Nonlinearity inH ∞-control theory, causality in the commutant lifting theorem, and extension of intertwining operators
Tóm tắt
The problems studied in this note have been motivated by our work in generalizing linearH
∞ control theory to nonlinear systems. These ideas have led to a design procedure applicable to analytic nonlinear plants. Our technique is a generalization of the linearH
∞ theory. In contrast to previous work on this topic ([9], [10]), we now are able to explicitly incorporate a causality constraint into the theory. In fact, we show that it is possible to reduce a causal optimal design problem (for nonlinear systems) to a classical interpolation problem solvable by the commutant lifting theorem [8]. Here we present the complete operator theoretical background of our research together with a short control theoretical motivation.
Tài liệu tham khảo
W. Arveson, “Interpolation problems in nest algebras,”J. Funct. Anal.,20 (1975), 208–233.
J. Ball and J.W. Helton, “Sensitivity bandwidth optimization for nonlinear feedback systems,” Technical Report, Department of Mathematics, University of California at San Diego, 1988.
J. Ball and J.W. Helton, “H ∞ control for nonlinear plants: connections with differential games,”IEEE Conference on Decision and Control Tampa, Florida, 1989, 956–962.
J. Doyle, B. Francis and A. Tannenbaum,Feedback Control Theory, MacMillan, New York, 1991.
C. Foias, “Contractive intertwining dilations and waves in layered media”Proc. International Congress Math., Helsinki2(1978), 605–613.
C. Foias and A. Frazho,The Commutant Lifting Approach to Interpolation Problems, Birkhauser-Verlag, Boston, 1990.
C. Foias, C. Gu and A. Tannenbaum, “Intertwining dilations, intertwining extensions and causality,”Acta Sci. Math. (Szeged),57(1993), 101–123.
C. Foias, C. Gu and A. Tannenbaum, “On a causal linear optimization theorem,” to appear inJ. of Math. Anal. and Appl..
C. Foias and A. Tannenbaum, “Iterated commutant lifting for systems with rational symbol,”Oper. Theory: Adv. and Appl.,41(1989), 255–277.
C. Foias and A. Tannenbaum, “Weighted optimization theory for nonlinear systems,”SIAM J. on Control and Optimiz.,27(1989), 843–860.
C. Foias and A. Tannenbaum, “On a causal commutant lifting theorem,”J. of Funct. Anal.,118(1993), 407–441.
B. Francis,A Course in H ∞ Control Theory, McGraw-Hill, New York, 1981.
B. Francis and A. Tannenbaum, “Generalized interpolation theory in control,”Mathematical Intelligencer,10(1988), 48–43.
D. Sarason, “Generalized interpolation inH ∞,”Trans. AMS,127(1967), 179–203.
B. Sz. Nagy and C. Foias,Harmonic Analysis of Operators on Hilbert Space, North-Holland Publishing Company, Amsterdam, 1970.
G. Zames, “Feedback and optimal sensitivity: model reference transformations, multiplicative semi-norms, and approximate inverses,”IEEE Trans. Auto. Control,AC-26(1981), 301–320.