National Instruments 370755B-01 Controller Reduction, Controller Robustness Result

Chapter 4 Frequency-Weighted Error Reduction

Xmath Model Reduction Module 4-2 ni.com

(so that ) is logical. However, a major use of weighting is in

controller reduction, which is now described.

Controller Reduction

Frequency weighted error reduction becomes particularly important in

reducing controller dimension. The LQG and design procedures lead to

controllers which have order equal to, or roughly equal to, the order of the

plant. Very often, controllers of much lower order will result in acceptable

performance, and will be desired on account of their greater simplicity.

It is almost immediately evident that (unweighted) additive approximation

of a controller will not necessarily ensure closeness of the behavior of the

two closed-loop systems formed from the original and reduced order

controller together with the plant. This behavior is dependent in part on the

plant, and so one would expect that a procedure for approximating

controllers ought in some way to reflect the plant. This can be done several

ways as described in the Controller Robustness Result section. The

following result is a trivial variant of one in [Vid85] dealing with robustness

in the face of plant variations.

Controller Robustness Result

Suppose that a controller C stabilizes a plant P, and that Cr is a (reduced

order) approximation to C with the same number of unstable poles. Then

Cr stabilizes P also provided

An extrapolation to this thinking [AnM89] suggests that Cr will be a good

approximation to C (from the viewpoint of some form of stability

robustness) if

VV*Φ=

H∞

Cjω()Crjω()–[]Pjω()ICjω()Pjω()+[]

1–

∞1<

IP+jω()Cjω()[]

1–Pjω()Cjω()Crjω()[]()

∞1<

EIS CC

–()PI CP+()

1–

∞

EIS CC

–()PI CP+()

1–

∞