# Asymptotic Hyperstability of a Class of Linear Systems under Impulsive Controls Subject to an Integral Popovian Constraint

Authors: M. De la Sen; A. Ibeas; S. Alonso-Quesada
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

This paper is focused on the study of the important property of the asymptotic hyperstability of a class of continuous-time dynamic systems. The presence of a parallel connection of a strictly stable subsystem to an asymptotically hyperstable one in the feed-forward loop is allowed while it has also admitted the generation of a finite or infinite number of impulsive control actions which can be combined with a general form of nonimpulsive controls. The asymptotic hyperstability property is guaranteed under a set of sufficiency-type conditions for the impulsive controls.


---

## Body

## 1. Introduction

The problems of absolute stability and hyperstability classically received much attention from the fifties in a wide class of problems of control theory and its applications, [1–10]. Some extensions or applications of hyperstability theory rely on the incorporation of the formalism of quantified controls [3] and multiple nonlinearities, [4]. Also, in some of the studies, the incorporation of extra subsystems to the linear positive real part, the presence of delays, and that of hybrid continuous-discrete systems has also been considered, [5–7]. Its extension to some models based on neural networks has been studied in [9] and references therein, while its usefulness for the design of adaptive schemes with good stability and transient properties was proposed in a set of results compiled in [10] and in more exhaustive later investigations performed along the eighties and nineties. The properties of parametrical stability of nonlinear systems with some robustness extensions have been investigated in [11, 12] and references therein. Also, applications of stability theory to base-isolated building structures including modelling delays and wireless control issues has been investigated in a number of papers. The main objective of the studies is to prevent the structures via active control designs against earthquake or heavy wind potential damages. See, for instance, [13, 14] and references therein. In those studies, Lyapunov stability and associated appropriate manipulations via linear matrix inequalities are used ad hoc for the problems at hand.Furthermore, the following Figures3 and 4 show, respectively, the output and control input to the system.The problem of absolute stability is basically stated as that of guaranteeing the global asymptotic stability of a control system consisting of a linear plant under a feedback nonlinear regulator which belongs to a certain class of regulators all satisfying either some sector-type or some integral-type constraints of Lurie’s or Popov’s type, respectively. See, for instance, [2–4, 15–17]. If global asymptotic stability is guaranteed for the whole class of regulators then it is typically said that the closed-loop system is absolutely stable for such a class of controllers. The main reason for setting this theory was its ability to guarantee the stabilization for a whole class of systems under potential dispersion of the values of the parameters in a serial controller construction process while keeping its stabilization capability. Thus, it is a robustly stable design for a characterization of the nonlinear feedback controller as belonging to a certain admissible class. Then, the absolute stability property is guaranteed for the feed-forward controlled object and the whole class of nonlinear controllers.The more general problem of asymptotic hyperstability extends that of absolute stability to classes of nonlinear regulators satisfying integral-type constraints for eventually time-varying nonlinearities which are not necessarily constrained to sectors. The asymptotic hyperstability property of a closed-loop system requires, in particular, that the feed-forward loop consists of a positive real transfer matrix (sometimes referred to as the asymptotic hyperstability of the linear feed-forward loop) or function, and that the nonlinear/time-varying controller belongs to a class satisfying an integral-type constraint, the so-called Popovian integral inequality, [15, 16]. Then, the property is guaranteed for such a linear feed-forward linear block and the whole class of nonlinear/time-varying controllers. Historically, the properties of absolute stability and hyperstability were of crucial interest by the time of World War II because the component dispersion in regulators controlling tanks often fail the blank targeting process, [16]. Such a practical problem motivated the convenience of formulating and designing stabilizing classes of regulators rather than individual stabilizing ones, [15, 16]. The related studies on this subject have been performed for the stabilization of both linear continuous-time and discrete dynamic systems.Asymptotic