CLF-Based Control for Hybrid Dynamical Systems
aa r X i v : . [ m a t h . O C ] S e p Pointwise Minimum Norm Control Laws forHybrid Systems
Ricardo G. Sanfelice
Abstract
Pointwise minimum norm control laws for hybrid dynamical systems are proposed. Hybrid systemsare given by differential equations capturing the continuous dynamics or flows , and by differenceequations capturing the discrete dynamics or jumps . The proposed control laws are defined as thepointwise minimum norm selection from the set of inputs guaranteeing a decrease of a control Lyapunovfunction. The cases of individual and common inputs during flows and jumps, as well as when inputsenter through one of the system dynamics, are considered. Examples illustrate the results.
I. I
NTRODUCTION
The construction of asymptotically stabilizing control laws from control Lyapunov functions(CLFs) has enabled the systematic design of feedback laws for nonlinear systems. Buildingfrom earlier results in [1], which revealed a key link between the availability of a controlLyapunov function and stabilizability (with relaxed controls), the construction of control lawsfrom Lyapunov inequalities was rendered as a powerful control design methodology (see also,e.g., [2], [3], for the connections between CLFs and asymptotic controllability to the origin).More importantly, design techniques that go beyond the possibility of determining the controllaw from the expression of the Lyapunov inequalities were proposed and employed in severalapplications. The control law introduced in [4], known as Sontag’s universal formula, providesa generic controller construction for nonlinear systems in affine form that (modulo some extraproperties at the origin) only requires the existence of a CLF. (Recent extensions to polynomial
R. G. Sanfelice is with the Department of Aerospace and Mechanical Engineering, University of Arizona, 1130 N. MountainAve, AZ 85721. Email: [email protected] . Research partially supported by NSF CAREER Grant no. ECS-1150306 and by AFOSR YIP Grant no. FA9550-12-1-0366.
Technical Report systems appeared in [5]). The constructions introduced in [6] have the extra property that theirpointwise norm is minimum (for a given CLF). More notably, as shown in [6] by making a linkbetween CLFs and the solution to a differential game, under additional properties, pointwiseminimum norm control laws guarantee robustness of the closed-loop system.In this paper, pointwise minimum norm control laws for hybrid dynamical systems are pro-posed. Hybrid dynamical systems are given by differential equations capturing the continuousdynamics or flows , and by difference equations capturing the discrete dynamics or jumps . Theconditions determining whether flows or jumps should occur are given in terms of both thestate and the inputs. For this class of hybrid systems, control Lyapunov functions are definedby continuously differentiable functions whose change, both along flows and jumps, is upperbounded by a negative definite function of the state. The proposed control law consists of apointwise minimum norm selection from the set of inputs that guarantees a decrease of theLyapunov function on each regime. We consider the case when the inputs acting during flowsare different than the inputs acting during jumps, the case when the inputs are the same, as wellas cases when inputs affect only the flows or the jumps. Conditions guaranteeing continuity andglobality of the proposed pointwise minimum norm control laws are also presented. Our resultsnot only recover the results in [7] when specialized to continuous-time systems, but also providethe discrete-time versions, which do not seem available in the literature.The remainder of the paper is organized as follows. Section II introduces the framework forhybrid systems, the notion of solution, and control Lyapunov functions. Section III presentsthe results on stabilization by pointwise minimum norm control laws. Examples in Section IVillustrate some of the results.
Notation: R n denotes n -dimensional Euclidean space, R denotes the real numbers. R ≥ denotesthe nonnegative real numbers, i.e., R ≥ = [0 , ∞ ) . N denotes the natural numbers including ,i.e., N = { , , . . . } . B denotes the closed unit ball in a Euclidean space. Given a set K , K denotes its closure. Given a set S , ∂S denotes its boundary. Given x ∈ R n , | x | denotes theEuclidean vector norm. Given a set K ⊂ R n and x ∈ R n , | x | K := inf y ∈ K | x − y | . Given x and y , h x, y i denotes their inner product. A function α : R ≥ → R ≥ is said to belong to class- K ∞ if itis continuous, zero at zero, strictly increasing, and unbounded. Given a closed set K ⊂ R n × U ⋆ with ⋆ being either c or d and U ⋆ ⊂ R m ⋆ , define Π( K ) := { x : ∃ u ⋆ ∈ U ⋆ s.t. ( x, u ⋆ ) ∈ K } Technical Report and Ψ( x, K ) := { u : ( x, u ) ∈ K } . That is, given a set K , Π( K ) denotes the “projection” of K onto R n while, given x , Ψ( x, K ) denotes the set of values u such that ( x, u ) ∈ K . Then, foreach x ∈ R n , define the set-valued maps Ψ c : R n ⇒ U c , Ψ d : R n ⇒ U d as Ψ c ( x ) := Ψ( x, C ) and Ψ d ( x ) := Ψ( x, D ) , respectively. Given a map f , its graph is denoted by gph ( f ) .II. P RELIMINARIES ON H YBRID S YSTEMS AND C ONTROL L YAPUNOV F UNCTIONS
In this section, we define control Lyapunov functions (CLFs) for hybrid systems H with data ( C, f, D, g ) and given by H ˙ x = f ( x, u c ) ( x, u c ) ∈ Cx + = g ( x, u d ) ( x, u d ) ∈ D, (1)where the set C ⊂ R n × U c is the flow set , the map f : R n × R m c → R n is the flow map , theset D ⊂ R n × U d is the jump set , and the map g : R n → R n is the jump map . The space for thestate is x ∈ R n and the space for the input u = ( u c , u d ) is U = U c × U d , where U c ⊂ R m c and U d ⊂ R m d . At times, we will require H to satisfy the following mild properties. Definition 2.1 (hybrid basic conditions):
