bert pluymers johan suykens, bart de moor department of electrotechnical engineering (esat) research...
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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
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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Constrained control ?Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
• 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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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not invariant invariant
Set Invariance
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Linear Parameter-Varying state space models with polytopic uncertainty description
LPV systems
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
LTI(L=1,n=2)
LPV(L>1, e.g. 2, n=2)
Straightforward extension towards LPV systems ?
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Algorithm (Pluymers et al., 2005, submitted to IEEE TAC)
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Algorithm (Pluymers et al., 2005, submitted to IEEE TAC)
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Example
Consider an LPV system with L=2 :
with feedback controller
and subject to constraints
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Example
Initialization
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Example
Iteration 10
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Example
Iteration 10 + garbage collection
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Example
Iteration 20
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Example
Iteration 20 + garbage collection
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Example
Final Result
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Example
Final Result
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complexity Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Example
Final Result
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complex. Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complex. Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
‘Branching’
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complex. Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Constraint tightening
• In case of branch splitting :
tighten one constraint in order to make the other redundant
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complex. Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
Scalability revisited
dim=3, nc=17
dim=4, nc=24
dim=5, nc=37
dim=6, nc=52
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Set Invariance –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complex. Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complex. Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complex. Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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 –
An Efficient Tool for Constrained Control
• Overview
• Motivation
• Set Invariance
• MAS for LPV systems
• Reduced Complex. Sets
• Open Research Issues
Signal processing Identification
System Theory Automation
Sde Boker workshop on Linear Systems Theory, 13 Sept. 2005, Ben Gurion Univ., [email protected]
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 !!!