On the Passivity and Positivity Properties in Dynamic Systems: Their Achievement under Control Laws and Their Maintenance under Parameterizations Switching

Abstract

This paper is devoted to discuss certain aspects of passivity results in dynamic systems and the characterization of the regenerative systems counterparts. In particular, the various concepts of passivity as standard passivity, strict input passivity, strict output passivity, and very strict passivity (i.e., joint strict input and output passivity) are given and related to the existence of a storage function and a dissipation function. Later on, the obtained results are related to external positivity of systems and positivity or strict positivity of the transfer matrices and transfer functions in the time-invariant case. On the other hand, how to achieve or how eventually to increase the passivity effects via linear feedback by the synthesis of the appropriate feed-forward or feedback controllers or, simply, by adding a positive parallel direct input-output matrix interconnection gain is discussed.

1. Introduction

This paper is devoted to discuss certain aspects of passivity results in dynamic systems and the characterization of the regenerative versus passive systems counterparts. In particular, the various concepts of passivity as standard passivity, strict input passivity, strict output passivity, and very strict passivity (i.e., joint strict input and output passivity) are given and related to the existence of a storage function and a dissipation function. Basic previous background concepts on passivity are given in [1–4] and some related references therein. More detailed generic results about passivity and positivity are given in [5–7]. Note, in particular, the use of passive devices is very relevant in certain physical and electronic applications. See, for instance, [8]. Later on, the obtained results are related to external positivity of systems and positivity or strict positivity of the transfer matrices and transfer functions in the time-invariant case. On the other hand, it is discussed and formalized how to proceed in the case of passivity failing. It is also analyzed the way of eventually increasing the passivity effects via linear feedback by the synthesis of the appropriate feed-forward or feedback controllers or, simply, by adding a positive parallel direct input-output matrix interconnection gain having a minimum positive lower-bounding threshold gain which is also a useful idea for asymptotic hyperstability of parallel disposals of systems, [9]. For the performed analysis, the concept of relative passivity index, which is applicable for both passive and nonpassive systems, is addressed and modified to a lower number by the use of appropriate feedback or feed-forward compensators. Finally, the concept of passivity is discussed for switched systems which can have both passive and nonpassive configurations which become active governed by switching functions. The passivity property is guaranteed by the switching law under a minimum residence time at passive active configurations provided that the first active configuration of the switched disposal is active and that there are no two consecutive active nonpassive configurations in operation. Some illustrative examples are also discussed. The so-called storage functions which play a relevant role in the study of passivity are Lyapunov functions. Lyapunov functions are commonly used in the background literature for stability analysis of deterministic and dynamic systems. See, for instance, [10–12].

1.1. Notation

(i), where , ,(ii) denotes that the real matrix is positive definite while denotes that it is positive semidefinite,(iii) and denote, respectively, the minimum and maximum eigenvalues of the real symmetric -matrix,(iv) denotes that the transfer matrix of a linear time-invariant system is positive real; that is, for all , and denotes that it is strictly positive real; that is, for all . The set of strongly positive real transfer matrices is the subset of of entries having relative degree zero so that they cannot diverge as . If the linear time –invariant system is a SISO one (i.e., it has one input and one output) then if for all and if for all ,(v)A dynamic system is positive (resp., externally positive) if all the state components (resp., if all the output components) are nonnegative for all time for any given nonnegative initial conditions and nonnegative input,(vi) is the complex unity,(vii) is the th identity matrix,(viii)The superscript stands for matrix transposition,(ix) is the Hardy space of all complex-valued functions of a complex variable which are analytic and bounded in the open right half-plane of norm (by the maximum modulus theorem) and is the subset of real-rational functions of .

2. Preliminaries

Consider a dynamic system with state , input , and output , where is the extended space of the Hilbert space endowed with the inner product from to consisting of the truncated functions for and ; >t and . If then

Definition 1 (see [2]). The above dynamic system is
(1) -stable if implies ;
(2) nonexpansive if and s.t. for all ,(3) passive if such that ;
(4) strictly input passive if and s.t. (5) strictly output passive if and s.t. (6) strictly input/output passive (or very strictly passive) if , and s.t.The constants , , and are, respectively, referred to as the passivity, input passivity, and output passivity constants.