A hybrid system H is said to satisfy the hybrid basicconditions if its data ( C, f, D, g ) is such that(A1) C and D are closed subsets of R n × U c and R n × U d , respectively;(A2) f : R n × R m c → R n is continuous;(A3) g : R n × R m d → R n is continuous.Solutions to hybrid systems H are given in terms of hybrid arcs and hybrid inputs on hybridtime domains. Hybrid time domains are subsets E of R ≥ × N that, for each ( T, J ) ∈ E , E ∩ ([0 , T ] × { , , ...J } ) can be written as ∪ J − j =0 ([ t j , t j +1 ] , j ) for some finite sequence of times t ≤ t ≤ t ... ≤ t J . A hybrid arc φ is a function on a hybrid time domain that, foreach j ∈ N , t φ ( t, j ) is absolutely continuous on the interval { t : ( t, j ) ∈ dom φ } , whilea hybrid input u is a function on a hybrid time domain that, for each j ∈ N , t u ( t, j ) is Lebesgue measurable and locally essentially bounded on the interval { t : ( t, j ) ∈ dom u } .Then, a solution to the hybrid system H is given by a pair ( φ, u ) , u = ( u c , u d ) , with dom φ =dom u (= dom( φ, u )) and satisfying the dynamics of H , where φ is a hybrid arc and u a hybrid This property is to hold at each ( T, J ) ∈ E , but E can be unbounded. Technical Report input. A solution pair ( φ, u ) to H is said to be complete if dom( φ, u ) is unbounded and maximal if there does not exist another pair ( φ, u ) ′ such that ( φ, u ) is a truncation of ( φ, u ) ′ to someproper subset of dom( φ, u ) ′ . For more details about solutions to hybrid systems, see [8].We introduce the concept of control Lyapunov function for hybrid systems H ; see [9] formore details and conditions on H guaranteeing its existence. Definition 2.2 (control Lyapunov function):
Given a compact set
A ⊂ R n and sets U c ⊂ R m c , U d ⊂ R m d , a continuous function V : R n → R , continuously differentiable on an open setcontaining Π( C ) is a control Lyapunov function with U controls for H if there exist α , α ∈ K ∞ and a positive definite function α such that α ( | x | A ) ≤ V ( x ) ≤ α ( | x | A ) ∀ x ∈ Π( C ) ∪ Π( D ) ∪ g ( D ) , (2) inf u c ∈ Ψ c ( x ) h∇ V ( x ) , f ( x, u c ) i ≤ − α ( | x | A ) ∀ x ∈ Π( C ) , (3) inf u d ∈ Ψ d ( x ) V ( g ( x, u d )) − V ( x ) ≤ − α ( | x | A ) ∀ x ∈ Π( D ) . (4)III. M INIMUM N ORM S TATE -F EEDBACK L AWS FOR H YBRID S YSTEMS
Given a hybrid system H satisfying the hybrid basic conditions, a compact set A , and a controlLyapunov function V satisfying Definition 2.2, define, for each r ∈ R ≥ , the set I ( r ) := { x ∈ R n : V ( x ) ≥ r } . Moreover, for each ( x, u c ) ∈ R n × R m c and r ∈ R ≥ , define the function Γ c ( x, u c , r ) := h∇ V ( x ) , f ( x, u c ) i + α ( | x | A ) if ( x, u c ) ∈ C ∩ ( I ( r ) × R m c ) , −∞ otherwiseand, for each ( x, u d ) ∈ R n × R m d and r ∈ R ≥ , the function Γ d ( x, u d , r ) := V ( g ( x, u d )) − V ( x ) + α ( | x | A ) if ( x, u d ) ∈ D ∩ ( I ( r ) × R m d ) , −∞ otherwise.Then, evaluate the functions Γ c and Γ d at points ( x, u c , r ) and ( x, u d , r ) where r = V ( x ) todefine the functions ( x, u c ) Υ c ( x, u c ) := Γ c ( x, u c , V ( x )) , ( x, u d ) Υ d ( x, u d ) := Γ d ( x, u d , V ( x )) (5) Following [7, Definition 4.1], (3) can be replaced by inf u c ∈ Ψ c ( x ) h∇ V ( x ) , f ( x, u c ) i < for all x ∈ Π( C ) \ A , since, then,[7, Proposition 4.3] guarantees the existence of a continuous positive definite function α satisfying (3) (similarly for (4)). Technical Report and the set-valued maps T c ( x ) := Ψ c ( x ) ∩ { u c ∈ U c : Υ c ( x, u c ) ≤ } , T d ( x ) := Ψ d ( x ) ∩ { u d ∈ U d : Υ d ( x, u d ) ≤ } . (6)Furthermore, define R c := Π( C ) ∩ { x ∈ R n : V ( x ) > } (7)and R d := Π( D ) ∩ { x ∈ R n : V ( x ) > } . (8)When, for each x , the functions u c Υ c ( x, u c ) and u d Υ d ( x, u c ) are convex, and the set-valued maps Ψ c and Ψ d have nonempty closed convex values on R c and R d , respectively, wehave that T c ( x ) and T d ( x ) have nonempty convex closed values on (7) and on (8), respectively(this follows from [7, Proposition 4.4]). Then, T c and T d have unique elements of minimumnorm on R c and R d , respectively, and their minimal selections ρ c : R c → U c , ρ d : R d → U d are given by ρ c ( x ) := arg min {| u c | : u c ∈ T c ( x ) } , (9) ρ d ( x ) := arg min {| u d | : u d ∈ T d ( x ) } . (10)Moreover, these selections are continuous under further properties of Ψ c and Ψ d .The hybrid system H under the effect of the control pair ( ρ c , ρ d ) in (9), (10) is given by e H ˙ x = e f ( x ) := f ( x, ρ c ( x )) x ∈ e Cx + = e g ( x ) := g ( x, ρ d ( x )) x ∈ e D (11)with e C := { x ∈ R n : ( x, ρ c ( x )) ∈ C } and e D := { x ∈ R n : ( x, ρ d ( x )) ∈ D } . The abovearguments and constructions enable the stabilization results in the following sections. Remark 3.1:
When bounds (3) and (4) hold for functions α ,c and α ,d , then a commonfunction α is given by α ( s ) = min { α ,c ( s ) , α ,d ( s ) } for all s ≥ . In such a case, theexpressions of the pointwise minimum norm control laws (9) and (10) could also be givenin terms of α ,c and α ,d by defining T c and T d in terms of α ,c and α ,d , respectively. Technical Report
A. Practical stabilization using min-norm hybrid control
