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

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.sub.0+] = [R.sub.+] [union] {0}, where [R.sub.+] = {r [member of] R : r > 0}, [bar.p] = {1, 2, ... p},

(ii) D > 0 denotes that the real matrix D is positive definite while D [greater than or equal to] 0 denotes that it is positive semi-definite,

(iii) [[lambda].sub.min](*) and [[lambda].sub.max](*) denote, respectively, the minimum and maximum eigenvalues of the real symmetric (*)-matrix,

(iv) [??] [member of] {PR} denotes that the transfer matrix [??](s) of a linear time-invariant system is positive real; that is, [??](s) + [[??].sup.T] (-s) [greater than or equal to] 0 for all Re s > 0, and [??] [member of] {SPR} denotes that it is strictly positive real; that is, [??](s) + [[??].sup.T](-s) > 0 for all Re s [greater than or equal to] 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 [absolute value of s] [right arrow] [infinity]. If the linear time -invariant system is a SISO one (i.e., it has one input and one output) then [??] [member of] {PR} if Re [??](s) [greater than or equal to] 0 for all Re s > 0 and [??] [member of] {SPR} if Re [??](s) > 0 for all Re s [greater than or equal to] 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 t [greater than or equal to] 0 for any given nonnegative initial conditions and nonnegative input,

(vi) i = [square root of -1] is the complex unity,

(vii) [I.sub.m] is the mth identity matrix,

(viii) The superscript T stands for matrix transposition,

(ix) [H.sub.[infinity]] is the Hardy space of all complex-valued functions F(s) of a complex variable s which are analytic and bounded in the open right half-plane Re s >0 of norm [[parallel]F[parallel].sub.[infinity]] = sup{[absolute value of F(s)] : Res > 0} = sup{[absolute value of F(i[omega]>)]: [omega] [member of] R} (by the maximum modulus theorem) and R[H.sub.[infinity]] is the subset of real-rational functions of [H.sub.[infinity]].

2. Preliminaries

Consider a dynamic system G : [H.sub.e] [right arrow] [H.sub.e] with state x [member of] [R.sup.n], input u [member of] [R.sup.m], and output y [member of] [R.sup.m], where [H.sub.e] is the extended space of the Hilbert space H endowed with the inner product (*, *) from [H.sub.e] x [H.sub.e] to R consisting of the truncated functions [u.sub.t]([tau]) = u([tau]) for [tau] [member of] [0, t] and [u.sub.t]([tau]) = 0; [for all]t, [tau] (>t) [member of] [R.sub.0+] and u : [R.sub.0+] [right arrow] [R.sup.m]. If u [member of] [H.sub.e] then

[mathematical expression not reproducible]. (1)

Definition 1 (see [2]). The above dynamic system is

(1) [L.sub.2]-stable if u [member of] [L.sup.m.sub.2] implies Gu [member of] [L.sup.m.sub.2];

(2) nonexpansive if [there exists][lambda] and [there exists][gamma] > 0 s.t. for all u [member of] [H.sub.e],

[mathematical expression not reproducible]; (2)

(3) passive if [there exists][epsilon] [greater than or equal to] 0 such that [mathematical expression not reproducible];

(4) strictly input passive if [there exists][epsilon] [greater than or equal to] 0 and [there exists][[epsilon].sub.u] > 0 s.t.

[mathematical expression not reproducible]; (3)

(5) strictly output passive if [there exists][beta] [greater than or equal to] 0 and [there exists][[epsilon].sub.y] > 0 s.t.

[mathematical expression not reproducible]; (4)

(6) strictly input/output passive (or very strictly passive) if [there exists][beta] [greater than or equal to] 0, [there exists][[epsilon].sub.u] > 0 and [there exists][[epsilon].sub.y] > 0 s.t.

[mathematical expression not reproducible]. (5)

The constants [epsilon], [[epsilon].sub.u], and [[epsilon].sub.y] 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 [epsilon] = 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 [??] [member of] {PR}. Then, the following properties hold:

(i) [mathematical expression not reproducible] and y(t)w(t) [greater than or equal to] 0; [for all]t [greater than or equal to] 0 and, furthermore, if u [member of] [L.sub.2] then y [member of] [L.sub.2]. Then, the system is passive.

(ii) Assume, in addition, that [??] [member of] {SPR}. Then[mathematical expression not reproducible] for any t [member of] (0, [infinity]] and some [gamma], [epsilon] [member of] [R.sub.0+].

(iii) If, furthermore, the system is externally positive then [mathematical expression not reproducible] for any given nonnegative initial conditions and nonnegative input.

(iv) Define [mathematical expression not reproducible] as the relative passivity index of the transfer function [mathematical expression not reproducible] and [??](s) feeing the numerator and denominator polynomials of [??](s)). Then, the constraint [mathematical expression not reproducible] is guaranteed for some [a.sub.G], [b.sub.G] ([greater than or equal to] [a.sub.G]) [member of] [R.sub.0+] if

[mathematical expression not reproducible]. (6)

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 [L.sub.2]. Therefore, Parseval's theorem can be applied to express [mathematical expression not reproducible] in the frequency domain. Take into account, in addition, that the hodograph of the frequency system's response [??](i[omega]) satisfies Re [??](i[omega]) = Re [??](-i[omega]) and Im [??](i[omega]) = -Im [??](-i[omega]) for all [omega] [member of] (-[infinity], [infinity]) and that [mathematical expression not reproducible] since [??] [member of] {PR}. Thus, the various expressions which follow hold under zero initial conditions of the dynamic system:

[mathematical expression not reproducible]. (7)

It has been proved, under zero initial conditions, that [mathematical expression not reproducible] and y(t)u(t) [greater than or equal to] 0; [for all]t [greater than or equal to] 0 and if u [member of] [L.sub.2] then [mathematical expression not reproducible] for some [beta] [member of] [0, [infinity]) independent of u (and independent of f). 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 [epsilon] [member of] R such that [mathematical expression not reproducible] since the initial conditions do not generate an unbounded homogeneous solution since [??] [member of] R[H.sub.[infinity]] since [??] [member of] {PR}. Property (i) has been proved. On the other hand, under any finite nonzero initial conditions [x.sub.0] [member of] [R.sup.n]:

[mathematical expression not reproducible]. (8)

for some uniformly bounded [lambda](t, [x.sub.0]) since [??](s) is stable, (perhaps including eventual single critical poles) since it is in {PR}. If, in addition, [??] [member of] {SPR} then it is strictly stable so that [mathematical expression not reproducible] and

[mathematical expression not reproducible], (9)

since u [member of] [L.sub.2], for any time t > 0, any given control and initial conditions, and some finite [[epsilon].sub.u] > 0 and [epsilon] [greater than or equal to] 0 with [lambda](t) = [c.sup.T][e.sup.At] [x.sub.0], where [c.sup.T] and A are the output vector and matrix of dynamics of a state-space realization of initial state [x.sub.0] so that [lambda](*, 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

[mathematical expression not reproducible] (10)

which proves Property (iii). To prove Property (iv), note that [mathematical expression not reproducible] holds if

[mathematical expression not reproducible] (11)

holds, that is, if

[mathematical expression not reproducible] (12)

which leads to (6).

Remark 3. The generalization of Theorem 2 to the multiinput multioutput (MIMO) case (i.e., m > 1) is direct by replacing the instantaneous power y(t)u(t) by the scalar product [y.sup.T] (t)u(t) 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 [[G].sub.[rho]D] of dynamic systems G([rho], D) : [H.sub.e] [right arrow] [H.sub.e], defined as G([rho], D) = [G.sub.0] + [rho]D for given [rho] [member of] R, D([member of] [R.sup.mxm]) [greater than or equal to] 0 and [G.sub.0] : [H.sub.e] [right arrow] [H.sub.e], such that G([alpha], D) [member of] [[G].sub.[rho]D] for any [alpha] [member of] [0, [rho]]. The following properties hold:

(i) Assume that G([rho], D) is very strictly passive, D > 0 and [[epsilon].sub.up] > [rho][[lambda].sub.max] (D + [D.sup.T])/2, where [[epsilon].sub.u[rho]] is the input passivity constant for G([rho], D). Then, G([alpha], D) is very strictly passive for all [alpha] [member of] [0, [rho]]. If [[epsilon].sub.u[rho]] = [rho][[lambda].sub.max](D + [D.sup.T])/2 then G([alpha], D) is very strictly passive for all [alpha] [member of] (0, [rho]] while [G.sub.0] is strictly output passive.

(ii) If [G.sub.0] is passive (resp., strictly output passive) then G([alpha], D) is strictly input passive (resp., very strictly passive) if D > 0 for any [alpha] [member of] [R.sub.+].

Proof. Since G([alpha], D) [member of] [[G].sub.[rho]D] for any given [alpha] [member of] [0, [rho]], [G.sub.0] = G(0, D) [member of] [[G].sub.[rho]D] (irrespective of D). Then, since G([rho], D) is very strictly passive [there exists][[epsilon].sub.[rho]] [greater than or equal to] 0, 3[[epsilon].sub.u[rho]] > 0 and 3[[epsilon].sub.y[rho]] > 0 s.t.

[mathematical expression not reproducible] (13)

for all t [greater than or equal to] 0 and any input-output pair (u, [y.sub.[alpha]]) with input and output being defined from [R.sub.0+] to [R.sup.m] for any [alpha] [member of] [0, [rho]]. Thus,

[mathematical expression not reproducible], (14)

where [[epsilon].sub.u0] = [[epsilon].sub.u[rho]] - ([rho]/2)[[lambda].sub.max(D] + [D.sup.T]) > 0 since D > 0 and [[epsilon].sub.u[rho]] > [rho][[lambda].sub.max](D + [D.sup.T])/2. Then, [G.sub.0] is very strictly passive with passivity, input passivity, and output passivity constants [mathematical expression not reproducible], and [[epsilon].sub.y0] = [[epsilon].sub.yp] > [degrees]. If [[epsilon].sub.u[rho]] = [rho][[lambda].sub.max](D + [D.sup.T])/2 then [G.sub.0] is guaranteed to be just output-strictly passive. Now, for any [alpha] [member of] [0, [rho]], one has

[mathematical expression not reproducible] (15a)

[mathematical expression not reproducible]; (15b)

[for all]t [greater than or equal to] 0. Then, G([alpha], D) is very strictly passive with passivity, input passivity, and output passivity constants [mathematical expression not reproducible] and [[epsilon].sub.y[alpha]] = [[epsilon].sub.y[rho]] > 0 for all [alpha] [member of] [0, [rho]] if [[epsilon].sub.u[rho]] > [rho][[lambda].sub.max](D + [D.sup.T])/2 while [[epsilon].sub.u[rho]] = [rho][[lambda].sub.max](D +[D.sup.T])/2 implies [[epsilon].sub.u0] = [[epsilon].sub.u[rho]] - ([rho]/2)[[lambda].sub.max](D+[D.sup.T]) = 0 so that [G.sub.0] is strictly output passive and G([alpha], D) is very strictly passive for [alpha] [member of] (0, [rho]]. Property (i) has been proved, Property (ii) follows from (15a) with [[epsilon].sub.u[alpha]] = [alpha].

