Page 761 - Mechanical Engineers' Handbook (Volume 2)

752 State-Space Methods for Dynamic Systems Analysis

is zero. Similarly,
The system is completely controllable except for the trivial case where R 3
if x or x is chosen as the only output, the system is completely observable as long as R 3
1
2
is nonzero. If the voltage across the resistor R is chosen as the output, the corresponding
3
output equation is
y(t) [R R ]
x
1
3
3
x
2 (70)
The observability matrix P in Eq. (65) is then nonsingular and the system is observable if
0
and only if
R 1 R 2
(71)
L 1 L 2
This observability condition is not an obvious one and is equivalent to the requirement that
the time constants associated with the two R–L pairs not be identical. However, it should be
noted that, given component tolerances in practice, the inequality (71) will be satisfied almost
always and the corresponding system will be observable. A discussion on conditions leading
to loss of controllability or observability is given by Friedland. 16
Despite the fact that lack of complete controllability or observability is infrequent for
SISO systems, these concepts have practical significance for SISO as well as MIMO systems
because of the relationship of these concepts to closed-loop control and state estimation
problems. Measures of the degree of controllability and observability can be defined for
time-invariant and time-varying systems.
The controllability and observability conditions in Tables 8 and 9 relate these properties
to the nonsingularity of square matrices for SISO systems. Measures of the degree of con-
trollability and observability are related to the closeness of these matrices to the singularity
condition. Such measures have been defined for time-invariant systems by Johnson 17 and
Friedland 18 and are significant for SISO as well as MIMO systems. A system with a better
degree of controllability can in general be controlled more effectively. Similarly, a better
degree of observability implies that state estimation can be performed more accurately. The
proposed measures of the degree of controllability and observability are not in common use
but have the potential to quantitatively evaluate proposed control strategies and measurement
schemes. 19
Additional concepts of degrees of controllability and observability for time-varying sys-
20
tems have been described by Silverman and Meadows and for MIMO systems by Kreindler
13
and Sarachik. Properties weaker than state controllability and observability have also been
defined 12 and are useful in ensuring that closed-loop control and state estimation problems
are well posed. A linear system is said to be stabilizable if the uncontrollable subsystems
S and S in the decomposition of Fig. 4 are stable. Similarly, if the unobservable subsystems
N O
S and S are stable, the system is said to be detectable. 12
C N
7 RELATIONSHIP BETWEEN STATE-SPACE AND
TRANSFER FUNCTION DESCRIPTIONS
The state-space representation of dynamic systems is an accurate representation of the in-
ternal structure of a system and its coupling to the system inputs and outputs. For LTI
systems, transfer functions (for SISO systems) or transfer function matrices (for MIMO
systems) are useful in practice since the dimensions of these matrices are invariably smaller
than the dimensions of the corresponding system matrices A or F in Eqs. (6) and (12).
Analysis and design procedures based on the transfer function matrix descriptions are there-