Page 789 - Mechanical Engineers' Handbook (Volume 2)

780 Control System Design Using State-Space Methods

similar equation describes the discrete-time observer characteristic equation. If CACSD pack-
ages 4–6 supporting only controller pole placement algorithms are available, the transforma-
tions of Eqs. (83) and (84) are needed to select observer gains using these algorithms.
The observers of Eqs. (74) and (78) are called full-order observers since their dimensions
are the same as those of the systems whose states are being estimated. Reduction of the
observer dimension can be achieved by using the fact that the output equation provides us
with p linear equations in the unknown state, where p is the number of output variables.
Therefore, the observer need only provide n p additional linear equations and thus need
only be of dimension n p. For time-invariant systems, the corresponding observer gain
matrix can be chosen to place the n p observer poles at any desired locations in the
complex plane if the original systems of Eqs. (6) and (7) or (12) and (13) in Chapter 17 are
completely observable. Equations for reduced-order observers for continuous-time systems
3
are given by Kwakernaak and Sivan and for discrete-time systems by Franklin and Powell. 11
In the presence of measurement noise, the state estimates x(t)or x(k) obtained using reduced-
order observers are more sensitive than those obtained using full-order observers.
The discrete-time observer of Eq. (78) is also referred to as a prediction estimator since
x(k 1) is ahead of the last measurements used, y(k) and u(k). A variation of this observer,
called the current estimator, is useful if the computation time associated with the observer
is very short compared to the sampling interval for the sampled-data system. The corre-
sponding observer equation for an LTI system is 11
ˆ x(k 1) Fˆx(k) Gu(k L[y(k 1) CFˆx(k) CBu(k) Du(k 1)] (88)
The state estimate ˆx(k 1) therefore depends on the current measurements y(k 1) and
u(k 1). In practice, however, the measurements precede the estimate by a very small
computation time. The observer gain formula for a single-output system is then given by

1
0
CF
CF
2
0
L (F) (89)
e
CF n 1
instead of Eq. (86). The term (F) has the same meaning as before.
e
For multioutput LTI systems, specification of the desired observer poles does not specify
the gain matrix L uniquely. The additional freedom in the gain matrix selection can be used
to assign the eigenvectors (or generalized eigenvectors) of the matrix A LC or F LC
in addition to the eigenvalues. The corresponding procedure would be very similar to that
used for gain matrix selection for multi-input systems and described in Section 2. The trans-
formations of Eqs. (83) and (84) can be used to adapt the controller gain selection procedure
to observer gain selection. Alternatively, the observer gains can be chosen to design observers
whose state estimates have low sensitivity to unmeasured disturbance inputs. 29
The observer designs described work well in the absence of significant levels of mea-
surement noise or unknown disturbance signals. The sensitivity of the state estimates to
measurement noise and unknown disturbance signals depends on the specified location of
the observer poles. If these pole locations are too far into the left half of the complex s-
plane or too close to the origin in the complex z-plane, the state estimates would be unduly
sensitive to measurement noise and disturbance signals. In the limiting case, the observers
can be shown to reduce to ideal differentiators or differencing devices. The observer pole
locations should therefore be chosen to avoid such high sensitivities of the state estimates
but at the same time ensure that the estimation error transients decay more rapidly than the
state variables being estimated. Specification of the observer poles in practice involves con-
siderable trial and error in much the same manner as specification of closed-loop poles does
for state feedback controller design. If some information is available concerning the distur-