Krasovskii and Shifted Passivity Based Control
Yu Kawano, Krishna Chaitanya Kosaraju, Jacquelien M. A. Scherpen
aa r X i v : . [ ee ss . S Y ] J a n Krasovskii and Shifted Passivity Based Control
Yu Kawano, Krishna Chaitanya Kosaraju, and Jacquelien M. A. Scherpen
Abstract —In this paper, our objective is to develop novelpassivity based control techniques by introducing a new passivityconcept named
Krasovskii passivity . As a preliminary step, weinvestigate properties of Krasovskii passive systems and estab-lish relations among four relevant passivity concepts includingKrasovskii passivity. Then, we develop novel dynamic controllersbased on Krasovskii passivity and based on extended shiftedpassivity. The proposed controllers are applicable to a class ofsystems for which the standard passivity based controllers maybe difficult to design.
I. I
NTRODUCTION
Passivity as a tool enables us to develop various types ofpassivity based control (PBC) techniques, and moreover as a property , it helps us to understand these techniques in thestandard engineering parlance. Lyapunov analysis discussesstability with respect to an equilibrium. However, notions likedifferential (or called contraction) analysis and incrementalstability [2]–[5] study the convergence between any pair oftrajectories. These differences have resulted in diverse stabil-ity definitions, which further resulted in disparate passivitydefinitions such as differential passivity and incremental pas-sivity [6]–[9].There are several papers that describe these relatively newdifferential passivity concepts [7], [8]. Apart from the eleganceof analysis, it is not well understood how differential passivitycan be used either as a tool or as a property although thereis a few differential passivity based control techniques [10]–[13]. This is because, generally, differential passivity can beinterpreted as a property of the variational system and doesnot give direct conclusions for the original system itself.In contrast, incremental passivity has been used as a toolor as a property via so called shifted passivity [14]–[16]. If asystem has an equilibrium point, incremental passivity resultsinto shifted passivity at the equilibrium point. Shifted passivitycan be interpreted as a generalization of standard passivity fora system whose equilibrium point is not necessarily the originand is applied to various situations, see, e.g. [14]–[16].Similarly, for differential passivity it can be of interest toconsider a passivity property at an equilibrium point that can
Y. Kawano is with the Graduate School of Engineering, Hiroshima Univer-sity, Japan (email: [email protected]).K.C. Kosaraju is with the Department of Electrical Engineering, Universityof Notre Dame, Notre Dame, IN 46556, USA (email: [email protected]).J.M.A. Scherpen is with the Jan C. Willems Center for Systems andControl and the Faculty of Science and Engineering, University of Groningen,Groningen, The Netherlands (email: [email protected]).This work of Kawano was supported in part by JSPS KAKENHI GrantNumber JP19K23517.This work of Kosaraju and Scherpen was supported in part by the Nether-lands Organisation for Scientific Research through Research ProgrammeENBARK+ under Project 408.urs+.16.005.The preliminary version of the paper was presented at the 11th IFACsymposium on nonlinear control systems in 2019 [1]. help with the analysis and control design. Motivated by this,we establish a new passivity concept, which we call
Krasovskiipassivity . First, we marshal the aforementioned relevant fourpassivity concepts: 1) differential passivity, 2) Krasovskii pas-sivity, 3) incremental passivity, and 4) shifted passivity. Thenwe establish a similar connection as the connection betweenincremental passivity and shifted passivity, for differentialpassivity and Krasovskii passivity. Furthermore, we show thatKrasovskii passivity implies shifted passivity, and differentialpassivity with respect to a constant metric implies incrementalpassivity. Next, we provide novel dynamic control techniquesbased on Krasovskii passivity, which also inspire a new shiftedpassivity based dynamic controller. The utility of the proposedcontrollers are illustrated by a DC-Zeta converter. It is worthmentioning that to the best of our knowledge for this converter,a passivity based controller has not been designed in literature.In the preliminary conference version [1], we have proposedKrasovskii passivity, provided sufficient conditions for port-Hamiltonian and gradient systems to be Krasovskii passive,and gave a brief introduction of Krasovskii passivity basedcontrol techniques. However, the considered storage functionis restricted into one with a constant metric. Also, incrementalpassivity and shifted passivity have not been considered in thepreliminary version. This paper contains the following newcontributions: • general storage functions are used for analysis and con-troller design; • a necessary and sufficient condition for Krasovskii pas-sivity is presented; • we establish relations among four types of passivityproperties; • we show an example of a Krasovskii passive system,which is not differentially passive with a quadratic posi-tive definite storage function; • the proposed Krasovskii passivity based dynamic con-troller is more general than the one in [1]. • we newly present a shifted passivity based dynamiccontroller.The remainder of this paper is organized as follows. InSection II, we define Krasovskii passivity and establish theconnection among the four passivity concepts. In Section III,we design two novel passivity based dynamic controllerson the basis of Krasovskii passivity and shifted passivity.In Section IV, the two provided dynamic controller designsare applied to solve the stabilization problem of a DC-Zetaconverter. Finally, Section V concludes this paper. Notation:
The set of real numbers and non-negative realnumbers are denoted by R and R + , respectively. For sym-metric matrices P, Q ∈ R n × n , P ≻ Q ( P (cid:23) Q ) meansthat P − Q is positive definite (semidefinite). For a vector x ∈ R n , define | x | P := √ x ⊤ P x , where P ∈ R n × n . If P is theidentity matrix, this is nothing but the Euclidean norm and issimply denoted by | x | . The open ball of radius r > centeredat p ∈ R n is denoted by B r ( p ) := { x ∈ R n : | x − p | < r } ;its dimension ( n in this case) is not stated explicitly becausethis is clear from the context.II. A NALYSIS OF P ASSIVITY P ROPERTIES
A. Preliminaries
Consider the following input-affine nonlinear system: ˙ x = f ( x, u ) := g ( x ) + m X i =1 g i ( x ) u i , (1)where x : R → R n and u = [ u , . . . , u m ] ⊤ : R → R m denotethe state and input, respectively. Functions g i : R n → R n , i = 0 , , . . . , m are of class C , and define g := [ g , . . . , g m ] by using the latter m vector valued functions.In some of our developments we assume an equilibriumpoint to exist, i.e., then we use the following assumption. Assumption 2.1:
For the system (1), the following set E := { ( x ∗ , u ∗ ) ∈ R n × R m : f ( x ∗ , u ∗ ) = 0 } is not empty. ⊳ In this paper, our objective is to design controllers basedon variants of passivity concepts. Although there are plentyof rich results based on standard passivity, there is a class ofsystems for which it is difficult to apply them, see e.g., [10],[12]. Our approach to deal with such systems is to investigatedifferent passivity properties which are defined by appropri-ately applying the following dissipativity concept.
Definition 2.2: [9], [17], [18] Suppose that Assumption 2.1holds. The system (1) is said to be dissipative (at x ∗ ) withrespect to a supply rate w : R n × R m → R if there exists anon-zero class C function S : R n → R such that S ( x ) ≥ , S ( x ∗ ) = 0 ,∂S ( x ) ∂x f ( x, u ) ≤ w ( x, u ) for all ( x, u ) ∈ R n × R m . The function S ( x ) is called a storagefunction . ⊳ If ( x ∗ , u ∗ ) = 0 , and the supply rate is w ( x, u ) = u ⊤ h ( x ) ,where h (0) = 0 , then dissipativity is nothing but the (standard) passivity of the system (1) with the output y = h ( x ) [9], [18].For passivity, the following necessary and sufficient condi-tion is well known. Its modifications are used in this paper. Proposition 2.3: [9], [18] Suppose that f (0 ,
0) = 0 . Asystem (1) is passive if and only if there exist h : R n → R m and non-zero class C function S : R n → R such that S ( x ) ≥ , S (0) = 0 , h (0) = 0 ,∂S ( x ) ∂x g ( x ) ≤ ,∂S ( x ) ∂x g ( x ) = h ⊤ ( x ) . for all x ∈ R n . ⊳ Remark 2.4:
For passivity analysis, f ( x ∗ , u ∗ ) = 0 , S ( x ∗ ) =0 , h ( x ∗ ) = 0 are not required in general; see e.g. [9]. However, Section IIIPBC DesignSection IIPassivity Relations
Section III.A Section III.BSection II.E(constant metric) Section II.CSection II.B Extended Shifted PBCKrasovskii PBC Extended Shifted PassivityExtended Incremental PassivityKrasovskii PassivityDifferential Passivity Section II.D
Fig. 1. Relationships among proposed passivity concepts and controllers. they are assumed for passivity based controller (PBC) design.Since our objective is to provide novel PBC techniques, wewill impose similar assumptions as in Proposition 2.3 for theease of discussions. For a similar reason, we assume S ( x ∗ ) =0 in Definition 2.2. ⊳ For a passive system, the controller u = − Kh ( x ) with K ≻ plays an important role. According to [9], [18], this con-troller achieves stabilization of the origin if S ( x ) is positivedefinite, and a passive system (1) is zero-state detectable, i.e., Definition 2.5:
Suppose that Assumption 2.1 holds. Thesystem (1) with the output y = h ( x ) , h ( x ∗ ) = 0 is said tobe detectable at ( x ∗ , u ∗ ) if u ( · ) = u ∗ and y ( · ) = 0 = ⇒ lim t →∞ x ( t ) = x ∗ . Detectability at (0 , is called zero-state detectability [9], [18]. ⊳ The rest of this section is dedicated to provide four passivityconcepts and investigate their relations. These relations aresummarized in Fig. 1.
B. Differential Passivity and Krasovskii Passivity
First, we show the definition of differential passivity [7].This is introduced by using the so-called prolonged systemconsisting of the nonlinear system (1) and its variationalsystem: ˙ δx = F ( x, u ) δx + m X i =1 g i ( x ) δu i , (2) F ( x, u ) := ∂g ( x ) ∂x + m X i =1 ∂g i ( x ) ∂x u i , where δx : R → R n and δu = [ δu , . . . , δu m ] ⊤ : R → R m denote the state and input of the variational system, respec-tively.Differential passivity is defined as a passivity property ofthe prolonged system. Definition 2.6: [7] The nonlinear system (1) is said tobe differentially passive if there exists h D : R n × R n → R m , h D ( · ,
0) = 0 such that at each ( x, u ) ∈ R n × R m , itsprolonged system is dissipative at ( x, δx ) = ( x, with respectto the supply rate δu ⊤ h D ( x, δx ) with a storage function of theform S D ( x, δx ) . ⊳ By applying Proposition 2.3 to the prolonged system, anecessary and sufficient condition is given by [7].
