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

Authors: M. De la Sen
Journal: Journal of Mathematics
 (2018)

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## 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.


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## Body

## 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)
R
0
+
=
R
+
∪
0, where R+=r∈R:r>0, p-=1,2,…,p,
(ii)
D
≻
0 denotes that the real matrix D is positive definite while D⪰0 denotes that it is positive semidefinite,
(iii)
λ
m
i
n
· and λmax· denote, respectively, the minimum and maximum eigenvalues of the real symmetric ·-matrix,
(iv)
G
^
∈
P
R denotes that the transfer matrix G^s of a linear time-invariant system is positive real; that is, G^s+G^T-s≥0 for all Re⁡s>0, and G^∈SPR denotes that it is strictly positive real; that is, G^s+G^T-s>0 for all Re⁡s≥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 s→∞. If the linear time –invariant system is a SISO one (i.e., it has one input and one output) then G^∈PR if Re⁡G^s≥0 for all Re⁡s>0 and G^∈SPR if Re⁡G^s>0 for all Re⁡s≥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 timet≥0 for any given nonnegative initial conditions and nonnegative input,
(vi)
i
=
-
1 is the complex unity,
(vii)
I
m is the mth identity matrix,
(viii)
The superscriptT stands for matrix transposition,
(ix)
H
∞ is the Hardy space of all complex-valued functions Fs of a complex variable s which are analytic and bounded in the open right half-plane Re⁡s>0 of norm F∞=supFs:Re⁡s>0=sup⁡Fiω:ω∈R (by the maximum modulus theorem) and RH∞ is the subset of real-rational functions of H∞.


## 1.1. Notation

(i)
R
0
+
=
R
+
∪
0, where R+=r∈R:r>0, p-=1,2,…,p,
(ii)
D
≻
0 denotes that the real matrix D is positive definite while D⪰0 denotes that it is positive semidefinite,
(iii)
λ
m
i
n
· and λmax· denote, respectively, the minimum and maximum eigenvalues of the real symmetric ·-matrix,
(iv)
G
^
∈
P
R denotes that the transfer matrix G^s of a linear time-invariant system is positive real; that is, G^s+G^T-s≥0 for all Re⁡s>0, and G^∈SPR denotes that it is strictly positive real; that is, G^s+G^T-s>0 for all Re⁡s≥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 s→∞. If the linear time –invariant system is a SISO one (i.e., it has one input and one output) then G^∈PR if Re⁡G^s≥0 for all Re⁡s>0 and G^∈SPR if Re⁡G^s>0 for all Re⁡s≥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 timet≥0 for any given nonnegative initial conditions and nonnegative input,
(vi)
i
=
-
1 is the complex unity,
(vii)
I
m is the mth identity matrix,
(viii)
The superscriptT stands for matrix transposition,
(ix)
H
∞ is the Hardy space of all complex-valued functions Fs of a complex variable s which are analytic and bounded in the open right half-plane Re⁡s>0 of norm F∞=supFs:Re⁡s>0=sup⁡Fiω:ω∈R (by the maximum modulus theorem) and RH∞ is the subset of real-rational functions of H∞.


## 2. Preliminaries

Consider a dynamic systemG:He→He with state x∈Rn, input u∈Rm, and output y∈Rm, where He is the extended space of the Hilbert space H endowed with the inner product ·,· from He×He to R consisting of the truncated functions utτ=uτ for τ∈0,t and utτ=0; ∀t,τ(>t)∈R0+ and u:R0+→Rm. If u∈He then(1)ut22=u,ut=ut,ut=∫0tuTτuτdτ=∫0∞utTτutτdτ;∀t≥0.Definition 1 (see [2]).
The above dynamic system is
(1)L2-stableif u∈L2m implies Gu∈L2m;
(2)nonexpansive if ∃λ and ∃γ>0 s.t. for all u∈He,(2)∫0tGuTτuτdτ≤λ+γ2∫0tuTτuτdτ;∀t≥0;
(3)passiveif ∃ε≥0 such that ∫0tyTτuτdτ≥-ε;∀t≥0;
(4)strictly input passive if ∃ε≥0 and ∃εu>0 s.t. (3)∫0tyTτuτdτ≥-ε+εu∫0tyTτuτdτ;∀t≥0;
(5)strictly output passive if ∃β≥0 and ∃εy>0 s.t. (4)∫0tyTτuτdτ≥-ε+εy∫0tyTτyτdτ;∀t≥0;
(6)strictly input/output passive (orvery strictly passive) if ∃β≥0, ∃εu>0 and ∃εy>0 s.t.(5)∫0tyTτuτdτ≥-ε+εu∫0tyTτuτdτ+εy∫0tyTτyτdτ;∀t≥0.The constants ε, εu, and εy 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 forε=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 single-input single-output (SISO) system by relating the time and frequency domains descriptions:Theorem 2.
Consider a linear time-invariant SISO (i.e.,m=1) system whose transfer function G^∈PR. Then, the following properties hold:
(i)∫0tyτuτdτ≥0 and ytut≥0;∀t≥0 and, furthermore, if u∈L2 then y∈L2. Then, the system is passive.
(ii) Assume, in addition, thatG^∈SPR. Then γ≥∫0tyτuτdτ≥εu∫0tuτ2dτ-ε for any t∈0,∞ and some γ,ε∈R0+.
(iii) If, furthermore, the system is externally positive thenγ≥∫0tyτuτdτ>0;∀t≥0 for any given nonnegative initial conditions and nonnegative input.
