Abstract
This paper overviews several results connected with using the properties of dissipativity and risk-sensitivity in control problems. It is demonstrated that these properties serve as powerful analysis and synthesis tools for deterministic and stochastic control systems. Finally, we substantiate the relevance of research focused on feasibility of combining the properties in question.
Similar content being viewed by others
References
Liptser, R.Sh. and Shiryaev, A.N., Teoriya martingalov (Martingale Theory), Moscow: Nauka, 1986.
Mazurov, A.Yu., Dissipativity of Stochastic Systems with Risk-Sensitive Storage Functions, Cand. Sci. (Phys.-Math.) Dissertation, Nizhni Novgorod: Alekseev State Technical Univ., 2010.
Pakshin, P.V., Dissipativity of Diffusion Itô Processes with Markovian Switching and Problems of Robust Stabilization, Autom. Remote Control, 2007, vol. 68, no. 9, pp. 1502–1519.
Pakshin, P.V., Exponential Dissipativeness of the Random-structure Diffusion Processes and Problems of Robust Stabilization, Autom. Remote Control, 2007, vol. 68, no. 10, pp. 1852–1871.
Polushin, I.G., Fradkov, A.L., and Hill, D.J., Passivity and Passification of Nonlinear Systems, Autom. Remote Control, 2000, vol. 61, no. 3, pp. 355–389.
Khas’minskii, R.Z., Ustoichivost’ sistem differentsial’nykh uravnenii pri sluchainykh vozmushcheniyakh ikh parametrov (Stability of Combined Differential Equations under Random Perturbations of Their Parameters), Moscow: Nauka, 1969.
Aliyu, M.D.S., Dissipative Analysis and Stability of Nonlinear Stochastic State-delayed Systems, Nonlinear Dynamics and Systems Theory, 2004, vol. 4, pp. 243–256.
Arslan, G. and Başar, T., Risk-sensitive Adaptive Trackers for Strict-feedback Systems with Output Measurements, IEEE Trans. Autom. Control, 2002, vol. 47, pp. 1754–1758.
Athans, M., The Role and Use of the Stochastic Linear-quadratic-Gaussian Problem in Control System Design, IEEE Trans. Autom. Control, 1971, vol. 16, pp. 529–552.
Başar, T. and Bernhard, P., H ∞-optimal Control and Related Minimax Design Problems: A Dynamic Game Approach, Boston: Birkhauser, 1995.
Bensoussan, A. and Van Schuppen, J.H., Optimal Control of Partially Observable Stochastic Systems with an Exponential-of-integral Performance Index, SIAM J. Control Optim., 1985, vol. 23, no. 4, pp. 599–613.
Borkar, V.S. and Mitter, S.K., A Note on Stochastic Dissipativeness, in Directions in Mathematical Systems Theory and Optim., Lecture Notes in Control Inform., vol. 286, Berlin: Springer, 2003, pp. 41–49.
Byrnes, C.I., Isidori, A., and Willems, J.C., Passivity, Feedback Equivalence, and the Global Stabilization of Minimum Phase Nonlinear Systems, IEEE Trans. Autom. Control, 1991, vol. 36, pp. 1228–1240.
Cavazos-Cadena, R. and Fernández-Gaucherand, E., Controlled Markov Chains with Risk-sensitive Criteria: Average Costs, Optimality Equations, and Optimal Solutions, Math. Methods Oper. Res., 1999, vol. 49, pp. 299–324.
Dawson, D.M., Nonlinear Control of Electric Machinery, London: Marcel Dekker, 1998.
Fleming, W.H. and Hernández-Hernández, D., Risk-sensitive Control of Finite State Machines on an Infinite Horizon I, SIAM J. Control Optim., 1997, vol. 35, no. 5, pp. 1790–1810.
Fleming, W.H. and Hernández-Hernández, D., Risk-sensitive Control of Finite State Machines on an Infinite Horizon II, SIAM J. Control Optim., 1999, vol. 37, pp. 1048–1069.
Fleming, W.H. and James, M.R., The Risk-sensitive Index and the H 2 and H ∞ Norms for Nonlinear Systems, Math. Control Signals Syst., 1995, vol. 8, pp. 199–221.
Fleming, W.H. and McEneaney, W.M., Risk-sensitive Control on an Infinite Time Horizon, SIAM J. Control Optim., 1995, vol. 33, no. 6, pp. 1881–1915.
Fleming, W.H. and McEneaney, W.M., Risk Sensitive Control and Differential Games, in Springer Lecture Notes in Control and Info. Sci., vol. 184, New York: Springer-Verlag, 1992, pp. 185–197.
Fleming, W.H. and Zhang, Q., Risk-sensitive Production Planning of a Stochastic Manufacturing System, SIAM J. Control Optim., 1998, vol. 36, pp. 1147–1170.
Florchinger, P., A Passive System Approach to Feedback Stabilization of Nonlinear Control Stochastic Systems, SIAM J. Control Optim., 1999, vol. 37, pp. 1848–1864.
Fossen, T., Nonlinear Backstepping Design: Application to Mechanical Systems and Ship Control, New York: Springer-Verlag, 1999.
Glover, K. and Doyle, J.C., State-space Formulae for All Stabilizing Controllers that Satisfy an H ∞-norm Bound and Relations to Risk Sensitivity, Syst. Control Lett., 1988, vol. 11, pp. 167–172.
Hernández-Hernández, D. and Marcus, S.J., Risk Sensitive Control of Markov Processes in Countable State Space, Syst. Control Lett., 1996, vol. 29, no. 3, pp. 147–155.
Howard, R.A. and Matheson, J.A., Risk-sensitive Markov Decision Processes, Manage Sci., 1972, vol. 18, pp. 357–370.
