Introduction Set invariance theory i-steps sets Robust invariant sets. Set Invariance. D. Limon A. Ferramosca E.F. Camacho

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1 Set Invariance D. Limon A. Ferramosca E.F. Camacho Department of Automatic Control & Systems Engineering University of Seville HYCON-EECI Graduate School on Control Limon, Ferramosca, Camacho Set Invariance Outline Some definitions One step set The reach set 4 Limon, Ferramosca, Camacho Set Invariance

2 Some definitions Set invariance Set invariance is a fundamental concept in design of controller for constrained systems. The reason: constraint satisfaction can be guaranteed for all time (and for all disturbances) if and only if the initial state is contained inside a (robust) control invariant set. The evolution of a constrained system is admissible if there exists an invariant set X, where X is the set where constraints on the state are fullfilled. Hence, if x X, then x k X, for all k. Limon, Ferramosca, Camacho Set Invariance Some definitions Set invariance Set invariance is strictly connected with stability. Lyapunov theory states that, if there exists a Lyapunov function V (x) such that: ΔV (x) then, for all x X, any set defined as: ={x R n : V (x) α} X is an invariant set for the system, and hence for any initial state x, the system fulfills the constraints and remains inside. Limon, Ferramosca, Camacho Set Invariance 4

3 Some definitions Positive invariant set Consider an autonomous system: x k+ = f (x k ), x k X Definition (Positive invariant set) R n is a positive invariant set if x X, x k, for all k. If the system reaches a positive invariant set, its future evolution remains inside this set. The maximum invariant set, max X, is the smallest positive invariant set that contains all the positive invariant sets contained in X. Limon, Ferramosca, Camacho Set Invariance 5 Some definitions Positive control invariant set Consider the system: x k R n and u k R m. x k+ = f (x k, uk), x k X, u k U Definition (Positive control invariant set) R n is a positive control invariant set if x X, there exists a control law u k = h(x k ) such that x k, for all k, and u k = h(x k ) U. Limon, Ferramosca, Camacho Set Invariance 6

4 Onestepset The reach set Onestepset Consider the system: x k+ = f (x k, uk), x k X, u k U x k R n and u k R m. Let f (, ) = be en equilibrium point. Definition (One step set) The set Q() is the set of states in R n for which an admissible control inputs exists which will guarantee that the system will be driven to in one step: Q() = {x k R n u k U : f (x k, u k ) } Limon, Ferramosca, Camacho Set Invariance 7 Onestepset The reach set Onestepset The previous definition is the same for a system controlled by a control law u = h(x): Q h () = {x k R n : f (x k, h(x k )) } Property: Monotonicity: consider sets, then: Q( ) Q( ) Limon, Ferramosca, Camacho Set Invariance 8

5 Onestepset The reach set Geometric invariance condition The one step set definition allow us to define a condition for guaranteing the invariance of a set. Geometric invariance condition: set is a control invariant set if and only if Q() Q().5.5 x.5 Q().5 x is not invariant is invariant Limon, Ferramosca, Camacho Set Invariance 9 The reach set Onestepset The reach set Definition (The reach set) The set R() is the set of states in R n to which the system will evolve at the next time step given any x k and admissible control input: R() = {z R n : x k, u k Us.t.z = f (x k, u k )} For closed-loop systems, R h () is the set of states in R n to which the system will evolve at the next time step given any x k : R h () = {z R n : x k, s.t.z = f (x k, h(x k ))} Limon, Ferramosca, Camacho Set Invariance

6 Definition The K i (X, ) is the set of states for which exists an admissible control sequence such that the system reaches the set X in exactly i steps, with an admissible evolution. K i (X, ) = {x X : k =,..., i, u k Us.t.x k Xandx i } This set represents the set of all states that can reach a given set in i steps, with an admissible evolution and an admissible control sequence. Limon, Ferramosca, Camacho Set Invariance Properties K i+ (X, ) = Q(K i (X, )) X, withk (X, ) =. K i (X, ) K i+ (X, ) iff is invariant. The set K (X, ) is finitely determined if and only if i N such that K (X, ) = K i (X, ). The smallest element i N such that K (X, ) = K i (X, ) is called the determinedness index. If j N such that K i+ (X, ) = K i (X, ), i j, then K (X, ) is finitely determined. For closed-loop systems: K h i (X, ) = {x X h : x k X h k =,..., i, andx i } where X h = {x X : h(x) U}. Limon, Ferramosca, Camacho Set Invariance