3. Some Passivity and Positivity Results: Passivity Achievement by Direct Input-Output Interconnection

Note that the above definitions can be expressed equivalently via an inner product notation. Note also that the above definitions are equivalent for to the corresponding positivity and strict positivity concepts [1] as mentioned in [2]. In particular, some relevant positivity and passivity properties are summarized in the following result for a single-input single-output (SISO) system by relating the time and frequency domains descriptions:

Theorem 2. Consider a linear time-invariant SISO (i.e., ) system whose transfer function . Then, the following properties hold:
(i) and and, furthermore, if then . Then, the system is passive.
(ii) Assume, in addition, that . Then for any and some .
(iii) If, furthermore, the system is externally positive then for any given nonnegative initial conditions and nonnegative input.
(iv) Define as the relative passivity index of the transfer function ( and being the numerator and denominator polynomials of ). Then, the constraint is guaranteed for some ifIf (resp., ) then (resp., ).

Proof. It turns out that the Fourier transforms (denoted with hats and the same symbols as their time functions counterparts) of the truncated input and output for any time exist since the truncated signals are in . Therefore, Parseval’s theorem can be applied to express ; in the frequency domain. Take into account, in addition, that the hodograph of the frequency system’s response satisfies and for all and that since . Thus, the various expressions which follow hold under zero initial conditions of the dynamic system:It has been proved, under zero initial conditions, that and ; and if then ; for some independent of (and independent of ). Since the zero state response generates and square-integrable output, since the input is square-integrable and since the zero input state is uniformly bounded as a result, the output is square-integrable for any square-integrable input. Also, the system is passive, since irrespective of the initial conditions, there exists some such that since the initial conditions do not generate an unbounded homogeneous solution since since . Property (i) has been proved. On the other hand, under any finite nonzero initial conditions :for some uniformly bounded since is stable, (perhaps including eventual single critical poles) since it is in . If, in addition, then it is strictly stable so that andsince , for any time , any given control and initial conditions, and some finite and with , where and are the output vector and matrix of dynamics of a state-space realization of initial state so that . Property (ii) has been proved. Finally, if the system is externally positive and the input is nonnegative for all time then, for any given set of nonnegative initial conditions, one has thatwhich proves Property (iii). To prove Property (iv), note that holds ifholds, that is, if which leads to (6).

Remark 3. The generalization of Theorem 2 to the multi-input multioutput (MIMO) case (i.e., ) is direct by replacing the instantaneous power by the scalar product in the corresponding expressions.

The following two results discuss how the basic passivity property can become a stronger property as, for instance, strict input passivity or very strict passivity, by incorporating to the input-output operator a suitable parallel static input-output interconnection structure.

Proposition 4. Consider a class of dynamic systems , defined as for given , and , such that for any . The following properties hold:
(i) Assume that is very strictly passive, and , where is the input passivity constant for . Then, is very strictly passive for all . If then is very strictly passive for all while is strictly output passive.
(ii) If is passive (resp., strictly output passive) then is strictly input passive (resp., very strictly passive) if for any .

Proof. Since for any given , (irrespective of ). Then, since is very strictly passive , and s.t.for all and any input-output pair with input and output being defined from to for any . Thus, where since and /2. Then, is very strictly passive with passivity, input passivity, and output passivity constants , , and . If then is guaranteed to be just output-strictly passive. Now, for any , one has Then, is very strictly passive with passivity, input passivity, and output passivity constants , and for all if while implies so that is strictly output passive and is very strictly passive for . Property (i) has been proved. Property (ii) follows from (15a) with .

Proposition 5. Assume that is passive and nonexpansive. Then: (i) is -stable and strictly input passive if ,(ii) is -stable if if ,(iii) is -stable if for any given .

Proof. Since is passive and nonexpansive, one hasThus, if thenThus, is -stable and strictly input passive if ; is -stable if if since the maximum and minimum eigenvalues of are distinct. If then for any real and is -stable.

Remark 6. It turns out through simple mathematical derivations that Propositions 12 still hold under the replacement , where is passive with associated constant for the properties to be extended from the case that and strictly input passive for those extended from the case when .

