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244 Computed-Torque Control

This is a complex multivariable design problem, for the feedback gain
matrix K is of dimension m×n. A classical controls approach might involve,
for instance, performing mn root locus designs to close the feedback loops one
at a time. On the other hand, a solution that guarantees stability can be found
using modern controls techniques simply by solving some standard matrix
design equations. This modern approach closes all the feedback loops
simultaneously and guarantees a good gain and phase margin.
The feedback matrix is found using modern control theory as follows.
First, define a quadratic performance index (PI) of the form

(4.6.4)

where Q is a symmetric positive semidefinite n×n matrix (denoted Q 0) and
R is a symmetric positive definite m×m matrix (R>0). That is, all eigenvalues
of R are greater than zero and those of Q are greater than or equal to zero. Q
is called the state-weighting matrix and R the control-weighting matrix. These
matrices are design parameters that are selected by the engineer depending,
for instance, on the desired form of the closed-loop time responses.
The optimal LQ feedback gain K is the one that minimizes the PI J. The
motivation follows. The quadratic terms x Qx and u Ru in the PI are
T
T
generalized energy functions (e.g., the energy in a capacitor is ½Cv , the
2
2 .
kinetic energy of motion is ½mv ) Suppose, then, that J is minimized in the
closed-loop system (4.6.3). This means that the infinite integral of
T
[x (t)Qx(t)+u (t)Ru(t)] is finite, so that this function of time goes to zero as t
T
becomes large. However,

with the square root of a matrix defined as . Since these
norms vanish with t and |R| 0, the functions and u(t) both go
to zero. Under the assumption that is observable [Kailath 1980], x(t)
goes to zero if y(t) does.
Therefore, the optimal gain K guarantees that all signals go to zero with
time in the closed-loop system (4.6.3). That is, K stabilizes (A-BK).
The determination of the optimal K is easy and is a standard result in
modern control theory (see, e.g., [Lewis 1986a], [Lewis 1986b]). The optimal
feedback gain is simply found by solving the matrix design equations

Copyright © 2004 by Marcel Dekker, Inc.