QUANTIZED CONTROL and GEOMETRIC OPTIMIZATION Francesco Bullo and

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

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

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

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 finite subset of Encoder Decoder QUANTIZER

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)

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

OBSTRUCTION to STABILIZATION Assume: fixed Asymptotic stabilization is usually lost

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

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

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

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

NONLINEAR SYSTEMS For linear systems, we saw that if gives then automatically gives when This is robustness to measurement errors 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

SUMMARY: PERTURBATION APPROACH 1. Design 2. View 3. Prove that this ignoring constraint as approximation still solves the problem Issue: error Need to be ISS w. r. t. measurement errors

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

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

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

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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 . . .

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

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