Proposition 5. Assume that [G.sub.0] is passive and nonexpansive. Then:

(i) [G.sub.[rho]] = [G.sub.0] + [rho]D is [L.sub.2]-stable and strictly input passive if [rho] [member of] [R.sub.+],

(ii) [G.sub.0] is [L.sub.2]-stable if [mathematical expression not reproducible],

(iii) [G.sub.0] is [L.sub.2]-stable if D = [I.sub.m] for any given [rho] [member of] R.

Proof. Since [G.sub.0] is passive and nonexpansive, one has

[mathematical expression not reproducible]. (6)

Thus, if u [member of] [L.sup.m.sub.2] then

[mathematical expression not reproducible]. (17)

Thus, [G.sub.[rho]] is [L.sub.2]-stable and strictly input passive if [rho] [member of] [R.sub.+]; [G.sub.0] is [L.sub.2]-stable if [mathematical expression not reproducible] since the maximum and minimum eigenvalues of D + [D.sup.T] are distinct, If D =[ .sub.Im] then [M.sub.0] = [[lambda].sub.0] + [[gamma].sup.2.sub.0] > 0 for any real [rho] [member of] R and [G.sub.0] is [L.sub.2]-stable.

Remark 6. It turns out through simple mathematical derivations that Propositions 1-2 still hold under the replacement D [right arrow] [G.sub.1], where [G.sub.1] : [H.sub.e] [right arrow] [H.sub.e] is passive with associated constant [[epsilon].sub.1] [less than or equal to] 0 for the properties to be extended from the case that D [greater than or equal to] 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 function [??](s), whose relative passivity index [Theorem 2(iv)] is [mathematical expression not reproducible], is controlled by a feedback controller of transfer function [[??].sub.1] (s) so that

[mathematical expression not reproducible], (18)

where [mathematical expression not reproducible] is the resulting closed-loop transfer function, The closed-loop relative passivity index is [mathematical expression not reproducible]. For any given [[??].sub.1](s) and associated [[??].sub.1](s), the controller transfer function is

[mathematical expression not reproducible]. (19)

(b) The controlled plant transfer function [??](s) is controlled by a feed-forward controller of transfer function [[??].sub.2](s) so that

[mathematical expression not reproducible], (20)

where [mathematical expression not reproducible] is the resulting closed-loop transfer function, The closed-loop relative passivity index is [mathematical expression not reproducible]. For any given [[??].sub.1](s) and associated [[??].sub.1](s), the controller transfer function is

[mathematical expression not reproducible]. (21)

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 that [??](s) is inversely stable with relative degree 0 or 1 while nonnecessarily in |P^| (or even nonnecessarily in R[H.sub.[infinity]]). Then, the following properties hold:

(i) A nonunique (state-space) realizable closed-loop transfer function [[??].sub.1] [member of] {PR}, or, respectively, [[??].sub.1] [member of] {SPP}, may be designed via a stable feedback controller of transfer function [[??].sub.1](s) (19) which is realizable if [??](s) and [[??].sub.1](s) have zero relative degree. In the above cases, [[??].sup.-1.sub.1] [member of] {PR}, or, respectively, [[??].sup.-1.sub.1] [member of] {SPR}.

(ii) A nonunique realizable closed-loop transfer function [[??].sub.2] [member of] {PP}, or [[??].sub.2] [member of] |SPP}, may be designed via a feed-forward controller of transfer function [[??].sub.2](s) via (21) which is realizable if the relative degree of the closed-loop transfer function [[??].sub.2](s) is not less than that of the plant transfer function [??](s). In the above cases, [[??].sup.-1.sub.2] [member of] {PR}, or, respectively, [[??].sup.-1.sub.2] [member of] {SPR}.

Proof. A nonunique realizable closed-loop transfer function can be targeted as design objective through a feedback controller of transfer function [[??].sub.1](s), (19), such that [mathematical expression not reproducible]. Since [mathematical expression not reproducible], it turns out that [[??].sub.1](s) is bounded real (i.e., Schur, i.e., with [H.sub.[infinity]]-norm not exceeding unity and with real numerator and denominator coefficients) so that [[??].sub.1] [member of] {PR} (since [[??].sub.1] [member of] {PR} if and only if [[??].sub.1](s) is bounded real, [4]) then with a relative degree (0 or 1) being identical to that of [??](s). The controller is realizable if [??](s) and [[??].sub.1](s) have zero relative degree. Then, [[??].sub.1](s) is stable and inversely stable. Since [??](s) is inversely stable and [[??].sup.-1.sub.1](s) is stable, since [[??].sup.-1.sub.1] [member of] {PR} since [[??].sub.1] [member of] {PR}, then it follows from (18)-(19) that [[??].sub.1] [member of] R[H.sub.[infinity]]. Also, [[??].sub.1](s) [member of] {SPR} if and only if [mathematical expression not reproducible], that is, if and only if [[??].sub.1](s) is strictly bounded real. Then, [[??].sub.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([D.sub.2]) [greater than or equal to] deg(D) + deg([N.sub.2]) - deg(N); that is, the relative degree of the closed-loop transfer function is not less than that of that of the plant [??](s).

