Further results on synergistic Lyapunov functions and hybrid feedback design through backstepping
aa r X i v : . [ m a t h . O C ] S e p Further results on synergistic Lyapunov functionsand hybrid feedback design through backstepping
Christopher G. Mayhew, Ricardo G. Sanfelice, and Andrew R. Teel
Abstract — We extend results on backstepping hybrid feed-backs by exploiting synergistic Lyapunov function and feedback(SLFF) pairs in a generalized form. Compared to existingresults, we delineate SLFF pairs that are “ready-made” anddo not require extra dynamic variables for backstepping. Froman (weak) SLFF pair for an affine control system, we constructan SLFF pair for an extended system where the control input isproduced through an integrator. The resulting hybrid feedbackasymptotically stabilizes the extended system when the “synergygap” for the original system is strictly positive. To highlight theversatility of SLFF pairs, we provide a result on the existenceof a SLFF pair whenever a hybrid feedback stabilizer exists.The results are illustrated on the “3D pendulum.”
I. I
NTRODUCTION
Hybrid feedbacks are commonly used to improve perfor-mance and achieve objectives that elude classical feedbackdesigns. Such objectives include global asymptotic stabiliza-tion of a point for a system evolving on a manifold that isnot topologically equivalent to a Euclidean space, or globalasymptotic stabilization of a disconnected set of points.In a recent series of results, synergistic potential functionsare developed and used to achieve robust, global asymptoticstabilization of planar orientation [1], orientation on the 2-sphere [2] (applied to the 3D pendulum in [3]), and rigid-body attitude [4], [5]. Synergistic potential functions areextended to synergistic Lyapunov function and feedback(SLFF) pairs in [6]. For a continuous-time system withembedded logic variables, a Lyapunov function and feedbackpair is synergistic when, at places in the state-space where thefeedback is ineffective, the logic variable can be switched todecrease the value of the Lyapunov function. The magnitudeof the available decrease is called the synergy gap . In [6],the synergy gap is defined as an infimum over an appropriatesubset of the state space and it is required to be positivefor control synthesis and backstepping. In this note, thesynergy gap is state dependent. It must be positive awayfrom a desired compact set and everywhere positive forbackstepping that achieves global asymptotic stability.Earlier control algorithms propose a similar scheme ex-ploiting multiple Lyapunov functions. Some have appearedin the context of adaptive control using hysteresis [7], [8]and supervisory control systems [9]. Applications using thisfeedback scheme have appeared for swing-up and stabiliza-tion of an inverted pendulum [9], [10] and for control of a
Dr. Mayhew is with the Robert Bosch Research and Technology Center,Palo Alto, CA, 94304. Dr. Sanfelice is with Department of Aerospaceand Mechanical Engineering, University of Arizona, Tucson, AZ 85721-0119. Dr. Teel is with the ECE Department, University of California, SantaBarbara, CA 93106-9560. Research supported in part by the AFOSR grantFA9550-09-1-0203 and the NSF grants ECCS-0925637, and CNS-0720842. double-tank system [11]. Multiple Lyapunov functions arealso proposed for control and analysis in [12].In Section III we construct a robustly globally asymptot-ically stabilizing hybrid feedback algorithm using an SLFFpair. In Sections V–VII, we broaden the applicability of(weak) SLFF pairs through backstepping. Starting from aweak SLFF pair for an affine control system, we constructan (non-weak) SLFF pair for an extended system where thecontrol is produced through an integrator. Results of this typefor continuous-time systems can be found in [13, Lemma2.8(ii)] and [14, Theorem 5.3]. Similar results for switchedsystems appear in [15]; however, the notion of synergism thatis crucial for ensuring global asymptotic stability does notappear in [15]. We provide a variety of backstepping results: • The backstepping algorithm resembles classical back-stepping when the assumed weak SLFF pair is pure and is ready-made relative to a quadratic function. An SLFF pairis pure when the Lyapunov function is non-increasing alongsolutions at every point in the state space when using thefeedback. An SLFF pair is ready-made when there is anappropriate relationship between the size of the jumps inthe feedback law and the synergy gap of the SLFF pair.These definitions are made precise in Section IV and thebackstepping algorithm is described in Section V-A. • If the weak SLFF pair is not pure, a backstepping resultcan still be obtained when the SLFF is ready-made relativeto a linear function. See the algorithm in Section V-B. • At times, backstepping may not be needed but it stillmay be desirable to smooth jumps in the control signal. Thissituation is addressed in Section VI, where the ideal feedbackis written in a form that is affine in a function of the logicmode; the latter is then treated as an ideal feedback andimplemented dynamically through backstepping. • For backstepping problems where the SLFF pair is notready-made, the extra dynamic variable described in thepreceding item can be exploited to achieve a backsteppingresult. See Section VII. This idea also appears in [6].