hyperstability has also received a very important attention in adaptive control designs such as model reference adaptive control or model reference adaptive systems. See, for instance, [17–25] and references therein. A wide representation of practical problems has been focused on this theoretical formalism including theoretical design of adaptive systems, adaptive control of manipulators and other robotic devices, adaptive neural networks, and synchronization. A main reason of the success of the hyperstability formal theory in the existing background of adaptive designs relies on the fact that the implementation of the estimation algorithms is flexible while offering the designer the chance of choosing wide ranges of admissible values of the free design parameters being compatible with accomplishing the required properties of convergence and closed-loop stabilization.Most of the stability problems can be extended to the use of switching laws acting on the controlled objects or on their controllers so that the dynamics can be changed to improve certain suitable performances like, for instance, suited settling times or transient errors, or to accommodate the system behavior to suitable transients around several operation points. This frequently happens in complex dynamic problems associated to distinct phases in some production chains like it occurs, for instance, in some chemical engineering processes. Some of such complex models might include point or distributed delays and/or hybrid structures consisting of continuous-time, discrete-time, and digitally modelled subsystems. The existing background literature on switching dynamics is exhaustive. See, for instance, [26–29] and references therein. Finally, recent work on absolute stability and hyperstability of complex nonlinear models, including absolute stability of hyperbolic systems and topics related to Ulam’s-type stability, has been reported in [30–32] and some of the references therein.Other topics of increasing interest in the last years and nowadays in stabilization and control of dynamic systems are that of the synthesis of impulsive controls and that of the switching conditions of the dynamics guaranteeing the closed-loop stabilization. In particular, the injection of impulsive controls lead to a discontinuous sudden change (in practice, in a very short time interval) in the state vector at impulsive time instants. Switched dynamic systems have several possible parameterizations while switches among them in such a way that the dynamic system is parameterized by one of such parameterizations in connected time-intervals before the next switching happens. It is of interest the design of switching laws which ensure closed-loop stabilization either conditionally, via appropriate rules either on the minimum residence time or on the averaging dwelling times, or under arbitrary switching. Exhaustive work has been made in this field. See, for instance, [26–30, 33–39] and references therein. Related studies are of interest, for instance, in synchronization, deterministic, and stochastic stabilization, impulsive vaccination in epidemic models, control of chemical process under their various dynamics, and so forth. Switched dynamic systems and impulsive control also appear in some combined problems. See, for instance, [27, 38–40] and references therein.This paper is focused on an extended study of asymptotic hyperstability in the continuous-time framework for a single-input single-output system. Basically, the undertaken generalizations are:(a)
it has admitted the presence of a parallel connection of a linear strictly proper and strictly stable subsystem to the standard linear asymptotically hyperstable subsystem in the feed-forward loop of the feedback system;(b)
the generation of a finite or infinite number of impulsive control actions is allowed while it can be combined with a general form of nonimpulsive controls. The control impulses can be state/output dependent or not.The asymptotic hyperstability property is guaranteed to hold under sufficiency-type conditions for the above proposed general structure. The property is guaranteed under conditions being dependent on certain combined constraints. Such constraints involve the values of the gain of the feed-forward strictly positive real transfer function at infinity, the maximum value of the modulus of its parallel-connected linear block, and the sign and gain bounds of the different impulses of the impulsive control. The paper is organized as follows. Section 2 is devoted to the problem statement and some preliminary results while the main asymptotic hyperstability results are stated and commented in Section 3. The proofs are given in Appendices. Section 4 considers some extensions for robust hyperstability due to the presence of state and measurement disturbances. Section 5 presents some illustrative simulated examples and, finally, conclusions end the main body of the paper.


