Quantitative design of robust multivariable control systems

We then have

v(x) = - \"G{X, 0Ja

= - \X2<b(y)Ja Ja

ry

dyfJa

fJa

(162)

CorrespondenceQUANTITATIVE DESIGN OF ROBUSTMULTIVARIABLE CONTROL SYSTEMS

In Paper 5725D [/££ Proc. D, Control Theory & Appl.,1988, 135, (1), pp. 57-66] the author claimed that the useof Perron roots has advantage over singular values as ameasurement of robustness because of the followinglemma (lemma 2):

'Let A be an n x n complex matrix and let XA and omax(A)be the Perron root and the maximum singular values ofA, respectively. Then kA < omax(A)\

I wish to point out that the above lemma is incorrect.The following serves as a simple counter example.

•C "0x(A) =which gives XA — 2 and o

Y.K. FOONanyang Technological InstituteSchool of Electrical & Electronic EngineeringNanyang AvenueSingapore 2263

26th April 1988

I wish to thank Dr. Foo for pointing out an apparenterror in the statement of the above lemma. The lemmawas stated in the context of semiconvergent matrices,that is, complex matrices A, whose spectral radii p{A)satisfy p(A) ̂ 1. This however was not explicit in thestatement.

The example following the lemma and the statementat the end of that example clearly indicate what wasimplied. Consider Dr. Foo's example:

•[: •:]This is not semiconvergent. If we now endow it with theregular splitting

A = D + C =- 1

0

and consider the 'error' matrix D lC, which is semi-convergent because p(D~1C)= 1, then it is clear that

^JD-1C)= 1.XD.lc ^ a^D

O.D.I. NWOKAHPurdue UniversitySchool of Mechanical EngineeringWest LafayetteIndiana 47907USA

24th May 1988

6226D

404 IEE PROCEEDINGS, Vol. 135, Pt. D, No. 6, NOVEMBER 1988