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

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.


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][2][3][4] and some related references therein.More detailed generic results about passivity and positivity are given in [5][6][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][11][12].(ii)  ≻ 0 denotes that the real matrix  is positive definite while  ⪰ 0 denotes that it is positive semidefinite, (iii)  min (⋅) and  max (⋅) denote, respectively, the minimum and maximum eigenvalues of the real symmetric (⋅)matrix, (iv) Ĝ ∈ {PR} denotes that the transfer matrix Ĝ() of a linear time-invariant system is positive real; that is, Ĝ() + Ĝ (−) ≥ 0 for all Re  > 0, and Ĝ ∈ {SPR} denotes that it is strictly positive real; that is, Ĝ() + Ĝ (−) > 0 for all Re  ≥ 0. The set of strongly positive real transfer matrices {SSPR} is the subset of {SPR} 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 Ĝ ∈ {PR} if Re Ĝ() ≥ 0 for all Re  > 0 and Ĝ ∈ {SPR} if Re Ĝ() > 0 for all Re  ≥ 0, (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  ≥ 0 for any given nonnegative initial conditions and nonnegative input, (vi) i = √ −1 is the complex unity, (vii)   is the th identity matrix, (viii) The superscript  stands for matrix transposition, (ix) H ∞ is the Hardy space of all complex-valued functions () of a complex variable  which are analytic and bounded in the open right half-plane Re  > 0 of norm ‖‖ ∞ = sup{|()| : Re  > 0} = sup{|(i)| :  ∈ R} (by the maximum modulus theorem) and RH ∞ is the subset of real-rational functions of H ∞ .
The constants ,   , and   are, respectively, referred to as the passivity, input passivity, and output passivity constants.

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  = 0 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 singleinput single-output (SISO) system by relating the time and frequency domains descriptions: Theorem 2. Consider a linear time-invariant SISO (i.e.,  = 1) system whose transfer function Ĝ ∈ {}.Then, the following properties hold: (i) ∫  0 ()() ≥ 0 and ()() ≥ 0; ∀ ≥ 0 and, furthermore, if  ∈  2 then  ∈  2 .Then, the system is passive.
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  2 .Therefore, Parseval's theorem can be applied to express ∫  0 ()() ≥ 0; ∀ ≥ 0 in the frequency domain.Take into account, in addition, that the hodograph of the frequency system's response Ĝ(i) satisfies Re Ĝ(i) = Re Ĝ(−i) and Im Ĝ(i) = − Im Ĝ(−i) for all  ∈ (−∞, ∞) and that min ∈R 0+ Re Ĝ(i) ≥ 0 since Ĝ ∈ {PR}.Thus, the various expressions which follow hold under zero initial conditions of the dynamic system: It has been proved, under zero initial conditions, that ∫  0 ()() ≥ 0 and ()() ≥ 0; ∀ ≥ 0 and if  ∈  2 then ∫  0 ()() ≥ ; ∀ ≥ 0 for some  ∈ [0,∞) 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  ∈ R such that ∫  0 ()() ≥  −   ≥ − since the initial conditions do not generate an unbounded homogeneous solution since Ĝ ∈ RH ∞ since Ĝ ∈ {PR}.Property (i) has been proved.On the other hand, under any finite nonzero initial conditions  0 ∈ R  : for some uniformly bounded (,  0 ) since Ĝ() is stable, (perhaps including eventual single critical poles) since it is in {PR}.If, in addition, Ĝ ∈ {SPR} then it is strictly stable so that min ∈R 0+ Re Ĝ(i) > 0 and since  ∈  2 , for any time  > 0, any given control and initial conditions, and some finite   > 0 and  ≥ 0 with () =      0 , where   and  are the output vector and matrix of dynamics of a state-space realization of initial state  0 so that (⋅, 0) = 0. 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 that which leads to (6).
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 inputoutput interconnection structure.
Proof.Since  0 is passive and nonexpansive, one has Thus, if  ∈   2 then Thus,   is  2 -stable and strictly input passive if Remark 6.It turns out through simple mathematical derivations that Propositions 1-2 still hold under the replacement  →  1 , where  1 : H  → H  is passive with associated constant  1 ≤ 0 for the properties to be extended from the case that  ⪰ 0 and strictly input passive for those extended from the case when  ≻ 0.

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 Ĝ(), whose relative passivity index [Theorem 2(iv)] is where For any given T1 () and associated M1 (), the controller transfer function is (b) The controlled plant transfer function Ĝ() is controlled by a feed-forward controller of transfer function K2 () so that where For any given T1 () and associated M1 (), the controller transfer function is The 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.
(ii) A nonunique realizable closed-loop transfer function T2 ∈ {}, or T2 ∈ {}, may be designed via a feed-forward controller of transfer function K2 () via ( 21) which is realizable if the relative degree of the closed-loop transfer function T2 () is not less than that of the plant transfer function Ĝ().In the above cases, T−1 ; that is, the relative degree of the closedloop transfer function is not less than that of that of the plant Ĝ().
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 Ĝ() 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 Ĝ ∈ RH ∞ is a transfer matrix of order  × .Then, Ĝ1 ∈ {}, where Ĝ1 () = Ĝ() +  with (ii) Assume that Ĝ() is an inversely stable transfer matrix order × controlled by a linear time-invariant feedback controller of transfer matrix K1 () of order  × .Then, T1 , T−1 (resp., the above inequality is strict).
(iii) Assume that Ĝ() is an inversely stable transfer matrix order  ×  being controlled by a linear time-invariant feedforward controller of transfer matrix K2 () of order  × .Then, T2 , T−1 Proof.The proof of Property (i) is direct from the conditions of positive and strictly positive realness for Ĝ1 ().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, T1 () and T−1 1 () (resp., T2 () and T−1 2 ()) are positive real 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.,  = 1) is min ∞ , where Re Ĝ(i) ≥ − Ĝ, if Ĝ() is stable and realizable (so that its  ∞ norm exists) and min ∞ .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  ≻ 0: Since   + −1 Ĝ(i) ≻ 0, ∀ ∈ R 0+ , the above matrix relation is equivalent to and to so that which yields, equivalently, Ĝ(i) + Ĝ (−i) ⪰ −( +   ); ∀ ∈ R 0+ .