## 2. Problem Statement and Some Preliminary Results

Consider the following linear and time-invariant system:(1)x˙(t)=Ax(t)+buim(t),x(0)=x0,(2)uim(t)=-φ(y(t),t),(3)y(t)=cTx(t)+d0uim(t),
which is subject to a potentially nonlinear time-varying controller device satisfying the Popovian type integral inequality below:
(4)∫0tφ(y(τ),τ)y(τ)dτ≥-γ2>-∞,∀t∈R0+,
wherex(t)∈Rnis then-state vector, andy(t)and uim(t) are the output and the (in general, impulsive) control which satisfies the Popovian integral constraint∫0tφ(y(τ),τ)y(τ)dτ≥-γ2>-∞forallt∈R0+and any nonzero  γ∈R. The, in general, nonlinear functionφ(y(t),t)used to generate the control input, subject to the integral constraint (4), defines a class of regulators via such a constraint which is admissible to control the linear plant (1) and (3). Thus, the hyperstability property to be addressed in the paper ensures the closed-loop stability for any member of such a class of regulators. Note that if (4) holds for any nonzero  γthen it also holds for anyγ′≥γ. The transfer function of the linear feed-forward loop (1)–(3) is decomposed in parallel blocks as follows:(5a)G(s)=cT(sI-A)-1b+d0=G*(s)+G~(s),(5b)G*(s)=c*T(sI-A*)-1b*+d*,where(i)
G~(s)is both strictly stable and strictly proper with poles in Res≤-σ<0for someσ∈R+and satisfying ReG~(iω)≥-K~>-∞for some nonnegative real constantK~  and allω∈R0+. Note that its real part is finitely lower-bounded since it is strictly stable.(ii)
G*(s)  is strictly positive real withd*=d0=limω→±∞G(iω)=limω→±∞G*(iω)>0, sincelimω→±∞G~(iω)=0, wherei=-1is the complex unit andc*T(sI-A*)-1b*is strictly proper and strictly stable sinceG*(s)is strictly positive real andd0is a positive real constant.Note thatG^*(s)=G*(s)-d0=c*T(sI-A*)-1b*is strictly proper and strictly stable sinceG*(s)is of zero relative degree (i.e., biproper) sinced*is nonzero. It is well-known that a transfer functionG*(s)  is strictly positive (resp., positive real) real if ReG(s)>0 (resp., ReG(s)≥0) for any complex number  s  with Res≥0(resp., for Res>0), and then, it has a relative degree equal to 0 or 1. The sets of strictly positive real and positive real transfer functions are simply denoted in the sequel by SPR and PR, respectively. Strictly positive real transfer functions are strictly stable while the positive real transfer functions are stable. Note that the set of positive real functions contains that of strictly positive real ones. If the transfer functions are realizable, then it is evident that they have relative degrees (also often referred to as relative orders, equivalently, and their pole-zero excess number) being 0 or 1. Now, assume that the impulsive control is of the following form:
(6)uim(t)=u(t)+∑j=1∞Kjg(tj)δ(t-tj),∀t∈R0+,
whereu(t)is the piecewise continuous impulse-free input,δ(t)is the Dirac distribution describing the impulsive actions of amplitudesKjg(tj)whereg(tj)=1 if the input impulse is independent of the output and g(tj)=y(tj)   in the case that it depends directly on such an output. The sequence{g(tj)}  can be, in general, any nonzero function sequence being dependent on the output and input sequences{y(tj)}and{u(tj)}  at the impulsive time instants. A typical case in many practical situations is that the input impulse is  u(tj+)=u(tj)+Kjy(tj), that is, g(tj)=y(tj). In the following, the subsequent notation is used:(i)
ft1,t2(τ)=f(t)ifτ∈[t1,t2)  andft1,t2(τ)=0, otherwise, it is a truncated function off(t)(f0,t(τ)≡ft(τ)  is used for notation abbreviation);(ii)
Im(t)={t1,t2,…,tp(t)}is the set of impulsive time instants until timet  with the conventiontp(t)<tso thatIm(t+)={t1,t2,…,tp+1(t)=t}  if  t, itself, is an impulsive time instant which is denoted byt∈Im(t+). Note that the convention implies that ift∉Im(t+)then Im(t+)=Im(t)and that  tk∉Im(tk). Let us also denoteIm∞≡⋃t∈R0+Im(t)as the whole set of impulsive time instants and cardIm∞ is its cardinal. The notationcardIm∞=χ0    (the usual symbol for an infinite cardinal of a numerable set) refers to the case when the set of impulses is infinity while cardIm∞<χ0indicates that there is a finite number of impulses.The notation “support sequence” for{g(tj)},tj∈Im∞  for some impulsive setIm∞is often used by obvious reasons for such a sequence  {g(tj)}.The limit notationlimt∈[tj,tj+1),tj→∞f(t)used in some of the obtained results stands for the limits as time tends to infinity within the interimpulse time intervals provided that they exist. In this context, note that (6) is very general so that the impulses can be state-dependent or not. Thus, such limits do not always exist as time tends to infinity to the left of the impulsive time instants. This motivate the use of such a notation.The following definitions are well-known from the background literature for the impulsive-free caseuim(t)=u(t) in (6).Definition 1.