4. Feed-Forward and Feedback Controllers and Closed-Loop Passivity

It is now discussed how the passivity properties can be improved via feedback with respect to an external reference input signal. Consider the following linear time-invariant SISO cases:

(a) The controlled plant transfer function , whose relative passivity index [Theorem 2(iv)] is , is controlled by a feedback controller of transfer function so that where is the resulting closed-loop transfer function. The closed-loop relative passivity index is . For any given and associated , the controller transfer function is

(b) The controlled plant transfer function is controlled by a feed-forward controller of transfer function so that where is the resulting closed-loop transfer function. The closed-loop relative passivity index is . For any given and associated , the controller transfer function isThe subsequent result uses the above considerations to rely on the property of linear time-invariant systems establishing that a positive real transfer function can be designed by using feedback or feed-forward control laws for the case when the plant transfer function is inversely stable even if it is not positive real or stable.

Theorem 7. Assume that is inversely stable with relative degree or 1 while nonnecessarily in (or even nonnecessarily in ). Then, the following properties hold:
(i) A nonunique (state-space) realizable closed-loop transfer function , or, respectively, , may be designed via a stable feedback controller of transfer function (19) which is realizable if and have zero relative degree. In the above cases, , or, respectively, .
(ii) A nonunique realizable closed-loop transfer function , or , may be designed via a feed-forward controller of transfer function via (21) which is realizable if the relative degree of the closed-loop transfer function is not less than that of the plant transfer function . In the above cases, , or, respectively, .

Proof. A nonunique realizable closed-loop transfer function can be targeted as design objective through a feedback controller of transfer function , (19), such that . Since , it turns out that is bounded real (i.e., Schur, i.e., with -norm not exceeding unity and with real numerator and denominator coefficients) so that (since if and only if is bounded real, [4]) then with a relative degree (0 or 1) being identical to that of . The controller is realizable if and have zero relative degree. Then, is stable and inversely stable. Since is inversely stable and is stable, since since , then it follows from (18)-(19) that . Also, if and only if , that is, if and only if is strictly bounded real. Then, is strictly stable and inversely strictly stable. Property (i) has been proved. Property (i) is proved in a similar way via (20)-(21) with the controller realizability constraint ; that is, the relative degree of the closed-loop transfer function is not less than that of that of the plant .

In the light of Propositions 4 and 5 and Remark 6, it turns out that real positivity of a time-invariant system can be achieved by modifying a stable transfer matrix with the incorporation of an input-output interconnection gain being at least positive semidefinite. Similar conclusions follow by the use of close arguments to those in Theorem 7 on the inverse of a transfer matrix to achieve positive real closed-loop transfer matrices under appropriate feedback and feed-forward controllers. The results can be extended to the discrete case [13]. The subsequent result follows related to these comments:

Theorem 8. The following properties hold:
(i) Assume that is a transfer matrix of order . Then, , where with if ; . If then the condition becomes ; with . If ; and then which becomes if ; .
(ii) Assume that is an inversely stable transfer matrix order controlled by a linear time-invariant feedback controller of transfer matrix of order . Then, (resp., ) if(resp., the above inequality is strict).
(iii) Assume that is an inversely stable transfer matrix order being controlled by a linear time-invariant feed-forward controller of transfer matrix of order . Then, (resp., ) if (resp., the inequality is strict).

Proof. The proof of Property (i) is direct from the conditions of positive and strictly positive realness for . Inspired by the definitions of positive realness and Theorem 7 for the SISO case, Properties (ii)-(iii) are proved as follows. By using the feedback and feed-forward controllers, the following respective closed-loop transfer matrices are obtained:with inverses Then, and (resp., and ) are positive real ifwhich is guaranteed if , respectively, ifwhich is guaranteed if . Strict positive realness in each of both cases is guaranteed under the corresponding strict inequalities in (26)-(27).

Note that a sufficient condition for (26) to hold for the SISO case (i.e., ) is , where , if is stable and realizable (so that its norm exists) and .

The proof of Theorem 8(i) can be also addressed from the fact that the inverse of a positive real matrix is positive real and the subsequent derivations if : and , then , ifSince , , the above matrix relation is equivalent toand toso thatwhich yields, equivalently, ; .

