This chapter investigates sensitivity limitations in multivariable linear control. There are different ways of extending the scalar results to a multivariable setting. We follow here two approaches, namely, one that considers integral constraints on the singular values of the sensitivity functions, and a second that develops integral constraints on sensitivity vectors. These approaches complement each other, in the sense that they find application in different problems, and hence both are needed to obtain a general view of multivariable design limitations imposed by ORHP zeros and poles. In order to avoid repetition, we use the first approach to derive the multivariable version of Bode’s integral theorems, whilst the second approach is taken to obtain the multivariable extension of the Poisson integrals. Both approaches emphasize the multivariable aspects of the problem by taking into account, in addition to location, the directions of zeros and poles.
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Notes and References
Bode Integral Formulae
- §4.1 and §4.2 are largely based on Chen (1995). This work also obtained Poisson integral constraints on the singular values of the sensitivity function.Google Scholar
Poisson Integral Formulae
- §4.3 is based on Gómez & Goodwin (1995). This latter work also presented design limitations for distributed systems, which allows one to consider transfer matrices with different time delays affecting each entry. The extension to this class of systems required additional technical results regarding zeros and the behavior at infinity of entire functions. The examples in §4.3.6 were also taken from Gómez & Goodwin (1995).Google Scholar
- Other results on design limitations for multivariable systems were given by Boyd & Desoer (1985). This work obtained inequality versions of the Bode and Poisson integral formulae based on the recognition that the logarithm of the largest singular value of an analytic transfer function is a subharmonic function.Google Scholar
- Related work has been reported by Freudenberg & Looze who convert the multivariable problem into a scalar one by pre and post multiplying the sensitivity function by vectors (Freudenberg & Looze 1985) or by the use of determinants (Freudenberg & Looze 1987). Similar ideas appear in the work of Sule & Athani (1991), who use directions associated with poles and zeros of the system, resulting in a directional study of trade-offs.Google Scholar
- §4.4 follows Gómez & Goodwin (1995). Alternative approaches to the problem of integral constraints for MIMO discrete systems can be found in Chen & Nett (1995), and Hara & Sung (1989).Google Scholar