Converse passivity theorems
aa r X i v : . [ m a t h . O C ] J a n Converse passivity theorems ⋆ Sei Zhen Khong ∗ Arjan van der Schaft ∗∗∗
Institute for Mathematics and its Applications, The University ofMinnesota, Minneapolis, MN 55455, USA. (e-mail: [email protected] ) ∗∗ Johann Bernoulli Institute for Mathematics and Computer Science,University of Groningen, Groningen 9747 AG, Netherlands. (email: [email protected] ). Abstract:
Passivity is an imperative concept and a widely utilized tool in the analysis andcontrol of interconnected systems. It naturally arises in the modelling of physical systemsinvolving passive elements and dynamics. While many theorems on passivity are known inthe theory of robust control, very few converse passivity results exist. This paper establishesvarious versions of converse passivity theorems for nonlinear feedback systems. In particular,open-loop passivity is shown to be necessary to ensure closed-loop passivity from an input-output perspective. Moreover, the stability of the feedback interconnection of a specific systemwith an arbitrary passive system is shown to imply passivity of the system itself.
Keywords:
Passivity, feedback, robustness1. INTRODUCTIONPassivity has emerged to be a crucial concept and toolin the analysis and control of feedback systems, andinterconnected systems in general; see e.g. Willems (1972);Moylan and Hill (1978); Vidyasagar (1981); Megretski andRantzer (1997); van der Schaft (2016). A salient result isthe passivity theorem , stating (in various versions) that thestandard feedback interconnection of two passive systemsis again passive (and hence stable in a certain sense).While passivity theory is deep-rooted in physical systemsmodeling and synthesis (such as electrical network theory)based on the essential notions of power and energy, itsunderlying concepts and results have turned out to beequally valuable in the broad field of control, ranging fromadaptive control to stabilization of nonlinear systems.While the passivity theorem pervades large parts of sys-tems and control theory, the converse versions of thepassivity theorem seem to be much less recognized andappreciated. The simplest version of a converse passivitytheorem, stating that the feedback interconnection of twosystems is passive if and only if the two (open-loop) sys-tems are passive, was previously noted and proved withinthe state-space context in Kerber and van der Schaft(2011) (see also van der Schaft (2016)), and an easy proofin the nonlinear input-output map setting will be providedin Section 3.The main part of the paper (Section 4) is concernedwith a different, and more involved, converse passivitytheorem, stating that if the feedback interconnection ofa system with an arbitrary passive system is stable (to bespecified later on), then the system is necessarily passive.This basic idea is, sometimes implicitly, underlying a large ⋆ The authors gratefully acknowledge the support of the Institutefor Mathematics and its Applications, where this work was initiatedduring the 2015-2016 program on Control Theory and its Applica-tions. part of literature on robotics and impedance control; seee.g. Stramigioli (2015). In fact, a version of this resultwas proved for linear single-input single-output systems inColgate and Hogan (1988) using arguments from Nyquiststability theory, exactly with this motivation in mind. Inrobotics the motivation for this converse passivity theoremcan be formulated as follows. Consider a controlled robotinteracting in operation with its environment (the normalscenario). In many cases the environment is largely un-known, while at the same time the stability of the robotinteracting with its environment can be often considered asa sine qua non . Since the interaction of the robot with itsenvironment typically takes place via the conjugated vari-ables of (generalized) velocity and force, whose