Proposition 3.2 below establishes that the pointwise minimum norm controller in (9)-(10)asymptotically stabilizes the compact set A r := { x ∈ R n : V ( x ) ≤ r } (12)for the hybrid system restricted to I ( r ) . More precisely, given r > , we restrict the flow andjump sets of the hybrid system H by the set I ( r ) , which leads to H I ˙ x = f ( x, u c ) ( x, u c ) ∈ C ∩ ( I ( r ) × R m c ) x + = g ( x, u d ) ( x, u d ) ∈ D ∩ ( I ( r ) × R m d ) . Proposition 3.2: Given a compact set
A ⊂ R n and a hybrid system H = ( C, f, D, g ) satisfyingthe hybrid basic conditions, suppose there exists a control Lyapunov function V with U controlsfor H . Furthermore, suppose the following conditions hold:(M1) The set-valued maps Ψ c and Ψ d are lower semicontinuous with convex values.(M2) For every r > and every x ∈ Π( C ) ∩ I ( r ) , the function u c Γ c ( x, u c , r ) is convexon Ψ c ( x ) and, for every r > and every x ∈ Π( D ) ∩ I ( r ) , the function u d Γ c ( x, u d , r ) is convex on Ψ d ( x ) .Then, for every r > , the state-feedback law pair ρ c : R c ∩ I ( r ) → U c , ρ d : R d ∩ I ( r ) → U d defined as ρ c ( x ) := arg min {| u c | : u c ∈ T c ( x ) } ∀ x ∈ R c ∩ I ( r ) , (13) ρ d ( x ) := arg min {| u d | : u d ∈ T d ( x ) } ∀ x ∈ R d ∩ I ( r ) (14) renders the compact set A r asymptotically stable for H I . Furthermore, if the set-valued maps Ψ c and Ψ d have closed graph then ρ c and ρ d are continuous. A compact set A is said to be asymptotically stable for a closed-loop system (e.g., e H in (11)) if: • for each ε > thereexists δ > such that each maximal solution φ starting from A + δ B satisfies φ ( t, j ) ∈ A + ε B for each ( t, j ) ∈ dom φ , and • each maximal solution is bounded and the complete ones satisfy lim t + j →∞ | φ ( t, j ) | A = 0 . A set-valued map S : R n ⇒ R m is lower semicontinuous if for each x ∈ R n one has that lim inf x i → x S ( x i ) ⊃ S ( x ) , where lim inf x i → x S ( x i ) = { z : ∀ x i → x, ∃ z i → z s.t. z i ∈ S ( x i ) } is the inner limit of S (see [10, Chapter 5.B]). Technical Report
Proof:
Given the CLF V for H , by Definition 2.2, we have (2) and, for every r > , inf u c ∈ Ψ c ( x ) h∇ V ( x ) , f ( x, u c ) i ≤ − α ( | x | A ) ∀ x ∈ Π( C ) ∩ I ( r ) , (15) inf u d ∈ Ψ d ( x ) V ( g ( x, u d )) − V ( x ) ≤ − α ( | x | A ) ∀ x ∈ Π( D ) ∩ I ( r ) , (16)from where Γ c and Γ d are defined. Using the continuity properties of f and g obtained from (A2)and (A3) of the hybrid basic conditions, and continuous differentiability of V , it follows that, forevery r ≥ , Γ c and Γ d are continuous on C ∩ ( I ( r ) × R m c ) and on D ∩ ( I ( r ) × R m d ) , respectively.Since C and D are closed by (A1) of the hybrid basic conditions and V is continuous, the sets C ∩ ( I ( r ) × R m c ) and D ∩ ( I ( r ) × R m d ) are closed for each r . By the closedness propertyof C and D along with assumption (M1), the set-valued maps Ψ c and Ψ d have nonemptyclosed convex values on R c and R d , respectively. Using (M2), the functions u c Υ c ( x, u c ) and u d Υ d ( x, u c ) defined in (5) are convex on R c and R d , respectively. Then, [7, Proposition 4.4]implies that T c and T d are lower semicontinuous with nonempty closed convex values on R c and R d , respectively. Moreover, T c and T d have unique elements of minimum norm, and theirminimal selections ρ c : R c → U c (17) ρ d : R d → U d (18)on (7) and on (8) (respectively) are, by definition, given by (13) and (14) (respectively). Then,from (13) and (14), we have ρ c ( x ) ∈ Ψ c ( x ) , Υ c ( x, ρ c ( x )) ≤ ∀ x ∈ R c ∩ I ( r ) ρ d ( x ) ∈ Ψ d ( x ) , Υ d ( x, ρ d ( x )) ≤ ∀ x ∈ R d ∩ I ( r ) . Using the definitions of Ψ c , Ψ d and Υ c , Υ d , we have h∇ V ( x ) , f ( x, ρ c ( x )) i ≤ − α ( | x | A ) ∀ x ∈ Π( C ) ∩ I ( r ) , (19) V ( g ( x, ρ d ( x ))) − V ( x ) ≤ − α ( | x | A ) ∀ x ∈ Π( D ) ∩ I ( r ) . (20)Then, for every r > , we have a state-feedback pair ( ρ c , ρ d ) that renders the compact set A r asymptotically stable for H I . This property follows from an application of the Lyapunov stabilityresult in [11, Theorem 3.18]. Technical Report If Ψ c and Ψ d have closed graph, then we have that the graph of T c and T d are closed since,for ⋆ = c, d , gph ( T ⋆ ) = gph (Ψ ⋆ ( x )) ∩ gph ( { u ⋆ ∈ U ⋆ : Υ ⋆ ( x, u ⋆ ) ≤ } ) , where the first graph is closed by assumption while the second one is closed by the closednessand continuity properties of U ⋆ and Υ ⋆ , respectively. Then, using [6, Proposition 2.19], theminimal selections ρ c : R c → U c , ρ d : R d → U d on (7) and on (8), which are given by (13) and (14), respectively, are continuous. Remark 3.3:
The state-feedback law (13)-(14) asymptotically stabilizes A r for H I (but notnecessarily for H as without an appropriate extension of these laws to Π( C ) and Π( D ) , respec-tively, there could exist solutions to the closed-loop system that jump out of A r ). This pointmotivates the following result on stabilization by a control law that has pointwise minimumnorm at points in I ( r ) , but not everywhere, and the global stabilization result in the next section.Finally, note that the assumptions placed on H , such as the existence of a CLF, can be relaxedby imposing them on H I instead. Theorem 3.4: Under the conditions of Proposition 3.2, for every r > there exists a state-feedback law pair ρ ′ c : R c → U c , ρ ′ d : R d → U d defined on R c ∩ I ( r ) and R d ∩ I ( r ) as ρ ′ c ( x ) := arg min {| u c | : u c ∈ T c ( x ) } ∀ x ∈ R c ∩ I ( r ) , (21) ρ ′ d ( x ) := arg min {| u d | : u d ∈ T d ( x ) } ∀ x ∈ R d ∩ I ( r ) (22) respectively, that renders the compact set A r asymptotically stable for H . Furthermore, if theset-valued maps Ψ c and Ψ d have closed graph then ρ ′ c and ρ ′ d are continuous on R c ∩ I ( r ) and R d ∩ I ( r ) , respectively. Note that by the hybrid basic conditions of H , continuity of ρ c and ρ d , and closedness of I ( r ) , the hybrid system H I withthe control laws (13) and (14) applied to it satisfies the hybrid basic conditions. Technical Report
