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

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

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

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 Encoder Decoder QUANTIZER finite subset of

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: is the range, is the quantization error bound For, 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 Quantized control law: where is quantization error Closed-loop system: is asymptotically stable 9 Lyapunov function

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

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 To have the same result, need to assume when

SUMMARY: PERTURBATION APPROACH 1.Design ignoring constraint 2.View as approximation 3.Prove that this 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 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

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

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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 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

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RESEARCH DIRECTIONS Robust control design Locational optimization Performance Applications