The closed-loop system (1)–(4) is said to be asymptotically hyperstable (resp.,  hyperstable) for the impulsive-free case  uim(t)=u(t) for all t∈R0+  if it is globally asymptotically stable (resp., globally stable) for any control device satisfying Popov’s integral inequality∫0tφ(y(τ),τ)y(τ)dτ≥-γ2>-∞forallt∈R0+and any nonzeroγ∈R+.It is well-known that the system (1)–(4) is asymptotically hyperstable if and only if its associate transfer functionG(s)∈SPR. Note that ifG(s)is strongly strictly positive real, sinceG~(s)is strictly proper, then
(7)d0=limω→±∞ReG(iω)>0,d=minω∈R0+ReG(iω)=d0+minω∈R0+Re(cT(iωI-A)-1b)>0.The following preliminary technical results to be then used to establish the main asymptotic hyperstability results hold.Lemma 2.
The input-output energy measure on the time intervals[0,t), respectively, on  [0,t],E(t)=∫0ty(τ)uim(τ)dτ, respectively, on[0,t] is uniformly bounded for all time while it satisfies the following relationships:(8)γ2≥E(t)≥(d-K~)∫-∞∞ut2(τ)dτ=(d-K~)∫0tu2(τ)dτ+∑tj∈Im(t)Kjg(tj)y(tj),∀t∈R+,(9)γ2≥E(t+)=E(t)+∫tp(t)t+yt(τ)u(τ)dτ≥(d-K~)∫0tu2(τ)dτ+∑tj∈Im(t+)Kjg(tj)y(tj),∀t∈R+,
whered=minω∈R0+ReG(iω).Proof.
See AppendixA.A useful result based on Lemma2 follows below so as to get the main stability result after establishing the subsequent definition.Definition 3.
Any sequence of impulsive gains{Kj}  subject to the following constraints:
(10)γ2-(d-K~)∫0tku2(τ)dτ-∑tj∈Im(tk)Kjg(tj)y(tj)≥Kkg(tk)y(tk)≥m(tk)-(d-K~)∫0tku2(τ)dτ-∑tj∈Im(tk)Kjg(tj)y(tj),
for some given real functionm(t), which satisfies0<m(t)<γ2forallt∈R+, and anytk∈Im∞≡⋃t∈R0+Im(t)such thaty(tk)≠0, is said to be of the classIgain=Igain(Im∞,γ,g)for some givenγ∈R+ and for the support sequence{g(tj)}, tj∈Im∞.The feature that if (4) holds for any nonzeroγthen it also holds for anyγ′≥γ  makes that the particular value of any finite nonzero γ in (10) can be increased to upper-bound the admissible impulsive gains for different controls while keeping the validity of the formulated asymptotic hyperstability theorem for the impulsive case. See Theorem 10 in Section 3.Definition 4.
An integral Popovian inequality (4) for a nonzero real constantγ  and an impulsive control is said to be impulsive control-compatible if the sequence of impulsive gains{Kj}is of the classIgain(Im∞,γ,g)for some  γ∈R+.Lemma 5.
Assume that the sequence of impulsive gains for system (1)–(4) is of the classIgain=Igain(Im∞,γ,g), then, the following properties hold:
(11)|d-K~|∫0tu2(τ)dτ<∞,|∑tj∈Im(t)Kjg(tj)y(tj)|<∞,∀t∈R+,
if cardIm∞<χ0, and
(12)∞>γ2≥(d-K~)∫0tu2(τ)dτ+∑tj∈Im(t)Kjg(tj)y(tj)>0,∀t∈R+,
ifcardIm∞≤χ0 so that the input-output energy is positive and uniformly upper-bounded by a finite bound for all nonzero time.Proof.
See AppendixB.Remark 6.
It is then proven in Theorem10 that (11) holds even ifcardIm∞=χ0and d=K~.Lemma5 is subject to the subsequent important necessary condition.Lemma 7.
Assume thatd<K~then a necessary condition for Lemma 5 to hold is that the classIgain=Igain(Im∞,γ,g)is such that  t1=0∈Im(t)⊂Im∞forallt∈R0+, and its associate impulsive gain satisfies the constraint:
(13)|d-K~|∫0t1u2(τ)dτ-γ2g(t1)y(t1)≤K1≤γ2+|d-K~|∫0t1u2(τ)dτg(t1)y(t1).
Another necessary condition for Lemma 5 to hold is that the classIgain=Igain(Im∞,γ,g)has an impulsive set of time instants Im∞of infinite cardinal.Proof.
Proceed by contradiction by assuming thatd<K~,Im∞∋t1>0, and Lemma 5 holds so that (12) holds. Then,
(14)∞>γ2≥-|d-K~|∫0t1u2(τ)dτ+K1g(t1)y(t1)>0,∞>γ2≥0>-|d-K~|∫0t1u2(τ)dτ>0,∀t∈[0,t1),
and the previous second relationship is clearly a contradiction tot1>0. On the other hand, if there is a finite number of impulsive time instants then the following contradiction follows from (12):
(15)∞>γ2≥limt→∞(d-K~)∫0tu2(τ)dτ+∑tj∈Im(t)Kjg(tj)y(tj)=∞.
Hence, the proof of the result is complete.To the light of Definition3, the hyperstability concepts of Definition 1 are extended so as to consider also certain classes on impulsive controls which lead to the global asymptotic stability of the closed-loop system as follows.Definition 8.
The impulsive closed-loop system (1)–(4) is said to be impulsive asymptoticallyIgain(Im∞,γ,g)-hyperstable (resp., impulsiveIgain(Im∞,γ,g)-hyperstable) if it is globally asymptotically stable (resp., globally stable) for any sequence of impulsive gains{Kj}of the classIgain=Igain(Im∞,γ,g) for a given nonzeroγ∈R+, a given support sequence{g(tj)},tj∈Im∞, and any control device satisfying an impulsive control compatible Popov’s integral inequality∫0tφ(y(τ),τ)y(τ)dτ≥-γ2>-∞forallt∈R0+.