The result follows using Proposition 3.2 and the fact that, from the definition of CLF inDefinition 2.2, since the right-hand side of (3) is negative definite with respect to A (respectively,(4)) the state-feedback ρ c (respectively, ρ d ) in (9) (respectively, (10)) can be extended – notnecessarily as a pointwise minimum norm law – to every point in Π( C ) ∩ A r (respectively, Π( D ) ∩ A r ) and guarantee that V is nonincreasing. The asymptotic stability of A r for H thenfollows from an application of [11, Theorem 3.18]. Finally, as the definition of T c and T d suggest,the norm-minimality of ρ c and ρ d are functions of V and α , and different such choices wouldgive different pointwise minimum norm control laws. B. Global stabilization using min-norm hybrid control
The result in the previous section guarantees a practical stability property through the use of apointwise minimum norm state-feedback control law. Now, we consider the global stabilizationof a compact set via continuous state-feedback laws ( ρ c , ρ d ) with pointwise minimum norm. Forsuch a purpose, extra conditions are required to hold nearby the compact set. For continuous-time systems, such conditions correspond to the so-called continuous control property and smallcontrol property [4], [6], [12]. To that end, given a compact set A and a control Lyapunovfunction V satisfying Definition 2.2, for each x ∈ R n , define T ′ c ( x ) := Ψ c ( x ) ∩ S ′ c ( x, V ( x )) , (23) T ′ d ( x ) := Ψ d ( x ) ∩ S ′ d ( x, V ( x )) , (24)where, for each x ∈ R n and each r ≥ , S ′ c ( x, r ) := S ◦ c ( x, r ) if r > ,ρ c, ( x ) if r = 0 , S ′ d ( x, r ) := S ◦ d ( x, r ) if r > ,ρ d, ( x ) if r = 0 , (25) S ◦ c ( x, r ) = { u c ∈ U c : Γ c ( x, u c , r ) ≤ } if x ∈ Π( C ) ∩ I ( r ) , R m c otherwise ,S ◦ d ( x, r ) = { u d ∈ U d : Γ d ( x, u d , r ) ≤ } if x ∈ Π( D ) ∩ I ( r ) , R m d otherwise , and the feedback law pair ρ c, : R n → U c , ρ d, : R n → U d Technical Report0 induces (strong) forward invariance of A , that is,(M3) Every maximal solution t φ ( t, to ˙ x = f ( x, ρ c, ( x )) , x ∈ Π( C ) ∩ A satisfies | φ ( t, | A = 0 for all ( t, ∈ dom φ ;(M4) Every maximal solution j φ (0 , j ) to x + = g ( x, ρ d, ( x )) , x ∈ Π( D ) ∩ A satisfies | φ (0 , j ) | A = 0 for all (0 , j ) ∈ dom φ .Under the conditions in Proposition 3.2, the maps in (25) are lower semicontinuous for every r > . To be able to make continuous selections at A , these maps are further required to belower semicontinuous for r = 0 . These conditions resemble those already reported in [6] forcontinuous-time systems. Theorem 3.5: Given a compact set
A ⊂ R n and a hybrid system H = ( C, f, D, g ) satisfyingthe hybrid basic conditions, suppose there exists a control Lyapunov function V with U controlsfor H . Moreover, suppose that conditions (M1)-(M2) of Proposition 3.2 hold. If the feedbacklaw pair ( ρ c, : R n → U c , ρ d, : R n → U d ) is such that conditions (M3) and (M4) hold, and(M5) The set-valued map T ′ c in (23) is lower semicontinuous at each x ∈ Π( C ) ∩ I (0) ,(M6) The set-valued map T ′ d in (24) is lower semicontinuous at each x ∈ Π( D ) ∩ I (0) hold, then the state-feedback law pair ρ c : Π( C ) → U c , ρ d : Π( D ) → U d defined as ρ c ( x ) := arg min {| u c | : u c ∈ T ′ c ( x ) } ∀ x ∈ Π( C ) (26) ρ d ( x ) := arg min {| u d | : u d ∈ T ′ d ( x ) } ∀ x ∈ Π( D ) (27) renders the compact set A globally asymptotically stable for H . Furthermore, if the set-valuedmaps Ψ c and Ψ d have closed graph and ( ρ c, , ρ d, )( A ) = 0 then ρ c and ρ d are continuous.Proof: The proof follows the ideas of the proof of [7, Proposition 7.1]. Proceeding as in theproof of Proposition 3.2, using (M5) and (M6), we have that T ′ c and T ′ d are lower semicontinuouswith nonempty closed values on Π( C ) and Π( D ) , respectively. Then, T ′ c and T ′ d have uniqueelements of minimum norm, and their minimal selections ρ c : Π( C ) → U c (28) ρ d : Π( D ) → U d (29) Technical Report1 on Π( C ) and Π( D ) (respectively) are given by (26) and (27) (respectively). Then, from (26)and (27), we have ρ c ( x ) ∈ Ψ c ( x ) , Γ c ( x, ρ c ( x ) , V ( x )) ≤ ∀ x ∈ Π( C ) ρ d ( x ) ∈ Ψ d ( x ) , Γ d ( x, ρ d ( x ) , V ( x )) ≤ ∀ x ∈ Π( D ) . Using the definitions of Ψ c , Ψ d and Γ c , Γ d , we have h∇ V ( x ) , f ( x, ρ c ( x )) i ≤ − α ( | x | A ) ∀ x ∈ Π( C ) ,V ( g ( x, ρ d ( x ))) − V ( x ) ≤ − α ( | x | A ) ∀ x ∈ Π( D ) . Then, the set A is globally asymptotically stable for the closed-loop system e H by an applicationof the Lyapunov stability theorem for hybrid systems [11, Theorem 3.18].When the set-valued maps Ψ c and Ψ d have closed graph, from Proposition 3.2 we have that ρ c and ρ d are continuous on Π( C ) \A and on Π( D ) \A , respectively. Moreover, if ( ρ c, , ρ d, )( A ) = 0 ,[9, Theorem 4.5] implies that there exists a continuous feedback pair ( κ c , κ d ) – not necessarily ofpointwise minimum norm – asymptotically stabilizing the compact set A and with the property ( κ c , κ d )( A ) = 0 (the pair ( κ c , κ d ) vanishes on A due to the fact that the only possible selectionfor r = 0 is the pair ( ρ c, , ρ d, ) , which vanishes at such points). Since ρ c and ρ d have pointwiseminimum norm, we have ≤ | ρ c ( x ) | ≤ | κ c ( x ) | ∀ x ∈ Π( C ) (30) ≤ | ρ d ( x ) | ≤ | κ d ( x ) | ∀ x ∈ Π( D ) . (31)Then, since κ c and κ d are continuous and vanish at points in A , the laws ρ c and ρ d are continuouson Π( C ) and Π( D ) , respectively. C. The case when the inputs affect only flows or only jumps
