Abstract
For the multivariable control system described by\(\dot x = Ax + Bu,y = Cx,z = Dx\) (wherey is the measured output andz the output to be regulated) conditions are given for the existence of a controller which preserves output regulation and loop stability, in the presence of small parameter variations in controller and plant. Under mild conditions, such a “strong” synthesis is shown to exist if and only if the regulator problem with internal stability (RPIS) is well-posed. Synthesis is achieved by means of a feedback configuration which in general incorporates an invariant, and suitably redundant, copy of the dynamic model adopted for the exogenous disturbance and reference signals which the system is required to process.
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References
W. M. Wonham andJ. B. Pearson, Regulation and internal stabilization in linear multivariable systems, Control Systems Report No. 7212, Dept. of Electrical Engineering, Univ. of Toronto, August, 1972; to appear,SIAM J. Control 12 (1), 1974.
E. Fabian andW. M. Wonham, Generic solvability of the decoupling problem, Control Systems Report No. 7301, Dept. of Electrical Engineering, Univ. of Toronto, January 1973; to appear,SIAM J. Control 12 (4), 1974.
A. S. Morse, Structural invariants of linear multivariable systems,SIAM J. Control 11 (3), 1973, 446–465.
M. L. J. Hautus, Controllability and observability conditions of linear autonomeus systems,Proc. Kon. Ned. Akad. Wetensch. ser. A,72, 1969, 443–448.
D. G. Luenberger, Observers for multivariable systems,I.E.E.E. Trans. AC-11 (2), 1966, 190–197.
W. M. Wonham, Dynamic observers: geometric theory,I.E.E.E. Trans. Aut. Control AC-15 (2), 1970, 258–259.
I. M. Horowitz,Synthesis of Feedback Systems, Academic Press, New York, 1963.
R. C. Conant, The information transfer required in regulatory processes,I.E.E.E. Trans. Systems Science and Cybernetics SSC-5 (4), 1969, 334–338.
O. J. M. Smith,Feedback Control Systems, McGraw-Hill, New York, 1958.
E. J. Davison, The output control of linear time-invariant multivariable systems with unmeasurable arbitrary disturbances,I.E.E.E. Trans. Aut. Control AC-17 (5), 1972, 621–630.
J. von Neumann, Probabilistic logics and the synthesis of reliable organisms from unreliable components, in C. E. Shannon and J. McCarthy (Eds.),Automata Studies, Annals of Math. Stud. No. 34, Princeton Univ. Press, Princeton, N. J., 1956; 43–98.
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This research was partially supported by The National Research Council of Canada, Grant No. A-7399. A shortened version entitled “Synthesis of Multivariable Regulators” has appeared in Proc. Eleventh Annual Allerton Conf. on Circuit and System Theory, Univ. of Illinois, October 1973.
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Francis, B., Sebakhy, O.A. & Wonham, W.M. Synthesis of multivariable regulators: The internal model principle. Appl Math Optim 1, 64–86 (1974). https://doi.org/10.1007/BF01449024
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DOI: https://doi.org/10.1007/BF01449024