product isequal to power , it is not completely unreasonable to assumethat the environment, seen from the interaction port withthe robot, is, although unknown, a passive system . Thusthe converse passivity theorem treated in Section 4 gives aclear rationale for the often expressed design and controlprinciple (Stramigioli, 2015; Colgate and Hogan, 1988)that a controlled robot should be passive at its interactionport with the unknown environment . Differently fromColgate and Hogan (1988); Stramigioli (2015), the proof ofthe general nonlinear converse passivity theorem treated inSection 4 will be based on the S-procedure lossless theorem(see Megretski and Treil (1993) or (J¨onsson, 2001, Thm.7)). 2. NOTATION AND PRELIMINARIESDenote by L n the set of R n -valued Lebesgue square-integrable functions: The second author would like to thank Stefano Stramigioli formany inspiring conversations on this and related subjects. n := n v : [0 , ∞ ) → R n : k v k = h v, v i := Z ∞ v ( t ) T v ( t ) dt < ∞ o . Define the truncation operator( P T v )( t ) := (cid:26) v ( t ) t ∈ [0 , T )0 otherwise , and the extended spaces L n e := { v : [0 , ∞ ) → R n : P T v ∈ L ∀ T ∈ [0 , ∞ ) } . In what follows, the superscript n is often suppressed fornotational simplicity. Define the shift operator ( S T u )( t ) = u ( t − T ) for T ≥ I : L e → L e . A system ∆ : L e → L e is said to be causal if P T ∆ P T = P T ∆ for all T ≥
0. It is said to be time-invariant if S T ∆ = ∆ S T for all T >
0. A causal ∆is called bounded if its Lipschitz bound (Willems, 1971,Section 2.4) is finite, i.e. k ∆ k := sup T > k P T u k =0 k P T ∆ u k k P T u k = sup = u ∈ L k ∆ u k k u k < ∞ . ∆ is said to be passive (Willems, 1972; van der Schaft,2016) if inf T > ,u ∈ L e Z T u ( t ) T (∆ u )( t ) dt ≥ , (1) strictly passive if there exists ǫ > Z T u ( t ) T (∆ u )( t ) dt ≥ ǫ ( k P T u k + k P T ∆ u k ) ∀ u ∈ L e , T >
0, and output strictly passive if there existsan ǫ > Z T u ( t ) T (∆ u )( t ) dt ≥ ǫ k P T ∆ u k ∀ u ∈ L e , T > . Lemma 1.
If ∆ is bounded, then passivity is equivalent toinf u ∈ L Z ∞ u ( t ) T (∆ u )( t ) dt ≥ . (2)Similarly, strict passivity and output strict passivity of ∆are equivalent to Z ∞ u ( t ) T (∆ u )( t ) dt ≥ ǫ ( k u k + k ∆ u k ) ∀ u ∈ L and Z ∞ u ( t ) T (∆ u )( t ) dt ≥ ǫ k ∆ u k ∀ u ∈ L , respectively. Proof.
First, note that (2) can be obtained from (1) byrestricting u ∈ L and taking T to ∞ . Conversely, supposethat (1) does not hold, then there exist T > u ∈ L e such that R T u ( t ) T (∆ u )( t ) dt <
0. Let ¯ u ( t ) := u ( t ) for t ∈ [0 , T ) and ¯ u ( t ) := 0 for t ≥ T . Then ¯ u ∈ L and R ∞ ¯ u ( t ) T (∆¯ u )( t ) = R T u ( t ) T (∆ u )( t ) dt <
0. That is, (2) isviolated. This completes the proof for the first part of thelemma. The rest of the lemma can be shown in a similarfashion. ✷ The main object of study in this paper is the feedbackinterconnection of causal systems Σ : L e → L e andΣ : L e → L e described by Σ + + e e u u − y y Σ Fig. 1. Feedback configuration u = e − y ; u = e + y ; y = Σ u ; y = Σ u ; (3)see Figure 1. Definition 2.
The feedback interconnection Σ k Σ is saidto be well-posed if the map ( y , y ) ( e , e ) defined by(3) has a causal inverse F on L e . It is finite-gain stable if in addition to being well-posed, k F k < ∞ . A well-posed Σ k Σ is said to be passive if the map ( e , e ) ( y , y ) is passive. In the case where e = 0, the feedbackinterconnection is said to be finite-gain stable if it is well-posed and e y is bounded.3. FEEDBACK PASSIVITYIn this section, a simple proof that passivity of the closed-loop system implies passivity of the open-loop compo-nents is provided. Contrary to the state-space settingin Kerber and van der Schaft (2011), the result adoptsthe input-output perspective and is applicable to infinite-dimensional systems, such as those modelled by time-delayor partial differential equations. Theorem 3.
