Abstract
This paper presents a description of solution approaches to the problem of output feedback control under unknown but bounded disturbances with hard bounds on the controls and the uncertain items. The problem is treated within a finite horizon which requires to track the system dynamics throughout the whole time interval rather than through asymptotic properties. The demand for such solutions is motivated by increasing number of applications. The suggested approaches are designed as a combination of Hamiltonian techniques in the form of generalized dynamic programming with those of set-valued analysis and problems on minimax. The paper indicates the crucial role of properly selecting the on-line generalized state of the system in the form of information states or information sets. The description ranges from theoretical schemes to computational routes with emphasis on the possibility of treating the overall problem through only finite-dimensional methods. The results apply to nonlinear systems, with more details for linear models. It turns out that in the last case, while moving through calculations, one may avoid the fairly difficult stage of integrating HJB equations. The procedures are here confined to ordinary differential equations and ellipsoidal or polyhedral techniques. The suggested schemes are also quite appropriate for parallel computation.
Similar content being viewed by others
References
Wonham, M.: On the separation theorem of stochastic control. SIAM J. Control Optim. 312–326 (1968)
Davis, M.H.A., Varaiya, P.: Information states for linear stochastic systems. J. Math. Anal. Appl. 37, 384–402 (1972)
Astrom, K.J.: Introduction to Stochastic Control Theory. Academic Press, San Diego (1970)
Krasovski, N.N.: Control and stabilization under lack of information: Izvestiya RAN. Technicheskaya Kibernetika (In Russian. Translated as ‘Engineering Cybernetics’), No 1. 148–151 (1993)
Basar, T., Bernhard, P.: H ∞ Optimal Control and Related Minimax Design Problems. SCFA, 2nd edn. Birkhäuser, Basel (1995)
James, M.R., Baras, J.S.: Partially observed differential games, infinite-dimensional Hamilton–Jacobi–Isaacs equations and nonlinear H ∞ control. SIAM J. Control Optim. 34(4), 1342–1364 (1996)
Helton, J.W., James, M.R.: Extending H ∞ Control to Nonlinear Systems. SIAM, Philadelphia (1999)
Kurzhanski, A.B.: Control and Observation Under Uncertainty. Nauka, Moscow (1977). in Russian
Krasovski, A.N., Krasovski, N.N.: Control Under Lack of Information. SCFA. Birkhauser, Boston (1995)
Kurzhanski, A.A., Varaiya, P.: Ellipsoidal Toolbox. University of California, Berkeley (2007)
Kostousova, E.K., Kurzhanski, A.B.: Theoretical framework and approximation techniques for parallel computation in set-membership state estimation. In: CESA ’96 IMACS Multiconference Computational Engineering in Systems Applications, Lille, France, Symposium on Modelling, Analysis and Simulation, Proc. vol. 2, pp. 849–854. (1996)
Kostousova, E.K.: Control synthesis via parallelotopes: optimization and parallel computations. Optim. Methods Softw. 14, 267–310 (2001)
Kurzhanski, A.B.: The comparison principle for equations of the Hamilton–Jacobi type in control theory. Proc. Steklov Math. Inst. Suppl. 1 (2006)
Kurzhanski, A.B., Varaiya, P.: A comparison principle for equations of the Hamilton–Jacobi type in set-membership filtering. Commun. Inf. Syst. 6(3), 179–192 (2006)
Leitmann, G.: Optimality and reachability with feedback controls. In: Blaquiere, A. (ed.) Dynamical Systems and Microphysics, p. 119. Academic Press, New York (1982)
Kurzhanski, A.B., Varaiya, P.: The Hamilton–Jacobi Equations for Nonlinear Target Control and Their Approximation, Analysis and Design of Nonlinear Control Systems (in Honor of Alberto Isidori), pp. 77–90. Springer, Berlin (2007)
Baras, J.S., Kurzhanski, A.B.: Nonlinear filtering: the set-membership (bounding) and the H ∞ approaches. In: Proc. of the IFAC NOLCOS Conference, Tahoe, CA. Plenum, New York (1995)
Krasovski, N.N.: On theory of controllability and observability of linear dynamic systems. Prikl. Mat. Meh. 28(1), 13–14 (1964) Translated as J. Appl. Math. Mech.
