Finite-Horizon Separation-Based Covariance Control for Discrete-Time Stochastic Linear Systems

In this paper, we address a finite-horizon stochastic optimal control problem with covariance assignment and input energy constraints for discrete-time stochastic linear systems with partial state information. In our approach, we consider separation-based control policies that correspond to sequences of control laws that are affine functions of either the complete history of the output estimation errors, that is, the differences between the actual output measurements and their corresponding estimated outputs produced by a discrete-time Kalman filter, or a truncation of the same history. This particular feedback parametrization allows us to associate the stochastic optimal control problem with a tractable semidefinite (convex) program. We argue that the proposed procedure for the reduction of the stochastic optimal control problem to a convex program has significant advantages in terms of improved scalability and tractability over the approaches proposed in the relevant literature.

[1]  Stephen P. Boyd,et al.  Design of Affine Controllers via Convex Optimization , 2010, IEEE Transactions on Automatic Control.

[2]  Jason L. Speyer,et al.  Stochastic Processes, Estimation, and Control , 2008, Advances in design and control.

[3]  John Lygeros,et al.  Stochastic receding horizon control with output feedback and bounded controls , 2012, Autom..

[4]  James A. Primbs.  Stochastic Receding Horizon Control of Constrained Linear Systems with State and Control Multiplicative Noise , 2007, ACC.

[5]  Efstathios Bakolas,et al.  Covariance control for discrete-time stochastic linear systems with incomplete state information , 2017, 2017 American Control Conference (ACC).

[6]  Karolos M. Grigoriadis,et al.  Covariance Controllers: A New Parameterization of the Class of All Stabilizing Controllers , 1990, 1990 American Control Conference.

[7]  Ian R. Petersen,et al.  State Distributions and Minimum Relative Entropy Noise Sequences in Uncertain Stochastic Systems: The Discrete-Time Case , 2012, SIAM J. Control. Optim..

[8]  W. Wonham.  On the Separation Theorem of Stochastic Control , 1968 .

[9]  Tryphon T. Georgiou,et al.  Optimal Steering of a Linear Stochastic System to a Final Probability Distribution, Part II , 2014, IEEE Transactions on Automatic Control.

[10]  Martin Connors,et al.  Optimization Models , 2014 .

[11]  Efstathios Bakolas,et al.  Stochastic linear systems subject to constraints , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[12]  Tryphon T. Georgiou,et al.  Optimal Steering of a Linear Stochastic System to a Final Probability Distribution, Part I , 2016, IEEE Transactions on Automatic Control.

[13]  John Lygeros,et al.  On convexity of stochastic optimization problems with constraints , 2009, 2009 European Control Conference (ECC).

[14]  Robert Skelton,et al.  A covariance control theory , 1985, 1985 24th IEEE Conference on Decision and Control.

[15]  Tryphon T. Georgiou,et al.  Optimal Steering of a Linear Stochastic System to a Final Probability Distribution—Part III , 2014, IEEE Transactions on Automatic Control.

[16]  B. Jamison,et al.  Reciprocal Processes: The Stationary Gaussian Case , 1970 .

[17]  Bernard C. Levy,et al.  Discrete-time Gauss-Markov processes with fixed reciprocal dynamics , 1997 .

[18]  John Lygeros,et al.  Stochastic Receding Horizon Control With Bounded Control Inputs: A Vector Space Approach , 2009, IEEE Transactions on Automatic Control.

[19]  Karolos M. Grigoriadis,et al.  Minimum energy covariance controllers , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[20]  E. Bakolas.  Constrained minimum variance control for discrete-time stochastic linear systems , 2018, Syst. Control. Lett..

[21]  Efstathios Bakolas.  Optimal covariance control for stochastic linear systems subject to integral quadratic state constraints , 2016, 2016 American Control Conference (ACC).

[22]  Panagiotis Tsiotras,et al.  Finite-horizon covariance control of linear time-varying systems , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[23]  Robert E. Skelton,et al.  An improved covariance assignment theory for discrete systems , 1992 .

[24]  Basil Kouvaritakis,et al.  Model Predictive Control: Classical, Robust and Stochastic , 2015 .