Given causal Σ and Σ for which Σ k Σ iswell posed, Σ k Σ is passive if, and only if, Σ and Σ arepassive. Proof. (= ⇒ ) By hypothesis,inf T > ,e ,e ∈ L e Z T e ( t ) T y ( t ) + e ( t ) T y ( t ) dt ≥ . Using (3) and the well-posedness of Σ k Σ , this is equiv-alent to0 ≤ inf T > ,u ,u ∈ L e Z T ( u ( t ) + y ( t )) T y ( t )+( u ( t ) − y ( t )) T y ( t ) dt = inf T > ,u ,u ∈ L e Z T u ( t ) T y ( t ) + u ( t ) T y ( t ) dt This implies thatinf
T > ,u ∈ L e Z T u ( t ) T y ( t ) dt ≥ u = 0) andinf T > ,u ∈ L e Z T u ( t ) T y ( t ) dt ≥ u = 0), which are equivalent to passivity ofΣ and Σ , respectively.( ⇐ =) This direction is well-known in the literature andcan be shown by reversing the arguments above. ✷ . PASSIVITY AS A NECESSARY AND SUFFICIENTCONDITION FOR STABLE INTERACTIONIn this section we show that a necessary and sufficientcondition in order that the closed-loop system arisingfrom interconnecting a given system to an unknown, butpassive, system is stable, is that the system is passive itself.We will formulate three slightly different versions of thismain result. Theorem 4.
Given a bounded time-invariant Σ , the feed-back interconnection Σ k Σ is finite-gain stable for allbounded passive Σ if, and only if, Σ is strictly passive. Proof.
Sufficiency is well known in the literature. Indeed,by Lemma 1, strict passivity of Σ and passivity of Σ yield ǫ ( k y k + k u k ) ≤ h u , y i + h u , y i = h e − y , y i + h e + y , y i = h e , y i + h e , y i . Therefore, k y k + h e − y , e − y i ≤ ǫ ( h e , y i + h e , y i )or k y k + k y k − h e , y i + k e k ≤ ǫ ( h e , y i + h e , y i )It follows that k y k ≤ h e , y i + 1 ǫ h e, y i ≤ (cid:18) ǫ (cid:19) k e k k y k , where y := ( y , y ) T and e := ( e , e ) T and the Cauchy-Schwarz inequality has been used.To show necessity, define H := { h = ( u , u , e , e ) ∈ L | u = e + Σ u } . Note that if h ∈ H , then S T h ∈ H for all T ≥ . Define the quadratic forms σ i : H → R , i = 0 ,
1, as σ ( u , u , e , e ) := * u u e e , I I − γI
00 0 0 − γI u u e e + σ ( u , u , e , e ) := * u u e e , − I − I I I u u e e + . By Lemma 1, stability of Σ k Σ for all bounded passiveΣ implies the existence of γ > σ ( u , u , e , e ) ≤ ∀ ( u , u , e , e ) ∈ H such that σ ( u , u , e , e ) ≥ . This is equivalent, via the S-procedure lossless theorem(see Megretski and Treil (1993) or (J¨onsson, 2001, Thm.7)), to the existence of τ ≥ σ ( u , u , e , e ) + τ σ ( u , u ,e , e ) ≤ ∀ ( u , u , e , e ) ∈ H . In the subset { ( u , u , , ∈ L | u = Σ u } ⊂ H , thisimplies that k Σ u k + k u k − τ h u , Σ u i ≤ ∀ u ∈ L . Equivalently, τ h u , Σ u i ≥ k Σ u k + k u k ∀ u ∈ L . It is obvious from the inequality above that τ = 0, hence h u , Σ u i ≥ τ ( k Σ u k + k u k ) ∀ u ∈ L , i.e. Σ is strictly passive, by Lemma 1. ✷ Certainly from a state space point of view the abovetheorem has the drawback of relying on strict passivity,since it is known that input strict passivity can onlyoccur for systems with direct feedthrough terms. Thus,it excludes a large class of physical systems. The followingalternative version avoids this problem by relying only on output strict passivity. Σ + + ˜ e e u u − y Σ ǫ I ǫ I + − y + − ˜ Σ ˜ Σ ˜ u ˜ y Fig. 2. Loop transformation
Theorem 5.
Given a bounded time-invariant ˜Σ , the feed-back interconnection ˜Σ k ˜Σ is finite-gain stable for alloutput strictly passive ˜Σ if, and only if, ˜Σ is outputstrictly passive. Proof.