Notation and terminology: R ( R ≥ ) denotes the (nonnega-tive) real numbers, and R n denotes n -dimensional Euclideanspace. Given x ∈ R n , | x | denotes its Euclidean norm. Theunit n -sphere is S n = { x ∈ R n +1 : | x | = 1 } . A function iscalled smooth if a sufficient number of its derivatives existand are continuous so that the derivations make sense. Anonnegative-valued function is said to be positive definitewith respect to a set if the function is zero if and only if itsargument belongs to the set. For a closed set X ⊂ Q × R n ,where Q ⊂ R is a finite set, and a smooth function V : X → R , we use ∇ V ( q, z ) to denote gradient of V relativeo z ∈ R n , with q ∈ Q considered to be constant. Given asmooth function κ : X → R m , we use D κ ( q, z ) to denotethe Jacobian matrix of κ relative to z , i.e., D κ ( q, z ) is an R m × n matrix with ij -th entry given as ∂κ i ( q, z ) /∂z j . Asin [16], a hybrid system with state x ∈ R n is described byflow and jump sets C, D ⊂ R n and set-valued flow and jumpmaps F, G : R n ⇒ R n . It satisfies the basic conditions [16]if C and D are closed, F and G are outer semicontinuousand locally bounded, F ( x ) is nonempty and convex for all x ∈ C , and G ( x ) is nonempty for all x ∈ D .II. SLFF PAIRS
We extend the definition of a synergistic Lyapunov func-tion and feedback pair defined in [6]. Consider the system ˙ q = 0˙ z = f ( q, z, ω ) ) ( q, z ) ∈ X , (1)where f : X × R m → R n is continuous, ω ∈ R m is thecontrol variable, the set X ⊂ Q × R n is closed and the set Q ⊂ R finite. Let A ⊂ Y ⊂ X be such that A is compactand Y is closed. We define the set B := { ( q, z ) ∈ X : ∃ s ∈ Q, ( s, z ) ∈ A} . (2)A C function V : X → R ≥ and a continuous function κ : X → R m form a synergistic Lyapunov function andfeedback (SLFF) pair candidate relative to ( A , Y ) if • { ( q, z ) ∈ X : V ( q, z ) ≤ c } is compact for each c ≥ ; • V ( q, z ) = 0 if and only if ( q, z ) ∈ A , • For all ( q, z ) ∈ Y , h∇ V ( q, z ) , f ( q, z, κ ( q, z )) i ≤ . (3)Given an SLFF pair candidate ( V, κ ) , define E := { ( q, z ) ∈ Y : h∇ V ( q, z ) , f ( q, z, κ ( q, z )) i = 0 } (4)and let Ψ ⊂ E be the largest weakly invariant set [17] for ˙ q = 0˙ z = f ( q, z, κ ( q, z )) ) ( q, z ) ∈ E . (5)For each ( q, z ) ∈ X , define µ V ( q, z ) := V ( q, z ) − min s ∈ Q V ( s, z ) . (6)The pair ( V, κ ) is called a synergistic Lyapunov function andfeedback pair relative to ( A , Y ) if µ V ( q, z ) > ∀ ( q, z ) ∈ (cid:16) Ψ ∪ X \ Y (cid:17) \ A , (7)in which case µ V ( q, z ) is called the synergy gap at ( q, z ) .Given a continuous function δ : X → R ≥ , when µ V ( q, z ) > δ ( q, z ) ∀ ( q, z ) ∈ (cid:16) Ψ ∪ X \ Y (cid:17) \ A (8)we say that the synergy gap exceeds δ . When δ satisfies µ V ( q, z ) > δ ( q, z ) ∀ ( q, z ) ∈ (cid:16) Ψ ∪ X \ Y ∪ B (cid:17) \ A , (9) we say that the synergy gap totally exceeds δ . Where thesynergy gap is positive, we can change q to reduce V , whichis desirable at points in Ψ \ A , where the value of V couldget stuck during flows, at points in (cid:16) X \ Y (cid:17) \ A , where the q -th feedback function is not effective, and possibly at pointsin B \ A to ensure that the set B is stabilized. Proposition 1: The synergy gap is a continuous function.If ( V, κ ) is an SLFF pair, then there exists a continuousfunction δ : X → R ≥ that is positive on X \ A suchthat the synergy gap (totally) exceeds δ . If the synergy gapfor ( V, κ ) (totally) exceeds the function δ then, for eachsmooth K ∞ -function ρ having a positive, nondecreasingderivative denoted ρ ′ , the pair ( ρ ( V ) , κ ) is an SLFF withsynergy gap (totally) exceeding the function ˜ δ defined as ˜ δ ( q, z ) := ρ ′ ( cV ( q, z ))(1 − c ) δ ( q, z ) , where c can be takenarbitrarily in the interval (0 , . We show that the existence of an SLFF pair relative to thecompact set A is equivalent to the existence of a feedback ω = α ( q, z ) ( q, z ) ∈ C ⊂ X q + ∈ G c ( q, z ) ⊂ Q ( q, z ) ∈ D ⊂ X (10)satisfying the basic conditions and the conditions A ⊂ C , C ∪ D = X , and rendering the compact set A globallyasymptotically stable for the system (1), (10), that is, for ˙ q = 0˙ z = f ( q, z, α ( q, z )) | {z } ( q, z ) ∈ C q + ∈ G c ( q, z ) z + = z | {z } ( q, z ) ∈ D. (11)We start by showing that this asymptotic stabilizabilityproperty implies the existence of an SLFF. The oppositeimplication is established in Theorem 2 given in Section III. Theorem 1: Suppose the data of (11) satisfies the basicconditions, the compact set A is globally asymptoticallystable for (11), A ⊂ C , and C ∪ D = X . Then there existsa smooth function V : X → R ≥ such that ( V, α ) is anSLFF pair relative to ( A , Y ) with Y = C and there exists ε > such that the synergy gap (totally) exceeds δ where δ ( q, z ) := εV ( q, z ) . If, in addition, D ∩ A = ∅ (and B \ A isclosed) then there exist ε > , ε > such that the synergygap (totally) exceeds δ with δ ( q, z ) := ε V ( q, z ) + ε . III. H
