Bert Pluymers Johan 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., Set Invariance An efficient tool for constrained control

1 Overview Set 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.,

2 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., © Copyright Ipcos N.V. Physical constraints on inputs and outputs Imposed (safety, environmental, economical) constraints

3 Constraint satisfaction 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., 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

4 “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., not invariant invariant Set Invariance

5 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., Set Invariance Useful tool for analysis of controllers for constrained systems Example : – linear system – linear controller – state constraints ‘feasible region’ of closed loop system

6 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., 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

7 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., 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)

8 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., Linear Parameter-Varying state space models with polytopic uncertainty description LPV systems

9 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., LTI (L=1,n=2) LPV (L>1, e.g. 2, n=2) Straightforward extension towards LPV systems ?

10 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., Ellipsoidal invariant sets for LPV systems (Kothare et al.,1996, Automatica) Constructed by solving semi-definite program (SDP) Conservative with respect to constraints

11 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., 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 :

12 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., 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

13 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., 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

14 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., 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.

15 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., 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. iteration 1 iteration 2 iteration 3iteration 4

16 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., Example Consider an LPV system with L=2 : with feedback controller and subject to constraints

17 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., Example Initialization

18 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., Example Iteration 10

19 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., Example Iteration 10 + garbage collection

20 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., Example Iteration 20

21 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., Example Iteration 20 + garbage collection

22 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., Example Final Result

23 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., Example Final Result

24 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., Example Final Result

25 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., 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, n c = 24 dim=4, n c = 47 dim=5, n c = 86dim=6, n c = 158

26 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., ‘Branching’

27 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., Constraint tightening In case of branch splitting : tighten one constraint in order to make the other redundant

28 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., Scalability revisited dim=3, n c =17 dim=4, n c =24 dim=5, n c =37 dim=6, n c =52

29 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., Test Case 2-dimensional projection of a 62-dimensional invariant set for the control of a chemical system Number of constraints : 642

30 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., Test Case 2-dimensional projection of a 62-dimensional invariant set for the control of a chemical system Ellipsoidal invariant set significantly smaller

31 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., 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…

32 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., 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

33 Thank you !!! you !!!