(iv) DefineRG^=1-G^iω/1+G^iω∞=supω∈R0+⁡1-G^iω/1+G^iω as the relative passivity index of the transfer function G^s=N^s/D^s∈RH∞ (N^s and D^s being the numerator and denominator polynomials of G^s). Then, the constraint aG≤RG^=D^Giω-N^Giω/D^Giω+N^Giω∞≤bG is guaranteed for some aG,bG(≥aG)∈R0+ if(6)1-aG221+aG2Re2⁡D^Giω+Re2⁡N^Giω≥Re⁡D^GiωRe⁡N^Giω+Im⁡D^GiωIm⁡N^Giω≥1-bG221+bG2Re2⁡D^Giω+Re2⁡N^Giω;∀ω∈R0+.If bG≤1 (resp., bG<1) then G^∈PR (resp., G^∈SPR).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 inL2. Therefore, Parseval’s theorem can be applied to express ∫0tyτuτdτ≥0; ∀t≥0 in the frequency domain. Take into account, in addition, that the hodograph of the frequency system’s response G^iω satisfies Re⁡G^iω=Re⁡G^-iω and Im⁡G^iω=-Im⁡G^-iω for all ω∈-∞,∞ and that minω∈R0+⁡Re⁡G^iω≥0 since G^∈PR. Thus, the various expressions which follow hold under zero initial conditions of the dynamic system:(7)∫0tyτuτdτ=∫0∞ytτutτdτ=12π∫-∞∞y^tiωu^t-iωdω=12π∫-∞∞G^iωu^tiω2dω=12πminω∈R0+⁡Re⁡G^iω∫-∞∞u^tiω2dω≥minω∈R0+⁡G^iω∫-∞∞u^tτ2dτ=minω∈R0+⁡Re⁡G^iω∫-∞∞utτ2dτ=minω∈R0+⁡Re⁡G^iω∫0tuτ2dτ≥0;∀t≥0.It has been proved, under zero initial conditions, that ∫0tyτuτdτ≥0 and ytut≥0; ∀t≥0 and if u∈L2 then ∫0tyτuτdτ≥β; ∀t≥0 for some β∈0,∞ independent of u (and independent of t). 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 ε∈R such that ∫0tyτuτdτ≥β-ε′≥-ε since the initial conditions do not generate an unbounded homogeneous solution since G^∈RH∞ since G^∈PR. Property (i) has been proved. On the other hand, under any finite nonzero initial conditions x0∈Rn:(8)∫0tyτuτdτ≥minω∈R0+⁡Re⁡G^iω∫0tuτ2dτ-∫0tλt-τ,x0uτdτ;∀t≥0for some uniformly bounded λt,x0 since G^s is stable, (perhaps including eventual single critical poles) since it is in PR. If, in addition, G^∈SPR then it is strictly stable so that minω∈R0+⁡Re⁡G^iω>0 and(9)+∞>γ≥∫0tyτuτdτ≥minω∈R0+⁡Re⁡G^iω∫0tuτ2dτ-∫0tλt-τ,x0uτdτ>εu∫0tuτ2dτ-ε,since u∈L2, for any time t>0, any given control and initial conditions, and some finite εu>0 and ε≥0 with λt=cTeAtx0, where cTand A are the output vector and matrix of dynamics of a state-space realization of initial state x0 so that λ·,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(10)+∞>γ≥∫0tyτuτdτ≥minω∈R0+⁡Re⁡G^iω∫0tuτ2dτ+∫0tλt-τ,x0uτdτ>εu∫0tuτ2dτ+ε≥0;∀t≥0which proves Property (iii). To prove Property (iv), note that aG≤RG^=(D^Giω-N^Giω)/(D^Giω+N^Giω)∞≤bG holds if(11)aGRe⁡D^Giω+N^Giω+iIm⁡D^Giω+N^Giω≤Re⁡D^Giω-N^Giω+iIm⁡D^Giω-N^Giω≤bGRe⁡D^Giω+N~Giω+iIm⁡D^Giω+N^Giω;∀ω∈R0+holds, that is, if (12)aG2Re2⁡D^Giω+N^Giω+Im2⁡D^Giω+N^Giω≤Re2⁡D^Giω-N^Giω+Im2⁡D^Giω-N^Giω≤bG2Re2⁡D^Giω+N^Giω+Im2⁡D^Giω+N^Giω;∀ω∈R0+which leads to (6).Remark 3.
The generalization of Theorem2 to the multi-input multioutput (MIMO) case (i.e., m>1) is direct by replacing the instantaneous power ytut by the scalar product yTtut 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 classGρD of dynamic systems Gρ,D:He→He, defined as Gρ,D=G0+ρD for given ρ∈R, D(∈Rm×m)⪰0 and G0:He→He, such that Gα,D∈GρD for any α∈0,ρ. The following properties hold:
(i) Assume thatG(ρ,D) is very strictly passive, D≻0 and εuρ>ρλmax(D+DT)/2, where εuρ is the input passivity constant for G(ρ,D). Then, G(α,D) is very strictly passive for all α∈[0,ρ]. If εuρ=ρλmax(D+DT)/2 then G(α,D) is very strictly passive for all α∈(0,ρ] while G0 is strictly output passive.
(ii) IfG0 is passive (resp., strictly output passive) then Gα,D is strictly input passive (resp., very strictly passive) if D≻0 for any α∈R+.Proof.
SinceGα,D∈GρD for any given α∈0,ρ, G0=G0,D∈GρD (irrespective of D). Then, since Gρ,D is very strictly passive ∃ερ≥0, ∃εuρ>0 and ∃εyρ>0 s.t.(13)∫0tyρTτuτdτ=∫0tGρuTτuτdτ=∫0tG0uTτ+ρuTτDTuτdτ≥-ερ+εuρ∫0tuTτuτdτ+εyρ∫0tyTτyτdτfor all t≥0 and any input-output pair u,yα with input and output being defined from R0+ to Rm for any α∈0,ρ. Thus, (14)∫0ty0Tτuτdτ=∫0tG0uTτuτ≥-ερ+∫0tuTτεuρIm-ρDTuτdτ+εyρ∫0tyTτyτdτ≥-ερ+∫0tuTτεuρ-ρ2λmaxD+DTuTτuτdτ+εyρ∫0tyTτyτdτ=-ερ+εu0∫0tuTτuτdτ+εyρ∫0tyTτyτdτ;∀t≥0,where εu0=εuρ-(ρ/2)λmaxD+DT>0 since D≻0 and εuρ>ρλmaxD+DT/2. Then, G0 is very strictly passive with passivity, input passivity, and output passivity constants ε0=ερ≥0, εu0=εuρ-(ρ/2)λmaxD+DT>0, and εy0=εyρ>0. If εuρ=ρλmaxD+DT/2 then G0 is guaranteed to be just output-strictly passive. Now, for any α∈[0,ρ], one has(15a)∫0tyαTτuτdτ=∫0tGαuTτuτdτ=∫0tG0uTτ+αuTτDTuτdτ=∫0ty0Tτuτ+αuTτDTuτdτ≥-ερ+∫0tuTτεuρ+α2λminD+DT-ρ2λmaxD+DTuTτuτdτ+εyρ∫0tyTτyτdτ(15b)≥-ερ+∫0tuTτεuρ-ρ2λmaxD+DTuTτuτdτ+εyρ∫0tyTτyτdτ;∀t≥0. Then, Gα,D is very strictly passive with passivity, input passivity, and output passivity constants εα=ερ≥0, εuα=εuρ>0 and εyα=εyρ>0 for all α∈0,ρ if εuρ>ρλmaxD+DT/2 while εuρ=ρλmaxD+DT/2 implies εu0=εuρ-(ρ/2)λmaxD+DT=0 so that G0 is strictly output passive and Gα,D is very strictly passive for α∈0,ρ. Property (i) has been proved. Property (ii) follows from (15a) with εuα=α.Proposition 5.
