1. CONTROL of NONLINEAR SYSTEMS with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

2. 0 Control objectives: stabilize to 0 or to a desired set containing 0, exit D through a specified facet, etc. CONSTRAINED CONTROL Constraint: – given control commands

3. LIMITED INFORMATION SCENARIO – partition of D – points in D, Quantizer/encoder: Control: for

4. MOTIVATION Limited communication capacity many systems/tasks share network cable or wireless medium microsystems with many sensors/actuators on one chip Need to minimize information transmission (security) Event-driven actuators PWM amplifier manual car transmission stepping motor EncoderDecoder QUANTIZER finite subset of

5. QUANTIZER GEOMETRY is partitioned into quantization regions uniform logarithmic arbitrary Dynamics change at boundaries => hybrid closed-loop system Chattering on the boundaries is possible (sliding mode)

6. QUANTIZATION ERROR and RANGE is the range, is the quantization error bound For, the quantizer saturates Assume such that: 1. 2.

7. OBSTRUCTION to STABILIZATION Assume: fixed Asymptotic stabilization is usually lost

8. 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?

9. 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?

10. STATE QUANTIZATION: LINEAR SYSTEMS Quantized control law: where is quantization error Closed-loop system: is asymptotically stable 9 Lyapunov function

11. LINEAR SYSTEMS (continued) Recall: Previous slide: Lemma: solutions that start in enter in finite time Combine:

12. 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. :

13. SUMMARY: PERTURBATION APPROACH 1.Design ignoring constraint 2.View as approximation 3.Prove that this still solves the problem (in a weaker sense) Issue: error Need to give ISS w.r.t. measurement errors

14. INPUT QUANTIZATION where Control law: Closed-loop system: Analysis – same as before Control law: where Need ISS with respect to actuator errors Closed-loop system:

15. 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?

16. LOCATIONAL OPTIMIZATION: NAIVE APPROACH This leads to the problem: for Also true for nonlinear systems ISS w.r.t. measurement errors Smaller => smaller Compare: mailboxes in a city, cellular base stations in a region

17. 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

18. 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

19. 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

20. DYNAMIC QUANTIZATION zoom in After ultimate bound is achieved, recompute partition for smaller region Zoom out to overcome saturation Can recover global asymptotic stability (also applies to input and output quantization) – zooming variable Hybrid quantized control: is discrete state zoom out

21. 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?

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

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

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

25. 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

26. 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

27. NONLINEAR SYSTEMS sampling times Example: Zoom out to get initial bound Between samplings

28. 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

29. 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 ?

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

31. RESEARCH DIRECTIONS ISS control design Locational optimization Performance and robustness Applications

32. REFERENCES Brockett & L, 2000 (IEEE TAC) Bullo & L, 2003, L & Hespanha, 2004 (http://decision.csl.uiuc.edu/~liberzon)