5. Regenerative versus Passive Systems

Note that passive systems are intrinsically stable and either consume or dissipate energy for all time. Looking at Definition 1, we can give an opposed one as follows:

Definition 9. A dynamic system is nonpassive (or active or, so-called, regenerative) if for some unbounded sequences , which satisfy the conditions
(1) ; for some positive bounded sequence ,
(2) ; for some positive bounded sequence ,
(3) , as .

The following result follows for a nonpassive system.

Theorem 10. If a dynamic system is nonpassive then .

Proof. Define such that . Thus, Subtracting the two above ones: since is unbounded but its associated incremental sequence is bounded, as ; then contradicts the above relations.

Remark 11. Note that a nonpassive system can reach an absolute infinity energy measure in finite time under certain atypical inputs as, for instance, a second-order impulsive Dirac input of appropriate component signs at some time instant with for . Then, .

The following result is concerned with passive versus nonpassive dynamic systems.

Theorem 12. The following properties hold:
(i) A passive system cannot be nonpassive in any time subinterval. A nonpassive system in some time interval cannot be a passive system.
(ii) A passive system is always stable and also dissipative (i.e., the dissipative energy function takes nonnegative values for all time) including the conservative particular case implying identically zero dissipation through time.
(iii) A nonpassive system can be stable or unstable (so, stable systems are nonnecessarily passive).

Proof. Property (i) is a direct consequence of Definitions 1 and 9 and Remark 11 since if the system is nonpassive so that it satisfies the constraint of Definition 9, it cannot satisfy a reversed passivity condition (for all time) of Definition 1 since is not compatible with the passivity condition. The converse statement is direct. We now prove Properties [(ii)-(iii)]. Note from Definition 9 that if is an energy measure storage state function, as for instance a Lyapunov function, and if the system is passive (resp., nonpassive) then there exists , respectively; there exists some unbounded sequences , andrespectively,with , , where is the dissipation function and is interpreted as the external environment supplied or consumed energy on the interval . Thus, the following cases are of interest.
Case  1 (passive system). One has from (23) thatis a condition of energy dissipation for all time which happens if either for all and for all which is also coherent with a null energetic interchange with the environment or if for all Then, one haswhich is coherent with a positive, negative, or null energetic interchange on with the environment satisfying ; ; and, in particular, with null such an energetic interchange, the dissipation function satisfies ; if .
Note that (38) implies that the passive system is also stable since ; for any finite state initial conditions. The so-called conservative system is described by the subcase of conditions (38) under the subsequent particular constraints which imply a constant storage energy defined for the given initial conditions and zero interchanged energy with the environment for any given time interval:Case  2 (nonpassive system in ). One has from (36) thatand which happens if either
(1)the system being stable since and which is coherent with a (negative) energetic supply on given to the environment, or if
(2)Then, one hasthe system being either stable if or unstable if and both situations are coherent with an energetic supply given to the environment on satisfying the constraint .
(3) Note that the situations (1) and (2) can coexist within the same interval for distinct disjoint time subintervals of nonzero measure if the control input is piecewise continuous and also if it is impulsive with a finite residence time interval in-between any two consecutive impulses. Note that a large amplitude control impulse can temporarily unstabilize a stable system or that a switched dynamic system can have switches between stable and unstable parameterizations for certain switching laws.

6. Passivity and Switching

Now consider a dynamic system subject to a switching law with finite or infinity eventual parameterizations , where the configurations (or parameterizations) are passive and the remaining ones are nonpassive, where is empty if and is empty if . The switching law is defined bywhere is an abbreviate notation for versus the explicit notation for the left limit with being the set of switching time instants, which can be either of infinity (denumerable) cardinal if the switching action never ends or finite if there is a finite final switching time. The minimum time interval in-between any two consecutive switching time instants is the minimum residence time at the active configuration. In summary, andIf , it is possible to describe each configuration by positive integer numbers by assigning and for and . Thus, the piecewise constant primary switching law is equivalently described in a simpler way by the piecewise constant switching law such that if if or if . Note thatif , that is, if the active configuration on the time interval , is passive, where and ; , withIf , that is, if the active configuration on the time interval , is nonpassive, where and ; .

Note that the whole set of switching time instants iswhere , and ; are, respectively, the whole set of time instants until for all and its disjoint subsets associated with the active passive and nonpassive configurations which occurred in the interval . Note that a system configuration is passive if ; for some real function and some .

