mpc invariant set
DESCRIPTION
Robust model predictive control of constrained linear systems with bounded disturbances using invariant setTRANSCRIPT
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
Set Invariance
D. Limon A. Ferramosca E.F. Camacho
Department of Automatic Control & Systems EngineeringUniversity of Seville
HYCON-EECI Graduate School on Control
Limon, Ferramosca, Camacho Set Invariance 1
IntroductionSet invariance theory
i-steps setsRobust invariant sets
Outline1 Introduction
Some definitions2 Set invariance theory
One step setThe reach set
3 i-steps setsi-steps controllable setMaximal Control invariant setMaximal stabilisable setExampleSome comments
4 Robust invariant sets
Limon, Ferramosca, Camacho Set Invariance 2
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
Some definitions
Set invariance
Set invariance is a fundamental concept in design ofcontroller for constrained systems.The reason: constraint satisfaction can be guaranteed forall time (and for all disturbances) if and only if the initialstate is contained inside a (robust) control invariant set.The evolution of a constrained system is admissible if thereexists an invariant set X , where X is the set whereconstraints on the state are fullfilled. Hence, if x0 X ,then xk X , for all k .
Limon, Ferramosca, Camacho Set Invariance 3
IntroductionSet invariance theory
i-steps setsRobust invariant sets
Some definitions
Set invariance
Set invariance is strictly connected with stability.Lyapunov theory states that, if there exists a Lyapunovfunction V (x) such that:
V (x) 0
then, for all x X , any set defined as:
= {x Rn : V (x) } X
is an invariant set for the system, and hence for any initialstate x0 , the system fulfills the constraints and remainsinside .
Limon, Ferramosca, Camacho Set Invariance 4
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
Some definitions
Positive invariant setConsider an autonomous system:
xk+1 = f (xk ), xk X
Definition (Positive invariant set) Rn is a positive invariant set if x0 X , xk , for allk 0.
If the system reaches a positive invariant set, its futureevolution remains inside this set.The maximum invariant set, max X , is the smallest positiveinvariant set that contains all the positive invariant setscontained in X .
Limon, Ferramosca, Camacho Set Invariance 5
IntroductionSet invariance theory
i-steps setsRobust invariant sets
Some definitions
Positive control invariant set
Consider the system:
xk+1 = f (xk , uk), xk X , uk U
xk Rn and uk Rm.
Definition (Positive control invariant set) Rn is a positive control invariant set if x0 X , thereexists a control law uk = h(xk ) such that xk , for all k 0,and uk = h(xk ) U.
Limon, Ferramosca, Camacho Set Invariance 6
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
One step setThe reach set
One step set
Consider the system:
xk+1 = f (xk , uk), xk X , uk U
xk Rn and uk Rm.
Let f (0, 0) = 0 be en equilibrium point.
Definition (One step set)The set Q() is the set of states in Rn for which an admissiblecontrol inputs exists which will guarantee that the system will bedriven to in one step:
Q() = {xk Rn|uk U : f (xk , uk ) }
Limon, Ferramosca, Camacho Set Invariance 7
IntroductionSet invariance theory
i-steps setsRobust invariant sets
One step setThe reach set
One step set
The previous definition is the same for a system controlled by acontrol law u = h(x):
Qh() = {xk Rn : f (xk , h(xk )) }
Property:Monotonicity: consider sets 1 2, then:
Q(1) Q(2)
Limon, Ferramosca, Camacho Set Invariance 8
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Our HomeSticky Note h k
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
One step setThe reach set
Geometric invariance condition
The one step set definition allow us to define a condition forguaranteing the invariance of a set.