Regenerative versus Passive Systems
Note that passive systems are intrinsically stable and either consume or dissipate energy for all time.Looking at Definition 1(3), we can give an opposed one as follows: Definition 9. A dynamic system is nonpassive (or active or, so-called, regenerative) if ,   → +∞ as  → +∞.
The following result follows for a nonpassive system.
Theorem 10.If a dynamic system is nonpassive 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(3) 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(3) since lim →∞ ∫  0   ()() = −∞ is not compatible with the passivity condition.The converse statement is direct.We now prove Properties [(ii)-(iii)].Note from Definition 9 that if  : R  × R 0+ → R 0+ is an energy measure storage state function, as for instance a Lyapunov function, and if the system is passive (resp., nonpassive) then there exists  ∈ R 0+ , respectively; there exists some unbounded sequences Then, one has which is coherent with a positive, negative, or null energetic interchange on [0, ] with the environment satisfying ∫  0   ()() ≥ −; ∀ ∈ R 0+ ; and, in particular, with null such an energetic interchange, the dissipation function satisfies 0 ≤ () < ; ∀ ∈ R 0+ if  ∈ R + .Note that (38) implies that the passive system is also stable since () ≤ (0) < +∞; ∀ ∈ R 0+ 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:  2) can coexist within the same interval [ 0 , ] 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.
Assumptions 13.Under switching, a system configuration is assumed passive if some  ∈ R 0+ : (1) (2) A system configuration of the switched law is nonpassive if for some  ∈ R 0+ 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  0(⋅) to modulate possible discrepancies of the necessary constants for distinct nonpassive configurations.The intuitive physical interpretation of Assumptions 13 is as follows if  0 = 0 ∈ SW  , (0, ) ≥ −; ∀ ∈ R 0+ as it follows for standard unswitched passivity.However, for   (>0) ∈ SW  and small enough time  ∈ [0,  min ), it can happen that (  , ) < − even if  −1 ∈ SW  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  ≥  min , the passivity property (  , ) ≥ −; ∀ ∈ R 0+ is recovered.For large enough , it is assumed that lim sup →∞   () ≤  *  ∈ [0, ∞).

Theorem 17. The following properties hold:
(i) Assume that Ĝ ∈ {}.Then, ,  ∈  2 and the system is globally asymptotically stable for any given finite initial conditions, irrespective of  ∈ {Φ}, so that it is asymptotically hyperstable.
(ii) Assume that Ĝ ∈ {} and that a frequency filter F(i) is used for the control inputs so that F(i)û(i) = 0 for any frequency interval, if any, such that min Re Ĝ(i) = 0 and it is of unity gain otherwise.Then, ,  ∈  2 and the system is globally asymptotically stable for any given finite initial conditions, irrespective of  ∈ {Φ}; then it is asymptotically hyperstable for the forward loop transfer function Ĝ() F().
Proof.Since Ĝ ∈ {SSPR} so that  > 0 and Ĝ ∈ {SPR}, that is, it is strongly positive real then, in addition, in RH ∞ (then with all its poles in Re  < 0) and having zero relative degree it follows in (74) that +∞ >  = ( 0 , ) > max(, ∫  0    (−)  0 ()) can be made and the Fourier transforms used in the Parseval's identity exist.Assume that () → ±∞ as  → ∞ then, there exists  ∈ R + such that since  2 ()/() → ∞ and   → 0 as  → ∞.Therefore,  ∈  2 , so that () → 0 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   () ≥ min ∈R 0+ Ĝ−1 (i) ∫  0  2 () with Ĝ−1 ∈ {SSPR} and is realizable, since Ĝ ∈ {SSPR} so it has relative degree zero.Thus,  ∈  2 and converges asymptotically to zero for any  ∈ {Φ}.Since Ĝ ∈ RH ∞ 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 that where with Ĝ(  ) (i) < 0. (80) The 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  ( 0 ) ∈ {} and that  (  ) ∈ RH ∞ and that, in the event that  (  ) ∈ RH ∞ \ {} for  ∈ Z + , then  ( +1 ) ∈ {}, and (81) 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  ∈ {Φ}, (72), and that Ĝ(  ) ∈ {}; ∀  ∈  with Ĝ( 0 ) ∈ {} and to zero state resetting at each switching time instant.Then, ,  ∈  2 and the system is globally asymptotically stable for any given finite initial conditions, irrespective of  ∈ {Φ}, so that it is asymptotically hyperstable.
(iv) Assume also that the constraints Ĝ( 0 ) ∈ {} and zero state switching resetting invoked in Property (iii) but frequency filters F(  ) (i) are used for the control inputs so that F(  ) (i)û(i) = 0 for any frequency interval, if any, such that min Re Ĝ(  ) (i) = 0 and it is of unity gain otherwise for   ∈ .Then, ,  ∈  2 and the system is globally asymptotically stable for any given finite initial conditions, irrespective of  ∈ Φ; then it is asymptotically hyperstable for the forward loop transfer function Ĝ() F().
Proof.Property (i) follows from (77a), (77b), and (78) and () =   () ≥ 0; ∀ ∈ R 0+ under zero state resetting at each switching time instant since (83) We 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 resetfree,  (⋅) is a Metzler matrix; (⋅) ≥ 0 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][17][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].