Assume thatG0 is passive and nonexpansive. Then: (i)
Gρ=G0+ρD is L2-stable and strictly input passive if ρ∈R+,(ii)
G0 is L2-stable if ρ>-2λ0+γ02/(λmaxD+DT-λminD+DT) if D(≠Im,0)⪰0,(iii)
G0 is L2-stable if D=Im for any given ρ∈R.Proof.
SinceG0 is passive and nonexpansive, one has(16)λ0+γ02∫0tuTτuτdτ+ρ2λmaxD+DT∫0tuTτuτdτ≥∫0tGρuTτuτdτ=∫0tG0+ρDuTτuτdτ≥∫0tG0uTτuτdτ+ρ2λminD+DT∫0tuTτuτdτ≥-ε0+ρ2λminD+DT∫0tuTτuτdτ;∀t≥0.Thus, if u∈L2m then(17)+∞>M=λ0+γ02+ρ2λmaxD+DT∫0tuTτuτdτ≥∫0tGρuTτuτdτ≥-ε0+ρ2λminD+DT∫0tuTτuτdτ;∀t≥0.+∞>M0=λ0+γ02+ρ2λmaxD+DT-λminD+DT∫0tuTτuτdτ≥∫0tG0uTτuτdτ≥-ε0;∀t≥0.Thus, Gρ is L2-stable and strictly input passive if ρ∈R+; G0 is L2-stable if ρ>-2λ0+γ02/(λmaxD+DT-λminD+DT) if D(≠Im,0)⪰0 since the maximum and minimum eigenvalues of D+DT are distinct. If D=Im then M0=λ0+γ02>0 for any real ρ∈R and G0 is L2-stable.Remark 6.
It turns out through simple mathematical derivations that Propositions 1-2 still hold under the replacement D→G1, where G1:He→He is passive with associated constant ε1≤0 for the properties to be extended from the case that D⪰0 and strictly input passive for those extended from the case when D≻0.


## 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 functionG^s, whose relative passivity index [Theorem 2(iv)] is RG^=supω∈R0+⁡(1-G^iω)/(1+G^iω), is controlled by a feedback controller of transfer function K^1s so that (18)M^1s=1-T^1s1+T^1s=1+G^sK^1s-11+G^sK^1s+1,where T^1s=G^s/(1+G^sK^1s) is the resulting closed-loop transfer function. The closed-loop relative passivity index is RT^1=supω∈R0+⁡(1-T^1iω)/(1+T^1iω)=supω∈R0+⁡M^1iω. For any given T^1s and associated M^1s, the controller transfer function is(19)K^1s=2G^s-1T^1s+12G^sT^1s=G^s1+M^1s+M^1sG^s1-M^1s.(b) The controlled plant transfer functionG^s is controlled by a feed-forward controller of transfer function K^2s so that (20)M^2s=1-T^2s1+T^2s=11+2G^sK^2s,where T^2s=G^sK^2s/(1+G^sK^2s) is the resulting closed-loop transfer function. The closed-loop relative passivity index is RT^2=supω∈R0+⁡(1-T^2iω)/(1+T^2iω)=supω∈R0+⁡M^2iω. For any given T^1s and associated M^1s, the controller transfer function is(21)K^2s=T^2sG^s1-T^2s=1-M^2s2G^sM^2s.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.Theorem 7.
Assume thatG^s is inversely stable with relative degree 0 or 1 while nonnecessarily in PR (or even nonnecessarily in RH∞). Then, the following properties hold:
(i) A nonunique (state-space) realizable closed-loop transfer functionT^1∈PR, or, respectively, T^1∈SPR, may be designed via a stable feedback controller of transfer function K^1s (19) which is realizable if G^s and T^1s have zero relative degree. In the above cases, T^1-1∈PR, or, respectively, T^1-1∈SPR.
(ii) A nonunique realizable closed-loop transfer functionT^2∈PR, or T^2∈SPR, may be designed via a feed-forward controller of transfer function K^2s via (21) which is realizable if the relative degree of the closed-loop transfer function T^2s is not less than that of the plant transfer function G^s. In the above cases, T^2-1∈PR, or, respectively, T^2-1∈SPR.Proof.
A nonunique realizable closed-loop transfer function can be targeted as design objective through a feedback controller of transfer functionK^1s, (19), such that RT^1≤1. Since RT^1≤1, it turns out that M^1s is bounded real (i.e., Schur, i.e., with H∞-norm not exceeding unity and with real numerator and denominator coefficients) so that T^1∈PR (since T^1∈PR if and only if M^1s is bounded real, [4]) then with a relative degree (0 or 1) being identical to that of G^s. The controller is realizable if G^s and T^1s have zero relative degree. Then, T^1s is stable and inversely stable. Since G^s is inversely stable and T^1-1s is stable, since T^1-1∈PR since T^1∈PR, then it follows from (18)-(19) that K^1∈RH∞. Also, T^1s∈SPR if and only if RT^1<1, that is, if and only if M^1s is strictly bounded real. Then, T^1 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 deg⁡D2≥deg⁡D+deg⁡N2-deg⁡N; that is, the relative degree of the closed-loop transfer function is not less than that of that of the plant G^s.In the light of Propositions4 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 G^s 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 thatG^∈RH∞ is a transfer matrix of order m×m. Then, G^1∈PR, where G^1s=G^s+D with D(⪰0)∈Rm×m if G^iω+G^T-iω⪰-D+DT; ∀ω∈R0+. If m=1 then the condition becomes Re⁡G^iω≥-d; ∀ω∈R0+ with d=D. If G^iω+G^T-iω≻-D+DT; ∀ω∈R0+ and then G^1∈SPR which becomes Re⁡G^iω>-d if m=1; ∀ω∈R0+.
