Abstract
It is proved that a multivariable distributed system, that has no poles on the boundary of the half-plane of analyticity of its transfer function, can be stabilized with a stable strictly proper lumped compensator if and only if the standard parity interlacing condition is satisfied. The problem is formulated and solved in a Banach algebra of transfer functions that are Laplace transforms of measures having a finite total variation with respect to some given submultiplicative weight function. In particular, this result can be applied to transfer functions in the Callier-Desoer class, and it seems to be new even for this class. The proofs are closely related to and based on the scalar case solved previously.
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References
F. M. Callier and C. A. Desoer, An algebra of transfer functions for distributed linear timeinvariant systems,IEEE Trans. Circuits and Systems,25 (1978), 651–662.
F. M. Callier and C. A. Desoer, Simplifications and clarifications on the paper, “An algebra of transfer functions for distributed linear time-invariant systems”,IEEE Trans. Circuits and Systems,27 (1980), 320–323.
F. M. Callier and C. A. Desoer, Stabilization, tracking and disturbance rejection in multivariable convolution systems,Ann. Soc. Sci. Bruxelles, Ser. I,94 (1980), 7–51.
I. M. Gel'fand, Über absolut konvergente trigonometrische Reihen und Integrale,Mat. Sb.,9 (1941), 51–66.
G. Gripenberg, S.-O. Londen, and O. J. Staffans,Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, 1990.
G. S. Jordan, O. J. Staffans, and R. L. Wheeler, Local analyticity in weighted L1-spaces and applications to stability problems for Volterra equations,Trans. Amer. Math. Soc.,274 (1982), 749–782.
G. S. Jordan, O. J. Staffans, and R. L. Wheeler, Convolution operators in a fading memory space: The critical case,SIAM J. Math. Anal.,18 (1987), 366–386.
O. J. Staffans, Feedback stabilization of a scalar functional differential equation,J. Integral Equations,10 (1985), 319–342.
M. Vidyasagar,Control System Synthesis: A Factorization Approach, MIT Press, Cambridge, MA, 1985.
N. Vidyasagar, H. Schneider, and B. A. Francis, Algebraic and topological aspects of feedback stabilization,IEEE Trans. Automat. Control,27 (1982), 880–894.
D. C. Youla, J. J. Bongiorno, Jr, and C. N. Lu, Single-loop feedback-stabilization of linear multivariable dynamic plants,Automatica,10 (1974), 159–173.
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Staffans, O.J. Stabilization of a distributed system with a stable compensator. Math. Control Signal Systems 5, 1–22 (1992). https://doi.org/10.1007/BF01211973
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DOI: https://doi.org/10.1007/BF01211973