The results in the previous sections also hold when inputs only affect either the flows orjumps, but not both. In particular, we consider the special case when u c is the only input, inwhich case H becomes H c ˙ x = f ( x, u c ) ( x, u c ) ∈ Cx + = g ( x ) x ∈ D (32) Technical Report2 with D ⊂ R n and g : R n → R n . When the only input is u d , H becomes H d ˙ x = f ( x ) x ∈ Cx + = g ( x, u d ) ( x, u d ) ∈ D (33)with, in this case, C ⊂ R n and f : R n → R n . The following results follow by combining theearlier results. Corollary 3.6: Given a compact set
A ⊂ R n and a hybrid system H c = ( C, f, D, g ) as in (32) satisfying the hybrid basic conditions, suppose there exists a control Lyapunov function V with U controls for H c . Furthermore, suppose the following conditions hold:(M1c) The set-valued map Ψ c is lower semicontinuous with convex values.(M2c) For every r > and every x ∈ Π( C ) ∩ I ( r ) , the function u c Γ c ( x, u c , r ) is convexon Ψ c ( x ) .Then, for every r > , there exists a state-feedback law ρ ′ c : Π( C ) → U c (34) defined on R c ∩ I ( r ) as in (21) that renders the compact set A r asymptotically stable for H c .Moreover, if the set-valued map Ψ c has a closed graph then ρ ′ c is continuous on Π( C ) ∩ I ( r ) .Furthermore, if the zero feedback law ρ c, : R n → { } ⊂ U c is such that condition (M3) holdsand if (M5) holds, then ρ c in (26) is globally asymptotically stabilizing. Furthermore, if theset-valued map Ψ c has closed graph then ρ c is continuous.Corollary 3.7: Given a compact set A ⊂ R n and a hybrid system H d = ( C, f, D, g ) as in (33) satisfying the hybrid basic conditions, suppose there exists a control Lyapunov function V with U controls for H d . Furthermore, suppose the following conditions hold:(M1d) The set-valued map Ψ d is lower semicontinuous with convex values.(M2d) For every r > and every x ∈ Π( D ) ∩ I ( r ) , the function u d Γ d ( x, u d , r ) is convexon Ψ d ( x ) .Then, for every r > , there exists a state-feedback law ρ ′ d : Π( D ) → U d (35) defined on R d ∩ I ( r ) as in (22) that renders the compact set A r asymptotically stable for H d .Moreover, if the set-valued map Ψ d has a closed graph then ρ ′ d is continuous on Π( D ) ∩ I ( r ) . Technical Report3
Furthermore, if the zero feedback law ρ d, : R n → { } ⊂ U d is such that condition (M4) holdsand if (M6) holds, then ρ d in (27) is globally asymptotically stabilizing. Furthermore, if theset-valued map Ψ d has closed graph then ρ d is continuous.D. The common input case When the input for flows and jumps are the same, i.e., u := u c = u d ( m := m c = m d ), thehybrid system H becomes H ˙ x = f ( x, u ) ( x, u ) ∈ Cx + = g ( x, u ) ( x, u ) ∈ D (36)and a common pointwise minimum norm control law exists when T ′ c ( x ) ∩ T ′ d ( x ) = ∅ ∀ x ∈ Π( C ) ∩ Π( D ) ∩ I ( r ) (37)for each r . A result paralleling Theorem 3.5 follows using T ′ ( x ) := T ′ c ( x ) if x ∈ (Π( C ) \ Π( D )) ∩ I ( r ) T ′ c ( x ) ∩ T ′ d ( x ) if x ∈ Π( C ) ∩ Π( D ) ∩ I ( r ) T ′ d ( x ) if x ∈ (Π( D ) \ Π( C )) ∩ I ( r ) R m otherwise , which, when further assuming (37), is lower semicontinuous and has nonempty, convex values.(The set valued map T can be defined similarly.) Corollary 3.8: Given a compact set
A ⊂ R n and a hybrid system H = ( C, f, D, g ) as in (36) satisfying the hybrid basic conditions, suppose there exists a control Lyapunov function V with U controls for H with input u = u c = u d ( m = m c = m d ). Suppose that conditions (M1)-(M2)of Proposition 3.2 and condition (37) hold. Then, for every r > , there exists a state-feedbacklaw ρ ′ : Π( C ) ∪ Π( D ) → U (38) defined on (Π( C ) ∪ Π( D )) ∩ I ( r ) as ρ ′ ( x ) := arg min {| u | : u ∈ T ( x ) } ∀ x ∈ (Π( C ) ∪ Π( D )) ∩ I ( r ) that renders the compact set A r asymptotically stable for H . Moreover, if the set-valued maps Ψ c and Ψ d have closed graph then ρ ′ is continuous on (Π( C ) ∪ Π( D )) ∩ I ( r ) . Furthermore, if Technical Report4 the zero feedback law ρ : R n → { } ⊂ U is such that (37) and (M3)-(M6) for r = 0 hold, thenthe state-feedback law ρ : Π( C ) ∪ Π( D ) → U (39) defined as ρ ( x ) := arg min {| u | : u ∈ T ′ ( x ) } ∀ x ∈ Π( C ) ∪ Π( D ) (40) renders the compact set A globally asymptotically stable for H . Furthermore, if the set-valuedmaps Ψ c and Ψ d have closed graph then ρ is continuous. IV. E
XAMPLES
Now, we present examples illustrating some of the results in the previous sections. Completedetails are presented for the first example.
Example 4.1 (Rotate and dissipate):
Given v , v ∈ R , let W ( v , v ) := { ξ ∈ R : ξ = r ( λv + (1 − λ ) v ) , r ≥ , λ ∈ [0 , } and define v = [1 1] ⊤ , v = [ − ⊤ , v = [1 − ⊤ , v = [ − − ⊤ . Let ω > and consider the hybrid system H ˙ x = f ( x, u c ) := u c ω − ω x ( x, u c ) ∈ C,x + = g ( x, u d ) ( x, u d ) ∈ D, (41) C := n ( x, u c ) ∈ R × R : u c ∈ {− , } , x ∈ b C o , b C := R \ ( W ( v , v ) ∪ W ( v , v )) ,D := (cid:8) ( x, u d ) ∈ R × R ≥ : u d ≥ γ | x | , x ∈ ∂ W ( v , v ) (cid:9) , for each ( x, u d ) ∈ R × R ≥ the jump map g is given by g ( x, u d ) := R ( π/ u d , R ( s ) = cos s sin s − sin s cos s , and γ > is such that exp( π/ (2 ω )) γ < . For each i ∈ { , } , the vectors v i , v i ∈ R are suchthat W ( v , v ) ∩ W ( v , v ) = { } . The set of interest is A := { } ⊂ R . Figure 1 depicts theflow and jump sets projected onto the x plane. Technical Report5