(ii) Assume thatG^s is an inversely stable transfer matrix order m×m controlled by a linear time-invariant feedback controller of transfer matrix K^1s of order m×m. Then, T^1,T^1-1∈PR (resp., T^1,T^1-1∈SPR) if(22)infω∈R0+λminK^1iω+K^1-iω≥-supω∈R0+⁡λmaxG^-1iω+G^-T-iω(resp., the above inequality is strict).
(iii) Assume thatG^s is an inversely stable transfer matrix order m×m being controlled by a linear time-invariant feed-forward controller of transfer matrix K^2s of order m×m. Then, T^2,T^2-1∈PR (resp., T^2,T^2-1∈SPR) if infω∈R0+⁡λminK^2iωG^-1iω+G^-TiωK^2T-iω≥-1 (resp., the inequality is strict).Proof.
The proof of Property (i) is direct from the conditions of positive and strictly positive realness forG^1s. 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:(23)T^1s=Im+G^sK^1s-1G^s=G^-1sIm+G^sK^1s-1=G^-1s+K^1s-1,(24)T^2s=Im+G^sK^2s-1G^sK^2s=K^2-1sG^-1sIm+G^sK^2s-1=Im+K^2-1sG^-1s-1with inverses (25)T^1-1s=G^-1s+K^1s;T^2-1s=Im+K^2-1sG^-1s.Then, T^1s and T^1-1s (resp., T^2s and T^2-1s) are positive real if(26)K^1iω+K^1T-iω⪰-G^-1iω+G^-T-iω;∀ω∈R0+which is guaranteed if infω∈R0+⁡λminK^1iω+K^1-iω≥-supω∈R0+⁡λmaxG^-1iω+G^-T-iω, respectively, if(27)K^2-1iωG^-1iω+G^-T-iωK^2-T-iω⪰-Im;∀ω∈R0+which is guaranteed if infω∈R0+⁡λminK^2iωG^-1iω+G^-TiωK^2T-iω≥-1. 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., m=1) is minω∈R0+⁡Re⁡K^1iω≥-dG^/supω∈R0+⁡G^∞2, where Re⁡G^iω≥-dG^, if G^s is stable and realizable (so that its H∞ norm exists) and minω∈R0+⁡Re⁡K^1iω≥-dG^/supω∈R0+⁡G^∞2.The proof of Theorem8(i) can be also addressed from the fact that the inverse of a positive real matrix is positive real and the subsequent derivations if D≻0: (28)G^-1s=G^s+D-1=DIm+D-1G^s-1=Im+D-1G^s-1D-1and G^-1∈PR, then G^∈PR, if(29)Im+D-1G^iω-1D-1+D-TIm+D-1G^iω-T⪰0;∀ω∈R0+.Since Im+D-1G^iω≻0, ∀ω∈R0+, the above matrix relation is equivalent to(30)D-1+Im+D-1G^iωD-TIm+D-1G^iω-T⪰0;∀ω∈R0+and to(31)D-1Im+D-1G^iωT+Im+D-1G^iωD-T⪰0;∀ω∈R0+so that(32)D-1G^iω+G^TiωD-T⪰-D-1+D-T;∀ω∈R0+which yields, equivalently, G^iω+G^T-iω⪰-D+DT; ∀ω∈R0+.


## 5. Regenerative versus Passive Systems

Note that passive systems are intrinsically stable and either consume or dissipate energy for all time. Looking at Definition1(3), we can give an opposed one as follows:Definition 9.
A dynamic system isnonpassive (oractiveor, so-called,regenerative) if ∫0tiyTτuτdτ+εti<0 for some unbounded sequences E=εti⊆R0+, T=ti⊆R0+ which satisfy the conditions
(1)0<δi-1≤ti+1-ti≤δi<∞; ∀i∈Z0+ for some positive bounded sequence Δ=δi,
(2)0<θi-1≤ε~ti=εti+1-εti≤θi<∞; ∀i∈Z0+ for some positive bounded sequence Θ=δi,
(3)εi, ti→+∞ as i→+∞.The following result follows for a nonpassive system.Theorem 10.
If a dynamic system is nonpassive thenlimt→∞⁡∫0tyTτuτdτ=-∞.Proof.
Defineε~ti∈R+ such that ∫0tiyTτuτdτ=-εti-ε~ti<-εti. Thus, (33)∫titi+1yTτuτdτ=-εti+1-ε~ti+1-∫0tiyTτuτdτ=εti+ε~ti-εti+1+ε~ti+1,∫0tyTτuτdτ=∫0tiyTτuτdτ+∫tityTτuτdτ=-εti-ε~ti+∫tityTτuτdτ,∫0tyTτuτdτ=∫0ti+1yTτuτdτ-∫tti+1yTτuτdτ=-εti+1-ε~ti+1-∫tti+1yTτuτdτ.Subtracting the two above ones: (34)εti+1-εti+ε~ti+1-ε~ti≥∫titi+1yTτuτdτ≥2max⁡∫tti+1yTτuτdτ,∫tityTτuτdτ,εti+ε~ti-εti+1+εti-ε~ti+1-ε~ti≤εti+ε~ti-∫tityTτuτdτ≤∫0tyTτuτdτ≤-εti-ε~ti+∫tityTτuτdτ≤-εti+ε~ti+12εti+1-εti+ε~ti+1-ε~tisince εti is unbounded but its associated incremental sequence ε~ti is bounded, 2εti-εti+1→∞ as ti→∞; then |∫0∞yTτuτdτ|<+∞ 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 instantt1<∞ with ut=0 for t>t1. Then, ∫0t1yTτuτdτ=limt→∞⁡∫0tyTτuτdτ=-∞.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 Definitions1(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 limt→∞⁡∫0tyTτuτdτ=-∞ is not compatible with the passivity condition. The converse statement is direct. We now prove Properties [(ii)-(iii)]. Note from Definition 9 that if V:Rn×R0+→R0+ is an energy measure storage state function, as for instance a Lyapunov function, and if the system is passive (resp., nonpassive) then there exists ε∈R0+, respectively; there exists some unbounded sequences E=εti⊆R0+, T=ti⊆R0+ and(35)∫0tyTτuτdτ=Vxt-Vx0+dt≥-ε;∀t∈R0+,respectively,(36)∫0tiyTτuτdτ=Vxt-Vx0+dt<-εtiwith E=εti⊆R0+, T=ti⊆R0+, where dt is the dissipation function and ∫0tyTτuτdτ;∀t∈R0+ is interpreted as the external environment supplied or consumed energy on the interval 0,t. Thus, the following cases are of interest.
