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4.6 Optimal Outer-Loop Design 245

(4.6.5)

(4.6.6)

where P is a symmetric n×n auxiliary design matrix on which the optimal
gain depends. The second of these is a nonlinear matrix quadratic equation
known as the Riccati equation; it is easy to solve this equation for the auxiliary
matrix P using standard routines in, for instance, MATLAB, [MATRIX x 1989],
and other software design packages.
The next result is of prime importance in modern control theory and
formalizes the stability discussion just given.

THEOREM 4.6–1: Let be observable and (A, B) be controllable.
Then:
(a) There is a unique positive definite solution P to the Riccati equation.
(b) The closed-loop system (A-BK) is asymptotically stable.
(c) The closed-loop system has an infinite gain margin and 60° of phase
margin.

Controllability was discussed in Chapter 1. Observability means roughly
speaking that all the system modes have an independent effect in the PI, so
that if J is bounded, all the modes independently go to zero with t. To verify
these properties is easy. The system is controllable if the controllability matrix

(4.6.7)

has full rank n. The system (A, C) is observable if the observability matrix

(4.6.8)

has full rank n. MATLAB, for instance, provides routines for these tests.
Therefore, the state-weighting matrix Q may be chosen to satisfy the
observability requirement.
The theorem makes this modern design approach very powerful. No matter
how many inputs and states, a stabilizing feedback gain can always be found
under the hypotheses that stabilizes the system. The gain is found by closing

Copyright © 2004 by Marcel Dekker, Inc.