Proposition 2.7: [7, Proposition 4.2] A system (1) is differ-entially passive if and only if there exist h D : R n × R n → R m and non-zero class C function S D : R n × R n → R such that S D ( x, δx ) ≥ , S D ( x,
0) = 0 , h D ( x,
0) = 0 ,∂S D ( x, δx ) ∂x f ( x, u ) + ∂S D ( x, δx ) ∂δx F ( x, u ) δx ≤ , (3) ∂S D ( x, δx ) ∂δx g ( x ) = h ⊤ D ( x, δx ) for all ( x, u ) ∈ R n × R m and δx ∈ R n . ⊳ It is worth mentioning that for general differentially passivesystems, a control design methodology has not been wellexplored yet. A bottle neck is that if one simply appliestechniques of passivity based control design, then a controlleris designed for the variational system, but not for the originalsystem. It is not obvious how to design a controller for theoriginal system from one designed for the variational system.In order to address this issue, we provide a new passivityconcept, which we call
Krasovskii passivity ; the reason forpicking this name will be explained in Remark 2.11. In fact, wewill show that every differentially passive system is Krasovskiipassive. This fact is helpful for stabilizing controller design ofdifferentially passive systems.The main idea of defining Krasovskii passivity comes fromthe fact that the pair ( f ( x, u ) , ˙ u ) satisfies the equation of thevariational system (2), namely df ( x, u ) dt = F ( x, u ) f ( x, u ) + g ( x ) ˙ u, (4)where recall that we consider input affine system (1). There-fore, it is expected that if a system is differentially passivefor the input port variable δu , then it is passive for the inputport variable ˙ u . In order to obtain this conclusion formally,we introduce the so called extended system [19]: (cid:26) ˙ x = f ( x, u ) , ˙ u = u d , (5)where ( x, u ) and u d : R → R m denote the state and input,respectively.We now define Krasovskii passivity as follows. Definition 2.8:
Suppose that Assumption 2.1 holds. Then,the nonlinear system (1) is said to be
Krasovskii passive at ( x ∗ , u ∗ ) if there exists h K : R n × R m → R m , h K ( x ∗ , u ∗ ) =0 such that the extended system (5) is dissipative at ( x ∗ , u ∗ ) with respect to the supply rate u ⊤ d h K ( x, u ) with a storagefunction S K ( x, u ) . ⊳ One notices that Krasovskii passivity can be viewed as thepassivity for the extended system (5) at a shifted equilibriumpoint ( x ∗ , u ∗ ) . Since this is a passivity property of the ex-tended system, it is possible to develop control methodologiesfor the extended system based on Krasovskii passivity, whichwill be investigated in Section III later.By applying Proposition 2.3, we have the following nec-essary and sufficient condition for Krasovskii passivity. Theproof follows from the fact that Krasovskii passivity is thepassivity for the extended system (5), and thus is omitted. Proposition 2.9:
Suppose that Assumption 2.1 holds. Asystem (1) is Krasovskii passive if and only if there ex-ist h K : R n × R m → R m and non-zero class C func-tion S K : R n × R m → R such that S K ( x, u ) ≥ , S K ( x ∗ , u ∗ ) = 0 , h K ( x ∗ , u ∗ ) = 0 ,∂S K ( x, u ) ∂x f ( x, u ) ≤ , (6) ∂S K ( x, u ) ∂u = h ⊤ K ( x, u ) (7)for all ( x, u ) ∈ R n × m . ⊳ We are now ready to show that differential passivity impliesKrasovskii passivity.
Theorem 2.10:
Under Assumption 2.1, if a system (1) is dif-ferentially passive, then it is Krasovskii passive at any ( x ∗ , u ∗ ) . Proof:
We show that S K ( x, u ) = S D ( x, f ( x, u )) ≥ and u ⊤ d h D ( x, f ( x, u )) are respectively a storage function andsupply rate for Krasovskii passivity by using Propositions 2.7and 2.9. First, from f ( x ∗ , u ∗ ) = 0 and S D ( x,
0) = 0 , itfollows that S K ( x ∗ , u ∗ ) = S D ( x ∗ , f ( x ∗ , u ∗ )) = S D ( x ∗ ,
0) = 0 . Similarly, h D ( x ∗ , f ( x ∗ , u ∗ )) = 0 .Next, (4) and Proposition 2.7 lead to ∂S K ( x, u ) ∂x f ( x, u )= (cid:18) ∂S D ( x, δx ) ∂x f ( x, u ) + ∂S D ( x, δx ) ∂δx F ( x, u ) δx (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δx = f ( x,u ) ≤ , and ∂S K ( x, u ) ∂u = (cid:18) ∂S D ( x, δx ) ∂δx g ( x ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δx = f ( x,u ) = h ⊤ D ( x, δx ) | δx = f ( x,u ) =: h ⊤ K ( x, u ) . Since all conditions in Proposition 2.9 hold, the system isKrasovskii passive.
