1. CONTROL of NONLINEAR SYSTEMS under COMMUNICATION CONSTRAINTS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign Caltech, Apr 1, 2005

2. LIMITED INFORMATION SCENARIO – partition of D – points in D, Quantizer: Control: for

3. OBSTRUCTION to STABILIZATION Asymptotic stabilization is usually lost

4. BASIC QUESTIONS What can we say about a given quantized system? How can we design the “best” quantizer for stability? What can we do with very coarse quantization? What are the difficulties for nonlinear systems?

5. STATE QUANTIZATION: LINEAR SYSTEMS Quantized control law: Closed-loop: 9 feedback gain & Lyapunov function quantization error

6. NONLINEAR SYSTEMS For nonlinear systems, GAS such robustness For linear systems, we saw that if gives then automatically gives when This is robustness to measurement errors This is input-to-state stability (ISS) for measurement errors when To have the same result, need to assume  pos.def. incr. :

7. LOCATIONAL OPTIMIZATION This leads to the problem: for Compare: mailboxes in a city, cellular base stations in a region Also true for nonlinear systems ISS w.r.t. measurement errors Small => small [Bullo-L]

8. MULTICENTER PROBLEM Critical points of satisfy 1. is the Voronoi partition : 2. This is the center of enclosing sphere of smallest radius Lloyd algorithm: Each is the Chebyshev center (solution of the 1-center problem). iterate

9. LOCATIONAL OPTIMIZATION: REFINED APPROACH only need this ratio to be small Revised problem:.............. Logarithmic quantization: Lower precision far away, higher precision close to 0 Only applicable to linear systems

10. WEIGHTED MULTICENTER PROBLEM This is the center of sphere enclosing with smallest Critical points of satisfy 1. is the Voronoi partition as before 2. Lloyd algorithm – as before Each is the weighted center (solution of the weighted 1-center problem) on not containing 0 (annulus) Gives 25% decrease in for 2-D example

11. DYNAMIC QUANTIZATION zoom in After ultimate bound is achieved, recompute partition for smaller region Can recover global asymptotic stability – zooming variable Hybrid quantized control: is discrete state Zoom out to overcome saturation zoom out

12. ACTIVE PROBING for INFORMATION PLANT QUANTIZER CONTROLLER dynamic (changes at sampling times) (time-varying) EncoderDecoder very small

13. LINEAR SYSTEMS (Baillieul, Brockett-L, Hespanha et. al., Nair-Evans, Petersen-Savkin, Tatikonda, and others)

14. LINEAR SYSTEMS sampling times Zoom out to get initial bound Example: Between sampling times, let

15. LINEAR SYSTEMS Consider is divided by 3 at the sampling time Example: Between sampling times, let grows at most by the factor in one period The norm

16. where is stable 0 LINEAR SYSTEMS (continued) Pick small enough s.t. sampling frequency vs. open-loop instability amount of static info provided by quantizer grows at most by the factor in one period is divided by 3 at each sampling time The norm

17. NONLINEAR SYSTEMS is divided by 3 at the sampling time Let Example: Between samplings grows at most by the factor in one period The norm on a suitable compact region

18. Pick small enough s.t. NONLINEAR SYSTEMS (continued) grows at most by the factor in one period is divided by 3 at each sampling time The norm What properties of guarantee GAS ?

19. ROBUSTNESS of the CONTROLLER ISS w.r.t. ISS w.r.t. measurement errors – quite restrictive... ISS w.r.t. Option 1. Option 2. [Hespanha-L] Look at the evolution of Easier to verify (e.g., GES & glob. Lip.)

20. SOME RESEARCH DIRECTIONS ISS control design ISS of impulsive systems (work with Hespanha, Teel) Performance and robustness (work with Nesic) Applications Other?