YBRID FEEDBACK FROM AN
SLFF
PAIR
Let ( V, κ ) denote the SLFF pair and let δ : X → R ≥ be continuous. We specify a hybrid controller to globallyasymptotically stabilize A (and B ) for (1) as C := { ( q, z ) ∈ X : µ V ( q, z ) ≤ δ ( q, z ) } ω := κ ( q, z ) D := { ( q, z ) ∈ X : µ V ( q, z ) ≥ δ ( q, z ) } G c ( z ) := { g c ∈ Q : µ V ( g c , z ) = 0 } (12)resulting in the closed-loop hybrid system ˙ q = 0˙ z = f ( q, z, κ ( q, z )) | {z } ( q, z ) ∈ C q + ∈ G c ( z ) z + = z | {z } ( q, z ) ∈ D. (13)ince δ and µ V are continuous, C and D are closed. Since µ V is continuous, G c is outer semicontinuous. Also, C ∪ D = X and G c ( z ) is non-empty for each z such that ( q, z ) ∈ X for some q ∈ Q , in particular, for ( q, z ) ∈ D . Theorem 2: Let
Y ⊂ X , let A ⊂ Y be compact, and let δ : X → R ≥ be continuous and positive on X \ A . If ( V, κ ) is an SLFF pair for (1) relative to ( A , Y ) with synergy gap(totally) exceeding δ , then A ⊂ C and A ( B ) is globallyasymptotically stable for the closed-loop system (13). IV. R
EFINEMENT OF
SLFF
PAIR PROPERTIES
A. Weak SLFF pairs for affine control systems
We introduce a weak synergistic Lyapunov function andfeedback pair (weak SLFF) for (1) when f ( q, z, ω ) = φ ( q, z ) + ψ ( q, z ) ω where φ and ψ are smooth. Given anSLFF pair candidate ( V, κ ) , with V and κ smooth, define W := (cid:8) ( q, z ) ∈ X : ∇ V ( q, z ) ⊤ ψ ( q, z ) = 0 (cid:9) . (14)Recall the definition of E in Section II and let Ω ⊂ E ∩ W denote the largest weakly invariant set for the system ˙ q = 0˙ z = φ ( q, z ) + ψ ( q, z ) κ ( q, z ) ) ( q, z ) ∈ E ∩ W . (15)The pair ( V, κ ) is called a weak synergistic Lyapunov func-tion and feedback pair relative to ( A , Y ) if µ V ( q, z ) > ∀ ( q, z ) ∈ (cid:16) Ω ∪ X \ Y (cid:17) \ A . (16)Given a continuous function δ : X → R ≥ , when µ V ( q, z ) > δ ( q, z ) ∀ ( q, z ) ∈ (cid:16) Ω ∪ X \ Y (cid:17) \ A , (17)we say that the synergy gap weakly exceeds δ . If δ satisfies µ V ( q, z ) > δ ( q, z ) ∀ ( q, z ) ∈ (cid:16) Ω ∪ X \ Y ∪ B (cid:17) \ A , (18)we say that the synergy gap weakly totally exceeds δ . Thenext lemma follows immediately from the fact that Ω ⊂ Ψ and then comparing (17) to (8). Lemma 1: If ( V, κ ) is a smooth SLFF pair with synergygap (totally) exceeding δ then it is also a weak SLFF pairwith synergy gap weakly (totally) exceeding δ .Example 1 (3-D Pendulum): The reduced dynamics ofthe 3-D pendulum are given in [18] as ˙ z = [ z ] × ω (19a) J ˙ ω = [ Jω ] × ω + mg [ ν ] × z + τ, (19b)where z ∈ S is the direction of gravity in the body-fixedframe, ω ∈ R is the angular velocity expressed in the body-fixed frame, m is the mass, g is the gravitational constant, ν is the vector from the pivot location to the center of mass, τ ∈ R is a vector of input torques, and for any x, y ∈ R , [ x ] × is the × skew-symmetric matrix that satisfies [ x ] × y = x × y , where × denotes the vector cross product.We now stabilize the “inverted” point ( z, ω ) = ( − ν/ | ν | , .Let Q be a finite set, X = Q × S , S ⊂ Q , and A = S × {− ν/ | ν |} . Let V : X → R be positive definite on X relative to A and define κ ( q, z ) = 0 . Clearly, wehave that (cid:10) ∇ V ( q, z ) , [ z ] × κ ( q, z ) (cid:11) = 0 for all ( q, z ) ∈ X so that Y = X = E and Ω = W = { ( q, z ) ∈ X : ∇ V ( q, z ) ⊤ [ z ] × = 0 } . The pair ( V , κ ) is then a weak SLFFpair for (19a) relative to ( A , X ) if inf ( q,z ) ∈ Ω \A µ V ( q, z ) > . (20)To satisfy (20), we may use the synergistic potential func-tions of [3], [2]. We henceforth assume that the synergy gapweakly totally exceeds a constant δ ( q, z ) = c > . (cid:3) B. Pure and Ready-made SLFF pairs
