On differential passivity of physical systems
aa r X i v : . [ c s . S Y ] S e p On differential passivity of physical systems
F. Forni, R. Sepulchre, A.J. van der Schaft
Abstract — Differential passivity is a property that allows tocheck with a pointwise criterion that a system is incrementallypassive, a property that is relevant to study interconnectedsystems in the context of regulation, synchronization, and esti-mation. The paper investigates how restrictive is the property,focusing on a class of open gradient systems encountered in thecoenergy modeling framework of physical systems, in particularthe Brayton-Moser formalism for nonlinear electrical circuits.
I. I
NTRODUCTION
Motivated by the differential Lyapunov framework pre-sented in [5] to study incremental stability, the recent papers[16] and [6] introduced the notion of differential dissipativityto study incremental dissipativity, the analog of incrementalstability for open systems. A related notion of tranverseincremental dissipativity is presented in [10] to study limitcycles. The interest for incremental notions of stability anddissipativity stems from analysis and design problems con-cerned with a distance between arbitrary solutions rather thana distance to a particular (equilibrium) solution : such prob-lems include regulation and tracking, estimation and observerdesign, or synchronization, coordination, and entrainment.The differential approach to study incremental propertiesis rooted in contraction theory, following the influential paperof [9] in control theory. In short, incremental properties ofdynamical systems can be studied differentially, through thevariational equations. The analysis of the variational equation(or more precisely of the prolonged system) is appealingbecause it leads to pointwise conditions to be verified on theprolonged vector field rather than on the solutions, in thespirit of Lyapunov theory. The approach is geometric andthe differential properties are potentially simpler to verifythan their incremental counterparts.The present paper pursues the developments of [16] and[6] to investigate how restrictive it is to check differentialpassivity on a given system. More fundamentally, we areinterested in which class of physical systems are differen-tially passive and what is the physical interpretation of theproperty, if any. The success of passivity as an analysis anddesign concept of system theory stems from its clear energyinterpretation in physical systems: passivity expresses that
This paper presents research results of the Belgian Network DYSCO (Dy-namical Systems, Control, and Optimization), funded by the InteruniversityAttraction Poles Programme, initiated by the Belgian State, Science PolicyOffice. The scientific responsibility rests with its authors.F. Forni is with the Department of Electrical Engineering and ComputerScience, University of Li`ege, 4000 Li`ege, Belgium, [email protected] .His research is supported by FNRS. R. Sepulchre is with the Uni-versity of Cambridge, Department of Engineering, Trumpington Street,Cambridge CB2 1PZ, and with the Department of Electrical Engineer-ing and Computer Science, University of Li`ege, 4000 Li`ege, Belgium, [email protected] . A.J. van der Schaft is with the JohannBernoulli Institute for Mathematics and Computer Science, University ofGroningen, 9700 AK, the Netherlands [email protected] . the increase of internally stored energy cannot exceed the en-ergy supplied by the environment. It is still unclear whethera similar interpretation exists for differential passivity.We provide geometric conditions that characterize dif-ferential passivity with respect to a quadratic storage