# Synthesis of Robust Controllers

• Uwe Mackenroth
Chapter

## Abstract

In the last chapter, we introduced the mathematical framework for the description of model uncertainty. As we have seen, the basic idea is to separate the uncertainty Δ from the plant such that the uncertain plant can be interpreted as a feedback system consisting of P and Δ. Herein, Pdenotes the nominal plant, which results from the original physical plant G by adding weights for performance and uncertainty. The basic block diagram which visualizes this situation is Fig. 12.9. The nominal feedback system is denoted by M and consists of P and the controller K. The uncertain feedback system is obtained if the input Wp is connected by feedback with the output z P using the uncertainty Δ (cf. Fig 12.10). As we have seen, concerning internal stability it suffices to use the block M 11 so that Fig. 12.11 becomes the fundamental block diagram with respect to robust stability. The question is now how large in the sense of the H norm the uncertainty may be so that the feedback system is stable. This question can completely be answered and is essentially the small gain theorem (cf. Fig. 12.11, where the uncertainty Δ can be formally viewed as a controller for the “plant” M 11). This result together with some applications is the content of Sect. 13.1 and Sect. 13.2. A sufficient condition for robust performance for such a type of uncertainty is also given in Sect. 13.2.

### Keywords

Stein Doyle Balas

## Preview

### Notes and References

1. [32]
Doyle, J. C.: Analysis of feedback systems with structured uncertainties, IEE Proc. D, Control Theory, 129, pp. 242–250, 1982.
2. [111]
Zames, G.: On the input-output stability of time-varying nonlinear feedback systems: Part I: Conditions derived using concepts of loop gain, conicity, and positivity, IEEE Trans. Autom. Control, 11, pp. 228–238, 1966.
3. [33]
Doyle, J. C, Wall, J. and Stein, G.: Performance and robustness analysis for structured uncertainty, Proc. 21st IEEE Conf. on Decision and Control, pp. 629–636, 1982.Google Scholar
4. [78]
Safonov, M. G.: Stability margins of diagonally perturbed multivariable feedback systems, IEE Proc. D, Control Theory Appl, 129, pp. 251–256, 1982.
5. [34]
Doyle, J. C.: Structured uncertainty in control system design, Proc. IEEE Conf. on Decision and Control, Ft. Lauderdale, 1985.Google Scholar
6. [39]
Fan, M. K. H., Tits, A. L. and Doyle, J. G: Robustness in the presence of mixed parametric uncertainty and unmodeled dyxnamics, IEEE Trans. Autom. Control, 36(1), pp. 25–38, 1991.
7. [110]
Young, P. M, Newlin, M. and Doyle, J. C: μ analysis with real parametric uncertainty, IEEE Proc. 30th Conf. on Decision and Control, pp. 1251–1235, England, 1991.Google Scholar
8. [72]
Packard, A. and Doyle, J. C: The complex structured singular value, Automatica, 29, pp. 71–109, 1993.
9. [7]
Balas, G., Doyle, J., Glover, K., Packard, A. and Smith, R.: μ Analysis and Synthesis Toolbox, The Mathworks Inc., 2001.Google Scholar