Page 775 - Mechanical Engineers' Handbook (Volume 2)

766 Control System Design Using State-Space Methods

1
T
T
K [R G (R P )G] G (R P )F (37)
s
2
s
s
1
1
T
P F (R P )(F GK ) (38)
s
1
s
s
The optimal-control law for the infinite-time LQR problem is
u(k) Kx(k) (39)
s
and requires only constant state feedback gains. The solution P of Eqs. (37) and (38) is
s
positive definite, and the optimal-control law results in an asymptotically stable closed-loop
system if:
1. The linear system of Eq. (12) in Chapter 17 is completely state controllable.
2. The pair F, H T is completely observable (reconstructible), where H is any matrix
r r
such that H H T equals R .
r r 1
Also, as in the case of continuous-time systems, the optimal feedback gains are nearly
constant even for finite-time LQR problems if the system matrices and weighting matrices
11
in J are constant. Finally, a number of techniques for solving the matrix algebraic equations
(37) and (38) and the matrix difference equations (33)–(35) are described by Kuo. 12
3.3 Stability and Robustness of the Optimal-Control Law
An important consideration in the practical usefulness of the optimal-control laws for the
LQR problems described is the implication of these laws for performance features of the
controlled systems not included in J, such as relative stability and sensitivity of the controlled
system to unmodeled dynamics or plant parameter variations. Reference has already been
made to the fact that the optimal-control laws for the continuous-time and discrete-time
infinite-time LQR problems described result in asymptotically stable closed-loop systems
provided that specified controllability and reconstructibility or observability conditions are
satisfied. Closed-loop systems with a prescribed degree of stability can be obtained by mod-
ifying the performance index J for linear, time-invariant, continuous-time systems: 10

T
T
2 t
J e (xR x uR u)dt (40)
2
1
0
where is a positive scalar constant. If the pair A, B is completely state controllable and
the pair A, H T r is completely observable where H H T r is equal to R , the solution to this
r
1
LQR problem results in a finite value of J. Hence, the transients decay at least as rapidly as
e t . Larger values of would therefore ensure a more rapid return of the system to equi-
librium. The corresponding algebraic Riccati equation is
T
R PBR B P AP PA 2 P 0 (41)
1
T
1
s
s
s
2
s
s
and the optimal-feedback-control law is given by Eq. (28). A similar procedure for discrete-
time LTI systems is described by Franklin and Powell. 11
Additional results concerning the stability properties of the optimal control law for
continuous-time, LTI systems described by Eq. (6) in Chapter 17 and employing only con-
stant weighting matrices in the index of performance, Eq. (24), are available and will be
summarized. Anderson and Moore 10 have shown that for single-input systems the optimal-
control law for the infinite-time LQR problem has 60 phase margin, an infinite gain
margin, and 50% gain reduction tolerance before the closed-loop system becomes unstable.
These results are best explained with the aid of Fig. 5a, where G (s) is normalized to be
p