## 3. Main Results on Asymptotic Hyperstability

It turns out that Lemma7 establishes that the origin has to be an impulsive time instant to keep the input-output energy to be positive and upper-bounded for all time. On the other hand, Lemma 5 leads to the subsequent main result of usefulness in the impulsive-free case.Theorem 9.
Assume thatIm∞=∅  (i.e., there are no impulsive controls). Then, the following properties hold.(i)
Assume also thatG*(s)∈SPRandd>K~. Then,∞>E(t)≥0for  allt∈R0+withE(0)=0, and the closed-loop system is asymptotically hyperstable for any nonlinear feedback deviceφ(y(t),t)satisfying Popov’s integral inequality∫0tφ(y(τ),τ)y(τ)dτ≥-γ2>-∞for  allt∈R0+. As a result,G(s)∈SPR,limt→∞u(t)=0,u(t), y(t)andx(t)are uniformly bounded for allt∈R0+, andlimt→∞y(t)=0andlimt→∞x(t)=0for  allx0∈Rn.(ii)
Assume thatReG*(iω)>0,ReG~(iω)>-ReG*(iω) for all ω∈R0+, andd=infω∈R0+ReG(iω)=limω→±∞ReG*(iω)=0.Then, Lemma 5 holds and the closed-loop system is asymptotically hyperstable for any nonlinear feedback deviceφ(y(t),t)satisfying Popov’s integral inequality ∫0tφ(y(τ),τ)y(τ)dτ≥-γ2>-∞ for all t∈R0+. The, limt→∞u(t)=0, u(t), y(t)andx(t)are uniformly bounded for all t∈R0+, and  limt→∞y(t)=0and  limt→∞x(t)=0 for all x0∈Rn.Proof.
It is given in AppendixC.Theorem9(i) addresses the asymptotic hyperstability for the impulsive-free case whenG*(s)∈SPRis strongly positive real sinceG*(iω)≥d>0forallω∈R0+. Theorem 9(ii) considers the same property for the case whenG*(s)∈SPRis weakly positive real sinceG*(iω)>0andd=limω→±∞G*(iω)=0, that is, the transfer function is strictly proper so that it is zero at infinity frequency. Lemma 5 leads to the subsequent main result of usefulness in the control impulsive case.Theorem 10.
Assume thatcardIm∞≤χ0 (i.e., there are impulsive controls withχ0being the infinity cardinal of a denumerable set) for any set of control impulses of amplitudes satisfying (10). Then, the following properties hold.(i)
Assume also thatG(s)∈SPRwith  d>K~. Then, ∞>E(t)≥0 for all t∈R0+withE(0)=0, and the closed-loop system is impulsive asymptoticallyIgain(Im∞,γ,g)-hyperstable for any nonlinear feedback deviceφ(y(t),t)satisfying a compatible Popov’s integral inequality∫0tφ(y(τ),τ)y(τ)dτ≥-γ2>-∞ for all t∈R0+. As a result,y(t) andx(t)are uniformly bounded for allt∈R0+ for all x0∈Rn,u(t)is bounded fort∈R0+∖Im∞, with  limt∈[tj,tj+1),tj→∞u(t)=0ifcardIm∞=χ0(i.e., there are infinitely many control impulsive time instants); andlimt→∞u(t)=0ifcardIm∞<χ0, that is, if there is a finite number of control impulsive time instants. Also
(16)∑tj∈Im(t)Kjg(tj)y(tj)≤γ2<∞,∑tj∈Im(t+)Kjg(tj)y(tj)≤γ2<∞,∀t∈R0+.(ii)
Assume thatReG*(iω)>0,  ReG~(iω)>-ReG*(iω) for all ω∈R0+andd=infω∈R0+ReG(iω)=limω→±∞ReG*(iω)=0.Then, Lemma 5 holds and the closed-loop system is asymptotically hyperstable for any nonlinear feedback deviceφ(y(t),t)satisfying Popov’s integral inequality  ∫0tφ(y(τ),τ)y(τ)dτ≥-γ2>-∞ for all t∈R0+. Then,y(t)  andx(t)are uniformly bounded for allt∈R0+, for all x0∈Rn,u(t)is bounded fort∈R0+∖Im∞, with limt∈[tj,tj+1),tj→∞u(t)=0  ifcardIm∞=χ0 (i.e., there are infinitely many control impulsive time instants); and  limt→∞u(t)=0,limt→∞x(t)=0and  limt→∞y(t)=0ifcardIm∞<χ0 (i.e., there is a finite number of control impulsive time instants) or ifcardIm∞=χ0(i.e., there are infinitely many control impulsive time instants) and Kj→0as j→∞. Also, the boundedness properties (16) for the impulsive contributions still hold.Proof.
It is given in AppendixD.Note that the constraint (10) invoked in Theorem 10 is guaranteed by amplitude constraints of each impulse attkdepending only on input and output values on[0,tk)for the case of signal-dependent control impulses.


