CONTROL with LIMITED INFORMATION

Daniel Liberzon

Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign

REASONS for SWITCHING

• Nature of the control problem

• Sensor or actuator limitations

• Large modeling uncertainty

• Combinations of the above

Plant

Controller

INFORMATION FLOW in CONTROL SYSTEMS

INFORMATION FLOW in CONTROL SYSTEMS

• Limited communication capacity • many control loops share network cable or wireless medium• microsystems with many sensors/actuators on one chip

• Need to minimize information transmission (security)

• Event-driven actuators

• Coarse sensing

• Theoretical interest

[Brockett, Delchamps, Elia, Mitter, Nair, Savkin, Tatikonda, Wong,…]

• Deterministic & stochastic models

• Tools from information theory• Mostly for linear plant dynamics

BACKGROUND

Previous work:

• Unified framework for• quantization

• time delays

• disturbances

Our goals:• Handle nonlinear dynamics

Caveat:

This doesn’t work in general, need robustness from controller

OUR APPROACH

(Goal: treat nonlinear systems; handle quantization, delays, etc.)

• Model these effects via deterministic error signals,

• Design a control law ignoring these errors,

• “Certainty equivalence”: apply control,

combined with estimation to reduce to zero

Technical tools:

• Input-to-state stability

(ISS)

• Lyapunov functions

• Small-gain theorems

• Hybrid systems

QUANTIZATION

Encoder Decoder

QUANTIZER

finite subset

of

is partitioned into quantization regions

QUANTIZATION and ISS

– assume glob. asymp. stable (GAS)

QUANTIZATION and ISS

QUANTIZATION and ISS

no longer GAS

QUANTIZATION and ISS

quantization error

Assume

class

Solutions that start in

enter and remain there

This is input-to-state stability (ISS) w.r.t. measurement errors

In time domain:

QUANTIZATION and ISS

quantization error

Assume

LINEAR SYSTEMS

Quantized control law:

9 feedback gain & Lyapunov function

Closed-loop:

(automatically ISS w.r.t. )

DYNAMIC QUANTIZATION

DYNAMIC QUANTIZATION

– zooming variable

Hybrid quantized control: is discrete state

DYNAMIC QUANTIZATION

– zooming variable

Hybrid quantized control: is discrete state

Zoom out to overcome saturation

DYNAMIC QUANTIZATION

– zooming variable

Hybrid quantized control: is discrete state

After ultimate bound is achieved,recompute partition for smaller region

DYNAMIC QUANTIZATION

– zooming variable

Hybrid quantized control: is discrete state

ISS from to ISS from to small-gain conditionProof:

Can recover global asymptotic stability

QUANTIZATION and DELAY

Architecture-independent approach

Based on the work of Teel

Delays possibly large

QUANTIZER DELAY

QUANTIZATION and DELAY

where

Can write

hence

SMALL – GAIN ARGUMENT

if

then we recover ISS w.r.t. [Teel ’98]

Small gain:

Assuming ISS w.r.t. actuator errors:

In time domain:

FINAL RESULT

Need:

small gain true

FINAL RESULT

Need:

small gain true

FINAL RESULT

solutions starting in

enter and remain there

Can use “zooming” to improve convergence

Need:

small gain true

EXTERNAL DISTURBANCES [Nešić–L]

State quantization and completely unknown disturbance

EXTERNAL DISTURBANCES [Nešić–L]

State quantization and completely unknown disturbance

Issue: disturbance forces the state outside quantizer range

Must switch repeatedly between zooming-in and zooming-out

Result: for linear plant, can achieve ISS w.r.t. disturbance

(ISS gains are nonlinear although plant is linear; cf. [Martins])

EXTERNAL DISTURBANCES [Nešić–L]

State quantization and completely unknown disturbance

After zoom-in:

NETWORKED CONTROL SYSTEMS [Nešić–L]

NCS: Transmit only some variables according to time scheduling protocol

Examples: round-robin, TOD (try-once-discard)

QCS: Transmit quantized versions of all variables

NQCS: Unified framework combining time scheduling and quantization

Basic design/analysis steps:

• Design controller ignoring network effects

• Apply small-gain theorem to compute upper bound on maximal allowed transmission interval (MATI)

• Prove discrete protocol stability via Lyapunov function

ACTIVE PROBING for INFORMATION

dynamic

dynamic

(changes at sampling times)

(time-varying)

PLANT

QUANTIZER

CONTROLLER

Encoder Decoder

very small

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

• grows at most by the factor in one period

The norm

on a suitable compact region(dependent on )

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

If this is ISS w.r.t. as before, then

ROBUSTNESS of the CONTROLLER

ISS w.r.t.

Same condition as before (restrictive, hard to check)

checkable sufficient conditions ([Hespanha-L-Teel])

ISS w.r.t.

Option 1.

Option 2. Look at the evolution of

LINEAR SYSTEMS

LINEAR SYSTEMS

[Baillieul, Brockett-L, Hespanha, Nair-Evans, Petersen-Savkin,Tatikonda]

Between sampling times,

where is Hurwitz0

• divided by 3 at each sampling time

• grows at most by in one period

amount of static infoprovided by quantizer

sampling frequency vs. open-loop instability

global quantity:

HYBRID SYSTEMS as FEEDBACK CONNECTIONS

continuous

discrete

[L–Nešić, ’05, ’06, L–Nešić–Teel ’14]

• Other decompositions possible

• Can also have external signals

SMALL – GAIN THEOREM

Small-gain theorem [Jiang-Teel-Praly ’94] gives GAS if:

• Input-to-state stability (ISS) from to :

• ISS from to :

• (small-gain condition)

SUFFICIENT CONDITIONS for ISS

[Hespanha-L-Teel ’08]

# of discrete events on is

• ISS from to if:

and

• ISS from to if ISS-Lyapunov function [Sontag ’89]:

LYAPUNOV – BASED SMALL – GAIN THEOREM

Hybrid system is GAS if:

•

and # of discrete events on is

•

•

SKETCH of PROOF

is nonstrictly decreasing along trajectories

Trajectories along which is constant? None!

GAS follows by LaSalle principle for hybrid systems [Lygeros et al. ’03, Sanfelice-Goebel-Teel ’07]

quantization error

Zoom in:

where

ISS from to with gain

small-gain condition!

ISS from to with some linear gain

APPLICATION to DYNAMIC QUANTIZATION

OTHER RESEARCH DIRECTIONS

• Modeling uncertainty (with L. Vu)

• Disturbances and coarse quantizers (with Y. Sharon)

• Multi-agent coordination (with S. LaValle and J. Yu)

• Quantized output feedback and observers (with H. Shim)

• Vision-based control (with Y. Ma and Y. Sharon)

• Performance-based design (with F. Bullo)

• Quantized control of switched systems