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

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.

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 Figures 3 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: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝑏𝑢_im(𝑡),𝑥(0)=𝑥₀,𝑢_im(𝑡)𝑦(𝑡),𝑦(𝑡)=−𝜑,𝑡=𝑐^𝑇𝑥(𝑡)+𝑑₀𝑢_im(𝑡), (1)(2)(3) which is subject to a potentially nonlinear time-varying controller device satisfying the Popovian type integral inequality below: ^𝑡₀𝜑𝑦(𝜏),𝜏𝑦(𝜏)𝑑𝜏≥−𝛾²>−∞,∀𝑡∈𝐑₀₊, (4) where𝑥(𝑡)∈𝐑^𝑛is the𝑛-state vector, and𝑦(𝑡)and 𝑢_im(𝑡) are the output and the (in general, impulsive) control which satisfies the Popovian integral constraint∫^𝑡₀𝜑(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:𝐺(𝑠)=𝑐^𝑇(𝑠𝐼−𝐴)⁻¹𝑏+𝑑₀=𝐺^∗(𝑠)+𝐺(𝑠),𝐺^∗(𝑠)=𝑐^∗𝑇𝑠𝐼−𝐴^∗−1𝑏^∗+𝑑^∗, (5a)(5b)

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=lim_𝜔→±∞𝐺(𝑖𝜔)=lim_𝜔→±∞𝐺^∗(𝑖𝜔)>0, sincelim_𝜔→±∞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: 𝑢_im(𝑡)=𝑢(𝑡)+^∞_𝑗=1𝐾_𝑗𝑔𝑡_𝑗𝛿𝑡−𝑡_𝑗,∀𝑡∈𝐑₀₊, (6) where𝑢(𝑡)is the piecewise continuous impulse-free input,𝛿(𝑡)is the Dirac distribution describing the impulsive actions of amplitudes𝐾_𝑗𝑔(𝑡_𝑗)where𝑔(𝑡_𝑗)=1 if the input impulse is independent of the output and 𝑔(𝑡_𝑗)=𝑦(𝑡_𝑗)   in the case that it depends directly on such an output. The sequence{𝑔(𝑡_𝑗)}  can be, in general, any nonzero function sequence being dependent on the output and input sequences{𝑦(𝑡_𝑗)}and{𝑢(𝑡_𝑗)}  at the impulsive time instants. A typical case in many practical situations is that the input impulse is  𝑢(𝑡⁺_𝑗)=𝑢(𝑡_𝑗)+𝐾_𝑗𝑦(𝑡_𝑗), that is, 𝑔(𝑡_𝑗)=𝑦(𝑡_𝑗). In the following, the subsequent notation is used:(i)𝑓_𝑡1,𝑡2(𝜏)=𝑓(𝑡)if𝜏∈[𝑡₁,𝑡₂)  and𝑓_𝑡1,𝑡2(𝜏)=0, otherwise, it is a truncated function of𝑓(𝑡)(𝑓_0,𝑡(𝜏)≡𝑓_𝑡(𝜏)  is used for notation abbreviation);(ii)Im(𝑡)={𝑡₁,𝑡₂,…,𝑡_𝑝(𝑡)}is the set of impulsive time instants until time𝑡  with the convention𝑡_𝑝(𝑡)<𝑡so thatIm(𝑡⁺)={𝑡₁,𝑡₂,…,𝑡_𝑝+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  𝑡_𝑘∉Im(𝑡_𝑘). 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(𝑡_𝑗)},𝑡_𝑗∈Im_∞  for some impulsive setIm_∞is often used by obvious reasons for such a sequence  {𝑔(𝑡_𝑗)}.

