Optimal Covariance Control for Stochastic Systems Under Chance Constraints

This letter addresses the optimal covariance control problem for stochastic discrete-time linear systems subject to chance constraints. To the best of our knowledge, covariance steering problems with probabilistic chance constraints have not been discussed previously in the literature, although their treatment seems to be a natural extension. In this letter, we first show that, unlike the case with no chance constraints, the covariance steering problem with chance constraints cannot be decoupled to mean and covariance steering sub-problems. We then propose an approach to solve the covariance steering problem with chance constraints by converting it to a convex programming problem. The proposed algorithm is verified using a numerical example.

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

[2]  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.

[3]  L. Blackmore,et al.  Convex Chance Constrained Predictive Control without Sampling , 2009 .

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

[5]  Marcello Farina,et al.  A probabilistic approach to Model Predictive Control , 2013, 52nd IEEE Conference on Decision and Control.

[6]  T. Iwasaki,et al.  Quadratic Optimization for Fixed Order Linear Controllers via Covariance Control , 1992, 1992 American Control Conference.

[7]  Karolos M. Grigoriadis,et al.  Minimum-energy covariance controllers , 1997, Autom..

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

[9]  Masahiro Ono,et al.  Chance-Constrained Optimal Path Planning With Obstacles , 2011, IEEE Transactions on Robotics.

[10]  Efstathios Bakolas,et al.  Finite-horizon covariance control for discrete-time stochastic linear systems subject to input constraints , 2018, Autom..

[11]  Emmanuel G. Collins,et al.  A theory of state covariance assignment for discrete systems , 1987 .

[12]  Yongxin Chen.  Modeling and control of collective dynamics: From Schrodinger bridges to Optimal Mass Transport , 2016 .

[13]  D. Bernstein.  Matrix Mathematics: Theory, Facts, and Formulas , 2009 .

[14]  Abhishek Halder,et al.  Finite horizon linear quadratic Gaussian density regulator with Wasserstein terminal cost , 2016, 2016 American Control Conference (ACC).

[15]  Aaron Becker,et al.  Algorithms For Shaping a Particle Swarm With a Shared Control Input Using Boundary Interaction , 2016, ArXiv.

[16]  András Prékopa,et al.  Boole-Bonferroni Inequalities and Linear Programming , 1988, Oper. Res..

[17]  Panagiotis Tsiotras,et al.  Uncertainty Quantication and Control During Mars Powered Descent and Landing using Covariance Steering , 2018 .

[18]  G. Prodi,et al.  Feedback cooling of the normal modes of a massive electromechanical system to submillikelvin temperature. , 2008, Physical review letters.

[19]  F. Borrelli,et al.  Stochastic Predictive Control of Autonomous Vehicles in Uncertain Environments , 2014 .

[20]  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.

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

[22]  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.

[23]  Pu Li,et al.  Advances and applications of chance-constrained approaches to systems optimisation under uncertainty , 2013, Int. J. Syst. Sci..

[24]  M Ono,et al.  Chance constrained finite horizon optimal control with nonconvex constraints , 2010, Proceedings of the 2010 American Control Conference.

[25]  Marcello Farina,et al.  Stochastic linear Model Predictive Control with chance constraints – A review , 2016 .

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

[27]  A. Mesbah,et al.  Stochastic Model Predictive Control: An Overview and Perspectives for Future Research , 2016, IEEE Control Systems.