Geometric invariance condition: set is a control invariantset if and only if
Q()
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Limon, Ferramosca, Camacho Set Invariance 9
IntroductionSet invariance theory
i-steps setsRobust invariant sets
One step setThe reach set
The reach set
Definition (The reach set)The set R() is the set of states in Rn to which the system willevolve at the next time step given any xk and admissiblecontrol input:
R() = {z Rn : xk ,uk Us.t .z = f (xk , uk )}
For closed-loop systems, Rh() is the set of states in Rn towhich the system will evolve at the next time step given anyxk :
Rh() = {z Rn : xk , s.t .z = f (xk , h(xk ))}
Limon, Ferramosca, Camacho Set Invariance 10
Our HomeSticky Note Q Q Q Q
Our HomeSticky Note Q
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
i-steps controllable setMaximal Control invariant setMaximal stabilisable setExampleSome comments
i-steps controllable set
DefinitionThe i-steps controllable set Ki(X ,) is the set of states forwhich exists an admissible control sequence such that thesystem reaches the set X in exactly i steps, with anadmissible evolution.
Ki(X ,) = {x0 X : k = 0, ..., i 1, uk Us.t.xk Xandxi }
This set represents the set of all states that can reach a givenset in i steps, with an admissible evolution and an admissiblecontrol sequence.
Limon, Ferramosca, Camacho Set Invariance 11
IntroductionSet invariance theory
i-steps setsRobust invariant sets
i-steps controllable setMaximal Control invariant setMaximal stabilisable setExampleSome comments
Properties
Ki+1(X ,) = Q(Ki(X ,)) X , with K0(X ,) = .Ki(X ,) Ki+1(X ,) iff is invariant.The set K(X ,) is finitely determined if and only if i Nsuch that K(X ,) = Ki(X ,). The smallest elementi N such that K(X ,) = Ki(X ,) is called thedeterminedness index.If j N such that Ki+1(X ,) = Ki(X ,), i j , thenK(X ,) is finitely determined.
For closed-loop systems:K hi (X ,) = {x0 Xh : xk Xhk = 0, ..., i 1, andxi }
where X h = {x X : h(x) U}.Limon, Ferramosca, Camacho Set Invariance 12
Our HomeSticky Note i (invarinant set) . .
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Our HomeSticky Notek i
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
i-steps controllable setMaximal Control invariant setMaximal stabilisable setExampleSome comments
Maximal Control invariant set
DefinitionThe set C(X ) is the maximal control invariant set contained inX for system xk+1 = f (xk , uk ) if and only if C(X ) is a controlinvariant set and contains all the invariant sets contained inX .
C(X ) X
This set is derived from the definition of the i-steps admissibleset, Ci(X ), that is the set of states for which exists anadmissible control sequence such that the evolution of thesystem remains in X during the next i steps.
Ci(X ) = {x0 X : k = 0, ..., i 1,uk Us.t .xk+1 X}
Limon, Ferramosca, Camacho Set Invariance 13
IntroductionSet invariance theory
i-steps setsRobust invariant sets
i-steps controllable setMaximal Control invariant setMaximal stabilisable setExampleSome comments
Properties
Ci(X ) = Ki(X , X ).Ci+1(X ) Ci(X ).If x0 X Ci(X ), there not exists an admissible control lawwhich will ensure that the evolution of the system isadmissible for i steps.C(X ) is the set of all states for which there exists anadmissible control law which ensures the fulfillment of theconstraints for all time.C(X ) is finitely determined if and only if there exists anelement i N, such that Ci+1(X ) = Ci(X ), i i.Hence, Ci(X ) = C(X ).Ki(X ,) C(X ), i and X .
Limon, Ferramosca, Camacho Set Invariance 14
Our HomeSticky Note invarianet set
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Our HomeSticky Note ci(x) i
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
i-steps controllable setMaximal Control invariant setMaximal stabilisable setExampleSome comments
Maximal stabilisable set
DefinitionThe set S(X ,) is the maximal stabilisable invariant setcontained in X for system xk+1 = f (xk , uk ) if and only ifS(X ,) is the union of all i-step stabilisable sets contained inX .
This set is derived from the definition of the i-steps stabilisableset, Si(X ,), that is the set of states for which exists anadmissible control sequence that drive the system to theinvariant set in i steps with an admissible evolution.
Si(X ,) = {x0 X : k = 0, ..., i1,uk Us.t .xk Xandxi }
The only difference between Si(X ,) and Ki(X ,) is theinvariance condition for .