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C D x x W ( v , v ) W ( v , v ) Fig. 1. Sets for Example 4.1. The white region (and its boundary) corresponds to the flow set projected onto the x plane. Thedashed line represents D . To construct a state-feedback law for (41), consider the candidate control Lyapunov function V given by V ( x ) = exp( T ( x )) x ⊤ x ∀ x ∈ R , (42)where T denotes the minimum time to reach the set W ( v , v ) with the continuous dynamicsof (41) and u c ∈ {− , } . The function T is precisely defined as follows. It is defined as acontinuously differentiable function from R to [0 , π ω ] given as T ( x ) := ω arcsin (cid:16) √ | x | + x | x | (cid:17) on b C and zero for every other point in W ( v , v ) . The definition of V is such that (2) holdswith α ( s ) := s and α ( s ) := exp (cid:0) π ω (cid:1) s for each s ≥ .Next, we construct the set-valued maps Ψ c and Ψ d and then check (3) and (4). Note that Π( C ) = b C and Π( D ) = ∂ W ( v , v ) . For each x ∈ R , Ψ c ( x ) = {− , } if x ∈ b C ∅ otherwise, Ψ d ( x ) = { u d ∈ R ≥ : u d ≥ γ | x | } if x ∈ ∂ W ( v , v ) , ∅ otherwise . During flows, we have that h∇ V ( x ) , f ( x, u c ) i = h∇ T ( x ) , f ( x, u c ) i V ( x )= √ ω r − (cid:16) | x | + x | x | (cid:17) (cid:28) ∇ | x | + x | x | , f ( x, u c ) (cid:29) V ( x )= u c ω h x | x | − x | x | i ω − ω x V ( x ) Technical Report6 for all ( x, u c ) ∈ C . For x ∈ b C , x > , h∇ T ( x ) , f ( x, u c ) i = 1 when u c = 1 , and for x ∈ b C , x < , h∇ T ( x ) , f ( x, u c ) i = − when u c = − . Then inf u c ∈ Ψ c ( x ) h∇ V ( x ) , f ( x, u c ) i ≤ − x ⊤ x (43)for all x ∈ Π( C ) . During jumps, we have that, for each ( x, u d ) ∈ D , V ( g ( x, u d )) = exp( T ( g ( x, u d ))) g ( x, u d ) ⊤ g ( x, u d )= exp (cid:16) π ω (cid:17) u d . It follows that inf u d ∈ Ψ d ( x ) V ( g ( x, u d )) − V ( x ) ≤ inf u d ∈ Ψ d ( x ) exp (cid:16) π ω (cid:17) u d − exp( T ( x )) x ⊤ x ≤ − (cid:16) − exp (cid:16) π ω (cid:17) γ (cid:17) x ⊤ x for each x ∈ Π( D ) . Finally, both (3) and (4) hold with s α ( s ) := (cid:0) − exp (cid:0) π ω (cid:1) γ (cid:1) s .Then, V is a CLF for (41).Now, we determine an asymptotic stabilizing control law for the above hybrid system. First,we compute the set-valued map T c in (6). To this end, the definition of Γ c gives, for each r ≥ , Γ c ( x, u c , r ) = u c ω h x | x | − x | x | i ω − ω x V ( x ) + α ( | x | A ) if ( x, u c ) ∈ C ∩ ( I ( r ) × R m c ) , −∞ otherwisefrom where we get Υ c ( x, u c ) = Γ c ( x, u c , V ( x )) . Then, for each r > and ( x, u c ) ∈ C ∩ ( I ( r ) × R m c ) , the set-valued map T c is given by T c ( x ) = Ψ c ( x ) ∩ { u c ∈ U c : Υ c ( x, u c ) ≤ } = {− , } ∩ ( { x > } ∪ {− x < } ) , which reduces to T c ( x ) = x > − x < (44)for each x ∈ Π( C ) ∩ { x ∈ R : V ( x ) > } . Technical Report7
Proceeding in the same way, the definition of Γ d gives, for each r ≥ , Γ d ( x, u d , r ) = exp (cid:16) π ω (cid:17) u d − V ( x ) + α ( | x | A ) if ( x, u d ) ∈ D ∩ ( I ( r ) × R m d ) , −∞ otherwisefrom where we get Υ d ( x, u c ) = Γ d ( x, u d , V ( x )) . Then, for each r > and ( x, u d ) ∈ D ∩ ( I ( r ) × R m d ) , the set-valued map T d is given by T d ( x ) = Ψ d ( x ) ∩ { u d ∈ U d : Υ d ( x, u d ) ≤ } = { u d ∈ R ≥ : u d ≥ γ | x | } ∩ n u d ∈ R ≥ : exp (cid:16) π ω (cid:17) u d − exp( T ( x )) x ⊤ x + α ( | x | A ) ≤ o = { u d ∈ R ≥ : u d ≥ γ | x | } ∩ n u d ∈ R ≥ : exp (cid:16) π ω (cid:17) u d − x ⊤ x + α ( | x | A ) ≤ o and using the definition of α , we get T d ( x ) = { u d ∈ R ≥ : u d ≥ γ | x | } ∩ n u d ∈ R ≥ : exp (cid:16) π ω (cid:17) u d − exp (cid:16) π ω (cid:17) γ | x | ≤ o = { u d ∈ R ≥ : u d ≥ γ | x | } ∩ { u d ∈ R ≥ : − γ | x | ≤ u d ≤ γ | x | } = { u d ∈ R ≥ : u d = γ | x | } (45)for each x ∈ Π( D ) ∩ { x ∈ R : V ( x ) > } . Then, according to (9), from (44), for each x ∈ Π( C ) ∩ { x ∈ R : V ( x ) > } we can take the pointwise minimum norm control selection ρ c ( x ) := x > − x < According to (10), from (45), for each x ∈ Π( D ) ∩ { x ∈ R : V ( x ) > } we can take thepointwise minimum norm control selection ρ d ( x ) := γ | x | . Figure 2 depicts a closed-loop trajectory with the control selections above when the region ofoperation is restricted to { x ∈ R : V ( x ) ≥ r } , r = 0 . . Example 4.2 (Impact control of a pendulum):
Consider a point-mass pendulum impacting ona controlled slanted surface. Denote the pendulum’s angle (with respect to the vertical) by x and the pendulum’s velocity (positive when the pendulum rotates in the clockwise direction) by x . When x ≥ µ with µ denoting the angle of the surface, its continuous evolution is given by ˙ x = x , ˙ x = − a sin x − bx + τ, Technical Report8 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2−1.5−1−0.500.511.52
PSfrag replacements CD x x W ( v , v ) W ( v , v ) Fig. 2. Closed-loop trajectory to the system in Example 4.1 starting from x (0 ,