The limit notationlim_{𝑡∈[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 case𝑢_im(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  𝑢_im(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∫^𝑡₀𝜑(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 𝑑₀=lim_𝜔→±∞Re𝐺(𝑖𝜔)>0,𝑑=min_{𝜔∈𝐑0+}Re𝐺(𝑖𝜔)=𝑑₀+min_{𝜔∈𝐑0+}Re𝑐^𝑇(𝑖𝜔𝐼−𝐴)⁻¹𝑏>0. (7)

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)=∫^𝑡₀𝑦(𝜏)𝑢_𝑖𝑚(𝜏)𝑑𝜏, respectively, on[0,𝑡] is uniformly bounded for all time while it satisfies the following relationships:𝛾²≥𝐸(𝑡)≥𝑑−𝐾^∞_−∞𝑢²_𝑡(𝜏)=𝐾𝑑𝜏𝑑−^𝑡₀𝑢²(𝜏)𝑑𝜏+_{𝑡𝑗∈𝐼𝑚(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗,∀𝑡∈𝐑₊,𝛾²≥𝐸𝑡⁺=𝐸(𝑡)+^𝑡+_{𝑡𝑝(𝑡)}𝑦_𝑡(𝜏)𝑢(𝜏)≥𝐾𝑑𝜏𝑑−^𝑡₀𝑢²(𝜏)𝑑𝜏+_{𝑡𝑗∈𝐼𝑚(𝑡+)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗,∀𝑡∈𝐑₊, (8)(9) where𝑑=min_{𝜔∈𝐑0+}𝑅𝑒𝐺(𝑖𝜔).

Proof. See Appendix A.

A useful result based on Lemma 2 follows below so as to get the main stability result after establishing the subsequent definition.

Definition 3. Any sequence of impulsive gains{𝐾_𝑗}  subject to the following constraints: 𝛾²−𝑑−𝐾^𝑡𝑘₀𝑢²(𝜏)𝑑𝜏−_{𝑡𝑗∈Im(𝑡𝑘)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗≥𝐾_𝑘𝑔𝑡_𝑘𝑦𝑡_𝑘≥𝑚𝑡_𝑘−𝑑−𝐾^𝑡𝑘₀𝑢²(𝜏)𝑑𝜏−_{𝑡𝑗∈Im(𝑡𝑘)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗, (10) for some given real function𝑚(𝑡), which satisfies0<𝑚(𝑡)<𝛾²forall𝑡∈𝐑₊, and any𝑡_𝑘∈Im_∞≡⋃_{𝑡∈𝐑0+}Im(𝑡)such that𝑦(𝑡_𝑘)≠0, is said to be of the class𝐼_gain=𝐼_gain(Im_∞,𝛾,𝑔)for some given𝛾∈𝐑₊ and for the support sequence{𝑔(𝑡_𝑗)}, 𝑡_𝑗∈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{𝐾_𝑗}is of the class𝐼_gain(Im_∞,𝛾,𝑔)for some  𝛾∈𝐑₊.

Lemma 5. Assume that the sequence of impulsive gains for system (1)–(4) is of the class𝐼_{𝑔𝑎𝑖𝑛}=𝐼_{𝑔𝑎𝑖𝑛}(𝐼𝑚_∞,𝛾,𝑔), then, the following properties hold: 𝑑−𝐾^𝑡₀𝑢²(𝜏)𝑑𝜏<∞,_{𝑡𝑗∈𝐼𝑚(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗<∞,∀𝑡∈𝐑₊, (11) if 𝑐𝑎𝑟𝑑𝐼𝑚_∞<𝜒₀, and ∞>𝛾²≥𝑑−𝐾^𝑡₀𝑢²(𝜏)𝑑𝜏+_{𝑡𝑗∈𝐼𝑚(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗>0,∀𝑡∈𝐑₊, (12) if𝑐𝑎𝑟𝑑𝐼𝑚_∞≤𝜒₀ so that the input-output energy is positive and uniformly upper-bounded by a finite bound for all nonzero time.

Proof. See Appendix B.

Remark 6. It is then proven in Theorem 10 that (11) holds even ifcardIm_∞=𝜒₀and 𝑑=𝐾.

Lemma 5 is subject to the subsequent important necessary condition.

