Part A | 10.2
240 Part A Fundamentals
Consider, for example, the characteristic polynomial of the transfer function in (10.68). This one exhibits two sign changes and therefore obtains roots in
the right-hand complex plane. On the other hand, the
following third-order polynomial also obtains roots in
the right-hand complex plane, that is, with positive real
part, although no coefficient sign changes occur
p.s/ D s
3
C 0:5s
2
C s C 3
D .s C 1:3694/ .s 0:4347 C 1:4148i/
.s 0:4347 1:4148i/ :
(10.69)
10.2.3 Controllability and Observability
The concepts of controllability and observability are
fundamental in modern control system theory. For an
LTI system with m inputs, p outputs and n states, controllability, and observability may be investigated by
using its state-space description.
P
x D Ax C Bu ;
y D Cx C Du
(10.70)
The dimensions for matrix A are n n, for B n m, for
C p n, and for D p m.
State Controllability
State vector controllability is the situation where there
exists a control action vector that can drive the state
vector to any finite value in a finite time interval. The
formal definition is provided below.
The state vector x of an LTI system as in (10.70) is
controllable if there exists a piecewise continuous control signal u that can drive the state vector from any
initial condition x 0 to any final value x f in a finite time
interval (t f t 0 ).
In the case that system’s matrix A in (10.70) is diagonal, one can easily see that, by the definition, matrix B
must contain no rows that are identically zero in order
for the LTI system to be controllable.
In the general case that A is not diagonal, the criterion to check controllability of the state vector for an
LTI system, as in (10.70), is based on the rank of the
controllability matrix, C
, defined as
C
D
B AB A
2 B A
.n1/ B
:
(10.71)
If the rank of C
is no less than n then the system is controllable, otherwise not. It is reminded here that the rank
of a matrix is defined as the dimension of the vector
subspace spanned by either its row or its column vectors [10.5, 6]. In effect, for an LTI system as in (10.70)
to be controllable, its controllability matrix C
must
contain n linearly independent row vectors or n linearly
independent column vectors.
State Observability
State vector observability is the situation where the initial value, at any time instant, of the state vector may
be determined solely on the basis of the recorded inputs
(controls) and outputs (measurements). The formal definition is provided below.
The state vector x of an LTI system as in (10.70) is
observable in the time interval [t 0 , t f ] if it is possible to
determine the initial value of the state vector, x 0 D x.t 0 /,
when presented with the values of the control signal u
and output signal y in the finite time interval [t 0 , t f ].
In the case that system’s matrix A in (10.70) is diagonal, one can easily see that, by the definition, matrix
C must contain no columns that are identically zero in
order for the LTI system to be observable.
In the general case that A is not diagonal, the criterion to check observability of the state vector for an
LTI system, as in (10.70), is based on the rank of the
observability matrix, O
, defined as
O
D
2
6
6
6
6
6
4
C
CA
CA
2
: : :
CA
.n1/
3
7
7
7
7
7
5
:
(10.72)
If the rank of O
is no less than n then the system is
observable, otherwise not. In effect, for an LTI system
as in (10.70) to be observable, its observability matrix
O
must contain n linearly independent row vectors or
n linearly independent column vectors [10.5, 6].
Zero-Pole Cancellation
An important lemma connecting the transfer function
(or the transfer function matrix) with the concepts of
controllability and observability is given in this section [10.1, 2].
If the same system pole (at least one) is cancelled
by a zero in each and every one of the transfer function matrix elements, then the system in hand is either
uncontrollable or unobservable or both.
The lemma above is now investigated in depth for
the case of SISO systems. Consider an nth order, SISO
system with description in state space as follows
P
x D Ax C bu ;
y D c
T x :
(10.73)
240 Part A Fundamentals
Consider, for example, the characteristic polynomial of the transfer function in (10.68). This one exhibits two sign changes and therefore obtains roots in
the right-hand complex plane. On the other hand, the
following third-order polynomial also obtains roots in
the right-hand complex plane, that is, with positive real
part, although no coefficient sign changes occur
p.s/ D s
3
C 0:5s
2
C s C 3
D .s C 1:3694/ .s 0:4347 C 1:4148i/
.s 0:4347 1:4148i/ :
(10.69)
10.2.3 Controllability and Observability
The concepts of controllability and observability are
fundamental in modern control system theory. For an
LTI system with m inputs, p outputs and n states, controllability, and observability may be investigated by
using its state-space description.
P
x D Ax C Bu ;
y D Cx C Du
(10.70)
The dimensions for matrix A are n n, for B n m, for
C p n, and for D p m.
State Controllability
State vector controllability is the situation where there
exists a control action vector that can drive the state
vector to any finite value in a finite time interval. The
formal definition is provided below.
The state vector x of an LTI system as in (10.70) is
controllable if there exists a piecewise continuous control signal u that can drive the state vector from any
initial condition x 0 to any final value x f in a finite time
interval (t f t 0 ).
In the case that system’s matrix A in (10.70) is diagonal, one can easily see that, by the definition, matrix B
must contain no rows that are identically zero in order
for the LTI system to be controllable.
In the general case that A is not diagonal, the criterion to check controllability of the state vector for an
LTI system, as in (10.70), is based on the rank of the
controllability matrix, C
, defined as
C
D
B AB A
2 B A
.n1/ B
:
(10.71)
If the rank of C
is no less than n then the system is controllable, otherwise not. It is reminded here that the rank
of a matrix is defined as the dimension of the vector
subspace spanned by either its row or its column vectors [10.5, 6]. In effect, for an LTI system as in (10.70)
to be controllable, its controllability matrix C
must
contain n linearly independent row vectors or n linearly
independent column vectors.
State Observability
State vector observability is the situation where the initial value, at any time instant, of the state vector may
be determined solely on the basis of the recorded inputs
(controls) and outputs (measurements). The formal definition is provided below.
The state vector x of an LTI system as in (10.70) is
observable in the time interval [t 0 , t f ] if it is possible to
determine the initial value of the state vector, x 0 D x.t 0 /,
when presented with the values of the control signal u
and output signal y in the finite time interval [t 0 , t f ].
In the case that system’s matrix A in (10.70) is diagonal, one can easily see that, by the definition, matrix
C must contain no columns that are identically zero in
order for the LTI system to be observable.
In the general case that A is not diagonal, the criterion to check observability of the state vector for an
LTI system, as in (10.70), is based on the rank of the
observability matrix, O
, defined as
O
D
2
6
6
6
6
6
4
C
CA
CA
2
: : :
CA
.n1/
3
7
7
7
7
7
5
:
(10.72)
If the rank of O
is no less than n then the system is
observable, otherwise not. In effect, for an LTI system
as in (10.70) to be observable, its observability matrix
O
must contain n linearly independent row vectors or
n linearly independent column vectors [10.5, 6].
Zero-Pole Cancellation
An important lemma connecting the transfer function
(or the transfer function matrix) with the concepts of
controllability and observability is given in this section [10.1, 2].
If the same system pole (at least one) is cancelled
by a zero in each and every one of the transfer function matrix elements, then the system in hand is either
uncontrollable or unobservable or both.
The lemma above is now investigated in depth for
the case of SISO systems. Consider an nth order, SISO
system with description in state space as follows
P
x D Ax C bu ;
y D c
T x :
(10.73)