When Y = X , a (weak) SLFF pair is called a (weak) pure SLFF pair. A weak SLFF pair ( V, κ ) with synergy gapweakly (totally) exceeding δ : X → R ≥ is said to be type Iready-made with respect to the continuous, positive definitefunction σ : R m → R ≥ if there exists a continuous function ̺ : X → R ≥ such that, ∀ ( q, z ) ∈ X , s ∈ Q , and ω = κ ( q, z ) , σ (cid:0) ω − κ ( s, z ) (cid:1) − σ (cid:0) ω − κ ( q, z ) (cid:1) ≤ ̺ ( q, z ) (21)and, for all ( q, z ) ∈ (Ω \ A ) ∪ X \ Y , µ V ( q, z ) > δ ( q, z ) + ̺ ( q, z ) . (22)Since µ V ( q, z ) = 0 for ( q, z ) ∈ A , the type I ready-madeproperty implies that X \ Y ∩ A = ∅ . (23)If κ does not depend on q then, in (21), we can take ̺ ( q, z ) =0 for all ( q, z ) ∈ X . With this choice for ̺ , if (23) holdsthen (22) follows from (17). According to the last statementof Proposition 1, if δ is positive valued, V is radiallyunbounded, and the condition (23) holds, then the type Iready-made property is achievable for any σ by modifyingthe function V as ρ ( V ) with ρ chosen appropriately.A weak SLFF pair ( V, κ ) with synergy gap weakly(totally) exceeding δ : X → R ≥ is said to be type IIready-made with respect to the continuous, positive definitefunction σ : R m → R ≥ if there exists a continuous function ̺ : X → R ≥ such that, for all ( q, z ) ∈ X , s ∈ Q ,and ω ∈ R m , (21) holds and, moreover, (22) holds forall ( q, z ) ∈ (Ω \ A ) ∪ X \ Y . In particular, the differencebetween type I and type II ready-made is in the requirementon ω for which (21) holds: ω = κ ( q, z ) for type I and ω ∈ R m for type II. Clearly, if the SLFF pair is type IIready-made then it is type I ready-made. Like for the typeI case, if κ is independent of q then, in (21), we can take ̺ ( q, z ) = 0 for all ( q, z ) ∈ X . Example 2 (3-D pendulum):
The weak SLFF pair ( V , κ ) for (19a) with synergy gap weakly totallyexceeding c > is type I/II ready made with respect toany positive definite function σ and appropriate function ̺ ,since κ ( q, z ) ≡ does not depend on q . (cid:3) V. R
EADY - MADE BACKSTEPPING
The ensuing backstepping results are useful mainly for thecase where the SLFF pair for the reduced-order system has aynergy gap (totally) exceeding a positive-valued continuousfunction δ , i.e., δ : X → R > . Indeed, the nature of ourbackstepping results is that the extended system admits anSLFF pair with synergy gap (totally) exceeding the samefunction δ . If δ is not positive valued then, since it does notdepend on the extended state, there is no hope of it beingpositive valued away from the attractor in the extended statespace. In this case, the hybrid control construction based onan SLFF pair given in Theorem 2 would not be applicable. A. From a weak, pure, ready-made SLFF pair
We consider the control system ˙ q = 0˙ ζ = φ ( q, ζ ) + ψ ( q, ζ ) u (cid:27) ( q, ζ ) ∈ X (24)with u ∈ R m , where ζ = ( z ⊤ , ω ⊤ ) ⊤ , X = X × R m and φ ( q, ζ ) = (cid:20) φ ( q, z ) + ψ ( q, z ) ω (cid:21) , ψ ( q, ζ ) = (cid:20) (cid:21) . (25)We construct a (non-weak) SLFF pair with synergy gapexceeding a positive-valued function δ by supposing wehave a weak, pure, ready-made SLFF pair with synergy gapweakly (totally) exceeding δ for the reduced system ˙ q = 0˙ z = φ ( q, z ) + ψ ( q, z ) ω (cid:27) ( q, z ) ∈ X (26)with controls ω ∈ R m .Let A ⊂ X be compact. For the system (26), let ( V , κ ) be a weak SLFF pair relative to ( A , X ) , with synergy gapweakly (totally) exceeding the continuous function δ : X → R ≥ . Let Γ ∈ R m × m be a symmetric, positive definite matrixand suppose that the SLFF pair is type I ready-made relativeto σ ( v ) := v ⊤ Γ v . Define A := { ( q, ζ ) ∈ X : ( q, z ) ∈ A , ω = κ ( q, z ) } . (27)For each ( q, ζ ) ∈ X , define V ( q, ζ ) = V ( q, z ) + σ ( ω − κ ( q, z )) . (28)Let θ : R ≥ → R ≥ be a continuous, positive definitefunction, and let the smooth function Θ : R m → R m satisfy v ⊤ ΓΘ( v ) + Θ( v ) ⊤ Γ v ≤ − θ ( | v | ) ∀ v ∈ R m . (29)Define κ ( q, ζ ) := Θ( ω − κ ( q, z )) − Γ − ψ ( q, z ) ⊤ ∇ V ( q, z )+ D κ ( q, z )( φ ( q, z ) + ψ ( q, z ) ω ) . (30) Theorem 3: Let the compact set A and the smooth func-tions ( V , κ ) be given. Let the compact set A be definedas in (27) and let the pair ( V , κ ) be defined by (28)-(30). Suppose, for the system (26), that the pair ( V , κ ) is a weak SLFF relative to the pair ( A , X ) with synergygap weakly (totally) exceeding the continuous function δ : X → R ≥ and the SLFF pair is type I ready-made relativeto the function σ defined as σ ( v ) = v ⊤ Γ v where Γ is asymmetric positive definite matrix. Under these conditions,for the system (24)-(25), the pair ( V , κ ) is an (non-weak) SLFF pair relative to the pair ( A , X ) with synergy gap(totally) exceeding δ .Example 3 (3-D pendulum): Consider the input transfor-mation τ = − [ Jω ] × ω − mg [ ν ] × z + Ju , which rendersthe angular velocity dynamics (19b) as ˙ ω = u . We nowapply Theorem 3. Let σ ( ω ) = ω ⊤ Jω (i.e. Γ = J/ ) andlet Θ( ω ) = J − (cid:0) [ Jω ] × ω − Ξ( ω ) (cid:1) , where Ξ : R → R satisfies ω ⊤ Ξ( ω ) ≥ θ ( | ω | ) and θ : R → R is a continuous,positive definite function. Applying (30), we arrive at u ( q, z ) = J − (cid:0) [ Jω ] × ω − Ξ( ω ) (cid:1) − J − [ z ] ⊤× ∇ V ( q, z ) , which yields τ = κ ( q, z ) = − mg [ ν ] × z − Ξ( ω ) − [ z ] ⊤× ∇ V ( q, z ) . (31)This recovers the feedback of [3]. As a result of Theorem 3, itfollows that ( V , κ ) , with V ( q, z, ω ) = V ( q, z ) + ω ⊤ Jω ,is an SLFF pair for (19) relative to ( A , X ) , where X = Q × S × R and A = { ( q, z, ω ) ∈ X : q ∈ S, z = − ν/ | ν | , ω = 0 } , with gap totally exceeding δ ( q, z ) = c . (cid:3) B. From a weak, ready-made SLFF pair