Lemma 7. Assume that𝑑<𝐾then a necessary condition for Lemma 5 to hold is that the class𝐼_{𝑔𝑎𝑖𝑛}=𝐼_{𝑔𝑎𝑖𝑛}(𝐼𝑚_∞,𝛾,𝑔)is such that  𝑡₁=0∈𝐼𝑚(𝑡)⊂𝐼𝑚_∞𝑓𝑜𝑟𝑎𝑙𝑙𝑡∈𝐑₀₊, and its associate impulsive gain satisfies the constraint: 𝑑−𝐾∫^𝑡1₀𝑢²(𝜏)𝑑𝜏−𝛾²𝑔𝑡₁𝑦𝑡₁≤𝐾₁≤𝛾²+𝑑−𝐾∫^𝑡1₀𝑢²(𝜏)𝑑𝜏𝑔𝑡₁𝑦𝑡₁. (13) Another necessary condition for Lemma 5 to hold is that the class𝐼_{𝑔𝑎𝑖𝑛}=𝐼_{𝑔𝑎𝑖𝑛}(𝐼𝑚_∞,𝛾,𝑔)has an impulsive set of time instants 𝐼𝑚_∞of infinite cardinal.

Proof. Proceed by contradiction by assuming that  𝑑<𝐾,Im_∞∋𝑡₁>0, and Lemma 5 holds so that (12) holds. Then, ∞>𝛾²≥−𝑑−𝐾^𝑡1₀𝑢²(𝜏)𝑑𝜏+𝐾₁𝑔𝑡₁𝑦𝑡₁>0,∞>𝛾²≥0>−𝑑−𝐾^𝑡1₀𝑢²(𝜏)𝑑𝜏>0,∀𝑡∈0,𝑡₁, (14) 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): ∞>𝛾²≥lim_𝑡→∞𝑑−𝐾^𝑡₀𝑢²(𝜏)𝑑𝜏+_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗=∞. (15) Hence, the proof of the result is complete.

To the light of Definition 3, 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 asymptotically𝐼_gain(Im_∞,𝛾,𝑔)-hyperstable (resp., impulsive𝐼_gain(Im_∞,𝛾,𝑔)-hyperstable) if it is globally asymptotically stable (resp., globally stable) for any sequence of impulsive gains{𝐾_𝑗}of the class𝐼_gain=𝐼_gain(Im_∞,𝛾,𝑔) for a given nonzero𝛾∈𝐑₊, a given support sequence{𝑔(𝑡_𝑗)},𝑡_𝑗∈Im_∞, and any control device satisfying an impulsive control compatible Popov’s integral inequality∫^𝑡₀𝜑(𝑦(𝜏),𝜏)𝑦(𝜏)𝑑𝜏≥−𝛾²>−∞forall𝑡∈𝐑₀₊.

3. Main Results on Asymptotic Hyperstability

It turns out that Lemma 7 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)≥0  for  all   𝑡∈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∫^𝑡₀𝜑(𝑦(𝜏),𝜏)𝑦(𝜏)𝑑𝜏≥−𝛾²>−∞  for  all   𝑡∈𝐑₀₊. As a result,𝐺(𝑠)∈𝑆𝑃𝑅,lim_𝑡→∞𝑢(𝑡)=0,𝑢(𝑡), 𝑦(𝑡)and𝑥(𝑡)are uniformly bounded for all𝑡∈𝐑₀₊, andlim_𝑡→∞𝑦(𝑡)=0andlim_𝑡→∞𝑥(𝑡)=0  for  all   𝑥₀∈𝐑^𝑛. (ii)Assume that𝑅𝑒𝐺^∗(𝑖𝜔)>0,𝑅𝑒𝐺(𝑖𝜔)>−𝑅𝑒𝐺^∗(𝑖𝜔) for all 𝜔∈𝐑₀₊, and𝑑=inf_{𝜔∈𝐑0+}𝑅𝑒𝐺(𝑖𝜔)=lim_𝜔→±∞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 ∫^𝑡₀𝜑(𝑦(𝜏),𝜏)𝑦(𝜏)𝑑𝜏≥−𝛾²>−∞ for all 𝑡∈𝐑₀₊. The, lim_𝑡→∞𝑢(𝑡)=0, 𝑢(𝑡), 𝑦(𝑡)and𝑥(𝑡)are uniformly bounded for all 𝑡∈𝐑₀₊, and  lim_𝑡→∞𝑦(𝑡)=0and  lim_𝑡→∞𝑥(𝑡)=0 for all 𝑥₀∈𝐑^𝑛.

Proof. It is given in Appendix C.

