National Instruments 370755B-01 Singular Perturbation

Chapter 1 Introduction

This is almost the algorithm set out in Section II of [LHPW87]. The one

difference (and it is minor) is that in [LHPW87], lower triangular Cholesky

factors of P and Q are used, in place of UcSc1/2 and UOSO1/2 in forming H

in step 2. The grammians themselves need never be formed, as these

Cholesky factors can be computed directly from A, B, and C in both

continuous and discrete time; this, however, is not done in balmoore.

The algorithm has the property that:

Thus the diagonal entries of SH are the Hankel singular values.

The algorithm implemented in balance( ) is older, refer to [Lau80].

A lower triangular Cholesky factor Lc of P is found, so that LcLc′=P.

Then the symmetric matrix Lc′QLc is diagonalized (through a singular

value decomposition), thus L′cQLc=VRU′, with actually V=U. Finally,

the coordinate basis transformation is given by T=LcVR–1/4, resulting in

T′QT=T–1P(T–1)′=R1/2 .

Singular Perturbation

A common procedure for approximating systems is the technique of

Singular Perturbation. The underlying heuristic reasoning is as follows.

Suppose there is a system with the property that the unforced responses of

the system contain some modes which decay to zero extremely fast. Then

an approximation to the system behavior may be attained by setting state

variable derivatives associated with these modes to zero, even in the forced

case. The phrase “associated with the modes” is loose: exactly what occurs

is shown below. The phrase “even in the forced case” captures a logical

flaw in the argument: smallness in the unforced case associated with initial

conditions is not identical with smallness in the forced case.

Suppose the system is defined by:

(1-1)

T′QT′T1–PT1–

()′SH

·1

·2

A11 A12

A21 A22

u+=

yC1C2

Du+=