We again consider the control system (24)-(25). Let A ⊂ X be compact and let Y ⊂ X be closed. The results inthis section apply to the case where Y is not necessarilyequal to X . For the system (26), let ( V , κ ) be a weak,SLFF pair relative to ( A , Y ) with synergy gap weakly(totally) exceeding the continuous function δ : X → R ≥ . Inaddition, suppose the SLFF pair is type I ready-made relativeto σ ( v ) := L | v | where L > . Let Γ ∈ R m × m be a positivedefinite, symmetric matrix. Define σ ( v ) := v ⊤ Γ v for all v ∈ R m . Let ρ ∈ K ∞ be smooth, such that ρ ′ ( s ) > for all s ≥ , and such that ρ ◦ σ is globally Lipschitz with constantless than or equal to L . For example, pick ρ ( s ) = c ˜ ρ ( s ) where c > is sufficiently small and ˜ ρ ( s ) = s for s ∈ [0 , , ˜ ρ ( s ) = k √ s for s ≥ where k ≥ , and such that ˜ ρ ′ ( s ) > for s ∈ [1 , to smoothly connect the value at s = 1 tothe value k √ at the value s = 2 . This construction makesthe SLFF pair ( V , κ ) type II ready-made for the function v ρ ( σ ( v )) . Define Y := Y × R m , V ( q, ζ ) := V ( q, z ) + ρ ( σ ( ω − κ ( q, z ))) . (32)and κ ( q, ζ ) := 1 ρ ′ ( σ ( ω − κ ( q, z ))) (cid:20) Θ( ω − κ ( q, z )) − Γ − ψ ( q, z ) ⊤ ∇ V ( q, z )+ D κ ( q, z )( φ ( q, z ) + ψ ( q, z ) ω ) (cid:21) . (33) Theorem 4: Let the compact set A ⊂ X and the closedset Y ⊂ X be given. Let ρ and σ be such that σ ( v ) = v ⊤ Γ v for all v ∈ R m where Γ ∈ R m × m is a symmetric,positive definite matrix, ρ ∈ K ∞ is smooth, ρ ′ ( s ) > for all s ≥ , and v ρ ( σ ( v )) is globally Lipschitz with constantless than or equal to L > . Let the compact set A bedefined as in (27) and let the pair ( V , κ ) be defined by(32)-(33). Suppose, for the system (26), that the pair ( V , κ ) s a weak SLFF pair relative to ( A , Y ) with synergy gapweakly (totally) exceeding the continuous function δ : X → R ≥ and the SLFF pair is type I ready-made relative to thefunction σ given by σ ( v ) := L | v | for all v ∈ R m . Underthese conditions, for the system (24)-(25), the pair ( V , κ ) is a (non-weak) SLFF pair relative to ( A , Y ) with synergygap (totally) exceeding δ . VI. S
MOOTHING WITHOUT BACKSTEPPING
Now we consider the situation where the control does notenter through an integrator but we want to remove jumpsfrom the feedback. The ideas described here are also usedin Section VII for a backstepping algorithm that does notrequire the SLFF pair to be ready-made. Henceforth, wework with SLFF pairs having a synergy gap bounded awayfrom a function δ . The synergy gap is said to be (totally)bounded away from a continuous function δ : X → R ≥ ifthere exists a positive real number ε such that the energy gap(totally) exceeds the function ( q, z ) ˜ δ ( q, z ) := δ ( q, z ) + ε .We note that if the synergy gap is (totally) bounded awayfrom a continuous function δ : X → R ≥ then, because µ V ( q, z ) = 0 for ( q, z ) ∈ A , it follows that X \ Y ∩ A = ∅ .We start with the control system ˙ q = 0˙ z = φ ( q, z ) + ψ ( q, z ) ω (cid:27) ( q, z ) ∈ X (34)with controls ω ∈ R m for which we suppose we have a (non-weak) SLFF pair relative to the pair ( A , X ) where A ⊂ X is compact with synergy gap (totally) bounded away fromthe function δ : X → R ≥ . Let M be the projection of X in the z direction, i.e., M := { z ∈ R n : ( q, z ) ∈ X for some q ∈ Q } . Let N be the cardinality of Q , let r ≤ N , and let the smoothfunctions β : M → R m and ϑ : M → R m × r satisfy κ ( q, z ) = β ( z ) + ϑ ( z ) ς ( q ) ∀ ( q, z ) ∈ X , (35)where ς : Q → R r is some function of the variable q . Inturn, we see that, for the system ˙ q = 0˙ z = φ ( q, z ) + ψ ( q, z ) ( β ( z ) + ϑ ( z ) p ) (cid:27) ( q, z ) ∈ X with p as the control variable, the pair ( V , ς ) is an SLFFpair with respect to ( A , X ) with synergy gap (totally)bounded away from δ . Let ε > be such that the synergygap (totally) exceeds ˜ δ ( q, z ) := δ ( q, z ) + ε . Since the set Q is finite, we can easily find a positive definite, symmetricmatrix Γ such that, with σ ( v ) = v ⊤ Γ v , we have σ ( ς ( q ) − ς ( s )) ≤ ε ∀ ( q, s ) ∈ Q × Q. (36)This implies that the SLFF pair ( V , ς ) , with a synergy gap(totally) exceeding ˜ δ ( q, z ) := δ ( q, z ) + ε/ , is type I ready-made for backstepping relative to σ . Like in Section V-B,we can also find a function ρ so that, for all ( p, q, s ) ∈ R r × Q × Q , ρ ( σ ( p − ς ( s ))) − ρ ( σ ( p − ς ( q ))) ≤ ε . (37) In particular, this implies the SLFF pair ( V , ς ) , with asynergy gap (totally) exceeding ˜ δ ( q, z ) := δ ( q, z ) + ε/ ,is type II ready-made for backstepping relative to ρ ◦ σ .Now, using Lemma 1, and depending on whether theoriginal pair ( V , κ ) was pure or not, we can apply eitherTheorem 3 or Theorem 4 to construct a pair ( V , ς ) that isan SLFF pair with synergy gap (totally) exceeding ˜ δ for ˙ q = 0˙ z = φ ( q, z ) + ψ ( q, z ) ( β ( z ) + ϑ ( z ) p ) , ˙ p = α (cid:27) ( q, z, p ) ∈ X × R r . In particular, from the definition of ˜ δ , it follows that thesynergy gap is (totally) bounded away from δ .Note that if ( V , κ ) was a weak SLFF pair for the system(34), this fact would not necessarily guarantee that ( V , ς ) is a weak SLFF pair for (36), because of the ϑ termthat multiplies ψ to generate the input vector field. Thisobservation motivates assuming that ( V , κ ) is an (non-weak) SLFF pair for the system (34). In the next section, wewill want to allow ( V , κ ) to be a weak SLFF pair for thesystem (34) in anticipation of another backstepping result.We will be able to get away with this weakened assumptionbecause we will come back to the integral of ω , rather thanthe integral of p , as the control variable. Example 4 (3-D pendulum):
Let e i ∈ R N denote thevector with in the i th index and zeros elsewhere. Assumingthat (without loss of generality) Q = { , . . . , N } , κ —asdefined in (31)—can be written as (35). In particular, define V ( z ) = [ V (1 , z ) · · · V ( N, z )] ⊤ ϑ ( z ) = [ z ] × DV ( z ) ⊤ β ( z ) = − mg [ ν ] × z − Ξ( ω ) ς ( q ) = e q , which yields the closed-loop dynamics of (19) as ˙ z = [ z ] × ω J ˙ ω = [ Jω ] × ω − Ξ( ω ) + ϑ ( z ) e q . By replacing e q with a control variable p , we have that ( V , ς ) is a (non-weak) SLFF pair relative to ( A , X ) (with V , A and X defined in Example 3) with synergy gaptotally exceeding δ ( q, z, ω ) = c . Suppose also that thesynergy gap totally exceeds c + ǫ and let σ ( v ) = ǫ | v | sothat for all ( q, s ) ∈ Q × Q , σ ( e q − e s ) ≤ ǫ/ and ( V , ς ) isalso type I ready-made with respect to σ .Now, define V ( q, z, ω, p ) = V ( q, z, ω )+ σ ( p − e q ) , X = Q × S × R × R N , A = { ( q, z, ω, p ) ∈ X : q ∈ S, z = − ν/ | ν | , ω = 0 p = e q } , and γ ( q, z, ω, p ) = Θ( p − e q ) − DV ( z ) [ z ] × ω, where Θ : R N → R N satisfies (29) with Γ = I . It followsfrom Theorem 3 that ( V , γ ) is an SLFF pair relative to ( A , X ) with synergy gap totally exceeding c + ǫ/ for thesystem ˙ q = 0 ˙ z = [ z ] × ω ˙ p = α J ˙ ω = [ Jω ] × ω − Ξ( ω ) + ϑ ( z ) p with α as the control variable.Having input ( z, ω ) ∈ S × R , memory states ( q, p ) ∈ Q × R N , and output τ , the hybrid controller for the 3-Dendulum with smoothing is given as τ = β ( z ) + ϑ ( z ) p, ˙ q = 0˙ p = Θ( p − e q ) − DV ( z ) [ z ] × ω | {z } ( q, z, ω, p ) ∈ C q + = G ( z, ω, p ) p + = p | {z } ( q, z, ω, p ) ∈ D, where C = { ( q, z, ω, p ) ∈ X : µ V ( q, z, ω, p ) ≤ c + ǫ/ } D = { ( q, z, ω, p ) ∈ X : µ V ( q, z, ω, p ) ≥ c + ǫ/ } G ( z, ω, p ) = { g ∈ Q : µ V ( g, z, ω, p ) = 0 } . If V satisfies (20), this controller globally asymptoticallystabilizes B , where B is related to A through (2). (cid:3) VII. B
ACKSTEPPING WITHOUT BEING READY - MADE
While the backstepping constructions in this section useextra dynamic states, their advantage is that no preliminarystep is needed to make them ready-made for backstepping.Suppose we have a non-weak SLFF pair ( V , κ ) withsynergy gap (totally) bounded away from δ for ˙ q = 0˙ z = φ ( q, z ) + ψ ( q, z ) ω (cid:27) ( q, z ) ∈ X . (38)From the results of Section VI, the pair ( V , κ ) , of the form V ( q, z, p ) = V ( q, z ) + σ ( p − ς ( q )) κ ( q, z, p ) = β ( z ) + ϑ ( z ) p, (39)is a non-weak SLFF with synergy gap (totally) bounded awayfrom δ for the system ˙ q = 0˙ z = φ ( q, z ) + ψ ( q, z ) ω ˙ p = ς ( q, z, p ) ( q, z, p ) ∈ X × R r . (40)Moreover, the pair ( V , κ ) is both type I and type II ready-made with respect to any function. Indeed, since κ does notdepend on q , we can take ̺ ( q, z, p ) = 0 for all ( q, z, p ) ∈ X × R r in (21) and then, since (23) holds because thesynergy gap is (totally) bounded away from δ , (22) holds.Now we can apply Theorem 3 or, if the SLFF pair is not pure,Theorem 4 to generate a non-weak SLFF pair ( V , κ ) withsynergy gap (totally) bounded