Mpc Invariant set

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



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




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 .




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 .


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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 .




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.


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One step set


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 ) }





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)


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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()

3 2 1 0 1 2 32

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x 2

Q()

is not invariant

6 4 2 0 2 4 62

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Q()

is invariant





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 ))}


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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.





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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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}





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 .


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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 .





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




Examplei-steps controllable and stabilizable sets

not Invariant

6 4 2 0 2 4 63

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K1

Controllable setK1(X , ) = Q() X

Invariant

6 4 2 0 2 4 63

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S1

Stabilisable setS1(X , ) = Q() X






6 4 2 0 2 4 63

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K1 K2

Controllable setK2(X , ) = Q(K1) X

6 4 2 0 2 4 63

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S2 S1

Stabilisable setS2(X , ) = Q(S1) X






6 4 2 0 2 4 63

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K1 K2 K3

Controllable setK3(X , ) = Q(K2) X

6 4 2 0 2 4 63

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S3 S2 S1

Stabilisable setS3(X , ) = Q(S2) X






6 4 2 0 2 4 63

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K1 K2 K3 K4

Controllable setKi+1(X , ) = Q(Ki ) XKi (X , ) Ki+1 X

6 4 2 0 2 4 63

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S4 S3 S2 S1

Stabilisable setSi+1(X , ) = Q(Si ) XSi (X , ) Si+1 X






6 4 2 0 2 4 63

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x1

x 2

K1 K2 K3 K4

Controllable setKi+1(X , ) = Q(Ki ) XKi (X , ) Ki+1 X

6 4 2 0 2 4 63

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x1

x 2

S9 = S10= S

Stabilisable setSi+1(X , ) = Q(Si ) XSi (X , ) Si+1 XS finitely determined iffSi (X , ) = Si+1 X





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}).


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Robust positive control invariant set


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.




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}




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()




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 )}




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.




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}




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.




Bibliography

F. Blanchini. Set invariance in control. Automatica.35:1747-1767, 1999.

E. Kerrigan. Robust Constrained Satisfaction: InvariantSets and Predictive Control. PhD Dissertation.


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Mpc Invariant set