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 [??](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 that [??] [member of] R[H.sub.[infinity]] is a transfer matrix of order m x m. Then, [[??].sub.1] [member of] {PR}, where [[??].sub.1](s) = [??](s) + D with D([greater than or equal to] 0) [member of] [R.sup.mxm] if [mathematical expression not reproducible]. If m = 1 then the condition becomes Re [??](i[omega]) > -(D + [D.sup.T]); [for all][omega] [member of] [R.sub.0+] with d = D. If [mathematical expression not reproducible] and then [[??].sub.1] [member of] {SPR} which becomes Re [??](i[omega]) > -d if m = 1; [for all][omega] [member of] [R.sub.0+].

(ii) Assume that [??](s) is an inversely stable transfer matrix order mxm controlled by a linear time-invariant feedback controller of transfer matrix [[??].sub.1] (s) of order m x m. Then, [[??].sub.1], [[??].sup.-1.sub.1] [member of] {PR} (resp., [[??].sub.1], [[??].sup.-1.sub.1] [member of] (SPR}) if

[mathematical expression not reproducible] (22)

(resp., the above inequality is strict).

(iii) Assume that [??](s) is an inversely stable transfer matrix order m x m being controlled by a linear time-invariant feed-forward controller of transfer matrix [[??].sub.2](s) of order m x m. Then, [mathematical expression not reproducible] if [mathematical expression not reproducible] (resp., the inequality is strict).

Proof. The proof of Property (i) is direct from the conditions of positive and strictly positive realness for [[??].sub.1](s). 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:

[mathematical expression not reproducible], (23)

[mathematical expression not reproducible] (24)

with inverses

[mathematical expression not reproducible]. (25)

Then, [[??].sub.1](s) and [[??].sup.-1.sub.1](s) (resp., [[??].sub.2](s) and [[??].sup.-1.sub.2](s)) are positive real if

[mathematical expression not reproducible]. (26)

which is guaranteed if, respectively, if

[mathematical expression not reproducible] (27)

which is guaranteed if [mathematical expression not reproducible]. 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 [mathematical expression not reproducible], where [mathematical expression not reproducible] is stable and realizable (so that its [H.sub.[infinity]] norm exists) and [mathematical expression not reproducible].

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 D > 0:

[mathematical expression not reproducible] (28)

and [[??].sup.-1] [member of] {PR}, then [??] [member of] {PR}, if

[mathematical expression not reproducible]. (29)

Since [I.sub.m] + [D.sup.-1] [??](i[omega]) > 0, [for all][omega] [member of] [R.sub.0+], the above matrix relation is equivalent to

[mathematical expression not reproducible] (30)

and to

[mathematical expression not reproducible] (31)

so that

[mathematical expression not reproducible] (32)

which yields, equivalently, [??](i[omega]) + [[??].sup.T](-i[omega]) [greater than or equal to] -(D + [D.sup.T]); [for all][omega] [member of] [R.sub.0+].

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(3), we can give an opposed one as follows:

Definition 9. A dynamic system is nonpassive (or active or, so-called, regenerative) if [mathematical expression not reproducible] for some unbounded sequences [mathematical expression not reproducible] which satisfy the conditions

(1) 0 < [[delta].sub.i-1] [less than or equal to] [t.sub.i+1] - [t.sub.i] [less than or equal to] [[delta].sub.i] < [infinity]; [for all]i [member of] [Z.sub.0+] for some positive bounded sequence [NABLA] = {[[delta].sub.i]},

(2) [mathematical expression not reproducible] for some positive bounded sequence [THETA] = {[[delta].sub.i]},

(3) [[epsilon].sub.i], [t.sub.i] [right arrow] +[infinity] as i [right arrow] +[infinity].

The following result follows for a nonpassive system.

Theorem 10. If a dynamic system is nonpassive then [mathematical expression not reproducible].

Proof. Define [mathematical expression not reproducible] such that [mathematical expression not reproducible]. Thus,

[mathematical expression not reproducible]. (33)

Subtracting the two above ones:

[mathematical expression not reproducible] (34)

since {[mathematical expression not reproducible]} is unbounded but its associated incremental sequence {[mathematical expression not reproducible]} is bounded, [mathematical expression not reproducible] as [t.sub.i] [right arrow] [infinity]; then [mathematical expression not reproducible] 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 [t.sub.1] < [infinity] with u(t) = 0 for t > [t.sub.1]. Then, [mathematical expression not reproducible].

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 [mathematical expression not reproducible] 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 : [R.sup.n] x [R.sub.0+] [right arrow] [R.sub.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 [epsilon] [member of] [R.sub.0+], respectively; there exists some unbounded sequences [mathematical expression not reproducible] and

[mathematical expression not reproducible], (35)

respectively,

[mathematical expression not reproducible] (36)

with [mathematical expression not reproducible], where d(t) is the dissipation function and [mathematical expression not reproducible] 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

[mathematical expression not reproducible] (37)

is a condition of energy dissipation for all time which happens if either

[mathematical expression not reproducible] (38)

for all t [member of] [R.sub.0+] and [mathematical expression not reproducible] for all t [member of] [R.sub.0+] which is also coherent with a null energetic interchange with the environment or if for all t [member of] [R.sub.0+]

[mathematical expression not reproducible]. (39)

Then, one has

[mathematical expression not reproducible] (40)

which is coherent with a positive, negative, or null energetic interchange on [0, t] with the environment satisfying [mathematical expression not reproducible]; and, in particular, with null such an energetic interchange, the dissipation function satisfies 0 [less than or equal to] d(t) < [epsilon]; [for all]t [member of] [R.sub.0+] if [epsilon] [member of] [R.sub.+].

