Page 792 - The Mechatronics Handbook

0066_Frame_C24 Page 40 Thursday, January 10, 2002 3:45 PM

If M is any square matrix and we denote by Λ{M} the set of eigenvalues of M, then

{
{
{
{
Λ A{} = Λ A co} ∪ Λ A 22} ∪ Λ A 33} ∪ Λ A 44} (24.236)
where
Λ A} = eigenvalues of the system,
{
Λ A co} = eigenvalues of the controllable and observable subsystem,
{
{
Λ A 22} = eigenvalues of the controllable but unobservable subsystem,
{
Λ A 33} = eigenvalues of the uncontrollable but observable subsystem,
Λ A 44} = eigenvalues of the uncontrollable and unobservable subsystem.
{
We observe that controllability for a given system depends on the structure of the input ports, i.e.,
where, in the system, the manipulable inputs are applied. Thus, the states of a given subsystem may be
uncontrollable for a given input, but completely controllable for another. This distinction is of funda-
mental importance in control system design since not all plant inputs can be manipulated (consider, for
example, disturbances) and, therefore, cannot be used to steer the plant to reach certain states.
Similarly, the observability property depends on which outputs are being considered. Certain states
may be unobservable from a given output, but they may be completely observable from some other
output. This also has a significant impact on output feedback control systems, since some states may not
appear in the plant output being measured and feeded back. However, they may appear in crucial internal
variables and thus be important to the control problem.

PBH Test
An alternative test for controllability and observability is provided by the following lemma known as
PBH test.
Lemma 24.7 Consider a state space model (A, B, C). Then
n
(i) The system is not completely observable if and only if there exists a nonzero vector x ∈ and a scalar
l ∈ such that
Ax = lx, Cx = 0 (24.237)

n
(ii) The system is not completely controllable if and only if there exists a nonzero vector x ∈ and a
scalar l ∈ such that

x A = lx , x B = 0 (24.238)
T
T
T
24.7 State Observers

Basic Concepts

When the state variables have to be measured for monitoring, implementing control systems, or other
purposes, there are hard technical and economical issues to face. Observers are a way to estimate the
state variables based upon a system model, measurements of the plant output y(t), and measurements
of the plant input u(t). This problem is a generalization of that of indirectly measuring a system variable
using a system model and the measurement of some other easier-to-measure variable.

Observer Dynamics
Assume that the system has a state space model given by (24.42) and (24.43) with D = 0 (a strictly proper
system has been assumed). Then, the general structure of a classic observer for the system state is as
shown in Fig. 24.15, where the matrix J is the observer gain.

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