## 4. Some Extensions Concerning Robustness

Assume that (1) is modified in the presence of state and measurement disturbancesη(t)and  λ(t), of respective Laplace transformsN(s)and  Λ(s), as follows:
(17)x˙(t)=Ax(t)+buim(t)+η(t),x(0)=x0,y(t)=cTx(t)+d0uim(t)+λ(t).
Taking Laplace transforms under zero initial conditions in (17), one gets
(18)Y(s)=G(s)Uim(s)+cT(sI-A)-1N(s)+Λ(s)=cT(sI-A)-1(bUim(s)+N(s))+d0Uim(s)+Λ(s)=(G*(s)+G~(s))Uim(s)+cT(sI-A)-1N(s)+Λ(s)=(c*T(sI-A*)-1b*+d0)Uim(s)+G~(s)Uim(s)+cT(sI-A)-1N(s)+Λ(s),
whereAandA* are stability matrices withG*(s)=c*T(sI-A*)-1b*+d*∈SPR(then also strictly stable biproper ifd0=d*>0and strictly proper ifd0=0), andG~(s)being strictly stable and strictly proper. Now, the results of Sections 2 and 3 can be reformulated for this case with the replacement of the nominal outputy(t)by a modified output calculated as follows:
(19)y(t)⟶y(t)+y-(t),y-(t)=y-(η,λ,t)=∫0tg0T(t-τ)η(τ)dτ+λ(t),
whereg0T(t)is the impulse response ofcT(sI-A)-1. Define the robust impulsive class and its associated robust impulsive Irgain=Irgain(Im∞,γ,g,η,λ)-hyperstability and asymptotic Irgain=Irgain(Im∞,γ,g,η,λ)-hyperstability as follows.Definition 11.
Any sequence of impulsive gains{Kj}subject to the following constraints:
(20)γ2-(d-K~)∫0tku2(τ)dτ-∫0tky-(η,λ,τ)u(τ)dτ-∑tj∈Im(tk)Kjg(tj)y(tj)≥Kkg(tk)y(tk)≥m(tk)-(d-K~)∫0tku2(τ)dτ-∫0tky-(η,λ,τ)u(τ)dτ-∑tj∈Im(tk)Kjg(tj)y(tj),
for some given real functionm(t), which satisfies0<m(t)<γ2forallt∈R+, and anytk∈Im∞≡⋃t∈R0+Im(t), such thaty(tk)≠0, is said to be of the robust impulsive classIrgain=Irgain(Im∞,γ,g,η,λ)for some givenγ∈R+ and for the support sequence{g(tj)},tj∈Im∞.Definition 12.
The impulsive closed-loop system (17), subject to (2), is said to be robustly impulsive asymptoticallyIrgain(Im∞,γ,g,η,λ)-hyperstable (resp., impulsiveIrgain(Im∞,γ,g,η,λ)-hyperstable) if it is globally asymptotically stable (resp., globally stable) for any sequence of impulsive gains  {Kj}of the classIrgain(Im∞,γ,g,η,λ)for a given nonzero  γ∈R+, a given support sequence{g(tj)},tj∈Im∞  and any control device satisfying an impulsive control compatible Popov integral inequality∫0tφ(y(τ),τ)y(τ)dτ≥-γ2>-∞;forallt∈R0+.Now, the asymptotic hyperstability Theorem10 is directly extended for robust asymptotic hyperstability as follows.Theorem 13.
Assume thatcardIm∞≤χ0  for any set of control impulses of amplitudes satisfying (20). Assume also thaty-(t)is uniformly bounded for all  t∈R0+  (it suffices thatλ(t)  andη(t)be bounded for all time). Then, the following properties hold.(i)
Assume also thatG(s)∈SPRwith  d>K~. Then,∞>E(t)≥0 for all t∈R0+ with E(0)=0, and the closed-loop system is robustly impulsive asymptoticallyIrgain(Im∞,γ,g,η,λ)-hyperstable for any nonlinear feedback device φ(y(t),t) satisfying a compatible Popov’s integral inequality ∫0tφ(y(τ),τ)y(τ)dτ≥-γ2>-∞ for all t∈R0+. As a result,y(t)  andx(t)are uniformly bounded for allt∈R0+ for all x0∈Rn, u(t)is bounded fort∈R0+∖Im∞, with  limt∈[tj,tj+1),tj→∞u(t)=0 ifcardIm∞=χ0(i.e., there are infinitely many control impulsive time instants); and  limt→∞u(t)=0 if cardIm∞<χ0  (i.e., there is a finite number of control impulsive time instants).(ii)
Assume thatReG*(iω)>0,  ReG~(iω)>-ReG*(iω) for all ω∈R0+andd=infω∈R0+ReG(iω)=limω→±∞ReG*(iω)=0.
Then, Lemma5 holds and the closed-loop system is robustly asymptotically hyperstable for any nonlinear feedback device φ(y(t),t) satisfying Popov’s integral inequality ∫0tφ(y(τ),τ)y(τ)dτ≥-γ2>-∞ for all t∈R0+. Then, y(t)andx(t)are uniformly bounded for allt∈R0+, for all x0∈Rn, u(t) is bounded for t∈R0+∖Im∞, with limt∈[tj,tj+1),tj→∞u(t)=0 if cardIm∞=χ0 (i.e., there are infinitely many control impulsive time instants); and  limt→∞u(t)=0, limt→∞x(t)=0and limt→∞y(t)=0  if cardIm∞<χ0  (i.e., there is a finite number of control impulsive time instants) or if cardIm∞=χ0 (i.e., there are infinitely many control impulsive time instants) andKj→0 as j→∞.Proof (outline).
It is given in AppendixE.


