Finite-horizon covariance control of linear time-varying systems

We consider the problem of finite-horizon optimal control of a discrete linear time-varying system subject to a stochastic disturbance and fully observable state. The initial state of the system is drawn from a known Gaussian distribution, and the final state distribution is required to reach a given target Gaussian distribution, while minimizing the expected value of the control effort. We derive the linear optimal control policy by first presenting an efficient solution for the diffusion-less case, and we then solve the case with diffusion by reformulating the system as a superposition of diffusion-less systems. We show that the resulting solution coincides with a LQG problem with particular terminal cost weight matrix.

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

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

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

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

[5]  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.

[6]  Stephen S.-T. Yau,et al.  Optimal control of the Liouville equation , 2007 .

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

[8]  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.

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

[10]  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.

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

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

[13]  A. Beghi,et al.  On the relative entropy of discrete-time Markov processes with given end-point densities , 1996, IEEE Trans. Inf. Theory.

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

[15]  H. Kushner.  Introduction to stochastic control , 1971 .

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

[17]  D. Sworder,et al.  Introduction to stochastic control , 1972 .

[18]  John Bagterp Jørgensen,et al.  Efficient implementation of the Riccati recursion for solving linear-quadratic control problems , 2013, 2013 IEEE International Conference on Control Applications (CCA).

[19]  G. G. Hamedani,et al.  Certain Characterizations of Normal Distribution via Transformations , 2001 .