Assumptions 13. Under switching, a system configuration is assumed passive if for some :
(1)with ; with given by ; for some function ; .
(2) A system configuration of the switched law is nonpassive if for some with ; with (if ) given by ; for some constant and some function with as ; .
The fact that is common for all configurations is made with no loss in generality. If there is a set of such constants for the configurations, it would suffice to take the maximum of all of them as a common . The same value of is valid by reversing the inequality for nonpassive configuration since there are extra additive thresholds to modulate possible discrepancies of the necessary constants for distinct nonpassive configurations. The intuitive physical interpretation of Assumptions 13 is as follows if , ; as it follows for standard unswitched passivity. However, for >0 and small enough time , it can happen that even if because of the switching action and the possible change with jump of value at the switching time instant of storage function from the previous active configuration to the current one. For , the passivity property ; is recovered. For large enough , it is assumed that .

The following result holds.

Theorem 14. If Assumptions 13 hold a switched system is passive for all time under a switching law , with at least one configuration being active, if
(1) the first active configuration of the switching law on ;
(2) the switching law does not involve two consecutive active configurations being nonpassive;
(3) each active passive configuration respects a minimum residence time, quantified in the proof, which can exceed the minimum common residence time given by Assumptions 13.

Proof. The system is passive if ; for some real function and some . It turns out that a necessary condition for switched passivity is that the first switched configuration be passive (otherwise, the passivity condition fails for ). Define the active switching passivity binary indicator function , . Direct calculations for any on the input-output energy yield holds via complete induction if =0 (i.e., the first configuration of the switched disposal is passive) provided that for any if the subsequent constraint holds:The following cases can occur:
Case  1. . Thus, andwhich is guaranteed ifor ifCase  2. . Thus, andCase  3. . Thus, andyields closely to Case  1:

7. Examples

Example 15. Consider the following damped linear system:with being the position, being the velocity, defined to be the system output, being the external force, defined to be the control action, and and being the spring constant and the viscous coefficient, respectively. Define the state vector as and let be the storage function defined to be the kinetic energy plus the potential energy . The input-output energy at time iswhere the dissipation function at time is ; . If , then , and the system is a conservative system (a particular case of dissipative system) with constant stored energy, zero dissipation for all time, and zero input-output energy for all time which implies also that the system is globally stable (i.e., stable for any initial conditions). If there exists some real constant such that ; , that is, then the system is passive. In the case that the passive system is also globally asymptotically stable, , as and for all and . Sufficient conditions are obtained by giving conditions on the stored energy to be a Lyapunov function for global asymptotic stability as follows:if the external force is generated via feedback as follows:where is a design function so that if and only if . Thus, and as and the passive system is globally asymptotically stable.
In the case that the damping device has a nonlinear cubic effect (forced Duffing equation), the motion and Lyapunov function time-derivative satisfyifIf then and global stability holds if and ; since the exact output linearization dynamics is achieved under such a control.

Example 16. Consider a th order SISO time-invariant linear system whose transfer function is , where is a state-space realization of , and the state -vector has initial conditions . The output is given by a superposition of the unforced (i.e., zero -input) and forced (i.e., zero state) solutions and aswhereThen, by using Parseval’s theorem, we can equalize the input- output energy evaluation in the time domain to its evaluation in the frequency domain as follows provided that the Fourier transforms of the input and output signals exist:since for any hodograph . Assume that so that from (71). Assume also that the control is generated via negative output-feedback from any member of nonlinear and perhaps time-varying class of controllers which satisfies a Popov’s type inequality, namely, , where so that for any given positive real such that and any nonidentically zero controls, one getsso that for nonidentically zero controlsThe following result holds from (74).

Theorem 17. The following properties hold:
(i) Assume that . Then, and the system is globally asymptotically stable for any given finite initial conditions, irrespective of , so that it is asymptotically hyperstable.
(ii) Assume that and that a frequency filter is used for the control inputs so that for any frequency interval, if any, such that and it is of unity gain otherwise. Then, and the system is globally asymptotically stable for any given finite initial conditions, irrespective of ; then it is asymptotically hyperstable for the forward loop transfer function .