Case  1 (passive system). One has from (23) that(37)dt=∫0tyTτuτdτ+V0-Vt≥max⁡V0-ε+Vt,0≥0;∀t∈R0+is a condition of energy dissipation for all time which happens if either (38)V0-ε-Vt≥0⟺Vt≤V0-ε⟹ε=0∧Vt≤V0for all t∈R0+ and dt≥∫0tyTτuτdτ for all t∈R0+ which is also coherent with a null energetic interchange with the environment or if for all t∈R0+(39)0>V0-ε-Vt⟺Vt>V0-ε.Then, one has(40)0≤dt<ε+∫0tyTτuτdτ;∀t∈R0+which is coherent with a positive, negative, or null energetic interchange on 0,t with the environment satisfying ∫0tyTτuτdτ≥-ε; ∀t∈R0+; and, in particular, with null such an energetic interchange, the dissipation function satisfies 0≤dt<ε; ∀t∈R0+ if ε∈R+.
Note that (38) implies that the passive system is also stable since Vt≤V0<+∞; ∀t∈R0+ 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:(41)dt=∫0tyTτuτdτ=V0-Vt=0;∀t∈R0+.
Case  2 (nonpassive system in 0,ti). One has from (36) that(42)dti=∫0tiyTτuτdτ+V0-Vt<min⁡V0-εti+Vti,0≤0and limt→∞⁡∫0tyTτuτdτ=-∞ which happens if either
(1)(43)Vt0-εti-Vti≥0⟺Vti≤Vt0-εt⟹εt=0∧Vti≤Vt0,the system being stable since Vti≤V0 and 0>dti≥∫0tiyTτuτdτ which is coherent with a (negative) energetic supply on 0,ti given to the environment, or if
(2)(44)0>Vt0-εti-Vti⟺Vti>Vt0-εt.Then, one has(45)dti<εti+∫0tiyTτuτdτ≤0the system being either stable if Vt0≥Vti>Vt0-εti or unstable if Vti>Vt0>Vt0-ε and both situations are coherent with an energetic supply given to the environment on t0,t satisfying the constraint ∫0tiyTτuτdτ<-εti<0.
(3) Note that the situations (1) and (2) can coexist within the same intervalt0,t 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 lawσs:R0+→C with q finite or infinity eventual parameterizations C=P∪NP, where the p(≤q) configurations (or parameterizations) P=Pi:i∈p- are passive and the q-p remaining ones NP=NPi:i∈q-p¯ are nonpassive, where P is empty if p=0 and NP is empty if q=p. The switching law σs:R0+→C is defined by(46)σst=σsti+=Cti∈C;∀t∈ti,ti+1,∀ti∈SW,where ti is an abbreviate notation for ti+ versus the explicit notation for the left limit ti- with (47)SW=ti∈R0+:σsti-≠σsti,ti+1-ti≥T>0,t0=0,i∈S⊆Z0+being the set of switching time instants, which can be either of infinity (denumerable) cardinal χ0 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 T is the minimum residence time at the active configuration. In summary, cardSW≤χ0 and(48)0≤min⁡cardNP,cardP≤min⁡cardNP,cardP≤cardC≤χ0.If q<∞, it is possible to describe each configuration by positive integer numbers by assigning Pi→i and NPj→j-p for i∈p- and j∈q-∖p-. Thus, the piecewise constant primary switching law σs:R0+→C is equivalently described in a simpler way by the piecewise constant switching law σ:R0+×SW→1,2,…,p,p+1,…,q such that σt=σti=l(∈q-) if σsti=Cl if l≤p or σsti=Cl+i if p<l≤q. Note that(49)∫tityTτuτdτ≥-εti-∫0tiyTτuτdτ;∀t∈ti,ti+1if σti≤p, that is, if the active configuration on the time interval ti,ti+1, ti,ti+1∈SW is passive, where εti∈ΣP=εi:i∈p- and 0≤εi≤ε; ∀i∈p-, with(50)∫tityTτuτdτ=∫0tyTτuτdτ-∫0tiyTτuτdτ≥-ε-∫0tiyTτuτdτ;∀t∈ti,ti+1,∀ti,ti+1∈SW.If p<σti≤q, that is, if the active configuration on the time interval ti,ti+1, ti,ti+1∈SW is nonpassive, where εti∈ΣNP=εi:i∈p- and εi≥ε′>0; ∀i∈q-∖p-.Note that the whole set of switching time instantsSW is(51)SW=⋃k∈S⊆Z0+SWk=⋃k∈SSWPk∪SWNPk=SWPk∪SWNPk∪SWNP,where SWk=SWPk∪SWNPk, SWPk⊆SWP and SWNPk⊆SWNP; ∀k∈Z0+ are, respectively, the whole set of time instants ti:i∈k-∪0⊆SW until tk∈SW for all k∈Z0+ and its disjoint subsets associated with the active passive and nonpassive configurations which occurred in the interval 0,tk. Note that a system configuration is passive if E0,t=∫0tyTτuτdτ=-ε+ε~t≥-ε; ∀t∈R0+ for some real function ε~:R0+→R0+ and some ε∈R0+.Assumptions 13.