## 5. Numerical Examples

This section discusses some simulation examples illustrating the theoretical results stated in the previous ones. Two different examples will be provided. The first one corresponds to the impulsive-free case while the second one considers the effect of feedback impulses. The following transfer functions are used:(21)G*(s)=s+3s2+7s+10,G~(s)=2s2+6s+9.G*(s)is SPR since it is stable andReG*(jω)=(4ω2+30)/(ω4+29ω2+100)>0  for ω≥0. On the other hand, G~(s)is stable but is not SPR since its relative degree is two. Therefore, G(s)=G*(s)+G~(s)is a parallel connection of an SPR and a non-SPR transfer function as discussed in Section 2. In addition, a family of nonlinear feedback functions is given byφi(y,t)=tanh(γiy), whereγi∈{0.2,0.6,1.4}. These nonlinear functions satisfy the integral-type Popov’s constraint specified by (4) since tanh(γiy)y>0for y≠0, and tanh(γiy)y=0for y=0, implying that∫0ttanh(γiy)ydt≥0for anyt>0and alli=1,2,3. The next Section 5.1 shows the asymptotic hyperstability of the closed-loop for the SPR transfer functionG*(s)and for the non-SPR transfer functionG(s)in the absence of impulses. The state variables are always defined from the (minimal) canonical controllable state-space realizations of the transfer functions with the first state component being the output. The initial conditions are in all the casesx1(0)=1,x2(0)=2.5for the realization ofG*(s), andx1(0)=1,x2(0)=2.5,x3(0)=-1,x4(0)=2for the realization ofG(s). We point out that the hyperstability properties of strictly positive real transfer functions are independent of the chosen realization and on the fact that it is minimal or not. Note that, if the realization is nonminimal, it has to be stable and any potential zero-pole cancellations do not modify the frequency plots and then the positive realness properties and minimum gain values.


### 5.1. Asymptotic Hyperstability in the Impulsive-Free Case

Figures1 and 2 show the evolution of the state variables of the SPR systemG*(s)when members of the family of nonlinear static feedback devicesφi(y,t)=tanh(γiy) are used as controllers.Figure 1
Evolution of the state variablex1(t).Figure 2
Evolution of the state variablex2(t).Figure 3
System output.Figure 4
Control inputu(t)=-φ(y,t).Figures1–4 confirm the results of Theorem 9(i) guaranteeing the boundedness and convergence to zero of the state, the output and the control signal. Moreover, it can be appreciated in Figures 1–3 that the evolution of the system is very similar for the different considered values ofγi. This similar evolution is obtained at the expense of a different control action for eachγi, as Figure 4 depicts. On the other hand, the following Figures 5, 6, and 7 show the state evolution, the output and the control input for the non-SPR systemG(s). The particular value ofγi=0.6has been used in Figures 5 and 6 since the evolution of the system is very similar for all values ofγi. However, Figure 7 shows the control input for different values ofγisince its value varies depending onγi. Figures 5–7 confirm the results of Theorem 9(ii) guaranteeing the convergence to zero of the state, the output, and the control signal. The next Section 5.2 is devoted to the study of the impulsive control.Figure 5
Evolution of the state variables.Figure 6
Output of the system.Figure 7
Control input to the system,u(t)=-φ(y,t).


