Robust Control Theory pp 105-128 | Cite as

# On the Computation of Suboptimal *H*^{∞} Controllers for Unstable Infinite Dimensional Systems

## Abstract

In this paper we show how to compute suboptimal *H* ^{∞} controllers, for a class of (possibly) unstable and (possibly) infinite dimensional plants, from a finite set of linear equations. A solution to the *H* ^{∞} suboptimal control problem for infinite dimensional *stable* plants was obtained in [6]. Also, in [13] and [14] the *H* ^{∞} *optimal* control problem was solved for unstable distributed plants. Our solution for the *suboptimal* control problem of *unstable* distributed plants is based on the techniques developed in [6] and [13]. We obtain a computable expression for the suboptimal *H* ^{∞} controllers and identify their finite and infinite dimensional parts.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]V.M. Adamjan, D.Z. Arov, and M.G. Krein,
*Analytic properties*of*Schmidt pairs for a Hankel operator and generalized Shur-Takagi problem*, Math. USSR Sbornik**15**(1971), pp. 31–73.CrossRefGoogle Scholar - [2]H. Bercovici, C. Foias and A. Tannenbaum,
*On skew Toeplitz operators I*, Operator Theory: Advances and Applications 32 (1988), pp. 21–43.MathSciNetGoogle Scholar - [3]D.S. Flamm, Outer
*Factor ‘Absorption’ for H**∞**Control Problems*, ISS Report No. 55, July 31, 1990, Dept. of Elec. Eng., Princeton University, 1990.Google Scholar - [4]C. Foias and A.E. Frazho,
*The Commutant Lifting Approach to Interpolation Problems*, Birkhäuser, Basel, 1990.zbMATHGoogle Scholar - [5]C. Foias, H. Ozbay and A. Tannenbaum,
*Remarks on H*^{∞}*optimization of multi-vari ate distributed systems*, IEEE Conference on Decision and Control, Austin, Texas, December 1988, pp. 985–986.Google Scholar - [6]C. Foias and A. Tannenbaum,
*On the parametrization of the suboptimal solutions in generalized interpolation*, Linear Algebra and its Applications,**122/123/124**(1989), pp. 145–164.Google Scholar - [7]C. Foias and A. Tannenbaum,
*On the four block problem, I*, Operator Theory: Advances and Applications,**32**(1988), pp. 93–112.MathSciNetGoogle Scholar - [8]C. Foias and A. Tannenbaum,
*On the four block problem, II: the singular system*, Operator Theory and Integral Equations,**11**(1988), pp. 726–767.MathSciNetCrossRefGoogle Scholar - [9]C. Foias, A. Tannenbaum and G. Zames,
*Some explicit formulae for the singular values of a certain Hankel operators with factorizable symbol*, SIAM J. Math. Analysis,**19**(1988), pp. 1081–1091.MathSciNetzbMATHCrossRefGoogle Scholar - [10]J.K. Doyle, K. Glover, P.P. Khargonekar and B. Francis,
*State space solutions to standard H2 and H*^{∞}*control problems*, IEEE Transactions on Automatic Control,**26**(1981), pp. 4–16.zbMATHCrossRefGoogle Scholar - [11]H. Ozbay and A. Tannenbaum,
*A skew Toeplitz approach to the H*^{∞}*optimal control of multivariable distributed systems*, SIAM J. Control and Optimization,**28**(1990) pp. 653–670.MathSciNetCrossRefGoogle Scholar - [12]H. Ozbay and A. Tannenbaum,
*On the structure of suboptimal H’ controllers in the sensitivity minimization problem for distributed stable plants*, Automatica,**27**(2) March (1991), pp. 293–305.MathSciNetCrossRefGoogle Scholar - [13]H. Ozbay, M.C. Smith and A. Tannenbaum,
*Mixed sensitivity optimization for a class of unstable infinite dimensional systems, (to appear in) Linear Algebra and its Applications*. (A short version of the paper, tinder the title*Controller design for unstable distributed plants*, appears in the Proc. of the American Control Conference, San Diego CA, May 1990, pp. 1583–1588.Google Scholar - [14]H. Ozbay, M.C. Smith and A. Tannenbaum,
*On the optimal two block H*^{∞}*compensators for distributed unstable plants*, Proceedings of the American Control Conference, Chicago IL, June 1992, pp. 1865–1869.Google Scholar - [15]H. Ozbay,
*A simpler formula for the singular values of a certain Hankel operator*, Systems and Control Letters,**15**(5) (1990), pp. 381–390.MathSciNetCrossRefGoogle Scholar - [16]M.C. Smith,
*On stabilization and existence of coprime factorizations*, IEEE Transactions on Automatic Control, 1989, pp. 1005–1007.Google Scholar - [17]H. Tu,
*An H*^{∞}*Optimization Method and Matlab Program for Linear Distributed Systems*, M.S. Thesis, University of Minnesota-Duluth, 1992.Google Scholar