Under switching, a system configuration is assumed passive if for someε∈R0+:
(1)(52)Eti,θ=∫titi+θyTτuτdτ=-ε+ε~ti,θ,θ≥0;ti∈SWP,Eti,θ=∫titi+θyTτuτdτ=-ε+ε~ti,θ=-ε+θ-Tminδiθ≥-ε;θ≥Tmin,ti∈SWP,Eti,Ti=∫titi+1yTτuτdτ=-ε+ε~ti,Ti=-ε+Ti-TminδiTi≥-ε;Ti≥Tmin;ti∈SWPwith ε~ti,θ∈R; ∀θ∈R0+ with ε~ti,0∈R0+ given by ε~ti,θ=θ-Tminδiθ≥0; θ≥Tmin for some function δiθ∈0,∞; ∀ti∈SWP.
(2) A system configuration of the switched law is nonpassive if for someε∈R0+(53)Eti,θ=∫titi+θyTτuτdτ=-ε-ε~ti,θ,θ≥0;ti∈SWP,Eti,θ=∫titi+θyTτuτdτ=-ε-ε~ti,θ=-ε-θ-Tminδiθ-ε0i<-ε;θ≥Tmin,ti∈SWNP,Eti,Ti=∫titi+1yTτuτdτ=-ε-ε~ti,Ti=-ε-Ti-TminδiTi-ε0i<-ε;Ti≥Tmin;ti∈SWNPwith ε~ti,θ∈R; ∀θ∈R0+ with ε~0,0∈R0+ (if t0=0∈SWP) given by ε~ti,θ=θ-Tminδiθ≥0; θ≥Tmin for some constant ε0i∈R+ and some function δiθ with δit→-∞ as t→∞; ∀ti∈SWNP.
The fact thatε 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 ε. The same value of ε is valid by reversing the inequality for nonpassive configuration since there are extra additive thresholds ε0· to modulate possible discrepancies of the necessary constants for distinct nonpassive configurations. The intuitive physical interpretation of Assumptions 13 is as follows if t0=0∈SWP, E0,t≥-ε; ∀t∈R0+ as it follows for standard unswitched passivity. However, for ti(>0)∈SWP and small enough time θ∈0,Tmin, it can happen that Eti,θ<-ε even if ti-1∈SWP 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 θ≥Tmin, the passivity property Eti,θ≥-ε; ∀t∈R0+ is recovered. For large enough θ, it is assumed that ⁡lim⁡supθ→∞⁡δit≤δi∗∈0,∞.The following result holds.Theorem 14.
If Assumptions13 hold a switched system is passive for all time under a switching law σ:R0+×SW→1,2,…,p,p+1,…,q, with at least one configuration being active, if
(1) the first active configuration of the switching law ont0=0,t1∈SWP;
(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 Assumptions13.Proof.
The system is passive if∫0tyTτuτdτ=-ε+ε~t≥-ε; ∀t∈R0+ for some real function ε~:R0+→R0+ and some ε∈R0+. It turns out that a necessary condition for switched passivity is that the first switched configuration be passive (otherwise, the passivity condition fails for t∈0,t1). Define the active switching passivity binary indicator function ρt=1;∀t∈ti,ti+1,∀ti∈SWP, ρt=0;∀t∈ti,ti+1,∀ti∈SWNP. Direct calculations for any t∈tk,tk+1;∀tk∈SW on the input-output energy yield (54)E0,t=∫0tyTτuτdτ=∑i=0k-1∫titi+1yTτuτdτ+∫tk+1tyTτuτdτ=∑ti∈SWPk∫titi+1yTτuτdτ+∑ti∈SWNPk∫titi+1yTτuτdτ+∫tk+1tyTτuτdτ=∑ti∈SWPk-ε+Ti-TminδiTi-∑ti∈SWNPkε+Ti-TminδiTi+ε0i+∫tk+1tyTτuτdτ=-εcPk+1+∑ti∈SWPkTi-TminδiTi-εcNPk-∑ti∈SWNPkTi-TminδiTi+ε0i+∫tk+1tyTτuτdτ=-k+1ε+∑ti∈SWk+1ρtiTi-TminδiTi-1-ρtiTi-TminδiTi+ε0i+∫tk+1tyTτuτdτ=-kε+∑ti∈SWkρtiTi-TminδiTi-1-ρtiTi-TminδiTi+ε0i+ρtkTk-TminδkTk-1-ρtkTk-TminδkTik+ε0k+∫tk+1tyTτuτdτ=E0,t-Tk-ε+ρtkTk-TminδkTk-1-ρtkTk-TminδkTik+ε0k+∫tk+1tyTτuτdτ≥-2ε+ρtkTk-TminδkTk-1-ρtkTk-TminδkTik+ε0k+∫tk+1tyTτuτdτ≥-ε;∀t∈tk,tk+1holds via complete induction if t0(=0)∈SWP (i.e., the first configuration of the switched disposal is passive) provided that E0,t-Tk≥-ε for any t∈R0+t-Tk∈tk-1,tk1 if the subsequent constraint holds:(55)ρtkTk-TminδkTk≥ε+1-ρtkTk-TminδkTik+ε0k-∫tk+1tyTτuτdτ.The following cases can occur:
Case  1. tk,tk+1∈SWP. Thus, ρtk=1 and(56)Tk-TminδkTk≥2ε-Tk+1-Tminδk+1t-tk+1which is guaranteed if(57)Tk+1≥Tmin+min⁡2ε-Tk-TminδkTkinft∈tk,tk+1⁡δk+1t-tk+1,0or if(58)Tk+1≥Tmin+min⁡ε-inft∈tk,tk+1⁡E0,t-tk-Tk-TminδkTkinft∈tk,tk+1⁡δk+1t-tk+1,0.
Case  2. tk,tk+1∈SWNP. Thus, ρtk=0 and(59)0≥2ε+Tk-TminδkTik+ε0k+ε0,k+1+t-tk+1-Tminδk+1t-tk+1.
Case  3. tk∈SWNP,tk+1∈SWP. Thus, ρtk=0 and(60)0≥2ε+Tk-TminδkTik+ε0k-t-tk+1-Tminδk+1t-tk+1yields closely to Case  1:(61)Tk+1≥Tmin+min⁡2ε+ε0,k+1-Tk-TminδkTkinft∈tk,tk+1⁡δk+1t-tk+1,0.


## 7. Examples

Example 15.
