Page 769 - Mechanical Engineers' Handbook (Volume 2)

760 Control System Design Using State-Space Methods

1
(s) (sI A) B (7)
n
If the desired s are distinct, it will always be possible to find n linearly independent columns
i
as already described. Here e is defined as the j th column of the r r identity matrix I
j i i r
and is uniquely determined once j is determined. When repeated eigenvalues are desired,
i
the procedure for specifying n linearly independent generalized eigenvectors is different and
8
has been described by Brogan. The results given here are readily applicable to multi-input
discrete-time systems described by Eq. (12) in Chapter 17 and Eq. (1) in this chapter. The
specified eigenvalues are z instead of s and the A and B matrices are replaced by F and G,
i i
respectively. Also, s is replaced by z in Eq. (7). Alternative methods for gain matrix selection
for multi-input systems have been described by Kailath. 7
If a single-input, linear time-variant (LTV) system is completely state controllable, linear
state-variable feedback can be used to ensure that the closed-loop transition matrix corre-
sponds to that of atty desired nth-order linear differential equation with time-varying coef-
ficients, The state-variable feedback gains are time varying in general and can be computed
using a procedure described by Wiberg. 9
If the complete state is not available for feedback, linear instantaneous feedback of the
measured output can be used to place some of the closed-loop poles at specified locations
in the complex plane. If the continuous-time and discrete-time systems described in Chapter
17 by Eqs. (6), (7), (12), and (13), respectively, satisfy the output controllability conditions
listed in Table 8 in Chapter 17, then p of the n eigenvalues of the closed-loop system can,
approach arbitrarily specified values to within any degree of accuracy but not always exactly.
The control law is
u Ky (8)
8
where K is a r p gain matrix. Brogan has described an algorithm for computing K,given
the desired values of p closed-loop eigenvalues. The corresponding characteristic equation
is
det[sI A BK(I DK) C] 0 (9)
1
p
n
for continuous-time systems and
1
det[zI F GK(I DK) C] 0 (10)
n p
for discrete-time systems.
An alternative approach to control system design in the case of incomplete state mea-
surement is to use an observer or a Kalman filter for state estimation, The estimated state is
then used for feedback. This procedure is discussed in Section 5.
The advantages of the pole placement design method already described are that the
controller achieves desired closed-loop pole locations without using pole–zero cancellation
and without increasing the order of the system. The desired pole locations can be chosen to
ensure a desired degree of stability or damping and speed of response of the closed-loop
system. However, there is no convenient way to ensure a priori that the closed-loop system
satisfies other important performance specifications such as a desired level of insensitivity
to plant parameter variations, acceptable disturbance rejection, and compatibility of control
effort with actuator limitations. In addition, for single-input LTI systems, instantaneous state
feedback of the form given by Eq. (1) does not affect the locations of zeros of the transfer
3
functions between the system input and system outputs. Thus, the pole placement design
method does not afford complete control over the system response to the reference input or
disturbance inputs. For multi-input systems, the available freedom in the gain matrix selection
can be used to assign individual transfer function zeros, in addition to achieving desired
closed-loop pole locations. However, systematic procedures to do this are not available. The