0) = (2 , . and evolving within (cid:8) x ∈ R : V ( x ) ≥ r (cid:9) , r = 0 . . The lines at ± deg define the boundary of the flow and jump sets projected ontothe x plane. The r -contour plot of V is also shown. where a > , b ≥ capture the system constants (e.g., gravity, mass, length, and friction)and τ corresponds to torque actuation at the pendulum’s end. For simplicity, we assume that x ∈ [ − π , π ] and µ ∈ [ − π , . Impacts between the pendulum and the surface occur when x ≤ µ, x ≤ . (46)At such events, the jump map takes the form x +1 = x + e ρ ( µ ) x , x +2 = − e ( µ ) x , where the functions e ρ : [ − π/ , → ( − , and e : [ − π/ , → [0 , are continuous andcapture the effect of pendulum compression and restitution at impacts, respectively, as a functionof µ . The function e ρ captures rapid displacements of the pendulum at collisions while e modelsthe effect of the angle µ on energy dissipation at impacts. For a vertical surface ( µ = 0 ), e ρ is chosen such that e ρ (0) ∈ ( − , and e is chosen to satisfy e (0) = e , where e ∈ (0 , isthe nominal (no gravity effect) restitution coefficient. For slanted surfaces ( µ ∈ [ − π , ), whenconditions (46) hold, e ρ is chosen as x + e ρ ( µ ) x > x , e ρ ( µ ) ∈ ( − , , so that, after the impacts,the pendulum is pushed away from the contact condition, while the function e is chosen as a Technical Report9 nondecreasing function of µ satisfying e ≤ e ( µ ) < at such angles so that, due to the effectof the gravity force at impacts, less energy is dissipated as | µ | increases.The model above can be captured by the hybrid system H given by H ˙ x = x ˙ x = − a sin x − bx + u c, =: f ( x, u c )( x, u c ) ∈ C,x +1 = x + e ρ ( u d ) x x +2 = − e ( u d ) x =: g ( x, u d )( x, u d ) ∈ D, (47)where u c = [ u c, u c, ] ⊤ = [ τ µ ] ⊤ ∈ R × [ − π ,
0] =: U c , u d = µ ∈ [ − π ,
0] =: U d , C := n ( x, u c ) ∈ h − π , π i × R × U c : x ≥ u c, o ,D := n ( x, u d ) ∈ h − π , π i × R × U d : x ≤ u d , x ≤ o . Note that the definitions of C and D impose state constraints on the inputs.Let A = { (0 , } and consider the candidate control Lyapunov function with U controls for H given by V ( x ) = x ⊤ P x, P = . (48)During flows, we have that h∇ V ( x ) , f ( x, u c ) i = 4 x x + 2 x +2( − a sin x − bx + u c, )( x + x ) for all ( x, u c ) ∈ C . It follows that (3) is satisfied with α defined as α ( s ) := s for all s ≥ .In fact, note that, for each x ∈ R , Ψ c ( x ) = { u c : x ≥ u c, } = R × [ − π , min { x , } ] x ∈ [ − π , π ] ∅ x [ − π , π ] . and that Π( C ) = [ − π , π ] × R . Then inf u c ∈ Ψ c ( x ) h∇ V ( x ) , f ( x, u c ) i = − x ⊤ x Technical Report0 for all x ∈ Π( C ) such that x + x = 0 , while when x + x = 0 , we have inf u c ∈ Ψ c ( x ) h∇ V ( x ) , f ( x, u c ) i = −∞ . Note that, for each x ∈ R , we have Ψ d ( x ) = { u d : x ≤ u d } = [ x , x ∈ [ − π , , x ≤ ∅ otherwise , and that Π( D ) = [ − π , × ( −∞ , . Then, during jumps, we have inf u d ∈ Ψ d ( x ) V ( g ( x, u d )) − V ( x ) = V ( g ( x, x )) − V ( x ) ≤ − min { − (1 + e ρ ( x )) ) , − e ( x ) } x ⊤ x for all x ∈ Π( D ) . Then, condition (4) is satisfied with α defined as α ( s ) := λs for all s ≥ , λ := min x ∈ [ − π , { − (1 + e ρ ( x )) ) , − e ( x ) } . It follows that both (3) and (4) hold withthis choice of α .The definition of Γ c gives, for each r ≥ , Γ c ( x, u c , r ) = x x + 2 x + 2( − a sin x − bx + u c, )( x + x ) + α ( | x | A ) if ( x, u c ) ∈ C ∩ ( I ( r ) × R m c ) −∞ otherwisefrom where we get Υ c ( x, u c ) = Γ c ( x, u c , V ( x )) . Then, for each r > and ( x, u c ) ∈ C ∩ ( I ( r ) × R m c ) , the set-valued map T c is given by T c ( x ) = Ψ c ( x ) ∩ { u c ∈ U c : Υ c ( x, u c ) ≤ } = (cid:16) R × h − π , min { x , } i(cid:17) ∩ (cid:8) u c ∈ U c : 4 x x + 2 x + 2( − a sin x − bx + u c, )( x + x ) + α ( | x | A ) ≤ (cid:9) = n u c ∈ R × h − π , min { x , } i : 4 x x + 2 x + 2( − a sin x − bx + u c, )( x + x ) + λx ⊤ x ≤ o (49)for each x ∈ Π( C ) ∩ { x ∈ R : V ( x ) > } . Proceeding in the same way, the definition of Γ d gives, for each r ≥ , Γ d ( x, u d , r ) = − x (1 − (1 + e ρ ( u d )) ) − x (1 − e ( u d )) − x x (2 + e ρ ( u d )) e ( u d ) + α ( | x | A ) if ( x, u d ) ∈ D ∩ ( I ( r ) × R m d ) −∞ otherwise Technical Report1 from where we get Υ d ( x, u c ) = Γ d ( x, u d , V ( x )) . Then, for each r > and ( x, u d ) ∈ D ∩ ( I ( r ) × R m d ) , the set-valued map T d is given by T d ( x ) = Ψ d ( x ) ∩ { u d ∈ U d : Υ d ( x, u d ) ≤ } = n u d ∈ h − π , i : u d ∈ [ x , o ∩ (cid:8) u d ∈ R : − x (1 − (1 + e ρ ( u d )) ) − x (1 − e ( u d )) − x x (2 + e ρ ( u d )) e ( u d ) + λx ⊤ x ≤ (cid:9) = n u d ∈ h − π , i : − x (1 − (1 + e ρ ( u d )) ) − x (1 − e ( u d )) + λx ⊤ x ≤ o (50)where we dropped the term − x x (2 + e ρ ( u d )) e ( u d ) since on D we have that x x ≥ .Defining ψ ( x ) := 4 x x + 2 x + 2( − a sin x − bx )( x + x ) + λx ⊤ x , and ψ ( x ) := 2( x + x ) ,the (49) can be rewritten as T c ( x ) = n u c ∈ R × h − π , min { x , } i : ψ ( x ) + ψ ( x ) u c, ≤ o for each x ∈ Π( C ) ∩{ x ∈ R : V ( x ) > } . To determine the pointwise minimum norm controlselection according to (9), note that, when ψ ( x ) ≤ , then the pointwise minimum norm controlselection is u c, = 0 and that, when ψ ( x ) > , is given by − ψ ( x ) ψ ( x ) ψ ( x ) = − ψ ( x ) ψ ( x ) which leads to ψ ( x ) + ψ ( x ) u c, = 0 . Then, the pointwise minimum norm control selection isgiven by ρ c, ( x ) := − ψ ( x ) ψ ( x ) ψ ( x ) > ψ ( x ) ≤ ρ c, ( x ) := 0 on Π( C ) ∩ { x ∈ R : V ( x ) > } (see [6, Chapter 4]). According to (10), from (50), since e ρ maps to ( − , and e to (0 , , for each x ∈ Π( D ) ∩ { x ∈ R : V ( x ) > } , the pointwiseminimum norm control selection is given by ρ d ( x ) := 0 . Since ρ c, = ρ d , the selection above can be implemented.Figure 3 depicts a closed-loop trajectory on the plane with the control selections above whenthe region of operation is restricted to { x ∈ R : V ( x ) ≥ r } , r = 0 . . Figure 4 shows theposition and velocity trajectories projected on the t axis. The functions e ρ and e used in thesimulations are defined as e ρ ( s ) = 0 . s − . and e ( s ) = − . s + 0 . for each s ∈ [ − π/ , . Technical Report2 −0.5 0 0.5 1 1.5 2 2.5 3−10−505