away from δ for the extendedsystem ˙ q = 0˙ z = φ ( q, z ) + ψ ( q, z ) ω ˙ p = ς ( q, z, p )˙ ω = u ( q, z, p, ω ) ∈ X × R r × R m . (41)Finally, consider the case where ( V , κ ) is a weak (ratherthan non-weak) SLFF pair for (38). In this case it turns outthat the SLFF pair ( V , κ ) of the form (39) is a weak SLFFpair for the system (40). This fact is explained below. Fromhere, Theorem 3 or 4 can be applied as above to derive a non-weak SLFF pair ( V , κ ) for the system (41).Suppose ( V , κ ) is a weak SLFF pair for (38). Write the system (40) in the form ˙ q = 0˙ ζ = φ ( q, ζ ) + ψ ( q, ζ ) ω (cid:27) ( q, ζ ) ∈ X (42)where ζ := ( z ⊤ , p ⊤ ) ⊤ , X := X × R r , φ ( q, ζ ) := (cid:20) φ ( q, z ) ς ( q, z, p ) (cid:21) , ψ ( q, ζ ) := (cid:20) ψ ( q, z )0 (cid:21) . (43)It follows from the definitions that ∇ V ( q, ζ ) ⊤ ψ ( q, ζ ) = ∇ V ( q, z ) ⊤ ψ ( q, z ) . Also, it follows from the proof of Theorems 3 and 4 that h∇ V ( q, z ) , φ ( q, ζ ) + ψ ( q, ζ ) κ ( q, ζ ) i = 0= ⇒ (cid:26) h∇ V ( q, z ) , φ ( q, z ) + ψ ( q, z ) κ ( q, z ) i p = ς ( q ) . Therefore Ω = { ( q, ζ ) ∈ X : ( q, z ) ∈ Ω , p = ς ( q ) } . This relationship can be used to arrive at the conclusion that ( V , κ ) is a weak SLFF for the system (40) with synergygap (totally) bounded away from δ .R EFERENCES[1] C. G. Mayhew and A. R. Teel, “Hybrid control of planar rotations,” in
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VIII. A
PPENDICES
A. Proof of Proposition 1
The continuity of the synergy gap follows from the conti-nuity of V . Since Ψ is closed [17, Lemma 3.3], it is possibleto find a function δ that is positive on X \ A so that thesynergy gap exceeds δ . A possible function δ : X → R ≥ isgiven as δ ( q, z ) := inf ( s,ζ ) ∈ Ψ ∪ X \Y (cid:18) | ( q, z ) − ( s, ζ ) | + 0 . µ V ( s, ζ ) (cid:19) which is continuous, for all ( q, z ) ∈ (cid:16) Ψ ∪ X \ Y (cid:17) \ A satisfies δ ( q, z ) ≤ . µ V ( q, z ) < µ V ( q, z ) , and, using (7),the fact that Ψ is closed, and the continuity of µ V , satisfies δ ( q, z ) > if ( q, z ) / ∈ A . Let W ( q, z ) = ρ ( V ( q, z )) where ρ is a smooth K ∞ function and ρ ′ is positive andnondecreasing. It follows that ( W, κ ) is an SLFF and thesets E and Ψ are the same as those for ( V, κ ) . Moreover, µ W ( q, z ) = ρ ( V ( q, z )) − min s ∈ Q ρ ( V ( s, z )) . (44)Inspired by the calculations in [19], [20], we lower bound µ W ( q, z ) on the set (cid:16) Ψ ∪ X \ Y (cid:17) \ A by considering twocases: min s ∈ Q V ( s, z ) ≤ cV ( q, z ) and min s ∈ Q V ( s, z ) ≥ cV ( q, z ) where c ∈ (0 , . In the first case, using the mean-value theorem and the monotonicity of ρ ′ , µ W ( q, z ) ≥ ρ ( V ( q, z )) − ρ ( cV ( q, z )) ≥ ρ ′ ( cV ( q, z ))(1 − c ) V ( q, z ) ≥ ρ ′ ( cV ( q, z ))(1 − c ) (cid:18) V ( q, z ) − min s ∈ Q V ( s, z ) (cid:19) > ρ ′ ( cV ( q, z ))(1 − c ) δ ( q, z ) . In the second case, using the mean-value theorem and themonotonicity of ρ ′ , µ W ( q, z ) ≥ ρ ′ ( cV ( q, z )) (cid:18) V ( q, z ) − min s ∈ Q V ( s, z ) (cid:19) > ρ ′ ( cV ( q, z )) δ ( q, z ) > ρ ′ ( cV ( q, z ))(1 − c ) δ ( q, z ) . These bounds establish the final statement of the proposition. (cid:4)
B. Proof of Theorem 1
According to [21, Theorem 4.2], there exists a smoothfunction V : R n +1 → R ≥ that is radially unbounded andpositive definite with respect to the compact set A and suchthat, for all ( q, z ) ∈ C = Y , h∇ V ( q, z ) , f ( q, z, α ( q, z )) i ≤ − V ( q, z ) ≤ , (45)and, for all ( q, z ) ∈ D and s ∈ G c ( q, z ) , V ( s, z ) ≤ e − V ( q, z ) . (46)The properties of V together with (45) make ( V, α ) an SLFFpair candidate relative to ( A , Y ) . In addition, (45) guaranteesthat the set E defined in (4) satisfies E ⊂ A ; then, since Ψ ⊂ E , Ψ \ A = ∅ . Next, since C ∪ D = X and D is closed,it follows that X \ C ⊂ D . Then, since G c ( q, z ) ⊂ Q for all ( q, z ) ∈ D , it follows that for all ( q, z ) ∈ D , µ V ( q, z ) = V ( q, z ) − min s ∈ Q V ( s, z ) ≥ V ( q, z ) − max s ∈ G c ( q,z ) V ( s, z ) ≥ (1 − e − ) V ( q, z ) . (47)Since V is continuous and positive definite with respect to A ,it follows that ( V, α ) is an SLFF pair relative to ( A , Y ) withsynergy gap exceeding ε V ( q, z ) for any ε ∈ (0 , − e − ) .When D ∩ A = ∅ , since V is positive definite with respectto A and radially unbounded, there exists ρ > such that ( q, z ) ∈ D implies V ( q, z ) ≥ ρ . In this case, the synergy gapexceeds any continuous function δ satisfying δ ( q, z ) < (1 − e − )0 . ρ + V ( q, z )] . In particular, the synergy gap exceedsthe function δ given as δ ( q, z ) = ε V ( q, z ) + ε where ε ∈ (0 , . − e − )) and ε ∈ (0 , . − e − ) ρ ) . (cid:4) C. Proof of Theorem 2