andwe further investigate the general conditions for a classof gradient systems. Our motivation stems from the factthat a broad class of physical models admits a gradientrepresentation in the coenergy framework, see e.g. [8], [15],after the work of Brayton and Moser for nonlinear electricalcircuits.The paper provides a number of simple examples that il-lustrate that differential passivity may hold for a sizable classof physical models and that feedback can help achieving theproperty, as for passivity.The paper is organized as follows: we revisit the notion ofdifferential passivity in Section II, providing the definitionsof prolonged and variational system, differential storage, anddifferential supply rate. Geometric conditions for passivityare summarized in Section III. Section IV studies the dif-ferential passivity of gradient systems. Differential passivityfor Brayton-Moser systems is characterized in Section V. Notation : Given a manifold X , and a point x of X , T x X denotesthe tangent space of X at x . T X := S x ∈X { x }× T x X is the tangentbundle . Given two manifolds X and X and a mapping f : X →X , f is of class C k , k ∈ N , if its coordinate representation isa C k function. A curve γ on a given manifold X is a mapping γ : I ⊂ R → X . We sometime use ˙ γ ( t ) to denote ∂γ ( t ) ∂t . I n is the identity matrix of dimension n . Given a vector v , v T denotes the transpose vector of v . Given a matrix M we say that M ≥ or M ≤ if v T Mv ≥ or v T Mv ≤ , for each v , re-spectively. Given the vectors { v , . . . , v n } , Span( { v , . . . , v n } ) := { v | ∃ λ , . . . λ n ∈ R s.t. v = P ni =1 λ i v i } . In coordinates, wedenote the differential of a function f at x by ∂f ( x ) ∂x . The Hessianof f at x is denoted by ∂ f ( x ) ∂x .A distance d : X × X → R ≥ on a manifold X is a positivefunction that satisfies d ( x, y ) = 0 if and only if x = y , for each x, y ∈ X and d ( x, z ) ≤ d ( x, y ) + d ( y, z ) for each x, y, z ∈ X .A set S ⊂ X is bounded if sup x,y ∈S d ( x, y ) < ∞ for any givendistance d on X . A curve γ : I → X is bounded when its imageis bounded. Given a manifold X , a set of isolated points Ω ⊂ X satisfies: for any distance function d on X and any given pair x , x in Ω , there exists an ε > such that d ( x , x ) ≥ ε . II. D
IFFERENTIAL PASSIVITY
A. Prolonged systems
Consider the nonlinear system Σ with state space X ,and inputs and outputs spaces U ⊂ R m and Y ⊂ R m ,respectively, given by (cid:26) ˙ x = f ( x ) + g ( x ) uy = h ( x ) (1)here x ∈ X , and u ∈ U , and y ∈ Y . f and g i , i ∈ { . . . m } are vector fields. h : X → Y .Contraction analysis requires sufficient differentiability( C ) of the solutions ψ ( t, x ) to (1), from any initial con-dition x ∈ X (see, e.g. [9], [12]). To enforce the desiredregularity, we make the following standing assumption. Assumption 1: f and g i , i ∈ { . . . m } , are C vectorfields ( g i denotes the i -th column of g ). h : X → Y is a C function. The input signal u : R → U is a C function.To a system of the form (1) one can associate the varia-tional system given by ( ˙ δx = ∂f ( x ) ∂x δx + ∂g ( x ) u∂x δx + g ( x ) δuδy = ∂h ( x ) ∂x δx . (2)We call prolonged system the combination of (1) and (2),following [2], [16]. A coordinate free representation of theprolonged system is provided by the notions of complete andvertical lifts, as shown in [2], [16].Under Assumption 1, for every solution ( x, u, y )( · ) to(1), the solutions ( δx, δu, δy )( · ) to (2) represent infinitesimalvariations on ( x, u, y )( · ) , that is, the infinitesimal mismatchbetween ( x, u, y )( · ) and neighboring solutions. This intuitiverepresentation is clarified in Remark 1. Pursuing this intu-ition, if the dynamics of (2) guarantee that δx convergesto zero then, necessarily, the solutions to (1) must convergetowards each other. A Lyapunov-based analysis of the con-nection between contraction of δx and incremental stabilitycan be found in [5]. Remark 1:
