## Abstract

In the last chapter, we considered the problem of minimizing **F** _{zw} with respect to the *H* _{ 2 } norm. The performance specifications were given in the time domain. As we have seen in Chap. 3 for SISO problems, for specifications in the frequency domain the *H* _{ ∞ } norm is an adequate tool. In this way we are naturally lead to the question of how controllers can be characterized in a way which minimizes the closed-loop transfer function **F** _{zw} with respect to the *H* _{ ∞ } norm. There are two important methods for solving this problem. One is based on two Riccati equations similar to those used in the *H* _{ 2 } problem. It will be analyzed in this chapter, whereas the other method uses linear matrix inequalities and is presented in the next chapter.

### Keywords

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### Notes and References

- [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 positiv-ity,
*IEEE Trans. Autom. Control*, 11, pp. 228–238, 1966.CrossRefGoogle Scholar - [35]Doyle, J. C, Glover, K., Khargonakar, P. and Francis, B. A.: State-space solutions to standard
*H*_{2}and*H*_{∞}control problems,*IEEE Trans, on Autom. Control*, 34, pp. 831–847,1989.MATHCrossRefGoogle Scholar - [48]Glover, K., Limebeer, D. J. N., Doyle, J. C., Kasenally, E. .M. and Safanov, M. G.: A characterization of all solutions of the four block general distance problem,
*SIAM J. Control Optim.*, 29, pp. 283–324,1991.MathSciNetMATHCrossRefGoogle Scholar - [79]Safonov, M. G., Limebeer, D. J. N., and Chiang, R. Y.: Simplifying the
*H*_{∞}theory via loop shifting, matriy pencil and descriptor concepts,*Int. J. Control*, 50, pp. 2467–2488,1990.MathSciNetCrossRefGoogle Scholar - [112]Zhou, K., Doyle, J. and Glover, K.:
*Robust and Optimal Control*, Prentice Hall, Englewood Cliffs, New Yersey, 1995.Google Scholar - [85]Sefton, J. and Glover, K.: Pole/zero cancellations in the general
*H*_{∞}problem with reference to a two-block design,*Syst. Control Lett*, 14, pp. 295–306, 1990.MathSciNetMATHCrossRefGoogle Scholar