Note that (38) implies that the passive system is also stable since V(t) [less than or equal to] V(0) < +[infinity]; [for all]t [member of] [R.sub.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:

[mathematical expression not reproducible]. (41)

Case 2 (nonpassive system in [0, [t.sub.i]]). One has from (36) that

[mathematical expression not reproducible] (42)

and [mathematical expression not reproducible] which happens if either

(1) [mathematical expression not reproducible], (43)

the system being stable since V([t.sub.i]) [less than or equal to] V(0) and [mathematical expression not reproducible] which is coherent with a (negative) energetic supply on [0, [t.sub.i]] given to the environment, or if

(2)

[mathematical expression not reproducible]. (44)

Then, one has

[mathematical expression not reproducible] (45)

the system being either stable if [mathematical expression not reproducible], or unstable if V([t.sub.i]) > V([t.sub.0]) > V([t.sub.0]) - [member of] and both situations are coherent with an energetic supply given to the environment on [[t.sub.0], t] satisfying the constraint [mathematical expression not reproducible].

(3) Note that the situations (1) and (2) can coexist within the same interval [[t.sub.0], 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 law [[sigma].sub.s] : [R.sub.0+] [right arrow] C with q finite or infinity eventual parameterizations C = P [union] NP, where the p([less than or equal to] q) configurations (or parameterizations) P ={[P.sub.i] : i [member of] [bar.p]} are passive and the (q - p) remaining ones NP = {N[P.sub.i] : i [member of] [bar.q - p]} are nonpassive, where P is empty if p = 0 and NP is empty if q = p. The switching law [[sigma].sub.s] : [R.sub.0+] [right arrow] C is defined by

[mathematical expression not reproducible], (46)

where [t.sub.i] is an abbreviate notation for [t.sup.+.sub.i] versus the explicit notation for the left limit [t.sup.-.sub.i] with

[mathematical expression not reproducible] (47)

being the set of switching time instants, which can be either of infinity (denumerable) cardinal [[chi].sub.0] 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, card(SW) [less than or equal to] [[chi].sub.0] and

[mathematical expression not reproducible]. (48)

If q < [infinity], it is possible to describe each configuration by positive integer numbers by assigning [P.sub.i] [right arrow] i and N[P.sub.j] [right arrow] j - p for i [member of] [bar.p] and j [member of] [bar.q] \ [bar.p]. Thus, the piecewise constant primary switching law [[sigma].sub.s] : [R.sub.0+] [right arrow] C is equivalently described in a simpler way by the piecewise constant switching law [sigma] : [R.sub.0+] x SW [right arrow] {1, 2, ..., p, p + 1, ..., q} such that [sigma](t) = [sigma]([t.sub.i]) = l ([member of] [bar.q]) if [[sigma].sub.s] ([t.sub.i]) = [C.sub.l] if l [less than or equal to] p or [[sigma].sub.s]([t.sub.i])) = [C.sub.l+i] if p < l [less than or equal to] q. Note that

[mathematical expression not reproducible] (49)

if [sigma]([t.sub.i]) [less than or equal to] p, that is, if the active configuration on the time interval [[t.sub.i], [t.sub.i+1]), [t.sub.i], [t.sub.i+1] [member of] SW is passive, where [epsilon]([t.sub.i]) [member of] [[summation].sub.P] = {[[epsilon].sub.i] : i [member of] [bar.p]} and 0 [less than or equal to] [[epsilon].sub.i] [less than or equal to] [epsilon]; [for all]i [member of] [bar.p], with

[mathematical expression not reproducible]. (50)

If p < [sigma]([t.sub.i]) [less than or equal to] q, that is, if the active configuration on the time interval [[t.sub.i], [t.sub.i+1]), [t.sub.i], [t.sub.i+1] [member of] SW is nonpassive, where [[epsilon].sub.([t.sub.i])] [member of] [[summation].sub.NP] = {[[epsilon].sub.i]: i [member of] [bar.p]} and [[epsilon].sub.i] [greater than or equal to] [epsilon]' > 0; [for all]i [member of] [bar.q] \ [bar.p].

Note that the whole set of switching time instants SW is

[mathematical expression not reproducible], (51)

where SW(k) = S[W.sub.P](k) U S[W.sub.NP](k), S[W.sub.P](k) [subset or equal to] S[W.sub.P] and S[W.sub.NP](k) [subset or equal to] S[W.sub.NP]; [for all]k [member of] [Z.sub.0+] are, respectively, the whole set of time instants ([t.sub.i] : i [member of] [bar.k] [union] {0} [subset or equal to] SW until [t.sub.k] [member of] SW for all k [member of] [Z.sub.0+] and its disjoint subsets associated with the active passive and nonpassive configurations which occurred in the interval [0, [t.sub.k]). Note that a system configuration is passive if [mathematical expression not reproducible] for some real function [??]: [R.sub.0+] [right arrow] [R.sub.0+] and some [epsilon] [member of] [R.sub.0+].