Consider the following damped linear system:(62)x˙t=ptm;p˙t=mx¨t=Ft-kxt-λx˙t;x0=x0,x˙0=x˙0with xt being the position, yt=x˙t being the velocity, defined to be the system output, ut=Ft being the external force, defined to be the control action, and k and λ being the spring constant and the viscous coefficient, respectively. Define the state vector as zt=xt,x˙tT and let Vzt=(1/2)mx˙2t+(1/2)kx2t=Vkinx˙t+Vpotxt be the storage function defined to be the kinetic energy Vkinx˙t plus the potential energy Vpotxt. The input-output energy at time t∈R0+ is(63)Et=∫0tyτuτdτ=∫0tx˙τFτdτ=∫0tx˙τmx¨τ+kxτ+λx˙τdτ=Vzt-Vz0+dt,where the dissipation function at time t is dt=λ∫0tx˙2τdτ≥0; ∀t∈R0+. If λ=0, xt=x0,Ft=0,x˙t=x˙0;∀t∈R0+ then Vzt=Vz0=(1/2)mx˙02+(1/2)kx02, Et=dt=0;∀t∈R0+ and the system is aconservative 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 γ such that Vzt-Vz0+dt≥-γ2; ∀t∈R0+, that is, (64)+∞>dt+γ2≥Vz0-Vzt>-∞;∀t∈R0+,then the system is passive. In the case that the passive system is also globally asymptotically stable, zt→0, Vzt→V0=0 as t→∞ and 0≤dt≤dt′<+∞ for all t′∈t,∞;∀t∈R0+ and d∞=∫0∞x˙2tdt<+∞. Sufficient conditions are obtained by giving conditions on the stored energy to be a Lyapunov function for global asymptotic stability as follows:(65)V˙zt=ddt1/2mx˙2t+1/2kx2t=mx˙tx¨t+kxtx˙t=mx˙tFt-kxt-λx˙t+kxtx˙t≤0;∀t∈R0+if the external force is generated via feedback as follows:(66)Ft=m-1mkxtx˙t+λx˙2t-ζxt,x˙t,t=1mm-1V˙pott+λVkint-ζxt,x˙t,t;∀t∈R0+,where ζ:R×R0+×R×R0+×R0+→R0+ is a design function so that ζxt,x˙t,t=0 if and only if xt=0. Thus, Vzt≤Vz0;∀t∈R0+ and Vzt→0 as t→∞ 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 satisfy(67)mx¨t=Ft-kxt-βx3t-λx˙t;∀t∈R0+,V˙zt=mx˙tFt-kxt-βx3t-λx˙t+kxtx˙t≤0;∀t∈R0+if(68)Ft=m-1mk1+βkxtxt+λx˙tx˙t-ζxt,x˙t,t;∀t∈R0+.If Ft=σtxt+σdx˙t then V˙zt≤0 and global stability holds if σd=λ and σt=k((m-1)/m)-βx2t; ∀t∈R0+ since the exact output linearization dynamics is achieved under such a control.Example 16.
Consider anth order SISO time-invariant linear system whose transfer function is G^s=cTsIn-A-1b+d, where A,b,cT,d is a state-space realization of G^s, and the state n-vector has initial conditions x0=x0. The output is given by a superposition of the unforced (i.e., zero -input) and forced (i.e., zero state) solutions yunft and yft as(69)yt=yunft+yft=cTeAtx0+∫0te-Aτbuτdτ+dut,where(70)yft=yt-yunft=∫0tcTeAt-τbuτ+dut.Then, 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:(71)Eft=∫0tyfτuτdτ=∫0∞yfτutτdτ=12π∫0∞y^fiωu^t-iωdω=12π∫0∞y^tiωu^t-iωdω=12π∫-∞∞G^iωu^tiωu^t-iωdω=12π∫-∞∞G^iωu^tiω2dω≥12πminω∈R0+G^iω∫-∞∞u^tiω2dω=12πminω∈R0+G^iω∫-∞∞u^tiω2dω=minω∈R0+G^iω∫0∞utτ2dτ=minω∈R0+G^iω∫0tu2τdτ,∀t∈R0+since Im⁡G^iω=-Im⁡G^iω for any hodograph G^iω;∀ω∈R0+. Assume that G^∈PR so that Eft≥d∫0∞u2τdτ from (71). Assume also that the control is generated via negative output-feedback from any member ϕ∈Φ of nonlinear and perhaps time-varying class of controllers Φ which satisfies a Popov’s type inequality, namely, ut=-ϕyt,t, where (72)∫0tyτuτdτ=-∫0tyτϕyτ,τdτ=-Et≥-γγ>0so that for any given positive real γ- such that γ-≥γ and any nonidentically zero controls, one gets(73)+∞>γ-≥γ≥Et=∫0tyτuτdτ≥d∫0tu2τdτ>0;∀t∈R0+so that for nonidentically zero controls(74)+∞>γ--Eunft=γ--∫0tyunfτuτdτ≥γ-∫0tyunfτuτdτ=γ-∫0tcTeAt-τx0uτdτ=γ-∫0tcTeAt-τx0uτdτ≥∫0tyfτuτdτ≥d∫0tu2τdτ>0;∀t∈R0+.The following result holds from (74).Theorem 17.
The following properties hold:
(i) Assume thatG^∈SSPR. Then, u,y∈L2 and the system is globally asymptotically stable for any given finite initial conditions, irrespective of ϕ∈Φ, so that it is asymptotically hyperstable.
(ii) Assume thatG^∈SPR and that a frequency filter F^iω is used for the control inputs so that F^iωu^iω=0 for any frequency interval, if any, such that min⁡Re⁡G^iω=0 and it is of unity gain otherwise. Then, u,y∈L2 and the system is globally asymptotically stable for any given finite initial conditions, irrespective of ϕ∈Φ; then it is asymptotically hyperstable for the forward loop transfer function G^sF^s.Proof.