Limon, Ferramosca, Camacho Set Invariance 15
IntroductionSet invariance theory
i-steps setsRobust invariant sets
i-steps controllable setMaximal Control invariant setMaximal stabilisable setExampleSome comments
Properties
Si+1(X ,) = Q(Si(X ,)) X , with S0(X ,) = .Si(X ,) Si+1(X ,).Any Si(X ,) is a control invariant set.Consider 1 and 2 invariant sets, such that 1 2.Then Si(X ,1) Si(X ,2).Si(X , Sj(X ,)) = Si+j(X ,).S(X ,) is finitely determined if and only if there exists anelement i N such that Si+1(X ,) = Si(X ,), for anyi i. Furthermore, S(X ,) = Si(X ,), for any i i.
For closed-loop systems:Shi (X ,) = {x0 Xh : xk Xhk = 0, ..., i 1, andxi }
where X h = {x X : h(x) U}.Limon, Ferramosca, Camacho Set Invariance 16
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
i-steps controllable setMaximal Control invariant setMaximal stabilisable setExampleSome comments
Examplei-steps controllable and stabilizable sets
not Invariant
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K1
Controllable setK1(X , ) = Q() X
Invariant
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S1
Stabilisable setS1(X , ) = Q() X
Limon, Ferramosca, Camacho Set Invariance 17
IntroductionSet invariance theory
i-steps setsRobust invariant sets
i-steps controllable setMaximal Control invariant setMaximal stabilisable setExampleSome comments
Examplei-steps controllable and stabilizable sets
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Controllable setK2(X , ) = Q(K1) X
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Stabilisable setS2(X , ) = Q(S1) X
Limon, Ferramosca, Camacho Set Invariance 18
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
i-steps controllable setMaximal Control invariant setMaximal stabilisable setExampleSome comments
Examplei-steps controllable and stabilizable sets
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Controllable setK3(X , ) = Q(K2) X
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S3 S2 S1
Stabilisable setS3(X , ) = Q(S2) X
Limon, Ferramosca, Camacho Set Invariance 19
IntroductionSet invariance theory
i-steps setsRobust invariant sets
i-steps controllable setMaximal Control invariant setMaximal stabilisable setExampleSome comments
Examplei-steps controllable and stabilizable sets
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K1 K2 K3 K4
Controllable setKi+1(X , ) = Q(Ki ) XKi (X , ) Ki+1 X
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S4 S3 S2 S1
Stabilisable setSi+1(X , ) = Q(Si ) XSi (X , ) Si+1 X
Limon, Ferramosca, Camacho Set Invariance 20
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
i-steps controllable setMaximal Control invariant setMaximal stabilisable setExampleSome comments
Examplei-steps controllable and stabilizable sets
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Controllable setKi+1(X , ) = Q(Ki ) XKi (X , ) Ki+1 X
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S9 = S10= S
Stabilisable setSi+1(X , ) = Q(Si ) XSi (X , ) Si+1 XS finitely determined iffSi (X , ) = Si+1 X
Limon, Ferramosca, Camacho Set Invariance 21
IntroductionSet invariance theory
i-steps setsRobust invariant sets
i-steps controllable setMaximal Control invariant setMaximal stabilisable setExampleSome comments
Comments
The difference between the controllable set and the stabilisableset is the fact that the set is an invariant set. This difference isvery important in relation to the concept of stability: ifx0 Si (X ,), then there exists a control sequence such that thesystem is driven to , and there exist a control law such that thesystem remains inside . If is not invariant, then the systemmight evolve outside , hence loosing stability.S(X ,) C(X). Hence, x0 C(X) \ Si(X ,), there existsan admissible control law such that the system fulfills theconstraints, but there not exists a control law that drives thesystem to .Set {0} is an invariant set. Then, the set of states thatasymptotically stabilize the system at the origin in i steps isgiven by Si(X , {0}).
Limon, Ferramosca, Camacho Set Invariance 22
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
Robust positive control invariant set
Consider the system:
xk+1 = f (xk , uk , wk ), xk X , uk U, wk W
xk Rn, uk R
m, wk R
q.