Assumptions 13. Under switching, a system configuration is assumed passive if for some [epsilon] [member of] [R.sub.0+]:

(1)

[mathematical expression not reproducible] (52)

with [??]([t.sub.i], [theta]) [member of] R; [for all][theta] [member of] [R.sub.0+] with [??]([t.sub.i], 0) [member of] [R.sub.0+] given by [mathematical expression not reproducible] for some function [[delta].sub.i]([theta]) [member of] [0, [infinity]); [for all][t.sub.i] [member of] S[W.sub.P].

(2) A system configuration of the switched law is nonpassive if for some [epsilon] [member of] [R.sub.0+]

[mathematical expression not reproducible] (53)

with [mathematical expression not reproducible] with [mathematical expression not reproducible] given by [mathematical expression not reproducible] for some constant [[epsilon].sub.0i] [member of] [R.sub.+] and some function [[delta].sub.i]([theta]) with [[delta].sub.i] (t) [right arrow] -[infinity] as [mathematical expression not reproducible].

The fact that [epsilon] 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 [epsilon]. The same value of [epsilon] is valid by reversing the inequality for nonpassive configuration since there are extra additive thresholds [[epsilon].sub.0(*)] to modulate possible discrepancies of the necessary constants for distinct nonpassive configurations. The intuitive physical interpretation of Assumptions 13 is as follows if [t.sub.0] = 0 [member of] S[W.sub.P], E(0, t) [greater than or equal to] -[epsilon]; [for all]t [member of] [R.sub.0+] as it follows for standard unswitched passivity. However, for [t.sub.i] (>0) [member of] S[W.sub.P] and small enough time [theta] [member of] [0, [T.sub.min]), it can happen that E([t.sub.i], [theta]) < -[epsilon] even if [t.sub.i-1] [member of] S[W.sub.P] 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 [theta] [greater than or equal to] [T.sub.min], the passivity property E([t.sub.i], [theta]) [greater than or equal to] -[epsilon]; [for all]t [member of] [R.sub.0+] is recovered. For large enough [theta], it is assumed that lim [sup.sub.[theta][right arrow][infinity]] [[delta].sup.i](t) [less than or equal to] [[delta].sup.*.sub.i] [member of] [0, [infinity])).

The following result holds.

Theorem 14. If Assumptions 13 hold a switched system is passive for all time under a switching law a : [R.sub.0+] x SW [right arrow] [1, 2, ..., p, p + 1, ..., q], with at least one configuration being active, if

(1) the first active configuration of the switching law on [[t.sub.0] = 0, [t.sub.1]) [member of] S[W.sub.P];

(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 [mathematical expression not reproducible] for some real function [??] : [R.sub.0+] [right arrow] [R.sub.0+] and some [epsilon] [member of] [R.sub.0+]. 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 [member of] (0, [t.sub.1])). Define the active switching passivity binary indicator function [mathematical expression not reproducible]. Direct calculations for any t [member of] [[t.sub.k], [t.sub.k+1]); [for all][t.sub.k] [member of] SW on the input-output energy yield

[mathematical expression not reproducible] (54)

holds via complete induction if [t.sub.0] (=0) [member of] S[W.sub.P] (i.e., the first configuration of the switched disposal is passive) provided that E(0, t - [T.sub.k]) [greater than or equal to] -[epsilon] for any t [member of] [R.sub.0+] (t - [T.sub.k]) [member of] [[T.sub.k-1], [T.sub.k1]) if the subsequent constraint holds:

[mathematical expression not reproducible]. (55)

The following cases can occur:

Case 1. [t.sub.k], [t.sub.k+1] [member of] S[W.sub.P]. Thus, [rho]([t.sub.k]) = 1 and

[mathematical expression not reproducible] (56)

which is guaranteed if

[mathematical expression not reproducible] (57)

or if

[mathematical expression not reproducible]. (58)

Case 2. [t.sub.k], [t.sub.k+1] [member of] S[W.sub.NP]. Thus, [rho]([t.sub.k]) = 0 and

[mathematical expression not reproducible]. (59)

Case 3. [t.sub.k] [member of] S[W.sub.NP], [t.sub.k+1] [member of] S[W.sub.P]. Thus, [rho]([t.sub.k]) = 0 and

[mathematical expression not reproducible] (60)

yields closely to Case 1:

[mathematical expression not reproducible]. (61)

7. Examples

Example 15. Consider the following damped linear system:

[mathematical expression not reproducible] (62)

with x(t) being the position, y(t) = [??](t) being the velocity, defined to be the system output, u(t) = F(t) being the external force, defined to be the control action, and k and [lambda] being the spring constant and the viscous coefficient, respectively. Define the state vector as z(t) = [(x(t), [??](t)).sup.T] and let V(z(t)) = (1/2)m[[??].sup.2](t) + (1/2)k[x.sup.2](t) = [V.sub.kin]([??](t)) + [V.sub.pot](x(t)) be the storage function defined to be the kinetic energy [V.sub.kin]([??](t)) plus the potential energy [V.sub.pot](x(t)). The input-output energy at time t [member of] [R.sub.0+] is

[mathematical expression not reproducible], (63)

where the dissipation function at time t is [mathematical expression not reproducible]. If [mathematical expression not reproducible] then [mathematical expression not reproducible] 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 y such that V(z(t)) - V(z(0)) + d(t) [greater than or equal to] -[[gamma].sup.2]; [[for all].sup.2] [member of] [R.sub.0+], that is,

