QUANTIZED CONTROL and GEOMETRIC OPTIMIZATION

1. QUANTIZED CONTROL andGEOMETRIC OPTIMIZATION Francesco Bullo and Daniel Liberzon Coordinated Science Laboratory Univ. of Illinois at Urbana-Champaign U.S.A. CDC 2003

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

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

4. MOTIVATION finite subset of Encoder Decoder QUANTIZER • 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

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

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

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

8. BASIC QUESTIONS • What can we say about a given quantized system? • How can we design the “best” quantizer for stability?

9. BASIC QUESTIONS • What can we say about a given quantized system? • How can we design the “best” quantizer for stability?

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

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

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

13. Design ignoring constraint • View as approximation • Prove that this still solves the problem Issue: error Need to be ISS w.r.t. measurement errors SUMMARY: PERTURBATION APPROACH

14. BASIC QUESTIONS • What can we say about a given quantized system? • How can we design the “best” quantizer for stability?

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

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

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

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21. Robust control design • Locational optimization • Performance • Applications RESEARCH DIRECTIONS