Consider the synergistic Lyapunov function V and feed-back κ . We claim that ( C \ A ) ∩ [Ψ ∪ X \ Y ] = ∅ . (48)Indeed µ V ( q, z ) ≤ δ ( q, z ) ∀ ( q, z ) ∈ C (49)while µ V ( q, z ) > δ ( q, z ) ∀ ( q, z ) ∈ [Ψ ∪ X \ Y ] \ A . (50)These bounds establish (48).The condition (48) together with the fact that A ⊂ Y ⊂ X implies that C ⊂ Y . By assumption, (3) holds for all ( q, z ) ∈Y and thus (3) holds for all ( q, z ) ∈ C .By the construction of D and G c in (12), for all ( q, z ) ∈ D and g c ∈ G c ( z ) , we have V ( g c , z ) = min s ∈ Q V ( s, z ) = V ( q, z ) − µ V ( q, z ) ≤ V ( q, z ) − δ ( q, z ) . (51)In particular V ( g c , z ) − V ( q, z ) ≤ for all ( q, z ) ∈ D and g c ∈ G c ( z ) , and V ( g c , z ) − V ( q, z ) = 0 implies ( q, z ) ∈ A .sing the properties of V and δ , it follows that the set A isstable and all solutions are bounded. It remains to establishthat all complete solutions converge to A . Note that A ⊂ C since ( q, z ) ∈ A implies µ V ( q, z ) = 0 ≤ δ ( q, z ) . Then, bythe invariance principle in [22], all complete solutions to (13)converge to the largest weakly invariant set of ˙ q = 0˙ z = f ( q, z, κ ( q, z )) (cid:27) ( q, z ) ∈ E ∩ C. (52)According to the definition of Ψ , this weakly invariant setmust be contained in Ψ ∩ C . It follows from (48) that Ψ ∩ C ⊂A . Thus all complete solutions must converge to A . (cid:4) D. Proof of Theorem 3
For all ( q, ζ ) ∈ X , h∇ V ( q, ζ ) , φ ( q, ζ ) + ψ ( q, ζ ) κ ( q, ζ ) i≤ h∇ V ( q, z ) , φ ( q, z ) + ψ ( q, z ) ω i− θ ( | ω − κ ( q, z ) | ) − h∇ V ( q, z ) , ψ ( q, z )( ω − κ ( q, z )) i = h∇ V ( q, z ) , φ ( q, z ) + ψ ( q, z ) κ ( q, z ) i− θ ( | ω − κ ( q, z ) | ) ≤ . (53)Define E := { ( q, ζ ) ∈ X : h∇ V ( q, ζ ) , φ ( q, ζ ) + ψ ( q, ζ ) κ ( q, ζ ) i = 0 } , W := { ( q, ζ ) ∈ X : h∇ V ( q, ζ ) , ψ ( q, ζ ) i = 0 } . (54)Let E , W , and Ω come from the definitions in SectionIV for the weak SLFF pair ( V , κ ) for the system (26). Itfollows from (53), the properties of θ , the definition of ψ in (25), and the definition of V in (28) that E = { ( q, z ) ∈ E , ω = κ ( q, z ) } ⊂ W . (55)Let Ψ ⊂ X denote the largest weakly invariant set for thesystem ˙ q = 0˙ ζ = φ ( q, ζ ) + ψ ( q, ζ ) κ ( q, ζ ) (cid:27) ( q, ζ ) ∈ E . (56)It follows from the definition of κ in (30), the fact that ˙ ω = κ ( q, ζ ) and the characterization of E in (55) that Ψ = { ( q, ζ ) ∈ X : ( q, z ) ∈ Ω , ω = κ ( q, z ) } . (57)Then, it follows from (28) that µ V ( q, ζ ) ≥ µ V ( q, z )+ σ ( ω − κ ( q, z )) − max s ∈ Q σ ( ω − κ ( s, z )) . Note that X \ Y = ∅ and ( q, ζ ) ∈ Ψ \ A implies that ( q, z ) ∈ Ω \ A . Therefore, for ( q, ζ ) ∈ (cid:16) Ψ ∪ X \ Y (cid:17) \A , µ V ( q, ζ ) ≥ µ V ( q, z ) − max s ∈ Q σ ( κ ( q, z ) − κ ( s, z )) ≥ µ V ( q, z ) − ̺ ( q, z ) > δ ( q, z ) . Thus, ( V , κ ) is an SLFF pair with gap exceeding δ . (cid:4) E. Proof of Theorem 4
For all ( q, ζ ) ∈ Y , h∇ V ( q, ζ ) , φ ( q, ζ ) + ψ ( q, ζ ) κ ( q, ζ ) i≤ h∇ V ( q, z ) , φ ( q, z ) + ψ ( q, z ) ω i− θ ( | ω − κ ( q, z ) | ) −h∇ V ( q, z ) , ψ ( q, z )( ω − κ ( q, z )) i = h∇ V ( q, z ) , φ ( q, z ) + ψ ( q, z ) κ ( q, z ) i− θ ( | ω − κ ( q, z ) | ) ≤ . (58)Define E := { ( q, ζ ) ∈ Y : h∇ V ( q, ζ ) , φ ( q, ζ ) + ψ ( q, ζ ) κ ( q, ζ ) i = 0 } , W := { ( q, ζ ) ∈ Y : h∇ V ( q, ζ ) , ψ ( q, ζ ) i = 0 } . (59)Let E , W , and Ω come from the definitions in SectionIV for the weak SLFF pair ( V , κ ) for the system (26). Itfollows from (58), the properties of θ , the definition of ψ in (25), and the definition of V in (32) that E = { ( q, z ) ∈ E , ω = κ ( q, z ) } ⊂ W . (60)Let Ψ ⊂ X denote the largest weakly invariant set for thesystem ˙ q = 0˙ ζ = φ ( q, z ) + ψ ( q, z ) κ ( q, z ) (cid:27) ( q, ζ ) ∈ E . (61)It follows from the definition of κ in (33), the fact that ˙ ω = κ ( q, ζ ) and the characterization of E in (60) that Ψ = { ( q, ζ ) ∈ X : ( q, z ) ∈ Ω , ω = κ ( q, z ) } . (62)Note that ( q, ζ ) ∈ Ψ \ A implies ( q, z ) ∈ Ω \ A . Also X \ Y = X \ Y × R m so that (cid:16) Ψ ∪ X \ Y (cid:17) \ A ⊂ (cid:16) (Ω \ A ) ∪ X \ Y (cid:17) × R m . Then, it follows from (32) and the facts that ρ ◦ σ is globallyLipschitz with Lipschitz constant less than or equal to L > and V is type I ready-made relative to σ with σ ( v ) := L | v | for all v ∈ R m that, for ( q, ζ ) ∈ (cid:16) Ψ ∪ X \ Y (cid:17) \ A µ V ( q, ζ ) ≥ µ V ( q, z )+ ρ ( σ ( ω − κ ( q, z ))) − max s ∈ Q ρ ( σ ( ω − κ ( s, z ))) ≥ µ V ( q, z ) − max s ∈ Q L | κ ( s, z ) − κ ( q, z ) |≥ µ V ( q, z ) − ̺ ( q, z ) > δ ( q, z ) . Thus, ( V , κ ) is an SLFF pair with gap exceeding δ ..