Abstract
The regulator problem with the plant stability condition is investigated by using new tools of the geometric approach, i.e., self-bounded controlled and self-hidden conditioned invariants, recently introduced by the authors. In this framework, a constructive procedure which solves the problem without eigenspaces is obtained. This procedure also includes a set of necessary and sufficient conditions which, compared with those previously known in the literature, are significantly simpler in their statement and more easily mechanized.
Similar content being viewed by others
References
Wonham, W. M.,Tracking and Regulation in Linear Multivariable Systems, SIAM Journal on Control, Vol. 11, No. 3, pp. 424–437, 1973.
Wonham, W. M., andPearson, J. B.,Regulation and Internal Stabilization in Linear Multivariable Systems, SIAM Journal on Control, Vol. 12, No. 1, pp. 5–18, 1974.
Francis, B. A.,The Linear Multivariable Regulator Problem, SIAM Journal on Control and Optimization, Vol. 15, No. 3, pp. 486–505, 1977.
Schumacher, J. M. H.,Regulator Synthesis Using (C, A, B)-Pairs, IEEE Transactions on Automatic Control, Vol. AC-27, No. 6, pp. 1211–1221, 1982.
Schumacher, J. M.,The Algebraic Regulator Problem from the State-Space Point of View, Linear Algebra and Its Applications, Vol. 50, No. 4, pp. 487–520, 1983.
Cheng L., andPearson, J. B.,Frequency Domain Synthesis of Multivariable Linear Regulators, IEEE Transactions on Automatic Control, Vol. AC-23, No. 1, pp. 3–15, 1978.
Antoulas, A. C.,A New Approach to Synthesis Problems in Linear System Theory, IEEE Transactions on Automatic Control, Vol. AC-30, No. 5, pp. 465–473, 1985.
Antoulas, A. C.,On the Solvability of General Control Problems, International Journal on Control, Vol. 44, No. 5, pp. 1273–1284, 1986.
Vidyasagar, M.,Control System Synthesis: A Factorization Approach, MIT Press, Cambridge, Massachusetts, 1985.
Basile, G., andMarro, G.,Dual-Lattice Theorems in the Geometric Approach, Journal of Optimization Theory and Applications, Vol. 48, No. 2, pp. 229–244, 1986.
Basile, G., andMarro, G.,Self-Bounded Controlled Invariants versus Stabilizability, Journal of Optimization Theory and Applications, Vol. 48, No. 2, pp. 245–263, 1986.
Basile, G., Marro, G., andPiazzi, A.,A New Solution to the Disturbance Localization Problem with Stability and Its Dual, Proceedings of the International AMSE Conference on Modelling and Simulation, Athens, Greece, Vol. 1.2, pp. 19–27, 1984.
Basile, G., Marro, G., andPiazzi, A.,Stability without Eigenspaces in the Geometric Approach: Some New Results, Frequency Domain and State Space Method for Linear Systems, Edited by C. A. Byrnes and A. Lindquist, North-Holland, Amsterdams, Holland, pp. 441–450, 1986.
Basile, G., Marro, G., andPiazzi, A.,Revisiting the Regulator Problem in the Geometric Approach, Part 1: Disturbance Localization by Dynamic Compensation, Journal of Optimization Theory and Applications, Vol. 53, No. 1, pp. 9–22, 1987.
Basile, G., Marro, G., andPiazzi, A.,Revisiting the Regulator Problem in the Geometric Approach, Part 2: Asymptotic Tracking and Regulation in the Presence of Disturbances, Journal of Optimization Theory and Applications, Vol. 53, No. 1, pp. 23–36, 1987.
Basile, G., andMarro, G.,Self-Bounded Controlled Invariant Subspaces: A Straightforward Approach to Constrained Controllability, Journal of Optimization Theory and Applications, Vol. 38, No. 1, pp. 71–81, 1982.
Schumacher, J. M.,On a Conjecture of Basile and Marro, Journal of Optimization Theory and Applications, Vol. 41, No. 2, pp. 371–376, 1983.
Piazzi, A.,A New Geometric Approach to Some Problems in Linear System Theory, PhD Thesis, University of Bologna, Bologna, Italy, 1986.
Author information
Authors and Affiliations
Additional information
Communicated by G. Leitmann
This research was supported by the Italian Ministry of Education. This paper is an extended version of a communication presented at the 8th International Symposium on the Mathematical Theory of Networks and Systems, Phoenix, Arizona, 1987.
Rights and permissions
About this article
Cite this article
Basile, G., Marro, G. & Piazzi, A. Stability without eigenspaces in the geometric approach: The regulator problem. J Optim Theory Appl 64, 29–42 (1990). https://doi.org/10.1007/BF00940020
Issue Date:
DOI: https://doi.org/10.1007/BF00940020