[mathematical expression not reproducible], (64)

then the system is passive. In the case that the passive system is also globally asymptotically stable, z(t) [right arrow] 0, V(z(t)) [right arrow] V(0) = 0 as t [right arrow] [infinity] and 0 [less than or equal to] d(t) [less than or equal to] d(t') < +[infinity] for all t' [member of] [t, [infinity]); [for all]t [member of] [R.sub.0+] and [mathematical expression not reproducible]. Sufficient conditions are obtained by giving conditions on the stored energy to be a Lyapunov function for global asymptotic stability as follows:

[mathematical expression not reproducible] (65)

if the external force is generated via feedback as follows:

[mathematical expression not reproducible], (66)

where [zeta] : (R x [R.sub.0+]) x (R x [R.sub.0+]) x [R.sub.0+] [right arrow] [R.sub.0+] is a design function so that [zeta](x(t), [??](t), t) = 0 if and only if x(t) = 0. Thus, V(z(t)) [less than or equal to] V([z.sub.0]); [for all]t [member of] [R.sub.0+] and V(z(t)) [right arrow] 0 as t [right arrow] [infinity] 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

[mathematical expression not reproducible] (67)

if

[mathematical expression not reproducible]. (68)

If F(t) = [sigma](t)x(t) + [[sigma].sub.d][??](t) then [??](z(t)) [less than or equal to] 0 and global stability holds if [[sigma].sub.d] = [lambda] and [sigma](t) = k((m - l)/m) - [beta][x.sup.2](t); [for all]t [member of] [R.sub.0+] since the exact output linearization dynamics is achieved under such a control.

Example 16. Consider a nth order SISO time-invariant linear system whose transfer function is [??](s) = [c.sup.T][(s[I.sub.n] - A).sup.-1] b + d, where [A, b, [c.sup.T], d] is a state-space realization of [??](s), and the state n-vector has initial conditions x(0) = [x.sub.0]. The output is given by a superposition of the unforced (i.e., zero -input) and forced (i.e., zero state) solutions [y.sub.unf](t) and [y.sub.f](t) as

[mathematical expression not reproducible], (69)

where

[mathematical expression not reproducible]. (70)

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:

[mathematical expression not reproducible] (71)

since Im [??](i[omega]) = - Im [??](i[omega]) for any hodograph [??](i[omega]); [for all][omega] [member of] [R.sub.0+]. Assume that [??] [member of] {PR} so that [mathematical expression not reproducible] from (71). Assume also that the control is generated via negative output-feedback from any member [phi] [member of] {[PHI]} of nonlinear and perhaps time-varying class of controllers [O] which satisfies a Popov's type inequality, namely, u(t) = -[phi](y(t), t), where

[mathematical expression not reproducible] (72)

so that for any given positive real [bar.[gamma]] such that [bar.[gamma]] [greater than or equal to] [gamma] and any nonidentically zero controls, one gets

[mathematical expression not reproducible] (73)

so that for nonidentically zero controls

[mathematical expression not reproducible] (74)

The following result holds from (74).

Theorem 17. The following properties hold:

(i) Assume that [??] [member of] {SSPR}. Then, u, y [member of] [L.sub.2] and the system is globally asymptotically stable for any given finite initial conditions, irrespective of [phi] [member of] {[PHI]}, so that it is asymptotically hyperstable.

(ii) Assume that [??] [member of] {SPR} and that a frequency filter [??](i[omega]) is used for the control inputs so that [mathematical expression not reproducible] for any frequency interval, if any, such that min Re [??](i[omega]) = 0 and it is of unity gain otherwise. Then, u, y [member of] [L.sub.2] and the system is globally asymptotically stable for any given finite initial conditions, irrespective of [phi] [member of] {[PHI]}; then it is asymptotically hyperstable for the forward loop transfer function [mathematical expression not reproducible].

Proof. Since [??] [member of] {SSPR} so that d > 0 and [??] [member of] {SPR}, that is, it is strongly positive real then, in addition, in R[H.sub.[infinity]] (then with all its poles in Re s < 0) and having zero relative degree it follows in (74) that [mathematical expression not reproducible] can be made and the Fourier transforms used in the Parseval's identity exist. Assume that u(t) [right arrow] [+ or -][infinity] as t [right arrow] [infinity] then, there exists [alpha] [member of] [R.sub.+] such that

[mathematical expression not reproducible] (75)

since [u.sup.2](t)/u(t) [right arrow] and [e.sup.At] [right arrow] 0 as t [right arrow] 0. Therefore, u [member of] [L.sub.2], so that u(t) [right arrow] 0 as t [right arrow] [infinity] which contradicts that u(t) [right arrow] [+ or -][infinity] as t [right arrow] [infinity]. It can also be concluded that u(t) 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 [mathematical expression not reproducible] with [[??].sup.-1] [member of] {SSPR} and is realizable, since [??] [member of] {SSPR} so it has relative degree zero. Thus, y [member of] [L.sub.2] and converges asymptotically to zero for any [phi] [member of] {[PHI]}. Since G [member of] R[H.sub.[infinity]] 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

[mathematical expression not reproducible], (76)

where [mathematical expression not reproducible] since [[??].sub.j] (i[omega]) [member of] {PR} and [??](r) [subset or equal to] [bar.r] is the indicator set such that [d.sub.j] > 0 if and only if j [member of] [??](r). Thus, [u.sub.j] [member of] [L.sub.2], [u.sub.j] is bounded; then [u.sub.j](t) [right arrow] 0 as t [right arrow] [infinity] since [??] [member of] {SPR} for all j [member of] [??](r) and the total control input [mathematical expression not reproducible] is also bounded since [u.sub.j] is bounded by the filtering actions; [mathematical expression not reproducible] then Property (i) holds for this class of constrained control inputs.