Proof. Since so that and , that is, it is strongly positive real then, in addition, in (then with all its poles in ) and having zero relative degree it follows in (74) that can be made and the Fourier transforms used in the Parseval’s identity exist. Assume that as then, there exists such that since and as . Therefore, , so that as which contradicts that as . It can also be concluded that is bounded on any finite time interval and it can be infinity only on a set of zero measure (i.e., it can be eventually impulsive only at isolated time instants). Thus, one concludes that the input is almost everywhere bounded and it converges asymptotically to zero as time tends to infinity. Note that (71) implies that with and is realizable, since so it has relative degree zero. Thus, and converges asymptotically to zero for any . Since the system is globally asymptotically stable for any given initial conditions so that the state, control input, and output are uniformly bounded for all time and converge asymptotically to zero as time tends to infinity. Property (i) has been proved. To prove Property (ii), note thatwhere since and is the indicator set such that if and only if . Thus, , is bounded; then as since for all and the total control input is also bounded since is bounded by the filtering actions; . If then Property (i) holds for this class of constrained control inputs.

Example 18. Consider a th order SISO time-invariant linear system described by a finite or infinite set of transfer functions governed by a switching law , with , which selects the active one on a time interval where is a finite or infinite set of switching time instants. Stability preservation or achievement under switched laws governing switched parameterizations has been studied in the background literature. It is of interest to extend it to positivity and passivity properties under switching laws. See, for instance, [14]. Let a time instant . To simplify the exposition, consider the system under zero initial conditions. Then where a truncated interswitching input for and for has been defined andwithThe following related result holds.

Theorem 19. The following properties hold:
(i) Assume that the switching law incorporates a zero state resetting action at each . Then, the switched system is positive and passive for such a switching law if all the transfer functions involved by the switching law are positive real.
(ii) Assume that the switching law incorporates a zero state resetting action at each . Assume also that the switching law satisfies that and that and that, in the event that for , then , and Then, the switched system is strictly input passive for such a switching law.
(iii) Assume that the system is subject to a feedback control law , (72), and that with and to zero state resetting at each switching time instant. Then, and the system is globally asymptotically stable for any given finite initial conditions, irrespective of , so that it is asymptotically hyperstable.
(iv) Assume also that the constraints and zero state switching resetting invoked in Property (iii) but frequency filters are used for the control inputs so that for any frequency interval, if any, such that and it is of unity gain otherwise for . Then, and the system is globally asymptotically stable for any given finite initial conditions, irrespective of ; then it is asymptotically hyperstable for the forward loop transfer function .

Proof. Property (i) follows from (77a), (77b), and (78) and under zero state resetting at each switching time instant sincefor any , any and any . Property (ii) by taking into account (77a) and which guarantees that and that (77a) and (77b) holds since any active stable, irrespective of it is positive or nonpositive real, transfer function is in active operation a sufficiently small time interval to guarantee that under zero resetting of the state conditions at any switching time instant. Properties (iii)-(iv) follow from Property (iii) and Theorem 17.

Example 20. Use the convolution expression for the zero state energy measure in Example 18 to yieldWe find the following properties if the system is externally positive in the sense that for any nonnegative controls and initial conditions the output is nonnegative for all time:(i)If all the transfer functions of the switching law have a state-space representation where the impulse response is nonnegative for all time and the controls are also everywhere nonnegative in the definition domain then the output is nonnegative for all time and the input-output energy is also nonnegative for all time provided that the initial state is zero and subject to reset to zero at each switching time instant.(ii)If the initial conditions are nonnegative and reset-free, is a Metzler matrix; and and have nonnegative components; then the output and the energy are positive for all time for all input with nonnegative components. This property as the previous one would still be kept under eventual positive additional control impulses [15] since the whole control action will kept its positive nature.

Note that positivity properties in the time domain are very relevant in the study of certain dynamic systems, like biological or epidemic ones, which, by nature, cannot have negative solutions at any time. See, for instance, [16–18]. The above properties follow from the positivity properties of the unforced and forced output solution trajectory in externally positive systems. However, those properties do not imply passivity without invoking additional conditions since the externally positive system can be nonstable, [19].

Conflicts of Interest

The author declares that he does not have any conflicts of interest.

Acknowledgments

This research is supported by the Spanish Government and by the European Fund of Regional Development FEDER through Grant DPI2015-64766-R and by UPV/EHU by Grant PGC 17/33.