Abstract
We consider two-criteria control or filtering problems for linear systems, where one criterion is the level of suppression for Gaussian white noise with unknown covariance, and another is the level of suppression for a deterministic signal of bounded power. We define a new criterion, the level of suppression for stochastic and deterministic disturbances that act jointly in the general case on different inputs. This criterion is characterized in terms of solutions of Riccati equations or linear matrix inequalities. We establish that for the choice of optimal controller or filter with respect to this criterion relative losses with respect to each of the original criteria compared to Pareto optimal solutions do not exceed the value \(1 - {{\sqrt 2 } \mathord{\left/ {\vphantom {{\sqrt 2 } 2}} \right. \kern-\nulldelimiterspace} 2}\) . We extend these results to dual control and filtering problems for systems with one input and two outputs, generalize them to the case of N criteria with loss estimate \(1 - {{\sqrt N } \mathord{\left/ {\vphantom {{\sqrt N } N}} \right. \kern-\nulldelimiterspace} N}\), and also apply them for systems with external and initial disturbances. We show a numerical example.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Doyle, J.C., Glover, K., Khargonekar, P.P., and Francis, B.A., State Space Solutions to Standard H 2 and H ∞ Control Problems, IEEE Trans. Automat. Control, 1989, vol. 34, pp. 831–847.
Vladimirov, I.G., Kurdyukov, A.P., and Semenov, A.V., Stochastic H ∞-Optimization Problem, Dokl. Ross. Akad. Nauk, 1995, vol. 343, no. 5, pp. 607–609.
Chaikovskii, M.M. and Kurdyukov, A.P., Strict Boundedness Criterion for Anisotropic Norm with a Given Value in Terms of Linear Matrix Inequalities, Dokl. Ross. Akad. Nauk, 2011, vol. 441, no. 3, pp. 318–321.
Zhou, K., Glover, K., Bodenheimer, B., and Doyle, J., Mixed H 2 and H ∞ Performance Objectives I: Robust Performance Analysis, IEEE Trans. Automat. Control, 1994, vol. 39, pp. 1564–1574.
Doyle, J., Zhou, K., Glover, K., and Bodenheimer, B., Mixed H 2 and H ∞ Performance Objectives II: Optimal Control, IEEE Trans. Automat. Control, 1994, vol. 39, pp. 1575–1587.
Bernstein, D.S. and Haddad, W.M., LQG Control with an H ∞ Performance Bound: A Riccati Equation Approach, IEEE Trans. Automat. Control, 1989, vol. 34, pp. 293–305.
Haddad, W.M., Bernstein, D.S., and Mustafa, D., Mixed-Norm H 2/H ∞-Regulation and Estimation: The Discrete-Time Case, Syst. Control Lett., 1991, vol. 16, pp. 235–247.
Khargonekar, P.P. and Rotea, M.A., Mixed H 2/H ∞ Control: A Convex Optimization Approach, IEEE Trans. Automat. Control, 1991, vol. 36, pp. 824–831.
Yaesh, I. and Shaked, U., Game Theory Approach to State Estimation of Linear Discrete-Time Processes and Its Relation to H ∞-Optimal Estimation, Int. J. Control, 1992, vol. 55, no. 6, pp. 1443–1452.
Kaminer, I., Khargonekar, P.P., and Rotea, M.A., Mixed H 2/H ∞ Control for Discrete-Time Systems via Convex Optimization, Automatica, 1993, vol. 29, no. 1, pp. 57–70.
Chen, X. and Zhou, K., Multiobjective H 2/H ∞ Control Design, SIAM J. Control Optim., 2001, vol. 40, no. 2, pp. 628–660.
Davis, L.D., Collins, E.G., and Haddad, W.M., Discrete-Time Mixed-Norm H 2/H ∞ Controller Synthesis, Optim. Control Appl. Methods, 1996, vol. 17, pp. 107–121.
Muradore, R. and Picci, G., Mixed H 2/H ∞ Control: The Discrete-Time Case, Syst. Control Lett., 2005, vol. 54, pp. 1–13.
Scherer, C., Mixed H 2/H ∞ Control, in Trends in Control: A European Perspective, Isidori, A., Ed., Berlin: Springer-Verlag, 1995, pp. 173–216.
El Ghaoui, L. and Folcher, J.P., Multiobjective Robust Control of LTI Systems Subject to Unstructured Perturbations, Syst. Control Lett., 1996, vol. 28, pp. 23–30.