Remark 2.11:
In the proof of Theorem 2.10, we obtainthe storage function S K ( x, u ) by replacing δx by f ( x, u ) in S D ( x, δx ) . In fact, for stability analysis, the Lyapunovfunction constructed by Krasovskii’s method is obtained sim-ilarly from the differential Lyapunov function of contractionanalysis [4]. Because of this analogy, we name the proposedpassivity concept Krasovskii passivity. ⊳ Theorem 2.10 shows that differential passivity impliesKrasovskii passivity. The converse is not always true in a casewhere we aim for the standard class of storage functions forthe variational part, as is shown in the next example.
Example 2.12:
Consider the following system: (cid:20) ˙ x ˙ x (cid:21) = g ( x ) u, g ( x ) := " x / − x / , where E = { ( x ∗ , u ∗ ) ∈ R × R : x ∗ = 0 }∪ { ( x ∗ , u ∗ ) ∈ R × R : u ∗ = 0 } . First, we show that this system is Krasovskii passive withstorage function: S K ( x, u ) = (1 / | g ( x ) u | = ( x / + x / ) u / , where S K ( · , · ) ≥ and S K ( x ∗ , u ∗ ) = 0 for any ( x ∗ , u ∗ ) ∈ E .Compute ∂S K ( x, u ) ∂x g ( x ) u = 0 ,∂S K ( x, u ) ∂u = ( x / + x / ) u =: h K ( x, u ) , where h K ( x ∗ , u ∗ ) = 0 for any ( x ∗ , u ∗ ) ∈ E . Therefore,Proposition 2.9 implies that the system is Krasovskii passiveat any ( x ∗ , u ∗ ) ∈ E .Next, we show that the system does not admit a storagefunction for differential passivity in the form of S D ( x, δx ) =( δx ⊤ M ( x ) δx ) / with positive definite M ( · ) ≻ . This canbe done by showing non existence of such a storage functionsatisfying (3), i.e., u δx ⊤ (cid:18) ∂M ( x ) ∂x x / − ∂M ( x ) ∂x x / (cid:19) δx + u δx ⊤ M ( x ) " x − / − x − / δx ≤ . Since this needs to hold for all u ∈ R , the equality should hold.If we choose δx = [1 , ⊤ , then the following is required tohold for all x ∈ R , ∂M , ( x ) ∂x − x / x / ∂M , ( x ) ∂x = − x M , ( x ) . We solve this partial differential equation by using the methodof characteristics. Consider the following set of differentialequations, dx dx = − x / x / , dM , dx = − x M , . The solution to the first and second equations are x / + x / = c , i.e., x = ± ( c − x / ) / and M , = c x − / ,respectively, with integration constants c , c ∈ R . Therefore,we have M , (cid:18) x , ± (cid:16) c − x / (cid:17) / (cid:19) = c x − / . This is not defined at x = 0 unless c = 0 . If c =0 , then M ( x ) is not positive definite at x = ± ( c − x / ) / . Therefore, the system does not admit a storagefunction for differential passivity in the form of S D ( x, δx ) =( δx ⊤ M ( x ) δx ) / with positive definite M ( · ) ≻ . ⊳ C. Extended Incremental Passivity and Shifted Passivity
In contraction analysis, differential properties have strongconnections with the corresponding incremental propertiessuch as stability [4], [5]. Motivated by these analysis, we alsoconsider incremental passivity, which is defined by using apair (( x, u ) , ( x ′ , u ′ )) of the states and inputs of the system (1)as follows. Definition 2.13:
The system (1) is said to be extendedincrementally passive if there exist h I : R n × R n → R m and non-zero class C function S I : R n × R n → R such that S I ( x, x ′ ) ≥ , S I ( x, x ) = 0 , h I ( x, x ) = 0 ,∂S I ( x, x ′ ) ∂x f ( x, u ) + ∂S I ( x, x ′ ) ∂x ′ f ( x ′ , u ′ ) ≤ ( u − u ′ ) ⊤ h I ( x, x ′ ) for all (( x, u ) , ( x ′ , u ′ )) ∈ ( R n × R m ) × ( R n × R m ) . ⊳ Remark 2.14:
This incremental passivity is an extension ofthe concept introduced by [6], [16] as h I is not necessarily anincremental function h ( x ) − h ( x ′ ) of some function h : R n → R m . ⊳ As shown in the previous subsection, differential passivityimplies Krasovskii passivity. As a counterpart, we have a sim-ilar relation between incremental and shifted passivity prop-erties. Shifted passivity is introduced by substituting ( x ∗ , u ∗ ) into ( x ′ , u ′ ) of Definition 2.13 for incremental passivity, i.e., Definition 2.15:
Suppose that Assumption 2.1 holds. Then,the system (1) is said to be extended shifted passive at ( x ∗ , u ∗ ) if there exists h S : R n → R m , h S ( x ∗ ) = 0 such that it is dis-sipative at x ∗ with respect to the supply rate ( u − u ∗ ) ⊤ h S ( x ) . ⊳ Remark 2.16:
Again our shifted passivity is an extensionof [14], [20] as h S is not necessarily an incremental function. ⊳ Passivity is nothing but extended shifted passivity at (0 , .That is, extended shifted passivity is defined by shiftingan equilibrium point from (0 , to an arbitrary ( x ∗ , u ∗ ) .Therefore its necessary and sufficient condition can readilybe obtained by the slight modification of Proposition 2.3. Dueto limitations of space, we refer to a similar result in [14,Proposition 1].From their definitions, it follows that incremental passivityimplies shifted passivity; the proof follows from the substitu-tion ( x ′ , u ′ ) = ( x ∗ , u ∗ ) , which is omitted. Proposition 2.17:
Under Assumption 2.1, if a system (1)is extended incrementally passive, then it is extended shiftedpassive at any ( x ∗ , u ∗ ) . ⊳ D. Krasovskii Passivity and Extended Shifted Passivity
As mentioned before, Krasovskii passivity can be seen asextended shifted passivity at (( x ∗ , u ∗ ) , for the extendedsystem (5). In fact, extended shifted passivity for the extendedsystem implies that of the original system (1), i.e., Theorem 2.18:
Suppose that Assumption 2.1 holds,and system (1) is Krasovskii passive at ( x ∗ , u ∗ ) .If ( ∂ ⊤ S K ( x ∗ , u ∗ ) /∂x ) g ( x ∗ ) = 0 , then it is extendedshifted passive at ( x ∗ , u ∗ ) . Proof:
By using Proposition 2.9, we show that S S ( x ) = S K ( x, u ∗ ) and ( u − u ∗ ) ⊤ g ⊤ ( x )( ∂ ⊤ S K ( x, u ∗ ) /∂x ) are respec-tively a storage function and supply rate for extended shifted passivity. First, S S ( x ∗ ) = S K ( x ∗ , u ∗ ) = 0 . Next, from (1)and (6), it follows that ∂S S ( x ) ∂x f ( x, u )= ∂S K ( x, u ∗ ) ∂x f ( x, u )= ∂S K ( x, u ∗ ) ∂x f ( x, u ∗ ) + ∂S K ( x, u ∗ ) ∂x g ( x )( u − u ∗ ) ≤ ∂S K ( x, u ∗ ) ∂x g ( x )( u − u ∗ ) . Therefore, the system is extended shifted passive for h S ( x ) = g ⊤ ( x )( ∂ ⊤ S K ( x, u ∗ ) /∂x ) , where h S ( x ∗ ) = 0 from the as-sumption. Remark 2.19:
The assumption ( ∂ ⊤ S K ( x ∗ , u ∗ ) /∂x ) g ( x ∗ ) =0 is mild if we start from differential passivity. In contractionanalysis including differential passivity, a storage function inthe form of S D ( x, δx ) = ( δx ⊤ M ( x ) δx ) / , M ( · ) ≻ is typi-cally used. If a system is differentially passive, Theorem 2.10shows that S K ( x, u ) = ( f ⊤ ( x, u ) M ( x ) f ( x, u )) / is a storagefunction for Krasovskii passivity. Since f ( x ∗ , u ∗ ) = 0 , theassumption holds for this storage function. ⊳ As mentioned in Section II-B, we will develop a PBCtechnique on the basis of Krasovskii passivity in Section IIIlater. Theorem 2.18 suggests that the developed techniquecan be modified for extended shifted passivity. We will alsoinvestigate this.
E. Differential Passivity and Incremental Passivity
In the previous sections, we have investigated connectionsamong passivity concepts. Still the connection between differ-ential passivity and extended incremental passivity in Fig. 1 ismissing. This connection is not very clear except for a specificcase.
Theorem 2.20:
If system (1) is differentially passive with astorage function S D ( x, δx ) = ( δx ⊤ M δx ) / for constant M (cid:23) , then it is extended incrementally passive. Proof:
Denote γ ( s ) := sx + (1 − s ) x ′ or µ ( s ) := su +(1 − s ) u ′ , s ∈ [0 , by the straight line connecting x and x ′ or u and u ′ , respectively. Compute f ( x, u ) − f ( x ′ , u ′ )= Z df ( γ ( s ) , µ ( s )) ds ds = Z (cid:18) F ( γ ( s ) , µ ( s )) dγ ( s ) ds + g ( γ ( s )) dµ ( s ) ds (cid:19) ds, where dγ ( s ) /ds = x − x ′ and dµ ( s ) /ds = u − u ′ .By using this, we show that S I ( x, x ′ ) = S D (cid:18) x, dγ ( s ) ds (cid:19) = 12 ( x − x ′ ) ⊤ M ( x − x ′ ) is a storage function for extended incremental passivity.From (3) with ∂S D ( x, δx ) /∂x = 0 , it follows that ∂S I ( x, x ′ ) ∂x f ( x, u ) + ∂S I ( x, x ′ ) ∂x ′ f ( x ′ , u ′ )= ( x − x ′ ) ⊤ M ( f ( x, u ) − f ( x ′ , u ′ ))= Z ∂S D ∂δx (cid:18) F ( γ ( s ) , µ ( s )) dγ ( s ) ds + g ( γ ( s )) dµ ( s ) ds (cid:19) ds ≤ Z ∂S D ∂δx g ( γ ( s )) dµ ( s ) ds ds ≤ ( u − u ′ ) ⊤ Z g ⊤ ( γ ( s )) M ( x − x ′ ) ds, where the arguments of S D are ( x, dγ ( s ) /ds ) . Therefore,the system is extended shifted passive for h I ( x, x ′ ) = R g ⊤ ( γ ( s )) M ( x − x ′ ) .In the proof of the above theorem, we consider the straightline as a path connecting x and x ′ or u and u ′ . One canhowever use an arbitrary class C path. The integral dependson the considered path.As well known from [21] if g ⊤ i ( x ) M dx , i = 1 , . . . , m isan exact differential one-form, i.e., there exists a function h i : R n → R such that g ⊤ i ( x ) M = ∂h i ( x ) ∂x , (8)then the path integral does not depend on the choice of apath. In the exact case, h I becomes h I ( x, x ′ ) = [ h ( x ) − h ( x ′ ) , . . . , h m ( x ) − h m ( x ′ )] ⊤ , and our incremental passivitymatches the incremental passivity in literature [6].Moreover, h i ( x ) is a linear function. The partial derivativesof both sides of (8) with respect to x yields ∂ h i ( x ) ∂x = ∂ ⊤ g i ( x ) ∂x M = 12 (cid:18) ∂ ⊤ g i ( x ) ∂x M + M ∂g i ( x ) ∂x (cid:19) , where we use the fact that ∂ h i ( x ) /∂x is symmetric.The inequality (3) with S D ( x, δx ) = ( δx ⊤ M δx ) / im-plies ∂ h i ( x ) /∂x = 0 . Therefore, h i ( x ) can be describedas c ⊤ i x with c i ∈ R n . Furthermore, one notices that g i ( x ) , i =1 , . . . , m is also constant if M ≻ . Indeed, from (8), it followsthat g ⊤ i = c ⊤ i M − .III. P ASSIVITY BASED D YNAMIC C ONTROLLER D ESIGNS
A. Krasovskii Passivity based Controllers
As mentioned above, differential passivity does not pro-vide a controller design method. However, we have provedin Section II-B that differential passivity implies Krasovskiipassivity. In this section, we illustrate the utility of Krasovskiipassivity for controller design.For a Krasovskii passive system, we provide the followingdynamic controller.
Theorem 3.1:
Suppose that Assumption 2.1 holds, and thesystem (1) is Krasovskii passive at ( x ∗ , u ∗ ) with a storagefunction S K ( x, u ) . Consider the following dynamic controllerfor the extended system (5): K ˙ u d = ν − K u d − K ( u − u ∗ ) − ∂ ⊤ S K ( x, u ) ∂u , (9) where ν : R → R m , and K ≻ and K , K (cid:23) are freetuning parameters. Then, the following three statements hold:(a) The closed-loop system consisting of (5) and (9) is dissi-pative at ( x ∗ , u ∗ , with respect to the supply rate ν ⊤ u d .(b) Let ν = 0 . Define y := ∂S K ( x, u ) ∂x f ( x, u ) ,y := K ( u − u ∗ ) + ∂ ⊤ S K ( x, u ) ∂u . (10)Suppose that there exists r > such that S K ( x, u )+ | u − u ∗ | K / > for all ( x, u ) ∈ B r ( x ∗ , u ∗ ) \ { ( x ∗ , u ∗ ) } ,where B r is an open ball as defined in the notation part.Then, there exists ¯ r > such that any solution to theclosed-loop system starting from B ¯ r ( x ∗ , u ∗ , convergesto the largest invariant set contained in { ( x, u, u d ) ∈ B ¯ r ( x ∗ , u ∗ ,
0) : y = 0 , K u d = 0 } . (11)(c) Moreover, if K ≻ and the extended systemwith the outputs ( y , y ) is detectable at (( x ∗ , u ∗ ) , ,then ( x ∗ , u ∗ ) is asymptotically stable. Proof:
Consider the following storage function: S ( x, u, u d ) := S K ( x, u ) + 12 | u − u ∗ | K + 12 | u d | K , (12)which is positive semi-definite at ( x ∗ , u ∗ , . By using (5)and (9), the Lie derivative of S along the vector field of theclosed-loop system, simply denoted by dS /dt , is computedas dS dt = ∂S K ( x, u ) ∂x f ( x, u )+ u ⊤ d (cid:18) ∂ ⊤ S K ( x, u ) ∂u + K ˙ u d + K ( u − u ∗ ) (cid:19) = ∂S K ( x, u ) ∂x f ( x, u ) + u ⊤ d ( ν − K u d ) . Therefore, (a) follows from (6).Next, we show (b). From the assumption, S ( x, u, u d ) ispositive definite at ( x ∗ , u ∗ , in B r ( x ∗ , u ∗ ) × R m . Therefore,the statement follows from LaSalle’s invariance principle.Finally, one can show (c) in a similar manner as [9, Corollary4.2.2], where y comes from (9) with