### 5.2. Asymptotic Hyperstability in the Presence of Impulses

A number of impulses are added to the nonlinear feedback controller introduced in Section5.1. The support sequence is given byg(tk)=y(tk)and the impulsive gains constantKk=1. The impulsive time instants are multiples of 0.2 seconds; this means that the presence of impulses is periodic. The valueγi=0.6has been used for the sake of simplicity. Figure 8 depicts the evolution of the state variables of the SPR transfer functionG*(s)while Figure 9 shows the output. Finally, Figure 10 illustrates the control input with the impulsive action.Figure 8
Evolution of the state variables in the presence of impulses.Figure 9
System output in the presence of impulses.Figure 10
Control input with impulses.Figures8–10 confirm the statements of Theorem 10(i) showing the convergence to zero of the state, the output, and the control signal. In addition, these figures explicitly show the presence of impulses in different signals of the system. On the other hand, for the complete system,G(s), we obtain the following Figures 11, 12, 13, and 14 depicting the state evolution, the output, and the control input.Figure 11
Evolution of the state componentx1(t).Figure 12
Evolution of the state componentsx2(t),x3(t), andx4(t).Figure 13
System output.Figure 14
Control input with impulses.Figures11–14 illustrate the results compiled in Theorem 10(ii) since they show the convergence of the state, the output and the input to zero. Moreover, it can be appreciated the effect of adding impulses to the control signal by first comparing Figures 3 and 9 and, on the other hand, Figures 6 and 13. It can be deduced that the use of an impulsive controller makes the output to converge to zero faster than when a single continuous-time one is used. This fact highlights how the concept of impulsive asymptotic hyperstability is relevant since it allows ensuring the closed-loop stability for a whole family of nonlinear devices while accelerating the convergence of the output to zero.


## 5.1. Asymptotic Hyperstability in the Impulsive-Free Case

Figures1 and 2 show the evolution of the state variables of the SPR systemG*(s)when members of the family of nonlinear static feedback devicesφi(y,t)=tanh(γiy) are used as controllers.Figure 1
Evolution of the state variablex1(t).Figure 2
Evolution of the state variablex2(t).Figure 3
System output.Figure 4
Control inputu(t)=-φ(y,t).Figures1–4 confirm the results of Theorem 9(i) guaranteeing the boundedness and convergence to zero of the state, the output and the control signal. Moreover, it can be appreciated in Figures 1–3 that the evolution of the system is very similar for the different considered values ofγi. This similar evolution is obtained at the expense of a different control action for eachγi, as Figure 4 depicts. On the other hand, the following Figures 5, 6, and 7 show the state evolution, the output and the control input for the non-SPR systemG(s). The particular value ofγi=0.6has been used in Figures 5 and 6 since the evolution of the system is very similar for all values ofγi. However, Figure 7 shows the control input for different values ofγisince its value varies depending onγi. Figures 5–7 confirm the results of Theorem 9(ii) guaranteeing the convergence to zero of the state, the output, and the control signal. The next Section 5.2 is devoted to the study of the impulsive control.Figure 5
Evolution of the state variables.Figure 6
Output of the system.Figure 7
Control input to the system,u(t)=-φ(y,t).


## 5.2. Asymptotic Hyperstability in the Presence of Impulses

A number of impulses are added to the nonlinear feedback controller introduced in Section5.1. The support sequence is given byg(tk)=y(tk)and the impulsive gains constantKk=1. The impulsive time instants are multiples of 0.2 seconds; this means that the presence of impulses is periodic. The valueγi=0.6has been used for the sake of simplicity. Figure 8 depicts the evolution of the state variables of the SPR transfer functionG*(s)while Figure 9 shows the output. Finally, Figure 10 illustrates the control input with the impulsive action.Figure 8
Evolution of the state variables in the presence of impulses.Figure 9
System output in the presence of impulses.Figure 10
Control input with impulses.Figures8–10 confirm the statements of Theorem 10(i) showing the convergence to zero of the state, the output, and the control signal. In addition, these figures explicitly show the presence of impulses in different signals of the system. On the other hand, for the complete system,G(s), we obtain the following Figures 11, 12, 13, and 14 depicting the state evolution, the output, and the control input.Figure 11
Evolution of the state componentx1(t).Figure 12
Evolution of the state componentsx2(t),x3(t), andx4(t).Figure 13
System output.Figure 14
Control input with impulses.Figures11–14 illustrate the results compiled in Theorem 10(ii) since they show the convergence of the state, the output and the input to zero. Moreover, it can be appreciated the effect of adding impulses to the control signal by first comparing Figures 3 and 9 and, on the other hand, Figures 6 and 13. It can be deduced that the use of an impulsive controller makes the output to converge to zero faster than when a single continuous-time one is used. This fact highlights how the concept of impulsive asymptotic hyperstability is relevant since it allows ensuring the closed-loop stability for a whole family of nonlinear devices while accelerating the convergence of the output to zero.


## 6. Conclusions

This paper has studied the problem of asymptotic hyperstability under a regular controller combined with a class of impulsive controls provided that the feed-forward linear- loop is not asymptotically hyperstable, since it has a parallel component subsystem being strictly stable. Asymptotic hyperstability results with robustness extensions have also been discussed. The controller has been assumed to satisfy an integral Popovian-type inequality for all time. Also, examples have been given to illustrate the proposed study.



---

*Source: 382762-2013-10-02.xml*