Sufficiency is well known in the literature and canbe shown in a similar manner using the arguments inthe sufficiency proof for Theorem 4. For necessity, notethat any output strictly passive ˜Σ can be written as thenegative feedback interconnection of Σ and ǫI for somebounded passive Σ and ǫ >
0; see Figure 2. To see this, letΣ be bounded and observe by Lemma 1 that ˜Σ satisfies Z ∞ ˜ u ( t ) T y ( t ) dt ≥ ǫ k y k ∀ ˜ u ∈ L , whereby Z ∞ (˜ u ( t ) − ǫy ( t )) T y ( t ) dt = Z ∞ u ( t ) T y ( t ) dt ≥ . The last inequality holds for all u ∈ L , since given any u ∈ L , ˜ u := ( I + ǫ Σ ) u ∈ L yields the desired u .Therefore, it follows that Σ is passive. By the same token,the negative feedback interconnection of a bounded passiveΣ and ǫI with ǫ > := ˜Σ + ǫI as illustrated in Figure 2,one obtains the loop transformation configuration therein.Consequently, the finite-gain stability of the feedbacknterconnection ˜Σ k ˜Σ in Figure 2 is equivalent to thatof Σ k Σ in Figure 1 (Green and Limebeer, 1995, Section3.5). Application of Theorem 4 then yields that Σ isstrictly passive. For sufficiently small ǫ >
0, it then followsthat ˜Σ = Σ − ǫI is output strictly passive. ✷ Another feature of Theorem 4 is the fact that it requiresan external signal e , which is not the typical case inrobotics applications. This motivates the following versionof converse passivity theorem. Recall that an outputstrictly passive system has finite L -gain (van der Schaft,2016). Theorem 6.
Given a bounded time-invariant Σ , the feed-back interconnection Σ k Σ (with e = 0) has finite L -gain from e to y for all passive Σ if, and only if, Σ isoutput strictly passive. Proof.
Sufficiency is well known in the literature. Indeed,if Σ is output strictly passive and Σ is passive, then forsome ε > h e , y i = h u + y , y i = h u , y i + h y , y i = h u , y i + h u , y i ≥ ε k y k , showing that the closed-loop system is ε -output strictlypassive, and hence has L -gain ≤ ε .To show necessity, define H := { h = ( u , y , e ) ∈ L | y = Σ u } . Note that if h ∈ H , then S T h ∈ H for all T ≥ . Define the quadratic forms σ i : H → R , i = 0 ,
1, as σ ( u , y , e ) := 12 *" u y e , I
00 0 − γ I " u y e σ ( u , y , e ) := 12 *" u y e , " − I − I I I u y e . By Lemma 1, stability of Σ k Σ for all bounded passiveΣ implies the existence of γ such that σ ( u , y , e ) ≤ ∀ ( u , y , e ) ∈ H such that σ ( u , y , e ) ≥ . This is equivalent, via the S-procedure lossless theorem(see Megretski and Treil (1993) or (J¨onsson, 2001, Thm.7)), to the existence of τ ≥ σ ( u , y , e ) + τ σ ( u , y , e ) ≤ ∀ ( u , y , e ) ∈ H . This implies that − τ h u , y i + τ h e , y i + 12 k y k − γ k e k ≤ ∀ e ∈ L , and thus in the subset { ( u , y , ∈ L | y = − Σ u } ⊂H , this yields h u , y i ≥ τ k y k ∀ u ∈ L , i.e., Σ is output strictly passive, by Lemma 1. ✷ In the case of linear single-input single-output systems, aversion of the above theorem was proved before in Colgate and Hogan (1988), using an argument based on Nyquiststability theory . 5. CONCLUSIONSSeveral versions of converse passivity theorems for nonlin-ear systems are provided. Besides contributing to robustcontrol theory, these fundamental results have implicationsin the field of robotics, as described in the introduction.Future work will involve seeking similar results within thecontext of large-scale interconnected systems.REFERENCESColgate, J.E. and Hogan, N. (1988). Robust control ofdynamically interacting systems. International Journalof Control , 48(1), 65–88.Green, M. and Limebeer, D.J.N. (1995).
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