Theorem 9(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=lim_𝜔→±∞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 asymptotically𝐼_{𝑔𝑎𝑖𝑛}(𝐼𝑚_∞,𝛾,𝑔)-hyperstable for any nonlinear feedback device𝜑(𝑦(𝑡),𝑡)satisfying a compatible Popov’s integral inequality∫^𝑡₀𝜑(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∈𝐑^𝑛,𝑢(𝑡)is bounded for𝑡∈𝐑₀₊⧵𝐼𝑚_∞, with  lim_{𝑡∈[𝑡𝑗,𝑡𝑗+1),𝑡𝑗→∞}𝑢(𝑡)=0if𝑐𝑎𝑟𝑑𝐼𝑚_∞=𝜒₀(i.e., there are infinitely many control impulsive time instants); andlim_𝑡→∞𝑢(𝑡)=0if𝑐𝑎𝑟𝑑𝐼𝑚_∞<𝜒₀, that is, if there is a finite number of control impulsive time instants. Also _{𝑡𝑗∈𝐼𝑚(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗≤𝛾²<∞,_{𝑡𝑗∈𝐼𝑚(𝑡+)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗≤𝛾²<∞,∀𝑡∈𝐑₀₊. (16)(ii)Assume that𝑅𝑒𝐺^∗(𝑖𝜔)>0,  𝑅𝑒𝐺(𝑖𝜔)>−𝑅𝑒𝐺^∗(𝑖𝜔) for all 𝜔∈𝐑₀₊and𝑑=inf_{𝜔∈𝐑0+}𝑅𝑒𝐺(𝑖𝜔)=lim_𝜔→±∞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  ∫^𝑡₀𝜑(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∈𝐑^𝑛,𝑢(𝑡)is bounded for𝑡∈𝐑₀₊⧵𝐼𝑚_∞, with lim_{𝑡∈[𝑡𝑗,𝑡𝑗+1),𝑡𝑗→∞}𝑢(𝑡)=0  if𝑐𝑎𝑟𝑑𝐼𝑚_∞=𝜒₀ (i.e., there are infinitely many control impulsive time instants); and  lim_𝑡→∞𝑢(𝑡)=0,lim_𝑡→∞𝑥(𝑡)=0and  lim_𝑡→∞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 𝐾_𝑗→0as 𝑗→∞. Also, the boundedness properties (16) for the impulsive contributions still hold.

Proof. It is given in Appendix D.

Note that the constraint (10) invoked in Theorem 10 is guaranteed by amplitude constraints of each impulse at𝑡_𝑘depending only on input and output values on[0,𝑡_𝑘)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: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝑏𝑢_im(𝑡)+𝜂(𝑡),𝑥(0)=𝑥₀,𝑦(𝑡)=𝑐^𝑇𝑥(𝑡)+𝑑₀𝑢_im(𝑡)+𝜆(𝑡). (17) Taking Laplace transforms under zero initial conditions in (17), one gets 𝑌(𝑠)=𝐺(𝑠)𝑈_im(𝑠)+𝑐^𝑇(𝑠𝐼−𝐴)⁻¹𝑁(𝑠)+Λ(𝑠)=𝑐^𝑇(𝑠𝐼−𝐴)⁻¹𝑏𝑈_im(𝑠)+𝑁(𝑠)+𝑑₀𝑈_im(𝑠)(𝑠)=𝐺+Λ_∗(𝑠)+𝐺(𝑠)𝑈_im(𝑠)+𝑐^𝑇(𝑠𝐼−𝐴)⁻¹𝑁(𝑠)(𝑠)=𝑐+Λ_∗𝑇𝑠𝐼−𝐴^∗−1𝑏^∗+𝑑₀𝑈_im(𝑠)+𝐺(𝑠)𝑈_im(𝑠)+𝑐^𝑇(𝑠𝐼−𝐴)⁻¹𝑁(𝑠)+Λ(𝑠), (18) 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: 𝑦(𝑡)⟶𝑦(𝑡)+𝑦(𝑡),𝑦(𝑡)=𝑦𝜂,𝜆,𝑡=^𝑡₀𝑔^𝑇₀(𝑡−𝜏)𝜂(𝜏)𝑑𝜏+𝜆(𝑡), (19) where𝑔^𝑇₀(𝑡)is the impulse response of𝑐^𝑇(𝑠𝐼−𝐴)⁻¹. Define the robust impulsive class and its associated robust impulsive 𝐼_rgain=𝐼_rgain(Im_∞,𝛾,𝑔,𝜂,𝜆)-hyperstability and asymptotic 𝐼_rgain=𝐼_rgain(Im_∞,𝛾,𝑔,𝜂,𝜆)-hyperstability as follows.