Scherer, C., Gahinet, P., and Chilali, M., Multiobjective Output-Feedback Control via LMI Optimization, IEEE Trans. Automat. Control, 1997, vol. 42, pp. 896–911.
Mäkilä, P.M., On Muptiple Criteria Stationary Linear Quadratic Control, IEEE Trans. Automat. Control, 1989, vol. 34, pp. 1311–1313.
Khargonekar, P.P. and Rotea, M.A., Muptiple Objective Optimal Control of Linear Systems: The Quadratic Norm Case, IEEE Trans. Automat. Control, 1991, vol. 36, pp. 14–24.
Hindi, H.A., Hassibi, B., and Boyd, S.P., Multiobjective H 2/H ∞-Optimal Control via Finite Dimensional Q-Parametrization and Linear Matrix Inequalities, Proc. Am. Control Conf., Philadelphia, 1998, pp. 3244–3249.
Ostertag, E., An Improved Path-following Method for Mixed H 2/H ∞ Controller Design, IEEE Trans. Automat. Control, 2008, vol. 53, pp. 1967–1971.
Balandin, D.V. and Kogan, M.M., Pareto Suboptimal Solutions in Control and Filtering Problems under Multiple Deterministic and Stochastic Disturbances, Proc. Eur. Control Conf., Aalborg, 2016, pp. 2263–2268.
Kogan, M.M., Optimal Estimation and Filtration under Unknown Covariances of Random Factors, Autom. Remote Control, 2014, vol. 75, no. 11, pp. 1964–1981.
Kogan, M.M., LMI Based Minimax Estimation and Filtering under Unknown Covariances, Int. J. Control, 2014, vol. 87, no. 6, pp. 1216–1226.
Kogan, M.M., Optimal Discrete-Time H ∞/γ 0 Filtering and Control under Unknown Covariances, Int. J. Control, 2016, vol. 89, no. 4, pp. 691–700.
Khargonekar, P.P., Nagpal, K.M., and Poolla, K.R., H ∞ Control with Transients, SIAM J. Control Optim., 1991, vol. 29, no. 6, pp. 1373–1393.
Nagpal, K.M. and Khargonekar, P.P., Filtering and Smoothing in an H ∞ Setting, IEEE Trans. Autom. Control, 1991, vol. 36, no. 2, pp. 152–166.
Balandin, D.V. and Kogan, M.M., Generalized H ∞-Optimal Control as a Trade-Off between the H ∞- Optimal and γ-Optimal Controls, Autom. Remote Control, 2010, vol. 71, no. 6, pp. 993–1010.
Balandin, D.V. and Kogan, M.M., LMI-based H ∞-optimal Control with Transients, Int. J. Control, 2010, vol. 83, no. 8, pp. 1664–1673.
Balandin, D.V., Kogan, M.M., Krivdina, L.N., and Fedyukov, A.A., Design of Generalized Discrete- Time H ∞-optimal Control over Finite and Infinite Intervals, Autom. Remote Control, 2014, vol. 75, no. 1, pp. 1–17.
Boyd, S., El Ghaoui, L., Feron, E., et al., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.
Balandin, D.V. and Kogan, M.M., Sintez zakonov upravleniya na osnove lineinykh matrichnykh neravenstv (Design of Control Laws Based on Linear Matrix Inequalities), Moscow: Fizmatlit, 2007.
Wilson, D.A., Convolution and Hankel Operator Norms for Linear Systems, IEEE Trans. Automat. Control, 1989, vol. 34, pp. 94–97.
Rotea, M.A., The Generalized H 2 Control Problem, Automatica, 1993, vol. 29, no. 2, pp. 373–385.
Balandin, D.V. and Kogan, M.M., Linear-Quadratic and γ-Optimal Output Control Laws, Autom. Remote Control, 2008, vol. 69, no. 6, pp. 911–919.
Gelig, A.Kh., Leonov, G.A., and Yakubovich, V.A., Ustoichivost’ nelineinykh sistem s needinstvennym sostoyaniem ravnovesiya (Stability of Nonlinear Systems with Non-Unique Equilibria States), Moscow: Nauka, 1978.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © D.V. Balandin, M.M. Kogan, 2017, published in Avtomatika i Telemekhanika, 2017, No. 1, pp. 35–58.
This paper was recommended for publication by P.S. Shcherbakov, a member of the Editorial Board
Rights and permissions
About this article
Cite this article
Balandin, D.V., Kogan, M.M. On Pareto set in control and filtering problems under stochastic and deterministic disturbances. Autom Remote Control 78, 29–49 (2017). https://doi.org/10.1134/S0005117917010039
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117917010039