For each s ∈ [0 , let γ ( s ) be an initial con-dition for (1) and u ( · , s ) an input signal. Assume that γ ( · ) ∈ C and u ( · , · ) ∈ C . Then, for each s ∈ [0 , x ( · , s ) is asolution to (1) from the initial condition γ ( s ) under the actionof the input u ( · , s ) . Define the displacement δx ( t, s ) := ∂∂s x ( t, s ) and δu ( t, s ) := ∂∂s u ( t, s ) . Then, by chain rule anddifferentiability, we have that ddt δx ( t, s ) = ∂ ∂s∂t x ( t, s ) = ∂∂s [ f ( x ( t, s )) + g ( x ( t, s )) u ( t, s )] = ∂f ( x ( t,s )) ∂x δx ( t, s ) + ∂g ( x ( t,s )) u ( t,s ) ∂x δx ( t, s ) + g ( x ( t, s )) δu ( t, s ) . Thus, δx ( · , s ) isa solution to (2) from the initial condition ∂γ ( s ) ∂s under theaction of the input δu ( · , s ) . Moreover, the output signal δy ( t, s ) is given by ∂y ( t,s ) ∂s = ∂h ( x ( t,s )) ∂x δx ( t, s ) . y B. Differential passivity
Henceforth we provide the notion of differential storagefunction and differential passivity. These notions are takenfrom [6, Sections 3 and 4] and restrict the definitions in[16, Section 4] to the case in which the function P in [16,Definition 4.1 and Proposition 4.3] is a candidate Finsler-Lyapunov function [5]. This restriction makes possible theconnection between differential passivity and incrementalstability. Definition 1:
Let Ω be a set of isolated point in X . Foreach x ∈ X , suppose that T x X can be subdivided into a vertical distribution V x ⊂ T x XV x := Span( { v ( x ) , . . . , v r ( x ) } ) 0 ≤ r < d , (3)and a horizontal distribution H x ⊆ T x X complementary to V x , i.e. V x ⊕ H x = T x X , H x := Span( { h ( x ) , . . . , h q ( x ) } ) 0 < q ≤ d − r (4) where v i , i ∈ { , . . . , r } , and h i , i ∈ { , . . . , q } , are C vector fields.A function δS : T X → R ≥ is a differential storagefunction for the dynamical system Σ in (1) if there exist c , c ∈ R ≥ , p ∈ R ≥ , and a function F : T X → R ≥ such that, for each ( x, δx ) ∈ T X , c F ( x, δx ) p ≤ δS ( x, δx ) ≤ c F ( x, δx ) p . (5) δS and F must satisfy the following conditions. Given aset of isolated points Ω ⊂ X ,(i a ) δS and F are C , ∀ x ∈ X , ∀ δx ∈ H x \ { } ;(i b ) δS ( x, δx ) = δS ( x, δx h ) and F ( x, δx ) = F ( x, δx h ) , ∀ ( x, δx ) ∈ T X such that ( x, δx ) = ( x, δx h ) + ( x, δx v ) , δx h ∈ H x , and δx v ∈ V x ;(ii) F ( x, δx ) > , ∀ x ∈ X \ Ω ∀ δx ∈ H x \ { } ;(iii) F ( x, λδx ) = λF ( x, δx ) , ∀ λ > , ∀ x ∈ X , ∀ δx ∈ H x ;(iv) F ( x, δx + δx ) < F ( x, δx ) + F ( x, δx ) , ∀ x ∈ X \ Ω and ∀ δx , δx ∈ H x \ { } such that δx = λδx for any given λ ∈ R . y When V x = ∅ , F ( x, δx ) provides a non symmetric normon each tangent space T x X . A suggestive notation for F isgiven by | δx | x which combined to (5) provides an intuitiveinterpretation of the differential storage function δS as a localmeasure of the displacement length. For V x = ∅ , it mayoccur that δS ( x, δx ) = δS ( x, δx ) for = δx − δx ∈ V x .In such a case, δS measures the length of each δx by lookingonly at its horizontal component. An example of a differentialstorage with V x = 0 is provided by δS ( x, δx ) = δy T δy .It is worth to mention that a differential storage function δS is also a horizontal Finsler-Lyapunov function [5, SectionVIII]. Therefore, δS endows X with the structure of apseudo-metric space, connecting differential passivity andincremental stability [14], [1]. An extended discussion andexamples are provided in [5, Sections IV and VIII].The notion of differential passivity introduced below isjust passivity lifted to the tangent bundle. Definition 2:
The dynamical system Σ in (1) is differen-tially passive if there exists a differential storage function δS such that δS ( x ( t ) , δx ( t )) − δS ( x (0) , δx (0)) ≤ Z t δy ( τ ) T δu ( τ ) dτ (6)for all t ≥ and all solutions ( x, u, y, δx, δu, δy )( · ) to theprolonged system (1),(2). y The equivalent formulation ddt δS ( x ( t ) , δx ( t )) ≤ δy ( t ) T δu ( t ) coincides with [16, Definition 4.1]. Incomparison to passivity, differential passivity builds arelation between the energy - or cost - δS associatedto an infinitesimal variation of the solution x ( t ) , andthe energy associated to an infinitesimal variation onthe input/output signals. In comparison to incrementalpassivity [4], [13], δy T δu does not impose anyprescribed form ∆ y T ∆ u = ( y − y ) T ( u − u ) tothe input/output mismatch. Instead, following Remark1, given a parameterization u ( s ) , y ( s ) such that ( u (0) , y (0)) = ( u , y )( · ) and ( u (1) , y (1)) = ( u , y )( · ) e have that that ( y − y ) T ( u − u ) is replaced by R ∂y ( s ) ∂s T ∂u ( s ) ∂s ds . Note that R ∂y ( s ) ∂s T ∂u ( s ) ∂s ds = ∆ y T ∆ u only if y ( s ) = sy + y (1 − s ) and u ( s ) = su + u (1 − s ) .This is particularly relevant at integration along solutions,since an initial parameterization satisfying the identity aboveat time t = 0 does not preserve the identity for t > , ingeneral (on nonlinear models).We conclude the section by illustrating two basic resultsof differential passivity. The reader is referred to [6], [16]for further results. Theorem 1:
Let Σ in (1) be differentially passive with adifferential storage δS whose vertical distribution V x = 0 for each x ∈ X . Then, (1) is incrementally stable. y Proof:
For δu = 0 , differential passivity guaranteesthat ˙ δS ≤ . For V x = 0 , δS is a Finsler-Lyapunov function,thus incremental stability follows from [5, Theorem 1]. Theorem 2:
Let Σ and Σ be differentially passive dy-namical systems (1). Let ( u i , y i ) be the input and the outputof Σ i , for i = 1 , . Then, the dynamical system Σ arisingfrom the feedback interconnection u = − y + v , u = y + v , (7)is differentially passive from v = ( v , v ) ∈ U × U to y = ( y , y ) ∈ Y × Y . y Proof:
Take δS = δS + δS . ˙ δS ≤ δy δv + δy δv .III. T HE GEOMETRY OF DIFFERENTIAL PASSIVITY
For quadratic differential storage functions δS = δx T M ( x ) δx (Riemannian metrics), M ( x ) > , the differ-ential passivity of systems of the form (1) is characterizedgeometrically by the following conditions. For each x ∈ X and u ∈ U , M ( x ) ∂f ( x ) ∂x + ∂f ( x ) ∂x T M ( x ) + X i ∂M ( x ) ∂x i [ f ( x ))] i ≤ (8) M ( x ) ∂g ( x ) u∂x + ∂g ( x ) u∂x T M ( x ) + X i ∂M ( x ) ∂x i [ g ( x ) u ] i = 0 (9) ∂h ( x ) ∂x T = M ( x ) g ( x ) . (10)In fact, along the solutions to the prolonged system, thetime derivative of δS is given by ˙ δS = δx T ( m f ( x ) + m g ( x, u )) δx + δx T M ( x ) g ( x ) δx , where m f ( x ) and m g ( x, u ) are given by the left-hand sides of (8) and (9), respectively.