Definition 11. Any sequence of impulsive gains{𝐾_𝑗}subject to the following constraints: 𝛾²−𝑑−𝐾^𝑡𝑘₀𝑢²(𝜏)𝑑𝜏−^𝑡𝑘₀𝑦𝜂,𝜆,𝜏𝑢(𝜏)𝑑𝜏−_{𝑡𝑗∈Im(𝑡𝑘)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗≥𝐾_𝑘𝑔𝑡_𝑘𝑦𝑡_𝑘≥𝑚𝑡_𝑘−𝑑−𝐾^𝑡𝑘₀𝑢²(𝜏)−𝑑𝜏_𝑡𝑘0𝑦𝜂,𝜆,𝜏𝑢(𝜏)𝑑𝜏−_{𝑡𝑗∈Im(𝑡𝑘)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗, (20) for some given real function𝑚(𝑡), which satisfies0<𝑚(𝑡)<𝛾²forall𝑡∈𝐑₊, and any𝑡_𝑘∈Im_∞≡⋃_{𝑡∈𝑅0+}Im(𝑡), such that𝑦(𝑡_𝑘)≠0, is said to be of the robust impulsive class𝐼_rgain=𝐼_rgain(Im_∞,𝛾,𝑔,𝜂,𝜆)for some given𝛾∈𝐑₊ and for the support sequence{𝑔(𝑡_𝑗)},𝑡_𝑗∈Im_∞.

Definition 12. The impulsive closed-loop system (17), subject to (2), is said to be robustly impulsive asymptotically𝐼_rgain(Im_∞,𝛾,𝑔,𝜂,𝜆)-hyperstable (resp., impulsive𝐼_rgain(Im_∞,𝛾,𝑔,𝜂,𝜆)-hyperstable) if it is globally asymptotically stable (resp., globally stable) for any sequence of impulsive gains  {𝐾_𝑗}of the class𝐼_rgain(Im_∞,𝛾,𝑔,𝜂,𝜆)for a given nonzero  𝛾∈𝐑₊, a given support sequence{𝑔(𝑡_𝑗)},𝑡_𝑗∈Im_∞  and any control device satisfying an impulsive control compatible Popov integral inequality∫^𝑡₀𝜑(𝑦(𝜏),𝜏)𝑦(𝜏)𝑑𝜏≥−𝛾²>−∞;forall𝑡∈𝐑₀₊.

Now, the asymptotic hyperstability Theorem 10 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 asymptotically𝐼_{𝑟𝑔𝑎𝑖𝑛}(𝐼𝑚_∞,𝛾,𝑔,𝜂,𝜆)-hyperstable for any nonlinear feedback device 𝜑(𝑦(𝑡),𝑡) satisfying a compatible Popov’s integral inequality ∫^𝑡₀𝜑(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∈𝐑^𝑛, 𝑢(𝑡)is bounded for𝑡∈𝐑₀₊⧵𝐼𝑚_∞, with  lim_{𝑡∈[𝑡𝑗,𝑡𝑗+1),𝑡𝑗→∞}𝑢(𝑡)=0 if𝑐𝑎𝑟𝑑𝐼𝑚_∞=𝜒₀(i.e., there are infinitely many control impulsive time instants); and  lim_𝑡→∞𝑢(𝑡)=0 if 𝑐𝑎𝑟𝑑𝐼𝑚_∞<𝜒₀  (i.e., there is a finite number of control impulsive time instants). (ii)Assume that 𝑅𝑒𝐺^∗(𝑖𝜔)>0,  𝑅𝑒𝐺(𝑖𝜔)>−𝑅𝑒𝐺^∗(𝑖𝜔) for all 𝜔∈𝐑₀₊and𝑑=inf_{𝜔∈𝑅0+}𝑅𝑒𝐺(𝑖𝜔)=lim_𝜔→±∞𝑅𝑒𝐺^∗(𝑖𝜔)=0.
Then, Lemma 5 holds and the closed-loop system is robustly asymptotically hyperstable for any nonlinear feedback device 𝜑(𝑦(𝑡),𝑡) satisfying Popov’s integral inequality ∫^𝑡₀𝜑(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∈𝐑^𝑛, 𝑢(𝑡) is bounded for 𝑡∈𝐑₀₊⧵𝐼𝑚_∞, with lim_{𝑡∈[𝑡𝑗,𝑡𝑗+1),𝑡𝑗→∞}𝑢(𝑡)=0 if 𝑐𝑎𝑟𝑑𝐼𝑚_∞=𝜒₀ (i.e., there are infinitely many control impulsive time instants); and  lim_𝑡→∞𝑢(𝑡)=0, lim_𝑡→∞𝑥(𝑡)=0and lim_𝑡→∞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) and𝐾_𝑗→0 as 𝑗→∞.