Example 18. Consider a nth order SISO time-invariant linear system described by a finite or infinite set of transfer functions {[[??].sub.j](s) : j [member of] [bar.r]} governed by a switching law [sigma] : [[union].sub.k] [[t.sub.k], [t.sub.k+1]) [right arrow] [bar.r], with card[bar.r] [less than or equal to] [[chi].sub.0], which selects the active one on a time interval [[t.sub.k], [t.sub.k+1]) where SW = {[t.sub.k]} 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 [member of] ([t.sub.k+1], [t.sub.k+2]]. To simplify the exposition, consider the system under zero initial conditions. Then

[mathematical expression not reproducible] (77a)

[mathematical expression not reproducible], (77b)

where a truncated interswitching input [mathematical expression not reproducible] for t [member of] [[t.sub.a], [t.sub.b]] and [mathematical expression not reproducible] for t [not member of] [[t.sub.a], [t.sub.b]] has been defined and

[mathematical expression not reproducible], (78)

[mathematical expression not reproducible] (79)

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 [t.sub.j] [member of] 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 each [t.sub.j] [member of] SW. Assume also that the switching law satisfies that [mathematical expression not reproducible] and that [mathematical expression not reproducible] [member of] R[H.sub.[infinity]] and that, in the event that [mathematical expression not reproducible] {PR} for k [member of] [Z.sub.+], then [mathematical expression not reproducible], and

[mathematical expression not reproducible]. (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 [phi] [member of] {[PHI]}, (72), and that [mathematical expression not reproducible]; [for all][t.sub.j] [member of] SW with [mathematical expression not reproducible] and to zero state resetting at each switching time instant. Then, u, y [member of] [L.sub.2] and the system is globally asymptotically stable for any given finite initial conditions, irrespective of [phi] [member of] {[PHI]}, so that it is asymptotically hyperstable.

(iv) Assume also that the constraints [mathematical expression not reproducible] and zero state switching resetting invoked in Property (iii) but frequency filters [mathematical expression not reproducible] are used for the control inputs so that [mathematical expression not reproducible] for any frequency interval, if any, such that [mathematical expression not reproducible] and it is of unity gain otherwise for [t.sub.j] [member of] SW. Then, u, y [member of] [L.sub.2] and the system is globally asymptotically stable for any given finite initial conditions, irrespective of [phi] [member of] [PHI]; then it is asymptotically hyperstable for the forward loop transfer function [mathematical expression not reproducible].

Proof. Property (i) follows from (77a), (77b), and (78) and E(t) = [E.sub.f](t) [greater than or equal to] 0; [for all]t [member of] [R.sub.0+] under zero state resetting at each switching time instant since

[mathematical expression not reproducible] (82)

for any k [member of] [Z.sub.0+], any [t.sub.k] [member of] SW and any t [member of] ([t.sub.k], [t.sub.k+1]). Property (ii) by taking into account (77a) and [mathematical expression not reproducible] which guarantees that [E.sub.f](t) = E(t) > 0; [for all]t [member of] [R.sub.+] 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 E(t) = [E.sub.f](t) > 0; [for all]t [member of] [R.sub.+] 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 yield

[mathematical expression not reproducible]. (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 [mathematical expression not reproducible] 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.sub.(*)] is a Metzler matrix; d(*) [greater than or equal to] 0 and [c.sub.(*)] and [b.sub.(*)] 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].

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

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.

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[14] M. de la Sen and A. Ibeas, "On the global asymptotic stability of switched linear time-varying systems with constant point delays," Discrete Dynamics in Nature and Society, vol. 2005, no. 1, 67 pages, 2005.

[15] J. Tariboon, S. K. Ntouyas, and P. Agarwal, "New concepts of fractional quantum calculus and applications to impulsive fractional q- difference equations," Advances in Difference Equations, vol. 2015, article 18, 2015.

[16] M. De la Sen, "The generalized Beverton-Holt equation and the control of populations," Applied Mathematical Modelling, vol. 32, no. 11, pp. 2312-2328, 2008.

[17] M. De La Sen and S. Alonso-Quesada, "Model-matching-based control of the beverton-holt equation in ecology," Discrete Dynamics in Nature and Society, vol. 2008, Article ID 793512, 21 pages, 2008.

[18] M. De la Sen and S. Alonso-Quesada, "Control issues for the Beverton-Holt equation in ecology by locally monitoring the environment carrying capacity: non-adaptive and adaptive cases," Applied Mathematics and Computation, vol. 215, no. 7, pp. 2616-2633, 2009.

[19] M. De la Sen, "On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays," Applied Mathematics and Computation, vol. 190, no. 1, pp. 382-401, 2007

M. De la Sen (iD)

Institute of Research and Development of Processes (IIDP), University of the Basque Country, Campus of Leioa, PO. Box48940, Leioa, Bizkaia, Spain

Correspondence should be addressed to M. De la Sen; manuel.delasen@ehu.eus

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

Academic Editor: Viliam Makis
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Title Annotation:Research Article
Author:De la Sen, M.
Publication:Journal of Mathematics
Geographic Code:4EUGE
Date:Jan 1, 2018
Words:9504
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