(8) guarantees that the system is contracting for u =0 , thus incrementally stable with respect to the geodesicdistance induced by the metric M . The reader will notice that(8) is just the usual condition for passivity ∂S ( x ) ∂x f ( x ) ≤ lifted to the tangent bundle. In a similar way, (10) guaranteesthat δy = M ( x ) g ( x ) δx , thus enforcing a differential versionof the passivity condition ∂S ( x ) ∂x g ( x ) = h ( x ) T .A notable difference with respect to passivity is providedby condition (9), which requires the columns of g ( x ) to bekilling vector fields for the metric M ( x ) . This guarantees that u does not appear in the right-hand side of ˙ δS , as requiredby (6). In this sense, the input matrix g ( x ) restricts the classof metrics that one can use to establish differential passivity. For the case g ( x ) = B , for example, (9) restricts thedifferential storage within the class of metrics M ( x ) suchthat P i ∂M ( x ) ∂x i [ Bu ] i = 0 , which is satisfied by constantmetrics M ( x ) = P = P T ≥ . In comparison to passivity, M ( x ) = P is not an issue for linear systems (cid:26) ˙ x = Ax + Buy = Cx (11)( A ∈ R n × n , B ∈ R n × ν , and C ∈ R ν × n ). In fact, for passivelinear systems one can always find P = P T ≥ such that A T P + P A ≤ C T = P B , (12)which also establishes the equivalence between passivity anddifferential passivity for linear systems. But M ( x ) = P determines a limitation for the satisfaction of (8) on systemsof the form ˙ x = f ( x ) + Bu (13)since it reduces (8) to ∂f ( x ) ∂x T P + P ∂f ( x ) ∂x ≤ . This lastinequality coincides with the early convergence conditionof Demidovich [3]. See also [11, Theorem 2.29]. It alsoresembles a classical Lyapunov inequality based on quadraticLyapunov functions and linearized vector fields. In fact,in the neighborhood of stable equilibria x e passivity anddifferential passivity are related, since locally around x e passive systems satisfies ∂f ( x ) ∂x T P + P ∂f ( x ) ∂x ≤ locallyaround x e .The relevance of the condition enforced by (9) is readilyillustrated by the following example. Example 1:
Consider the simple dynamics on S given by ˙ x = − sin( x ) + g ( x ) u g ( x ) = 1 . (14)For g ( x ) = 1 , (9) allows for differential storages of the form δS = δx , for which ˙ δS = − cos( x ) δx + δxδu . Thus, (14)is differentially passive along solution curves whose rangebelongs to [ − π , π ] . In fact, (8) holds only for x ∈ [ − π , π ] .Using a non constant metric, (8) can be satisfied in thewhole set ( − π, π ) . Indeed, taking δS = M ( x ) δx M ( x ) = 11 + cos( x ) (15)(8) reads − x )1 + cos( x ) − sin( x ) (1 + cos( x )) = − . (16)However, (9) does not hold, unless the input matrix g ( x ) = 1 in (14) is replaced by g ( x ) = γ cos( x ) , where γ ∈ R . Insuch a case, following (10), (14) is differential passive withrespect to the output y = γ R x z )1+cos( z ) dz . y The discussion above makes clear that differential passiv-ity for nonlinear systems of the form (1) can be establishedonly for suitable pairs f ( x ) and g ( x ) . The latter, through(9), defines the class of feasible metrics. The former, through(8), is required to be a contractive vector field with respectto a feasible metric (see [5], [9]). Finally, in analogy withpassivity, the (differential) passivating output depends on thedifferential storage and on the input matrix, as establishedby (10).V. O PEN GRADIENT SYSTEMS
A. General formulation and prolonged system
Given a smooth manifold X , a Riemannian metric Q on X , and a potential function V : X → R , the local coordinatesrepresentation of a gradient system is given by Q ( x ) ˙ x = − ∂V ( x ) ∂x + Bu . (17)Following the discussion of the previous section, the study ofdifferential passivity for gradient systems amounts to verifythat f ( x ) := Q ( x ) − ∂V ( x ) ∂x and g ( x ) := Q ( x ) − B satisfy(8), (9) for some differential storage δS = δx T M ( x ) δx .The prolonged system is given by (17) and by the varia-tional system Q ( x ) ˙ δx = − (cid:20) ∂ V ( x ) ∂x δx + Bδu (cid:21) + Γ (cid:18) x, u, ∂V ( x ) ∂x (cid:19) δx (18)where the matrix Γ satisfies Γ (cid:18) x, u, ∂V ( x ) ∂x (cid:19) δx := − "X i ∂Q ( x ) ∂x i δx i ˙ x . Γ is homogeneous of degree one in ∂V ( x ) ∂x and u , thusconverges to zero as x approaches an extremal point of V and u converges to . Note that Γ = 0 when Q ( x ) is constant. B. Differential passivity via natural metric and convexity
