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QUANTIZED CONTROL and GEOMETRIC OPTIMIZATION

This research explores the intricacies of quantized control systems and geometric optimization techniques to enhance stability under constrained conditions. It addresses critical challenges in designing optimal quantizers for systems with limited communication capacity, where multiple tasks share resources. By investigating closed-loop linear and nonlinear systems, the study presents robust control strategies and solutions to minimize quantization error. Fundamental questions regarding system behavior and optimal quantization design are thoroughly examined, paving the way for advancements in robust control methodologies.

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QUANTIZED CONTROL and GEOMETRIC OPTIMIZATION

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  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

  17. Play movie: step3-animation.fli

  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

  20. Play movie: step5-animation.fli

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

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