Definition (Robust positive control invariant set) Rn is a robust positive control invariant set if x0 X ,there exists a control law uk = h(xk ) such that xk , for allk 0 and wk W, and uk = h(xk ) U.
Limon, Ferramosca, Camacho Set Invariance 23
IntroductionSet invariance theory
i-steps setsRobust invariant sets
Robust one step set
Definition (Robust one step set)The set Q() is the set of states in Rn for which an admissiblecontrol inputs exists which will guarantee that the system will bedriven to in one step, for any w W:
Q() = {xk Rn|uk U : f (xk , uk , wk ) wk W}
For closed-loop systems:
Qh() = {xk Rn : f (xk , h(xk ), wk ) wk W}
Limon, Ferramosca, Camacho Set Invariance 24
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
Properties
Monotonicity: consider sets 1 2, then:
Q(1) Q(2)
The one step set definition allow us to define a condition forguaranteing the invariance of a set.
Geometric robust invariance condition: set is a controlinvariant set if and only if
Q()
Limon, Ferramosca, Camacho Set Invariance 25
IntroductionSet invariance theory
i-steps setsRobust invariant sets
Robust reach set
Definition (Robust reach set)The set R() is the set of states in Rn to which the system willevolve at the next time step given any xk , any wk W andadmissible control input:
R() = {z Rn : xk , uk U, wk Ws.t.z = f (xk , uk , wk )}
For closed-loop systems, Rh() is the set of states in Rn towhich the system will evolve at the next time step given anyxk and any wk W:
Rh() = {z Rn : xk ,wk Ws.t .z = f (xk , h(xk ), wk )}
Limon, Ferramosca, Camacho Set Invariance 26
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
i-steps robust controllable set
DefinitionThe i-steps robust controllable set Ki(X ,) is the set of statesfor which exists an admissible control sequence such that thesystem reaches the set X in exactly i steps, with anadmissible evolution, for any wk W.
Ki (X , ) = {x0 X : k = 0, ..., i 1,uk Us.t.xk Xandxi wk W}
This set represents the set of all states that can reach a givenset in i steps, with an admissible evolution and an admissiblecontrol sequence.
Limon, Ferramosca, Camacho Set Invariance 27
IntroductionSet invariance theory
i-steps setsRobust invariant sets
Maximal robust control invariant set
DefinitionThe set C(X ) is the maximal control invariant set contained inX for system xk+1 = f (xk , uk , wk ) if and only if C(X ) is arobust control invariant set and contains all the robust invariantsets contained in X .
C(X ) X
This set is derived from the definition of the i-steps robustadmissible set, Ci(X ), that is the set of states for which existsan admissible control sequence such that the evolution of thesystem remains in X during the next i steps, for any wk W.
Ci(X) = {x0 X : uk Us.t.xk+1 X , wk W, k = 0, ..., i 1}
Limon, Ferramosca, Camacho Set Invariance 28
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IntroductionSet invariance theory
i-steps setsRobust invariant sets
Maximal robust stabilisable set
DefinitionThe set S(X ,) is the maximal robust stabilisable invariantset contained in X for system xk+1 = f (xk , uk , wk ) if and only ifS(X ,) is the union of all i-step stabilisable sets contained inX , for any wk W.
This set is derived from the definition of the i-steps robuststabilisable set, Si(X ,), that is the set of states for whichexists an admissible control sequence that drive the system tothe invariant set in i steps with an admissible evolution.Si (X ,) = {x0 X : k = 0, ..., i1, uk Us.t.xk Xandxi wk W}
All the properties of the nominal invariant sets are applicable tothe robust case.
Limon, Ferramosca, Camacho Set Invariance 29
IntroductionSet invariance theory
i-steps setsRobust invariant sets
Bibliography
F. Blanchini. Set invariance in control. Automatica.35:1747-1767, 1999.
E. Kerrigan. Robust Constrained Satisfaction: InvariantSets and Predictive Control. PhD Dissertation.
Limon, Ferramosca, Camacho Set Invariance 30