Output dynamic controller analysis for stochastic systems of multiplicative type. (English. Russian original) Zbl 1489.93119
Autom. Remote Control 83, No. 3, 343-354 (2022); translation from Avtom. Telemekh. 2022, No. 3, 54-68 (2022).
Summary: The most important unsolved problem of control theory is the problem of optimizing a multiplicative stochastic system in the class of noisy output signal controllers. The controller in the feedback loop must be feasible; i.e., it must ensure the stability of the closed-loop system and guarantee the required level of suppression of exogenous disturbances acting on the plant. In this article, within the framework of the stochastic \(H^2/H_{\infty } \)-control theory in the presence of noise, we solve the problem of analysis, i.e. finding conditions for the existence of feasible controllers. The results obtained can be used further in the synthesis problem for conditional optimization of a multiplicative stochastic system in the class of feasible dynamic controllers.
MSC:
| 93E03 | Stochastic systems in control theory (general) |
| 93B36 | \(H^\infty\)-control |
| 93B52 | Feedback control |
Keywords:
\( H^2/H_\infty \)-control theory; Itô diffusion equation; induced operator norm; multiplicative stochastic system; controlled output signal; dynamic output controllerReferences:
| [1] | McLane, P. J., Optimal stochastic control of linear systems with state and control-dependent disturbances, IEEE Trans. Autom. Control., AC-16, 292-299 (1971) |
| [2] | Willems, J. L.; Willems, J. S., Feedback stabilizability for stochastic systems with state and control dependent noise, Automatica, 12, 277-283 (1976) · Zbl 0329.93036 · doi:10.1016/0005-1098(76)90029-7 |
| [3] | Krylov, N. V., Introduction to the Theory of Diffusion Processes (1995), Providence, RI: Am. Math. Soc., Providence, RI · Zbl 0844.60050 |
| [4] | Krylov, N. V., Upravlyaemye protsessy diffuzionnogo tipa (Controlled Processes of Diffusion Type) (1977), Moscow: Nauka, Moscow · Zbl 0513.60043 |
| [5] | Doyle, J.; Glover, K.; Khargonekar, P. P.; Francis, B., State space solutions to standard \(H_2\) and \(H^{\infty }\) control problems, IEEE Trans. Autom. Control, AC-34, 831-847 (1989) · Zbl 0698.93031 · doi:10.1109/9.29425 |
| [6] | Gahinet, P.; Apkarian, P., A linear matrix inequality approach to \(H^{\infty }\) control, Int. J. Robust Nonlinear Control, 4, 421-448 (1994) · Zbl 0808.93024 · doi:10.1002/rnc.4590040403 |
| [7] | Hinrichsen, D.; Pritchard, A. J., Stochastic \(H_{\infty }\), SIAM J. Control Optim., 36, 5, 1504-1538 (1998) · Zbl 0914.93019 · doi:10.1137/S0363012996301336 |
| [8] | Petersen, I. R.; Ugrinovsky, V. A.; Savkin, F. V., Robust Control Design Using \(H_\infty \)-Methods (2006), London: Springer, London |
| [9] | Gershon, E.; Shaked, U.; Yaesh, I., \( H_{\infty } \)-control and Estimation of State-Multiplicative Linear Systems. Vol. 318 of Lect. Notes in Control and Information Sciences (2005), London: Springer, London · Zbl 1116.93003 |
| [10] | Balandin, D. V.; Kogan, M. M., Sintez zakonov upravleniya na osnove lineinykh matrichnykh neravenstv (Synthesis of Control Laws Based on Linear Matrix Inequalities) (2007), Moscow: Fizmatlit, Moscow · Zbl 1124.93001 |
| [11] | Polyak, B. T.; Khlebnikov, M. V.; Shcherbakov, P. S., Upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh. Tekhnika lineinykh matrichnykh neravenstv (Control of Linear Systems under Exogenous Disturbances. Technique of Linear Matrix Inequalities) (2014), Moscow: LENARD, Moscow |
| [12] | Dragan, V., Morozan, T., and Stoica, A.M., Mathematical methods in robust control of linear stochastic systems, in Mathematical Concepts and Methods in Science and Engineering, Berlin: Springer, 2006. · Zbl 1101.93002 |
| [13] | Shaikin, M. E., Design of optimal state controller robust to external disturbance for one class of nonstationary stochastic systems, Autom. Remote Control, 76, 7, 1242-1251 (2015) · Zbl 1320.93092 · doi:10.1134/S0005117915070097 |
| [14] | Shaikin, M. E., Multiplicative stochastic systems: optimization and analysis, Differ. Equations, 53, 3, 1-16 (2017) · Zbl 1364.93869 · doi:10.1134/S0012266117030090 |
| [15] | Limebeer, D. J.N.; Anderson, B. D.O.; Khargonekar; Green, M., A game theoretic approach to \(H_\infty\) control for time-varying systems, SIAM J. Control Optim., 30, 262-283 (1992) · Zbl 0791.93026 · doi:10.1137/0330017 |
| [16] | Mustafa, D.; Glover, K., Minimum Entropy \(H_{\infty }\) Control (1990), Berlin: Springer-Verlag, Berlin · Zbl 0701.93027 · doi:10.1007/BFb0008861 |
| [17] | Ugrinovskii, V. A., Robust \(H_{\infty }\)-control in the presence of stochastic uncertainty, Int. J. Control, 71, 2, 219-237 (1998) · Zbl 0953.93028 · doi:10.1080/002071798221849 |
| [18] | Shaikin, M. E., Multiplicative stochastic systems with multiple external disturbances, Autom. Remote Control, 79, 2, 300-310 (2018) · Zbl 1391.93300 · doi:10.1134/S0005117918020091 |
| [19] | Kallianpur, G., Stochastic Filtering Theory (1980), New York-Heidelberg-Berlin: Springer-Verlag, New York-Heidelberg-Berlin · Zbl 0458.60001 · doi:10.1007/978-1-4757-6592-2 |
| [20] | Bulinskii, A. V.; Shiryaev, A. N., Teoriya sluchainykh protsessov (Theory of Random Processes) (2003), Moscow: Fizmatlit, Moscow |
| [21] | Venttsel’, A. D., Kurs teorii sluchainykh protsessov (Course in the Theory of Random Processes) (1975), Moscow: Nauka, Moscow |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
