Bert Pluymers Johan Suykens, Bart De Moor Department of Electrotechnical Engineering (ESAT) Research Group SCD-SISTA Katholieke Universiteit Leuven, Belgium.

Bert PluymersJohan Suykens, Bart De Moor

Department of Electrotechnical Engineering (ESAT)Research Group SCD-SISTA

Katholieke Universiteit Leuven, Belgium

Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]

Set InvarianceAn efficient tool for constrained control

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OverviewSet Invariance –

An Efficient Tool for Constrained Control

• Overview

• Motivation

• Set Invariance

• MAS for LPV systems

• Reduced Complexity Sets

• Open Research Issues

• Motivation

• Set invariance




Signal processing Identification

System Theory Automation


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Constrained control ?Set Invariance –


• Overview

• Motivation

• Set Invariance







P

Fuel gas

FeedEDC

EDC / VC / HCl

CrackingFurnace

evaporator

superheater

waste gas

T

P

L

TF

H

F

condenser

© Copyright Ipcos N.V.

• Physical constraints on inputs and outputs

• Imposed (safety, environmental, economical) constraints

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Constraint satisfactionSet Invariance –


• Overview

• Motivation

• Set Invariance







• Car = system (position, speed = system state)

• Driver = controller (gas, brake, steering wheel = inputs)

• Road = constraint

instantaneous constraint satisfaction≠

‘dynamic’ constraint satisfaction

120 km/h

10 m

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“Given an autonomous dynamical system, then a set is (positive) invariant if it is guaranteed that if the current state lies within , all future states will also lie within .”

Set Invariance –


• Overview

• Motivation

• Set Invariance







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not invariant invariant

Set Invariance

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Set Invariance

• Useful tool for analysis of controllers for constrained systems• Example :

– linear system

– linear controller – state constraints

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‘feasible region’ of closed loop system

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Set Invariance

Consider an autonomous time-invariant system as defined previously

A set is …

… feasible iff

Problem :

Given an autonomous dynamical system subject to state constraints, find the feasible invariant set of maximal size.

… invariant iff

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Given an LTI system subject to linear constraints

then the largest size feasible invariant set can be found as

with a finite integer.

Invariant sets for LTI systems (Gilbert et al.,1991, IEEE TAC)

• is constructed by simple forward prediction• can be proven to be the largest feasible invariant set• is called the Maximal Admissible Set (MAS)

Given an LTI system subject to linear constraints

then the largest size feasible invariant set can be found as

with a finite integer.

Invariant sets for LTI systems (Gilbert et al.,1991, IEEE TAC)

• is constructed by simple forward prediction• can be proven to be the largest feasible invariant set• is called the Maximal Admissible Set (MAS)

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Linear Parameter-Varying state space models with polytopic uncertainty description

LPV systems

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Set Invariance –


• Overview

• Motivation

• Set Invariance







LTI(L=1,n=2)

LPV(L>1, e.g. 2, n=2)

Straightforward extension towards LPV systems ?

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Ellipsoidal invariant sets for LPV systems (Kothare et al.,1996, Automatica)

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S X

• Constructed by solving semi-definite program (SDP)• Conservative with respect to constraints

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Reformulated invariance condition (Pluymers et al., 2005, submitted to IEEE TAC)

A set is invariant with respect to a system defined by iff

with

Sufficient condition :

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Algorithm (Pluymers et al., 2005, submitted to IEEE TAC)

• Initialize

• iteratively add constraints from to until

Advantages :

• in step 2 only ‘significant’ constraints are added to :

significant insignificant

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Set Invariance –


• Overview

• Motivation

• Set Invariance








Advantages :

• prediction tree never explicitly constructed

• given a polyhedral set , it is straightforward to calculate :

• Initialize

• iteratively add constraints from to until

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Set Invariance –


• Overview

• Motivation

• Set Invariance








1. Initialize

2. Set

3. For each check whether constraint

is significant with respect to . If significant, add the constraint to

4. Set

5. If go to step 3., otherwise exit and return

Resulting set can be proven to satisfy and is feasible due to step 1.

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Garbage collection (Pluymers et al., 2005, submitted to IEEE TAC)

• Constraints added in previous iterations can become redundant with respect to the other constraints.

• Garbage collection : removal of redundant constraints.

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iteration 1 iteration 2

iteration 3 iteration 4

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Example

Consider an LPV system with L=2 :

with feedback controller

and subject to constraints

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Example

Initialization

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Example

Iteration 10

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Example

Iteration 10 + garbage collection

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Example

Iteration 20

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Example

Iteration 20 + garbage collection

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Example

Final Result

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Example

Final Result

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Example

Final Result

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Set Invariance –


• Overview

• Motivation

• Set Invariance


• Reduced Complex. Sets





Scalability• Efficient algorithm formulation through exploitation of structure of invariant set.

• Consecutive Linear Programming →

with the number of constraints

• However : typically epx.(dimension)

dim=3 , nc = 24

dim=4 , nc = 47

dim=5 , nc = 86

dim=6, nc = 158

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Set Invariance –


• Overview

• Motivation

• Set Invariance







‘Branching’

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Constraint tightening

• In case of branch splitting :

tighten one constraint in order to make the other redundant

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Scalability revisited

dim=3, nc=17

dim=4, nc=24

dim=5, nc=37

dim=6, nc=52

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Test Case• 2-dimensional projection of a 62-dimensional invariant set for the

control of a chemical system• Number of constraints : 642

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Test Case• 2-dimensional projection of a 62-dimensional invariant set for the

control of a chemical system• Ellipsoidal invariant set significantly smaller

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Conclusion

• Invariant sets useful tools for characterization of feasible regions

• Efficient algorithm for the construction of ‘robust’ invariant sets for LPV systems

• Improved scaling behavior for high-dimensional systems

• The odds have turned against ellipsoidal invariant sets…

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Set Invariance –


• Overview

• Motivation

• Set Invariance







Open research issues

• upper / lower bounds to achievable complexity reduction

• Robustness with respect to additive disturbances

• Minimal admissable sets

• Reduced complexity control-invariant sets

• Various other types of systems : PWA, Hybrid, NL

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ThankThank you !!!you !!!

Bert Pluymers Johan Suykens, Bart De Moor Department of Electrotechnical Engineering (ESAT) Research Group SCD-SISTA Katholieke Universiteit Leuven, Belgium.

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