u d ( · ) = 0 .The controller can be interpreted in terms of the transferfunction. From the definition of Krasovskii passivity, it followsthat h K ( x, u ) = ∂ ⊤ S K ( x, u ) ∂u . Take h K ( x, u ) as the output y of the system (1). If K s + K s + K is invertible, the Laplace transformation of thecontroller dynamics (9) can be computed as U ( s ) = ( K s + K s + K ) − ( V ( s ) − Y ( s )) , (13)where U ( s ) , Y ( s ) , V ( s ) denote the Laplace transformationsof u − u ∗ , y and ν , respectively. Therefore, the controller canbe viewed as a second order output feedback controller.In the above theorem, K is chosen to be positive definite.However, even if K = 0 , by assuming positive definitenessof K instead, it is possible to construct a first order controller. The following corollary is a generalization of a controllerfor boost converters in DC microgrids [10], [12] to generalnonlinear systems. The proof is similar to the proof that canbe found in [1] and is omitted. Corollary 3.2:
Suppose that Assumption 2.1 holds, and thesystem (1) is Krasovskii passive at ( x ∗ , u ∗ ) . Then, considerthe extended system (5) with the following controller: K u d = ν − K ( u − u ∗ ) − ∂ ⊤ S K ( x, u ) ∂u , (14)where ν : R → R m , and K ≻ and K (cid:23) are free tuningparameters. Then, the following three statements hold:(a) The closed-loop system consisting of (5) and (14) is dis-sipative at ( x ∗ , u ∗ ) with respect to the supply rate ν ⊤ u d .(b) Let ν = 0 . Suppose that there exists r > suchthat S K ( x, u ) + | u − u ∗ | K / > for all ( x, u ) ∈ B r ( x ∗ , u ∗ ) \ { ( x ∗ , u ∗ ) } . Then, there exists ¯ r > suchthat any solution to the closed-loop system startingfrom B ¯ r ( x ∗ , u ∗ ) converges to the largest invariant setcontained in { ( x, u ) ∈ B ¯ r ( x ∗ , u ∗ ) : y = 0 , y = 0 } for ( y , y ) in (10).(c) Moreover, if the extended system with the out-puts ( y , y ) is detectable at (( x ∗ , u ∗ ) , , then ( x ∗ , u ∗ ) is asymptotically stable. ⊳ B. Extended Shifted Passivity based Controllers
In this subsection, inspired by the Krasovskii passive casesin Theorems 2.18 and 3.1, we provide an extended shiftedpassivity based control design, where the controller is designedfor the original system (1) instead of its extended system (5).
Theorem 3.3:
Suppose that all assumptions in Theorem 2.18hold. Consider the original system (1) with the followingdynamic feedback controller: u = u ∗ − K g ⊤ ( x ) ∂ ⊤ S K ( x, u ∗ ) ∂x + K v,K ˙ v = ν − K g ⊤ ( x ) ∂ ⊤ S K ( x, u ∗ ) ∂x − K v, (15)where ν : R → R m , and K ≻ and K , K , K (cid:23) arefree tuning parameters. Then, the following three statementshold:(a) The closed-loop system consisting of (1) and (15) isdissipative with respect to the supply rate ν ⊤ v .(b) Let ν = 0 . Define y := ∂S K ( x, u ∗ ) ∂x f ( x, u ∗ ) ,y := g ⊤ ( x ) ∂ ⊤ S K ( x, u ∗ ) ∂x . Suppose that there exists r > such that S K ( x, u ∗ ) > for all x ∈ B r ( x ∗ ) \ { x ∗ } ; recall the definition of an openball B r in the notation part. Then, there exists ¯ r > such that any solution to the closed-loop system starting from B ¯ r ( x ∗ , converges to the largest invariant setcontained in { ( x, v ) ∈ B ¯ r ( x ∗ ,
0) : y = 0 , K y = 0 , K v = 0 } . (16)(c) Moreover, if K , K ≻ and the system with theoutputs ( y , y ) is detectable at ( x ∗ , u ∗ ) , then ( x ∗ , u ∗ ) is asymptotically stable. Proof:
Consider the following storage function: S ( x, v ) := S K ( x, u ∗ ) + 12 | v | K , (17)which is positive semi-definite at ( x ∗ , . In a similar manneras the proof of Theorem 2.18, by using (15), the Lie derivativeof S along the vector field of the closed-loop system, simplydenoted by dS /dt is computed as follows, dS dt = ∂S K ( x, u ∗ ) ∂x f ( x, u ∗ ) + ∂S K ( x, u ∗ ) ∂x g ( x )( u − u ∗ )+ v ⊤ K ˙ v = v ⊤ ν + ∂S K ( x, u ∗ ) ∂x f ( x, u ∗ ) − v ⊤ K v − ∂S K ( x, u ∗ ) ∂x g ( x ) K g ⊤ ( x ) ∂ ⊤ S K ( x, u ∗ ) ∂x . From (6), we obtain (a). Also, one can show (b) and (c) insimilar manners as the proof of Theorem 2.18.