For M ( x ) = Q ( x ) = P > (constant), the differentialstorage δS = δx T P δx guarantees that both (8) and (9)hold, provided that ∂ V∂x ≥ for all x ∈ X . In fact, alongthe solutions of the prolonged system, we have ˙ δS = − δx T ∂ V ( x ) ∂x δx | {z } ≥ + δx T Bδu . (19)Thus, the gradient system is differentialy passive with respectto the output y = B T x .The case of Q ( x ) non constant is more involved. For M ( x ) = Q ( x ) conditions (8) and (9) may not hold, ingeneral. In fact, along the solutions of the prolonged system,the differential storage δS = δx T Q ( x ) δx has derivative ˙ δS = − δx T ∂ V ( x ) ∂x δx + δx T Bδu + 12 δx T Γ (cid:18) x,u, ∂V ( x ) ∂x (cid:19) T δx + 12 δx T Γ (cid:18) x,u, ∂V ( x ) ∂x (cid:19) δx + 12 δx T Ω (cid:18) x,u, ∂V ( x ) ∂x (cid:19) δx (20)where Ω (cid:18) x, u, ∂V ( x ) ∂x (cid:19) := X i ∂Q ( x ) ∂x i ˙ x i ; (21)and (8) and (9) are equivalent to the following inequality δx T (cid:18) − ∂ V ( x ) ∂x + Γ T + Γ + Ω (cid:19) δx ≤ . (22)When (22) holds for each ( x, δx ) ∈ T X and u ∈ U , then (17)is differentially passive with respect to the output y = B T u . Example 2: [Example 1 revised] Taking V ( x ) = 1 − cos( x ) and g ( x ) = 1 the dynamics in (14) reads ˙ x = − ∂V ( x )) ∂x + u . (23)Note that V ( x ) is convex in the region [ − π , π ] since ∂ V ( x ) ∂x = cos( x ) ≥ for x ∈ [ − π , π ] . In fact, (23) isdifferentially passive with δS = δx and y = x .For g ( x ) = cos( x ) , define Q ( x ) = x ) and V ( x ) = − x ) . Then, (14) is well defined in ( − π, π ) and reads Q ( x ) ˙ x = − sin( x )cos (cid:0) x (cid:1) + u = − (cid:16) x (cid:17) + u = − ∂V ( x ) ∂x + u . (24) V ( x ) is convex in ( − π, π ) , however differential dissipativitycannot be achieved because the term Γ T + Γ + Ω in (22)shows a dependence on u . y Remark 2:
When (22) does not hold, we can still achievelocal differential passivity under the assumption of strictconvexity of V , for small signals u . Given a (sufficientlysmall) neighborhood C ( x e ) , ∂ V ( x ) ∂x > aI for x ∈ C ( x e ) ,while the last three terms in (22) are bounded by a functionof the form b ( x e ) | u || ∂V ( x ) ∂x | , by homogeneity. Thus, ˙ δS ≤ (cid:16) − a + b ( x e ) | u | (cid:12)(cid:12)(cid:12) ∂V ( x ) ∂x (cid:12)(cid:12)(cid:12)(cid:17) | δx | + δx T Bδu ≤ δx T Bδu for x ∈ C ( x e ) and for | u | and C ( x e ) sufficiently small. y C. Differential passivity beyond the natural metric
We consider the case of differential storage functions δS = δx T M ( x ) δx where M ( x ) = Q ( x ) P Q ( x ) for some givenmatrix P = P T ≥ . A first consequence of the definitionof M ( x ) is that Q can be relaxed to a pseudo-Riemannianmetrics, that is, Q ( x ) is not necessarily positive but stillinvertible. In contrast to this generalization effort, we restrict Q to the class of pseudo-metrics defined by Q ( x ) = ∂ q ( x ) ∂x ,where q is a function differentiable sufficiently many times.Under these assumptions, for y = C ∂q ( x ) ∂x (25)(8), (9), and (10) are equivalent to the following conditions. Theorem 3:
Consider q : X → R and Q ( x ) = ∂ q∂x ( x ) .Then (18) is differentially passive with respect to the output y = C ∂q ( x ) ∂x if there exists a matrix P = P T ≥ such thatfor all x ∈ X ∂ V ( x ) ∂x P Q ( x ) + Q ( x ) P ∂ V ( x ) ∂x ≥ (26a) C T = P B . (26b) δS = δx T Q ( x ) P Q ( x ) δx is the differential storage y (26a) is a generalized convexity property on V . We getclassical convexity when Q ( x ) = P = I . For P positivedefinite, the particular selection of the output y = C ∂q ( x ) ∂x guarantees that (17) has relative degree one. In fact, ˙ y = C ∂ q ( x ) ∂x ˙ x = C ( ∂V ( x ) ∂x ( x ) + Bu ) , where CB = B T P B .Finally, note that for q ( x ) = V ( x ) , the inequality in (26)is always satisfied. This is not surprising since, by defining e = ∂V ( x ) ∂x , (17) reads ˙ e = − e + Bu , y = Ce . roof of Theorem 3: Define f ( x ) := h ∂ q ( x ) ∂x i − ∂V ( x ) ∂x , g ( x ) := h ∂ q ( x ) ∂x i − Bu , and h ( x ) := C T ∂q ( x ) ∂x , and considerthe prolonged system (1),(2). By exploiting the differentia-bility of q , and using the chain rule, ˙ δS = δx T Q ( x ) P (cid:20) ∂ [ Q ( x ) f ( x )] ∂x δx + ∂ [ Q ( x ) g ( x ) u ] ∂x (cid:21) δx + δx T Q ( x ) P Q ( x ) g ( x ) δu = δx T Q ( x ) P ∂V ( x ) ∂x δx | {z } ≤ + δx T Q ( x ) P ∂ [ Bu ] ∂x | {z } =0 δx + δx T Q ( x ) P B |{z} C T δu ≤ δx T Q ( x ) C T δu = δy T δu . (27)(8), (9), (10) read δx T Q ( x ) P ∂V ( x ) ∂x δx ≤ , δx T Q ( x ) P ∂ [ Bu ] ∂x δx = 0 , and δx T Q ( x ) P B = δxQ ( x ) C T ,respectively. (cid:4) Remark 3:
Theorem 3 extends to systems of the form Q ( x ) ˙ x = A ( x ) + Bu (28)where A ( x ) is a vector field not derived from a potential. Inthis case, ∂ V ( x ) ∂x in (26a) is replaced by ∂A ( x ) ∂x . y Example 3: [Example 2 revised] Consider the system for-mulation given in (24) for the case g ( x ) = cos (cid:0) x (cid:1) . Take thedifferential storage δS = δx T Q ( x ) P Q ( x ) δx for P = 1 .Then, from Theorem 3, the inequality (26a) reads (cid:0) x (cid:1) cos (cid:0) x (cid:1) = 2 ≥ , (29)and (24) is differentially passive in ( − π, π ) with respectto the output y = R x Q ( z ) dz . Because (29) is strictlypositive, the system is incrementally asymptotically stable.The solutions converge to the unique steady-state solutioncompatible with the input signal u [5] (see Fig 1). y x t 0 2 4 6 8 10-3-2-10123 x t Fig. 1. Entrainment of (24) with g ( x ) = cos (cid:0) x (cid:1) for the (small) input u = 1 + 0 . πt ) , left, and the (large) input u = 1 + 5 sin( πt ) , right. V. B
RAYTON -M OSER SYSTEMS
A. Passivity conditions
The approach developed in the previous section allowsfor the analysis of the passivity of Brayton-Moser systems[7], [8], [15]. Brayton-Moser modeling of physical systemscharacterizes a class of gradient systems of the form Q ( z ) ˙ z = ∂V ( z, u ) ∂z , (30) where the state-s[ace is given by flow and efforts z = ( f, e ) , V is a the potential, and Q ( z ) satisfies Q ( z ) = " − ∂ H ∗ ( f,e ) ∂f ∂ H ∗ ( f,e ) ∂e . (31) H ∗ is the Legendre transform of the Hamiltonian H . Inrelation to the theory developed in the previous section, weassume that H ∗ has the following structure H ∗ ( f, e ) = H ∗ f ( f ) + H ∗ e ( e ) (32)which guarantees that Q ( z ) = ∂ [ − H ∗ f ( f )+ H ∗ e ( e )] ∂z . In a similarway, we assume that V has the form V ( z, u ) = p ( z ) + z T Bu . (33)Under these assumptions, (30) reads ∂ H ∗ ( z ) ∂z ˙ z = ∂p ( z ) ∂z + Bu . (34)From Theorem 3, the system (34) is differential passive withrespect to the output y = B T ∂H ∗ ( z ) ∂z , if ∂ H ∗ ( z ) ∂z ∂ p ( z ) ∂z + ∂ p ( z ) ∂z ∂ H ∗ ( z ) ∂z ≤ . (35)The reader will notice that the output y = B T ∂H ∗ ( z ) ∂z isnot the usual passive output y p = B T z . However, y and y p show an intriguing duality, through energy and co-energyformulation of the system [15, Section 4]. B. Differential passivity of a nonlinear RC circuit
