Journal of Mathematics

Volume 2018 (2018), Article ID 5298756, 16 pages

https://doi.org/10.1155/2018/5298756

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

Correspondence should be addressed to M. De la Sen

Received 4 August 2017; Accepted 28 November 2017; Published 24 January 2018

Academic Editor: Viliam Makis

Copyright © 2018 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 has