7
7
5
2
6
6
6
6
6
4
x 1
x 2
x 3
: : :
x n
3
7
7
7
7
7
5
C
2
6
6
6
6
6
4
0
0
0
: : :
1
3
7
7
7
7
7
5
u ;
y D
b 0 0 0 : : : 0
2
6
6
6
6
6
4
x 1
x 2
x 3
: : :
x n
3
7
7
7
7
7
5
:
(10.47)
This form of the state equations and especially of matrix
A is also known as the companion canonical form.
10.1.5 Nonlinear Systems and Linearization
Consider a nonlinear system with m inputs, p outputs,
and n state variables with the following description in
state space
P
x D f .x; u/ ;
y D g.x; u/ :
(10.48)
Significant progress has been achieved in recent years
in the analysis and controller synthesis for a wide class
of nonlinear systems, especially through the studies of
Isidori [10.7, 8] and Kokotovic et al. [10.9]. However,
the technique of linearization, the reduction to or approximation by a linear system is still extensively used.