The behavior of the nonlinear circuit represented in Figure2 is captured by the following equations: ˙ q = − i r + i, i r = R ( v ) , v = ∂h∂q ( q ) , q = C ( v ) = ∂h ∗ ∂v ( v ) . i c i r iv C ( v ) R ( v ) Fig. 2. V , I - external voltage and current. v c , i c - capacitor voltage andcurrent. v r , i r - resistor voltage and current. Defining Q ( v ) = d h ∗ dv ( v ) , we get the gradient system Q ( v ) ˙ v = − R ( v ) + i . (36)From Theorem 3, differential passivity can be achieved if Q ( v ) ∂R ( v ) ∂v ≥ . In fact, defining δS ( v, δv ) = ( Q ( v ) δv ) ,we have that ˙ S = − Q ( v ) ∂R ( v ) ∂v δv + Q ( v ) δvδi . (37)Therefore, if R ( v ) is not decreasing and ∂h ∗ ( v ) ∂v is strictlyincreasing, we get ˙ S ≤ Q ( v ) δvδi = δqδi . (38)For example, suppose that v can only take positive values,and take R ( v ) = v . R ( v ) models a nonlinear resistor v = v t 0 2 4 6 8 1000.511.52 v t Fig. 3. Contraction and nonlinear behavior of the nonlinear RC circuit.The left figure illustrates contraction for a broad range of initial conditions.The right figure illustrates the nonlinear response of the circuit to a largeharmonic input signal. ˜ R ( i ) i whose value ˜ R ( i ) decreases as i increases. For thecapacitor, consider the relation C ( v ) = ∂h ∗ ∂v ( v ) = log(1+ v ) ,to model a saturation effect on the capacitor plates, wherethe charge on the plates grows at sub-linear rate with respectto the voltage. Note that Q ( v ) = v > for v ≥ .The incremental stability property of the circuit is clearlyvisible in the left part of Figure 3. The steady-state behaviorof the circuit is independent from the initial condition,(nonlinear filter). C. Differential passivation of the rigid body
Let us consider the rigid-body dynamics given by (cid:20) I I
00 0 I (cid:21) ˙ w = (cid:20) I − I I − I
00 0 I − I (cid:21) ω ω ω ω ω ω + u (39)where ω k and I k are the angular velocities of the body withrespect to the axis of a frame fixed to the body, and theprinciple moments of inertia.Suppose that I > I > I and define I := diag( I , I , I )˜ Q := diag( I − I , I − I , I − I ) Q := I ˜ Q − p ( ω ) := ω ω ω (40)then we can rewrite the rigid body dynamics as follows Q ˙ ω = ∂p ( ω ) ∂ω + ˜ Q − u ( q ( ω ) = 12 ∂ ω T Qω∂ω ) . (41)Furthermore, let us consider a passivation design given by u = I ( − r ( ω ) + Gv ) , r ( ω ) := (cid:2) r ω r ω r ω (cid:3) T . (42)(41) becomes Q ˙ ω = ∂p ( ω ) ∂ω − Qr ( ω ) + QGv . (43)From Theorem 3, picking P = Q − , (26a) reads Q − ∂ p ( ω ) ∂ω + ∂ p ( ω ) ∂ω Q − − ∂r ( ω ) ∂ω ≤ (44)while condition (26a) becomes C T = Q − G . Therefore,differential passivity from v to y = G T ω can be guaranteedsemi-globally, since for any given compact region of veloci-ties, there exists a selection of r , r , r that guarantees (44)within that region. For I = 3 , I = 2 , I = 1 and r = r = r = 0 . ,to achieve a desired steady-state solution [ d ( t ) , , T it issufficient to define G = [1 , , T and v = r d ( t ) + ˙ d ( t ) , asshown on the left of Figure 4 for d ( t ) = 3 sin( πt ) . Usingdifferential passivity, we can improve the convergence rateby output feedback v = − . y + ( r + 0 . d ( t ) + ˙ d ( t ) , asshown in the simulation on the right. ω t 0 5 10 15 20-101234 ω t Fig. 4. The passivation design on the rigid body guarantees contraction.The left figure illustrates the contraction of the three states. Output injection y = G T ω improves the convergence rate, as illustrated by the right figure. VI. C
ONCLUSIONS
Building upon [6] and [16], we introduced the notion ofdifferential passivity and we proposed geometric conditionsfor differential passivity of gradient and Brayton-Mosersystems. The meaning and the feasibility of such conditionsis investigated through detailed discussion and several exam-ples. Examples suggests that differential passivity may holdfor a sizeable class of physical models.R
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