QUANTIZED SYSTEMS AND CONTROL Daniel Liberzon Coordinated - - PowerPoint PPT Presentation

QUANTIZED SYSTEMS AND CONTROL Daniel Liberzon

Coordinated Science Laboratory and

• Dept. of Electrical & Computer Eng.,
• Univ. of Illinois at Urbana-Champaign

DISC HS, June 2003

HYBRID CONTROL

Classical continuous feedback paradigm:

u y

P C

u y

P Plant: But logical decisions are often necessary: The closed-loop system is hybrid

u y C1 C2 l o g i c

P

REASONS for SWITCHING

• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above

REASONS for SWITCHING

• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above

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 sharing 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

• f

QUANTIZED CONTROL ARCHITECTURES

PLANT QUANTIZER CONTROLLER STATE PLANT QUANTIZER CONTROLLER OUTPUT PLANT QUANTIZER CONTROLLER INPUT PLANT QUANTIZER CONTROLLER INPUT QUANTIZER OUTPUT

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

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

EXAMPLES of QUANTIZERS

• Temperature sensor

normal too low too high

• Camera with zoom

Tracking a golf ball

• Coding and decoding
• A/D conversion

OBSTRUCTION to STABILIZATION

Assume: fixed

∆ , M

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?
• What can we do with very coarse quantization?
• What are the difficulties for nonlinear systems?

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?
• What are the difficulties for nonlinear systems?
• What are the difficulties for nonlinear systems?

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: Combine: Lemma: solutions that start in enter in finite time

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

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:

OUTPUT QUANTIZATION

Control law: Closed-loop system: Analysis – same as before (need a bound on initial state) Can also treat input and state/output quantization together

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?
• What are the difficulties for nonlinear systems?
• What are the difficulties for nonlinear systems?

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).

LOCATIONAL OPTIMIZATION: REFINED APPROACH

• nly 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)

• n

not containing 0 (annulus) Gives 25% decrease in for 2-D example

DYNAMIC QUANTIZATION: IDEA

zoom out zoom in

After ultimate bound is achieved, recompute partition for smaller region Zoom out to

• vercome saturation

Temperature sensor – can adjust threshold settings Digital camera – can zoom in and out Encoder – can change the coding mechanism Can recover global asymptotic stability (also applies to input and output quantization)

DYNAMIC QUANTIZATION: DETAILS

– zooming variable Hybrid quantized control: is discrete state

(More realistic, easier to design and analyze, robust to time delays)

We know: solutions starting in enter in finite time after units of time dwell time Increase fast enough until unknown

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?
• What are the difficulties for nonlinear systems?
• What are the difficulties for nonlinear systems?

ACTIVE PROBING for INFORMATION

PLANT QUANTIZER CONTROLLER

dynamic dynamic (changes at sampling times) (time-varying) Encoder Decoder very small

LINEAR SYSTEMS

sampling times

Zoom out to get initial bound Example: Between sampling times, let

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

where is Hurwitz

LINEAR SYSTEMS (continued)

Pick small enough s.t.

sampling frequency vs.

• pen-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

NONLINEAR SYSTEMS

sampling times

Example: Zoom out to get initial bound Between samplings

NONLINEAR SYSTEMS

• is divided by 3 at the sampling time

Let Example: Between samplings

where is Lipschitz constant of

• grows at most by the factor in one period

The norm

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 Need ISS w.r.t. measurement errors!

RESEARCH DIRECTIONS

• Robust control design
• Locational optimization
• Performance
• Applications

REFERENCES

Brockett & L, 2000 (IEEE TAC) Bullo & L, 2003 (submitted)