SinceG^∈SSPR so that d>0 and G^∈SPR, that is, it is strongly positive real then, in addition, in RH∞ (then with all its poles in Re⁡s<0) and having zero relative degree it follows in (74) that +∞>γ-=γ-x0,u>max⁡γ,∫0tcTeAt-τx0uτdτ can be made and the Fourier transforms used in the Parseval’s identity exist. Assume that ut→±∞ as t→∞ then, there exists α∈R+ such that (75)γ≥lim⁡supt→∞d+α∫0tu2τdτ≥lim⁡supt→∞d∫0tu2τdτ+∫0tyunfτuτdτsince u2t/ut→∞ and eAt→0 as t→∞. Therefore, u∈L2, so that ut→0 as t→∞ which contradicts that ut→±∞ as t→∞. It can also be concluded that ut 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 Eft≥minω∈R0+⁡G^-1iω∫0ty2τdτ with G^-1∈SSPR and is realizable, since G^∈SSPR so it has relative degree zero. Thus, y∈L2 and converges asymptotically to zero for any ϕ∈Φ. Since G^∈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(76)Eft≥12πminω∈R0+⁡G^iω∫-∞∞u^tiω2dω=12π∑j=1rminω∈ωj,ωj+1⁡G^jiω∫ωjωj+1u^tjiω2dω≥∑j=1rminω∈ωj,ωj+1⁡G^jiω∫0∞utj2τdτ=∑j∈ϑrdj∫0∞utj2τdτ=∑j∈ϑrdj∫0tuj2τdτ,where dj=minω∈ωj,ωj+1⁡G^jiω≥0 since G^∈PR and ϑr⊆r- is the indicator set such that dj>0 if and only if j∈ϑr. Thus, uj∈L2, uj is bounded; then ujt→0 as t→∞ since G^∈SPR for all j∈ϑr and the total control input ut=∑j∈ϑrujt+∑j∈r-∖ϑrujt is also bounded since uj is bounded by the filtering actions; ∀j∈r-∖ϑr. If uj≡0;∀j∈r-∖ϑr then Property (i) holds for this class of constrained control inputs.Example 18.
Consider anth order SISO time-invariant linear system described by a finite or infinite set of transfer functions G^js:j∈r- governed by a switching law σ:⋃ktk,tk+1→r-, with cardr-≤χ0, which selects the active one on a time interval tk,tk+1 where SW=tk 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 t∈tk+1,tk+2. To simplify the exposition, consider the system under zero initial conditions. Then (77a)Eft=∑j=0k∫tjtj+1yτuτdτ+∫tk+1tyτuτdτ=∑j=0k∫0∞yτutj,tj+1τdτ+∫0∞yτutk+1,tτdτ≥12π∑j=0kminω∈R0+⁡G^σtkiω∫-∞∞u^tj,tj+1iω2dω+minω∈R0+⁡G^σtkiω∫-∞∞u^tk,tiω2dω=∑j=0kminω∈R0+⁡G^σtkiω∫0tutj,tj+12τdτ+minω∈R0+⁡G^σtkiω∫tk+1tutk+1,t2τdτ(77b)≥max⁡ζ1t,ζ2t,where a truncated interswitching input uta,tbt=ut for t∈ta,tb and uta,tbt=0 for t∉ta,tb has been defined and(78)ζ1t=∑j=0kminω∈R0+⁡G^σtjiω+minω∈R0+⁡G^σtk+1iωmin⁡minj∈k-∪0⁡∫0tυjutj,tj+12τdτ,∫0tυk+1utk+1,t2τdτ=∑j=0kminω∈R0+⁡G^σtjiω+minω∈R0+⁡G^σtkiωmin⁡minj∈k-∪0⁡∫tjtj+1υju2τdτ,∫tk+1tυku2τdτ,(79)ζ2t=min⁡minj∈k-∪0⁡minω∈R0+⁡G^σtj,minω∈R0+⁡G^σtkiω∫0∞u2τdτwith(80)υj=1ifminω∈R0+⁡G^σtjiω>00ifminω∈R0+⁡G^σtjiω=0-1ifminω∈R0+⁡G^σtjiω<0.The following related result holds.Theorem 19.
The following properties hold:
(i) Assume that the switching law incorporates a zero state resetting action at eachtj∈SW. 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 eachtj∈SW. Assume also that the switching law satisfies that Gσt0∈SPR and that Gσtk∈RH∞ and that, in the event that Gσtk∈RH∞∖PR for k∈Z+, then Gσtk+1∈SPR, and (81)tk+2<inf⁡t>tk+1:∫tk+1tu2τdτ≤∑j=0kminω∈R0+⁡G^σtkiω∫tjtj+1u2τdτmaxω∈R0+⁡G^σtkiω.Then, the switched system is strictly input passive for such a switching law.
(iii) Assume that the system is subject to a feedback control lawϕ∈Φ, (72), and that G^σtj∈SPR;∀tj∈SW with G^σt0∈SSPR and to zero state resetting at each switching time instant. Then, u,y∈L2 and the system is globally asymptotically stable for any given finite initial conditions, irrespective of ϕ∈Φ, so that it is asymptotically hyperstable.
(iv) Assume also that the constraintsG^σt0∈SSPR and zero state switching resetting invoked in Property (iii) but frequency filters F^σtjiω are used for the control inputs so that F^σtjiωu^iω=0 for any frequency interval, if any, such that min⁡Re⁡G^σtjiω=0 and it is of unity gain otherwise for tj∈SW. Then, u,y∈L2 and the system is globally asymptotically stable for any given finite initial conditions, irrespective of ϕ∈Φ; then it is asymptotically hyperstable for the forward loop transfer function G^sF^s.Proof.
Property (i) follows from (77a), (77b), and (78) and Et=Eft≥0;∀t∈R0+ under zero state resetting at each switching time instant since(82)∑j=0kminω∈R0+⁡G^σtkiω+minω∈R0+⁡G^σtkiω≥0for any k∈Z0+, any tk∈SW and any t∈tk,tk+1. Property (ii) by taking into account (77a) and Gσt0∈SPR which guarantees that Eft=Et>0;∀t∈R+ 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 Et=Eft>0;∀t∈R+ 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 Example18 to yield(83)Eft=∫0t∫0τGτ-θuθdθuτdτ=∑j=0k∑l=0kτ∫tjtj+1∫tltl+1Glτ-θulθujτdθdτ+∑j=0k∫tjtj+1∫tkτ+1τGtkτ+1τ-θutkτ+1θujτdθdτ+∑l=0kτ∫tk+1t∫tltl+1Glτ-θulθuk+1τdθdτ+∫tk+1t∫tkτ+1τGtkτ+1τ-θutkτ+1θuk+1τdθdτ.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 responsec·TeA·tb·+d· 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,A· is a Metzler matrix; d·≥0 and c· and b· 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].



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