Optimal Stabilization Problem for the Quasilinear System with Controllable Parameters

The main concern of this paper is the problem of optimal stabilization of a quasilinear stochastic system with controllable parameters. Systems of this type are described by linear stochastic differential equations with multiplicative noises whose matrices, in general case, are nonlinear functions of control. The performance criterion is a modification of the classic quadratic performance cost. The goal is to minimize the criterion on the set of admissible control processes. This formulation of the problem is interesting because it allows us to study a wide range of optimization problems of linear systems with multiplicative perturbations, including: optimization of design parameters of the system, the problem of optimal stabilization under constraints on the gain matrix of the linear regulator in the form of inequalities, the problem of optimal stabilization of linear stochastic systems under information constraints. The main result of this paper is the necessary conditions for the optimal vector in the problem of stabilization of a quasilinear stochastic system with controllable parameters.The numerical gradient-type procedure for synthesis of the optimal stabilizing vector is also proposed. In addition, using obtained results we construct the algorithm for synthesis of a suboptimal time-dependent control. The result of the proposed algorithm is piecewise constant control, which gives the value of the criterion is guaranteed not worse than for the optimal stabilizing vector. This algorithm is relatively simple and one may use it for calculations in real time. The obtained results are applied to the problem of optimal stabilization under information constraints, in which the necessary optimality conditions are also obtained and the gradient-type procedure for the synthesis of the optimal control is proposed. The use of the obtained results is demonstrated by a model example.

continuous stochastic systems, systems with controllable parameters, quadratic criterion, optimal stabilization, linear systems with multiplicative noise, information constraints

1. Wonham W. M. Optimal Stationary Control of a Linear System with State-dependent Noise, SIAM Journal on Control, 1967, vol. 5, iss. 3, pp. 486—500.

2. Damm T. Rational Matrix Equations in Stochastic Control. Springer, Berlin Heidelberg, 2004.

3. McLane P. J. Optimal Stochastic Control of Linear Systems with State- and Control-dependent Disturbances, IEEE Transactions on Automatic Control, 1971, vol. 16, iss. 6, pp. 793—798.

4. Haussmann U. G. Optimal Stationary Control with State Control Dependent Noise, SIAM Journal on Control, 1971, vol. 9, iss. 2, pp. 184—198.

5. Khrustalyov M. M. Avtomatika i Telemekhanika, 2011, no. 11, pp. 174—190 (in Russian).

6. Khalina A. S., Khrustalyov M. M. Optimizaciya oblika i stabilizaciya upravlyaemyh kvazilinejnyh stohasticheskih sistem, funkcioniruyushchih na neogranichennom intervale vremeni, Izvestiya Rossijskoj akademii nauk. Teoriya i sistemy upravleniya, 2017, no. 1, pp. 65—88 (in Russian).

7. Mil’shtejn G. N. Avtomatika i Telemekhanika, 1976, no. 8, pp. 48—53 (in Russian).

8. McLane P. J. Linear Optimal Stochastic Control Using Instantaneous Output Feedback, International Journal of Control, 1971, vol. 13, iss. 2, pp. 383—396.

9. Trushkova E. A. Avtomatika i Telemekhanika, 2011, no. 6, pp. 151—159 (in Russian).

10. Tsar’kov K. A., Khrustalev M. M., Rumyantsev D. S. Optimization of Quasilinear Stochastic Control-nonlinear Diffusion Systems, Automation and Remote Control, 2017, vol. 78, iss. 6, pp. 1028—1045.

11. Onegin E., Khrustalev M. The Optimal Disturbance Suppression Problem on the Infinite Time Interval for Quasilinear Stochastic Systems with Output Feedback, In Proc. 13th Int. Conf. Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conf.), 2016, available at: https://ieeexplore.ieee.org/abstract/ document/7541193.

12. Onegin E., Khrustalev M. Optimal Stabilisation of a Quasilinear Stochastic System with Controllable Parameters, In Proc. 14th Int. Conf. "Stability and Oscillations of Nonlinear Control Systems" (Pyatnitskiy’s Conf.), 2018, available at: https://ieeexplore. ieee.org/document/8408384.

13. Khrustalyov M. M. Izvestiya Rossijskoj akademii nauk, Part 1, 1995, no. 6, pp. 194—208; Part 2, 1996, no. 1, pp. 72—79 (in Russian).

14. Øksendal B. Stochastic Differential Equations. An Introduction with Applications, Springer, Berlin Heidelberg, 2003.

15. Etienne de Klerk. Aspects of Semidefinite Programming, Springer US, 2002.