PSfrag replacements CD x x W ( v , v ) W ( v , v ) Fig. 3. Closed-loop trajectory to the system in Example 4.2 on the plane starting from x (0 ,
0) = (2 , − and evolving within (cid:8) x ∈ R : V ( x ) ≥ r (cid:9) , r = 0 . . PSfrag replacements CD t [ sec ] t [ sec ] x x W ( v , v ) W ( v , v ) Fig. 4. Closed-loop position ( x ) and velocity ( x ) to the system in Example 4.2 starting from x (0 ,
0) = (2 , − and evolvingwithin (cid:8) x ∈ R : V ( x ) ≥ r (cid:9) , r = 0 . . Example 4.3 (Desynchronization of coupled timers with controlled resets):
Consider the hy-
Technical Report3 brid system with state x := τ τ ∈ P := [0 , ¯ τ ] × [0 , ¯ τ ] , with x , x being timer states with threshold ¯ τ > . The state x evolves continuously accordingto the flow map f ( x ) := when x ∈ C := P (51)The state x jumps when any of the timers expires. Defining inputs affecting the jumps by u d = ( u d, , u d, ) ∈ P , jumps will be triggered when ( x, u d ) ∈ D := { ( x, u d ) ∈ P × P : max { τ , τ } = ¯ τ } . (52)At jumps, if a timer x i reached the threshold ¯ τ , then it gets reset to the value of the respectiveinput component of u d,i , while if x j , j = i , did not reach the threshold then it gets reduced bya fraction of its value. More precisely, the jump map is given by g ( x, u d ) = g ( x , x , u d, ) g ( x , x , u d, ) ∀ ( x, u d ) ∈ D, where g is defined as g ( s , s , s ) = (1 + ε ) s if s < ¯ τ , s = ¯ τs if s = ¯ τ , s < ¯ τ { (1 + ε ) s , s } if s = ¯ τ , s = ¯ τ ∀ ( s , s ) ∈ Π( D ) , s ∈ P with parameter ε ∈ ( − , .We are interested in the asymptotic stabilization of the set A := { x ∈ P : | x − x | = k } , k > , (53)which, for an appropriate k , would correspond to the two timers being desynchronized sinceasymptotic stability of A would imply lim ( t,j ) ∈ dom x, t + j →∞ | x ( t, j ) − x ( t, j ) | = k > Technical Report4 for every complete solution x . Let k = ε +1 ε +2 ¯ τ , which for ε ∈ ( − , is such that k ∈ (0 , ¯ τ ) .Consider the candidate control Lyapunov function V : P → R given by V ( x ) = min {| x − x + k | , | x − x − k |} (54)Defining e A = e ℓ ∪ e ℓ ⊃ A , where e ℓ = { x : ¯ τ ¯ τε +2 + t ∈ P ∪ √ τ B , t ∈ R } , e ℓ = { x : ¯ τε +2 ¯ τ + t ∈ P ∪ √ τ B , t ∈ R } . (55)Note that e A is an inflation of A and is such that V ( x ) = | x | e A on P .Next, we construct the set-valued map Ψ d , and then check (4). Note that Π( D ) = { x : max { x , x } = ¯ τ } .For each x ∈ R , Ψ d ( x ) = P if x ∈ Π( D ) ∅ otherwise,We have the following properties. For all x ∈ C where V is differentiable, we obtain h∇ V ( x ) , f ( x ) i = 0 (56)For each ( x, u d ) ∈ D , we have that there exists i ∈ { , } such that x i = ¯ τ and x j ≤ ¯ τ . Withoutloss of generality, suppose that i = 1 and j = 2 . Then, η ∈ g ( x, u d ) , is such that η = u d, and η = (1 + ε ) x if x < ¯ τ , while η ∈ { (1 + ε ) x , u d, } and η ∈ { (1 + ε ) x , u d, } if x = ¯ τ .Then, for each ( x, u d ) ∈ D , V ( η ) − V ( x ) = min {| η − η + k | , | η − η − k |} − min {| x − ¯ τ + k | , | x − ¯ τ − k |} = min {| η − η + k | , | η − η − k |} − | x − ¯ τ + k | . Using the fact that k = ε ε ¯ τ , it follows that for every x ∈ Π( D ) , x = ¯ τ , x ≤ ¯ τ , η ∈ g ( x, u d ) ,we have inf u d ∈ Ψ d ( x ) V ( η ) − V ( x ) ≤ ε (cid:12)(cid:12)(cid:12)(cid:12) x − ¯ τ ε (cid:12)(cid:12)(cid:12)(cid:12) = ε || x − x | − k | = ε | x | e A . (57) Technical Report5
Proceeding similarly for every other point in D , we have that (4) holds with s α ( s ) := − εs .Now, we determine an asymptotic stabilizing control law for the above hybrid system. Wecompute the set-valued map T d in (6). To this end, the definition of Γ d gives, for each r ≥ , Γ d ( x, u d , r ) = max η ∈ g ( x,u d ) V ( η ) − V ( x ) + α ( | x | A ) if ( x, u d ) ∈ D ∩ ( I ( r ) × R m d ) −∞ otherwisefrom where we get Υ d ( x, u d ) = Γ d ( x, u d , V ( x )) . Then, for each r > and ( x, u d ) ∈ D ∩ ( I ( r ) ∩ R m d ) , the set-valued map T d is given by T d ( x ) = Ψ d ( x ) ∩ { u d ∈ U d : Υ d ( x, u d ) ≤ } = (cid:26) u d ∈ P : max η ∈ g ( x,u d ) V ( η ) − V ( x ) − ε | x | e A ≤ (cid:27) To determine the pointwise minimum norm control u d , consider again x = ¯ τ and x ≤ ¯ τ ,which implies that η ∈ g ( x, u d ) is such that η = u d, and η = (1 + ε ) x if x < ¯ τ , while η ∈ { (1 + ε ) x , u d, } and η ∈ { (1 + ε ) x , u d, } if x = ¯ τ . Then if x < ¯ τ T d ( x ) = (cid:8) u d ∈ P : min {| η − η + k | , | η − η − k |} − | x − ¯ τ + k | − ε | x | e A ≤ (cid:9) = (cid:8) u d ∈ P : min {| (1 + ε ) x − u d, + k | , | (1 + ε ) x − u d, − k |} − | x − ¯ τ + k | − ε | x | e A ≤ (cid:9) For each x < ¯ τ , ( u d, , u d, ) with u d, = 0 belongs to T d ( x ) since min {| (1 + ε ) x − k | , | (1 + ε ) x − − k |} − | x − ¯ τ + k | − ε | x | e A (58) = | (1 + ε ) x − k | − | x − ¯ τ + k | − ε || x − x | − k | (59) = | (1 + ε ) x − k | − (1 + ε ) | x − ¯ τ + k | (60) = (1 + ε ) (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) x − k ε (cid:12)(cid:12)(cid:12)(cid:12) − | x − ¯ τ + k | (cid:19) (61) = (1 + ε ) (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) x − k ε (cid:12)(cid:12)(cid:12)(cid:12) − | x − ¯ τ + k | (cid:19) (62) = 0 (63)since k ε = ¯ τ ε and − ¯ τ + k = − ¯ τ ε . When x = ¯ τ , then η = u d, and η = u d, are possiblevalues of η , in which case u d, = u d, = 0 belong to T d ( x ) . The same property holds for everyother possibility of η .Then, according to (10), for each x ∈ Π( D ) we can take the pointwise minimum norm controlselection ρ d ( x ) := 0 . Technical Report6 R EFERENCES [1] Z. Artstein. Stabilization with relaxed controls.
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