Kurzhanski, A.B.: Differential games of observation. Sov. Math. Dokl. 13(6), 1556–1560 (1972)
Schweppe, F.C.: Uncertain Dynamic Systems. Prentice Hall, Englewood Cliffs (1973)
Bertsekas, D.P., Rhodes, I.B.: Recursive state estimation for a set-membership description of uncertainty. IEEE Trans. Autom. Control AC-16, 117–128 (1971)
Chernousko, F.L.: State Estimation for Dynamic Systems, CDC Press (1993)
Milanese, M., Norton, J., Piet-Lahanier, H., Walter, E. (eds.): Bounding Approach to System Identification. Plenum Press, New York (1995)
Krasovski, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer, New York (1998)
Kurzhanski, A.B.: On the problem of measurement feedback control: ellipsoidal techniques. In: Abed, E.H. (ed.) Advances in Control, Communication Networks and Transportation Systems (In Honor of Pravin Varaiya). SCFA, pp. 21–38. Birkhauser, Boston (2005)
Aubin, J.-P.: Viability Theory. Birkhauser, Boston (1991)
Blanchini, F., Miani, S.: Set-Theoretic Methods in Control. Birkhauser, Boston (2008)
Isidori, A., Astolfi, A.: Disturbance attenuation and H ∞ control via measurement feedback in nonlinear systems. IEEE Trans. Autom. Control AC-37, 1283–1293 (1992)
Isidori, A., Marconi, L.: Asymptotic analysis and observer design in the theory of nonlinear output regulation. In: Nonlinear Observers and Applications. LNCIS, vol. 363, pp. 181–210. Springer, Berlin (2007)
Lygeros, J., Tomlin, C., Sastry, S.: Controllers for reachability specifications for hybrid systems. Automatica 35(3), 349–370 (1999)
Kurzhanski, A.B., Filippova, T.F.: On the theory of trajectory tubes: a mathematical formalism for uncertain dynamics, viability and control. In: Kurzhanski, A.B. (ed.) Advances in Nonlinear Dynamics and Control. PSCT, vol. 17, pp. 122–188. Birkhauser, Boston (1993)
Kurzhanski, A.B., Mitchell, I., Varaiya, P.: Optimization problems for state-constrained control and obstacle problems. J. Optim. Theory Appl. 128(3), 488–521 (2006)
Krener, A.: The convergence of the extended Kalman filter. In: Rantzer, A., Byrnes, C. (eds.) Directions in Mathematical Systems Theory and Optimization, pp. 73–182. Springer, Berlin (2003)
Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–42 (1983)
Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (1993)
Subbotin, A.I.: Generalized Solutions of First-Order PDE’s. The Dynamic Optimization Perspective. Birkhauser, Boston (1995)
Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)
Kurzhanski, A.B., Valyi, I.: Ellipsoidal Calculus for Estimation and Control. SCFA. Birkhauser, Boston (1997)
Kurzhanski, A.B., Varaiya, P.: Ellipsoidal techniques for reachability analysis. Part I: External approximations. Part II: Internal approximations. Box-valued constraints. Optim. Methods Softw. 17, 177–237 (2002)
Kurzhanski, A.B., Varaiya, P.: Ellipsoidal techniques for reachability under state constraints. SIAM J. Control Optim. 45(4), 1369–1394 (2006)
Kurzhanski, A.B., Varaiya, P.: Reachability analysis for uncertain systems—the ellipsoidal technique. Dyn. Contin. Discrete Impuls. Syst., Ser. B 9(3), 347–367 (2002)
Kurzhanski, A.B., Nikonov, O.I.: Evolution equations for tubes of trajectories of synthesized control systems. Russ. Acad. Sci. Math. Dokl. 48(3), 606–611 (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Leitmann.
Rights and permissions
About this article
Cite this article
Kurzhanski, A.B., Varaiya, P. Optimization of Output Feedback Control Under Set-Membership Uncertainty. J Optim Theory Appl 151, 11–32 (2011). https://doi.org/10.1007/s10957-011-9861-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-011-9861-z