Robust Control of a Class of Feedback Systems Subject to Limited Capacity Constraints

1. Robust Control of a Class of Feedback Systems Subject to Limited Capacity Constraints Alireza Farhadi School of Information Technology and Engineering, University of Ottawa, Ottawa, Canada C. D. Charalambous Electrical and Computer Engineering Department University of Cyprus, Nicosia, Cyprus

2. Overview • Uncertain Control System • Reliable Communication for Uncertain Sources Described via a Relative Entropy Constraint • Uncertain Fully Observed Controlled System

3. Problem Formulation

4. Problem Formulation • Control/communication system subject to the uncertainty in the source

5. Problem Formulation • Information Source: The information source is described by the probability measure which depends on the control sequence as shown in the Figure. It is assumed that the density function belongs to the class

6. Problem Formulation • Example: Uncertain class of fully observed Gauss Markov systems where is i.i.d. is the perturbed noise random process and is the signal to be controlled. [Pra-Meneghini-Runggaldier 96, Ugrinovskii-Petersen 99]

7. Problem Formulation • Nominal system: • It can be shown that for the sequence

8. Problem Formulation • The objective is to design encoding/decoder and stabilizing controller which guarantee uniform mean square reconstruction and stability. • Uniform mean square reconstruction: • For :

9. Problem Formulation • Shannon: Consider a communication system without feedback. A necessary condition for mean uniform reconstruction of is given by • Above result can be extended to channels and sources which use feedback from the output of the decoder to the input of the encoder ( the control/communication system), provided 1) The capacity with and without feedback are the same. 2) The rate distortion of a source without using feedback from the decoder to the encoder is the same as the one that uses such feedback, and the reconstruction kernel is causal. (generalizations [Charalambous 2006]

10. Uncertain Source Described Via Relative Entropy Constraint

11. Rate Distortion for a Class of Sources • Information Source: Consider a class of sources which produce orthogonal zero mean output process described via the following relative entropy constraint where the density function is Gaussian. That is, • The rate distortion for this class is defined by the following minimax problem where

12. Rate Distortion for a Class of Sources • It can be shown that above minimax problem is equivalent to the following maximin problem • Consider the case of (the vector case was treated in the paper). • Under assumption of , we have where is unique solution to the following equation

13. Computation of Robust Shannon Lower Bound • It can be shown that when and , the robust Shannon lower bound is given by • Thus, for , the robust Shannon lower bound is an exact approximation of . That is,

14. Realization of a Communication Link Matched to the Uncertain Source • Next, consider the following AWGN channel. • Under assumption of and , it can be shown that if the encoder multiplies by ( ) and transmits it under transmission power constraint , where is the unique solution of the following equation , we have • On the other hand, if the decoder multiplies the channel outputs by , we have an end to end transmission with distortion

15. Uncertain Fully Observed Controlled Gauss Markov System

16. Uncertain and Nominal System • Consider the following uncertain system • The corresponding nominal system is • The uncertain system is subject to the following sum quadratic uncertainty constraint.

17. Computing the Robust Entropy • For and , consider the following robust entropy problem • Suppose , then above problem is equivalent to the following robust entropy problem

18. Computing the Robust Entropy • Theorem: Consider the robust entropy problem. Let for some . Then, i) where is a real symmetric solution of and is the minimizing solution of the following equation

19. Computing the Robust Entropy • Theorem (Continued): The robust entropy rate is given by where is the solution of the following Algebraic Riccati equation appearing in the estimation and control problems

20. Optimal Controller Design • Let The objective is to design an encoder, decoder and controller for mean square stability subject to the following cost functional. where

21. Optimal Controller Design • Below figure illustrates encoding and stabilizing schemes for uniform observability and robust stability. The encoder, decoder and controller will be an extension of the results of previous section and [1].

22. Conclusion • Our motivation for considering the relative entropy uncertainty description is that it gives as a special case a constraint on the energy of the uncertainty. Such uncertainty description has been considered in [2]. • Under certain conditions the robust Shannon lower bound is a tight bound for uniform observability. • For future work • Separation Theorem • Uncertain Channels and Sources

23. References [1] I. R. Petersen, M. R. James and P. Dupuis, Minimax Optimal Control of Stochastic Uncertain Systems with Relative Entropy Constraints, IEEE Transactions on Automatic Control, vol. 45, No. 3, pp. 398-412, March 2000. [2] C. D. Charalambous and Alireza Farhadi, Stochastic Control of General Discrete Time Partially Observed Systems over Finite Capacity Communication Channels: LQG Optimality and Separation Principle, submitted to Automatica, July 2007, under review.

24. References for Control/Communication Tatikonda, Mitter, Sahai, Elia, Nuno, Dahleh, Yuksel, Basar, Girish, Dey, Evans, Liberzon, …