Jacobson, D.H., Optimal Stochastic Linear Systems with Exponential Performance Criteria and Their Relation to Deterministic Differential Games, IEEE Trans. Autom. Control, 1973, vol. 18, pp. 124–131.
James, M.R. and Baras, J.S., Partially Observed Differential Games, Infinite-dimensional Hamilton-Jacobi-Isaacs Equations, and Nonlinear H ∞ Control, SIAM J. Control Optim., 1996, vol. 34, pp. 1342–1364.
James, M.R., Asymptotic Analysis of Nonlinear Stochastic Risk-sensitive Control and Differential Games, Math. Control Signals Syst., 1992, vol. 5, pp. 401–417.
Janković, M., Janković, M., and Kolmanovsky, I., Constructive Lyapunov Control Design for Turbocharged Diesel Engines, Proc. 17th Am. Control Conf., Philadelphia, 1998, pp. 1389–1394.
Kaise, H. and Nagai, H., Bellman-Isaacs Equations of Ergodic Type Related to Risk-sensitive Control and Their Singular Limits, Asympt. Anal., 1998, vol. 16, pp. 347–362.
Krainak, J., Speyer, J., and Marcus, S., Static Team Problems-Part I: Sufficient Conditions and the Exponential Cost Criterion, IEEE Trans. Autom. Control, 1982, vol. 27, pp. 839–848.
McEneaney, W.M., Connections between Risk-sensitive Stochastic Control, Differential Games and H ∞-control: The Nonlinear Case, PhD Dissertation, R.I.: Brown Univ., 1993.
Ortega, R. and Spong, M.W., Adaptive Motion Control of Rigid Robots: A Tutorial, Automatica, 1989, vol. 25, no. 6, pp. 877–888.
Pan, Z. and Başar, T., Backstepping Controller Design for Nonlinear Stochastic Systems under a Risksensitive Cost Criterion, SIAM J. Control Optim., 1999, vol. 37, pp. 957–995.
Pogromsky, A.Yu., Fradkov, A.L., and Hill, D.J., Passivity Based Damping of Power System Oscillations, Proc. 35th IEEE Conf. on Decision and Control, Kobe, 1996, pp. 3876–3881.
Runolfsson, T., The Equivalence between Infinite-horizon Optimal Control of Stochastic Systems with Exponential-of-integral Performance Index and Stochastic Differential Games, IEEE Trans. Autom. Control, 1994, vol. 39, no. 8, pp. 1551–1563.
Runolfsson, T., Stationary Risk-sensitive LQG Control and Its Relation to LQG and H ∞-control, Proc. 29th IEEE Conf. on Decision and Control, 1990, pp. 1018–1023.
Runolfsson, T., Risk-sensitive Control of Stochastic Hybrid Systems on Infinite Time Horizon, Math. Problems in Engineering, 2000, vol. 5, pp. 459–478.
Shaked, U. and Berman, N., H ∞ Control for Nonlinear Stochastic Systems: The Output-feedback Case, Preprints 16th IFAC World Congr., Prague, 2005, CD-ROM, pp. 1–6.
Sira-Ramires, H. and Angulo-Núñez, M.I., Passivity-based Control of Nonlinear Chemical Processes, Int. J. Control, 1997, vol. 68, pp. 971–996.
Sira-Ramires, H. and Ortega, R., Passivity-based Control of DC to DC Converters, Proc. 34th IEEE Conf. on Decision and Control, New Orleans, 1995, pp. 3471–3476.
Speyer, J.L., Deyst, J., and Jacobson, D.H., Optimization of Stochastic Linear Systems with Additive Measurement and Process Noise Using Exponential Performance Criteria, IEEE Trans. Autom. Control, 1974, vol. 19, pp. 358–366.
Speyer, J.L., An Adaptive Terminal Guidance Scheme Based on an Exponential Cost Criterion with Applications to Homing Missile Guidance, IEEE Trans. Autom. Control, 1976, vol. 21, pp. 371–375.
Thygesen, U.H., Robust Performance and Dissipation of Stochastic Control Systems, PhD Dissertation, Copenhagen: Technical Univ. of Denmark, 1998.
Van der Ploeg, F., Economic Policy Rules for Risk-sensitive Decision Making, Zeitschrift für National ökonomie, 1984, vol. 44, pp. 207–235.
Whittle, P., Risk-sensitive Linear/quadratic/Gaussian control, Adv. Appl. Prob., 1981, vol. 13, pp. 764–777.
Whittle, P., A Risk-sensitive Maximum Principle: The Case of Imperfect State Observation, IEEE Trans. Autom. Control, 1991, vol. 36, no. 7, pp. 793–801.
Whittle, P., A Risk-sensitive Maximum Principle, Syst. Control Lett., 1990, vol. 15, pp. 183–192.
Willems, J.C., Dissipative Dynamical Systems. Part I: General Theory, Arch. Rational Mech. Analysis, 1972, vol. 45, pp. 321–351.
Willems, J.C., Dissipative Dynamical Systems. Part II: Linear Systems with Quadratic Supply Rates, Arch. Rational Mech. Analysis, 1972, vol. 45, pp. 352–393.
Zhang, W. and Chen, B.-S., State Feedback H ∞ Control for a Class of Nonlinear Stochastic Systems, SIAM J. Control Optim., 2006, vol. 44, pp. 1973–1991.
Author information
Authors and Affiliations
Additional information
Original Russian Text © A.Yu. Mazurov, 2010, published in Upravlenie Bol’shimi Sistemami, 2010, No. 29, pp. 42–67.
Rights and permissions
About this article
Cite this article
Mazurov, A.Y. Dissipativity and risk-sensitivity in control problems. Autom Remote Control 72, 2196–2209 (2011). https://doi.org/10.1134/S0005117911100183
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117911100183