7 Definition The set C (X ) is the maximal control invariant set contained in X for system x k+ = f (x k, u k ) if and only if C (X) is a control invariant set and contains all the invariant sets Φ contained in X. Φ C (X) X This set is derived from the definition of the i-steps admissible set, C i (X ), that is the set of states for which exists an admissible control sequence such that the evolution of the system remains in X during the next i steps. C i (X )={x X : k =,..., i, u k Us.t.x k+ X} Limon, Ferramosca, Camacho Set Invariance Properties C i (X )=K i (X, X ). C i+ (X ) C i (X ). If x XC i (X ), there not exists an admissible control law which will ensure that the evolution of the system is admissible for i steps. C (X ) is the set of all states for which there exists an admissible control law which ensures the fulfillment of the constraints for all time. C (X ) is finitely determined if and only if there exists an element i N, such that C i+ (X) =C i (X), i i. Hence, C i (X )=C (X). K i (X, ) C (X ), i and X. Limon, Ferramosca, Camacho Set Invariance 4

8 Definition The set S (X, ) is the maximal stabilisable invariant set contained in X for system x k+ = f (x k, u k ) if and only if S (X, ) is the union of all i-step stabilisable sets contained in X. This set is derived from the definition of the i-steps stabilisable set, S i (X, ), that is the set of states for which exists an admissible control sequence that drive the system to the invariant set in i steps with an admissible evolution. S i (X, ) = {x X : k =,..., i, u k Us.t.x k Xandx i } The only difference between S i (X, ) and K i (X, ) is the invariance condition for. Limon, Ferramosca, Camacho Set Invariance 5 Properties S i+ (X, ) = Q(S i (X, )) X, withs (X, ) =. S i (X, ) S i+ (X, ). Any S i (X, ) is a control invariant set. Consider and invariant sets, such that. Then S i (X, ) S i (X, ). S i (X, S j (X, )) = S i+j (X, ). S (X, ) is finitely determined if and only if there exists an element i N such that S i+ (X, ) = S i (X, ), for any i i. Furthermore, S (X, ) = S i (X, ), for any i i. For closed-loop systems: S h i (X, ) = {x X h : x k X h k =,..., i, andx i } where X h = {x X : h(x) U}. Limon, Ferramosca, Camacho Set Invariance 6

9 i-steps controllable and stabilizable sets not Invariant Invariant S x x K x x Controllable set K (X, ) = Q() X Stabilisable set S (X, ) = Q() X Limon, Ferramosca, Camacho Set Invariance 7 i-steps controllable and stabilizable sets x x S S K K x x Controllable set K (X, ) = Q(K ) X Stabilisable set S (X, ) = Q(S ) X Limon, Ferramosca, Camacho Set Invariance 8

10 K K i-steps controllable and stabilizable sets S S S x x K x x Controllable set K (X, ) = Q(K ) X Stabilisable set S (X, ) = Q(S ) X Limon, Ferramosca, Camacho Set Invariance 9 i-steps controllable and stabilizable sets S 4 S S S x x K K K K x x Controllable set K i+ (X, ) = Q(K i ) X K i (X, ) K i+ X Stabilisable set S i+ (X, ) = Q(S i ) X S i (X, ) S i+ X Limon, Ferramosca, Camacho Set Invariance