Remark 3.4:
Similar conclusions as Theorem 3.3 hold ifAssumption 2.1 holds, and the original system (1) is extendedshifted passive with a storage function S S ( x ) . In this case, asimilar controller as (15) is obtained by replacing S K ( x, u ∗ ) with S S ( x ) . ⊳ From the definition of the shifted passivity, it follows that h S ( x ) = g ⊤ ( x ) ∂ ⊤ S K ( x, u ∗ ) ∂x . Take h S ( x ) as the output y of the system (1). If K s + K is invertible, the Laplace transformation of the controllerdynamics (15) can be computed as U ( s ) = − K Y ( s ) + K ( K s + K ) − ( V ( s ) − K Y ( s )) . where U ( s ) , Y ( s ) , V ( s ) denote the Laplace transformationsof u − u ∗ , y and ν , respectively. Therefore, the controllercan be viewed as a proper output feedback controller and isdifferent from (13). If K = 0 , one has a structure of the lowpass filter. If K = 0 , one has a standard-type PBC. If K =0 , one has a PI feedback controller, which is an extensionof one presented in [14] as h S ( x, x ∗ ) is not necessarily anincremental function h ( x ) − h ( x ∗ ) .IV. E XAMPLE
In this example, we consider the average model of a DC-Zeta converter. It has the capability of both buck and boostconverters, i.e., it can amplify and reduce the supply voltagewhile maintaining the polarity. The schematic of the Zetaconverter is given in Fig. 2. As shown, it contains fourenergy storage elements, namely two inductors L , L and twocapacitors C , C , an ideal switching element u and an idealdiode. Further, V s and G denote the constant supply voltage uV s L C L C G Fig. 2. Electrical scheme of the Zeta converter. and the load, respectively. The objective of the converter is tomaintain a desired voltage v ∗ across the load G . After somechanges of state and time variables, one obtains the followingnormalized model for the converter; for more details aboutchanges of variables, see [22, Chapter 2.8]. ˙ x = − x x − x /α ( x − x /α ) /α + x − ( x + x )(1 + x ) /α u, (18)where α , α and α are positive constants depending on thesystem parameters. It is worth pointing out that a (standard)PBC has not been provided for this class of systems becauseit is difficult (or maybe impossible) to find a storage functionfor passivity. However, we demonstrate that our proposed twoPBC techniques are useful for controller design.For this system, the set E is obtained as E v = (cid:26) ( x ∗ , u ∗ ) ∈ R × R : x ∗ = (cid:18) ( v ∗ ) α , v ∗ , v ∗ α , v ∗ (cid:19) , u ∗ = v ∗ v ∗ + 1 , (cid:27) , v ∗ ∈ R + . One notices that E v has a unique element for any v ∗ ∈ R + .The parameters are chosen as a = a = a = 1 and v ∗ = 1 / which determines x ∗ = [1 / , / , / , / and u ∗ = 1 / .First, we illustrate the Krasovskii PBC (9) in Theorem 3.1.One can confirm that the DC-Zeta converter (18) satisfies theconditions of Proposition 2.9 for S K ( x, u ) = | f ( x, u ) | M / , M = diag { , , α , α } . (19)Then, h K ( x, u ) = ∂S K ( x, u ) ∂u = (cid:2) x − ( x + x ) 1 + x (cid:3) ˙ x. If x and ˙ x are measured, the controller (9) does not requirethe information of parameters α , α , α for the DC-Zetaconverter.It is possible to show that the storage function S ( x, u, u d ) in (12) is radially unbounded, and the largest invariant setcontained in the set (11) is nothing but E v . Therefore, forany v ∗ ∈ R + , any trajectory of the closed-loop systemconverges to E v . Figure 3 shows the trajectories of the closed-loop system starting from several initial states, where ν = 0 and K = K = K = 1 . Fig. 3. Closed-loop trajectories when controlled by the Krasovskii PBC.
Fig. 4. Closed-loop trajectories when controlled by the extended shifted PBC.
Second, we illustrate the extended shifted PBC (15) inTheorem 3.3, where ∂S K ( x, u ∗ ) ∂x g ( x )= 12 f ⊤ ( x, u ∗ ) (1 − u ∗ )( x + x )(1 − u ∗ + u ∗ /α )(1 + x ) − u ∗ ( x + x )(1 /α )(1 + x ) . Again, it is possible to show that the storage function S ( x, v ) in (17) is radially unbounded, and the largest invariant setcontained in the set (16) is nothing but E v . Therefore, forany v ∗ ∈ R + , any trajectory of the closed-loop systemconverges to E v . Figure 4 shows the trajectories of the closed-loop system starting from several initial states, where ν = 0 and K = K = K = K = 1 .Finally, simulation results indicate that the Krasovskii PBCachieves convergence to E v with less oscillations than theshifted PBC, whereas the extended shifted PBC convergeswith relatively lower amplitudes of the oscillations than theKrasovskii PBC case. V. C ONCLUSION
In this paper, we have introduced the concept of Krasovskiipassivity for addressing the difficulty of differential passivitybased controller design. First, we have established relationsamong the relevant four passivity concepts: 1) differentialpassivity, 2) Krasovskii passivity, 3) extended incrementalpassivity, and 4) extended shifted passivity. Next, we haveproposed Krasovskii/extended shifted passivity based dynamic controllers. The utility of the proposed controllers has beenillustrated by the DC-Zeta converter for which traditional PBCtechniques are hard to use. Future work includes studying con-trol methodologies for networked Krasovskii passive systemsas done for shifted passivity-short systems [23].R
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