11 i-steps controllable and stabilizable sets S 9 = S = S x x K K K K x x Controllable set K i+ (X, ) = Q(K i ) X K i (X, ) K i+ X Stabilisable set S i+ (X, ) = Q(S i ) X S i (X, ) S i+ X S finitely determined iff S i (X, ) = S i+ X Limon, Ferramosca, Camacho Set Invariance Comments The difference between the controllable set and the stabilisable set is the fact that the set is an invariant set. This difference is very important in relation to the concept of stability: if x S i (X, ), then there exists a control sequence such that the system is driven to, and there exist a control law such that the system remains inside. If is not invariant, then the system might evolve outside, hence loosing stability. S (X, ) C (X). Hence, x C (X) \ S i (X, ), there exists an admissible control law such that the system fulfills the constraints, but there not exists a control law that drives the system to. Set {} is an invariant set. Then, the set of states that asymptotically stabilize the system at the origin in i steps is given by S i (X, {}). Limon, Ferramosca, Camacho Set Invariance

12 Robust positive control invariant set Consider the system: x k+ = f (x k, uk, w k ), x k X, u k U, w k W x k R n, u k R m, w k R q. Definition (Robust positive control invariant set) R n is a robust positive control invariant set if x X, there exists a control law u k = h(x k ) such that x k, for all k and w k W, and u k = h(x k ) U. Limon, Ferramosca, Camacho Set Invariance Robust one step set Definition (Robust one step set) The set Q() is the set of states in R n for which an admissible control inputs exists which will guarantee that the system will be driven to in one step, for any w W: Q() = {x k R n u k U : f (x k, u k, w k ) w k W} For closed-loop systems: Q h () = {x k R n : f (x k, h(x k ), w k ) w k W} Limon, Ferramosca, Camacho Set Invariance 4

13 Properties Monotonicity: consider sets, then: Q( ) Q( ) The one step set definition allow us to define a condition for guaranteing the invariance of a set. Geometric robust invariance condition: set is a control invariant set if and only if Q() Limon, Ferramosca, Camacho Set Invariance 5 Robust reach set Definition (Robust reach set) The set R() is the set of states in R n to which the system will evolve at the next time step given any x k, anyw k Wand admissible control input: R() = {z R n : x k, u k U, w k Ws.t.z = f (x k, u k, w k )} For closed-loop systems, R h () is the set of states in R n to which the system will evolve at the next time step given any x k and any w k W: R h () = {z R n : x k, w k Ws.t.z = f (x k, h(x k ), w k )} Limon, Ferramosca, Camacho Set Invariance 6

14 i-steps robust controllable set Definition The i-steps robust controllable set K i (X, ) is the set of states for which exists an admissible control sequence such that the system reaches the set X in exactly i steps, with an admissible evolution, for any w k W. K i (X, ) = {x X : k =,..., i, u k Us.t.x k Xandx i w k W} This set represents the set of all states that can reach a given set in i steps, with an admissible evolution and an admissible control sequence. Limon, Ferramosca, Camacho Set Invariance 7 Maximal robust control invariant set Definition The set C (X ) is the maximal control invariant set contained in X for system x k+ = f (x k, u k, w k ) if and only if C (X) is a robust control invariant set and contains all the robust invariant sets Φ contained in X. Φ C (X) X This set is derived from the definition of the i-steps robust admissible set, C i (X ), that is the set of states for which exists an admissible control sequence such that the evolution of the system remains in X during the next i steps, for any w k W. C i (X) ={x X : u k Us.t.x k+ X, w k W, k =,..., i } Limon, Ferramosca, Camacho Set Invariance 8

15 Maximal robust stabilisable set Definition The set S (X, ) is the maximal robust stabilisable invariant set contained in X for system x k+ = f (x k, u k, w k ) if and only if S (X, ) is the union of all i-step stabilisable sets contained in X, for any w k W. This set is derived from the definition of the i-steps robust stabilisable set, S i (X, ), that is the set of states for which exists an admissible control sequence that drive the system to the invariant set in i steps with an admissible evolution. S i (X, ) = {x X : k =,..., i, u k Us.t.x k Xandx i w k W} All the properties of the nominal invariant sets are applicable to the robust case. Limon, Ferramosca, Camacho Set Invariance 9 Bibliography F. Blanchini. Set invariance in control. Automatica. 5: , 999. E. Kerrigan. Robust Constrained Satisfaction: Invariant Sets and Predictive Control. PhD Dissertation. Limon, Ferramosca, Camacho Set Invariance