Proof (outline). It is given in Appendix E.

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: 𝐺^∗(𝑠)=𝑠+3𝑠²,𝐺(𝑠)=2+7𝑠+10𝑠².+6𝑠+9 (21)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 by𝜑_𝑖(𝑦,𝑡)=tanh(𝛾_𝑖𝑦), where𝛾_𝑖∈{0.2,0.6,1.4}. These nonlinear functions satisfy the integral-type Popov’s constraint specified by (4) since tanh(𝛾_𝑖𝑦)𝑦>0for 𝑦≠0, and tanh(𝛾_𝑖𝑦)𝑦=0for 𝑦=0, implying that∫^𝑡₀tanh(𝛾_𝑖𝑦)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

Figures 1 and 2 show the evolution of the state variables of the SPR system𝐺^∗(𝑠)when members of the family of nonlinear static feedback devices𝜑_𝑖(𝑦,𝑡)=tanh(𝛾_𝑖𝑦) 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 input 𝑢(𝑡)=−𝜑(𝑦,𝑡).

Figures 1–4 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 of𝛾_𝑖. This similar evolution is obtained at the expense of a different control action for each𝛾_𝑖, 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 of𝛾_𝑖=0.6has been used in Figures 5 and 6 since the evolution of the system is very similar for all values of𝛾_𝑖. However, Figure 7 shows the control input for different values of𝛾_𝑖since its value varies depending on𝛾_𝑖. 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 Section 5.1. The support sequence is given by𝑔(𝑡_𝑘)=𝑦(𝑡_𝑘)and the impulsive gains constant𝐾_𝑘=1. The impulsive time instants are multiples of 0.2 seconds; this means that the presence of impulses is periodic. The value𝛾_𝑖=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.

Figures 8–10 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.

Figures 11–14 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.

Appendices

A. Proof of Lemma 2

The Popovian integral constraints (8)-(9) and Parseval’s theorem while absorbing the delta distribution describing impulses from the time-integral or frequency-integral of the input-output energy measures are used. Then, one gets the following sets of inequalities and equalities by taking into account the use of truncated signals to convert time-integral expressions in frequency ones and vice versa as well as the symmetry of the frequency-response hodographs 𝐺^∗(𝑖𝜔) and 𝐺(𝑖𝜔): 𝛾²≥𝐸(𝑡)=−^𝑡₀𝜑𝑦(𝜏)𝑦(𝜏)𝑑𝜏=^𝑡₀𝑦(𝜏)𝑢_im(𝜏)=𝑑𝜏_∞−∞𝑦_𝑡(𝜏)𝑢_im𝑡(𝜏)𝑑𝜏=^∞_−∞𝑦_𝑡(𝜏)𝑢_im(𝜏)=𝑑𝜏_∞−∞𝑦(𝜏)𝑢_𝑡(𝜏)𝑑𝜏+_{𝑡𝑖∈Im(𝑡)}^𝑡+𝑖+1_𝑡𝑖𝐾_𝑗𝑔𝑡_𝑖𝑦_𝑡(𝜏)𝛿𝜏−𝑡_𝑖𝑑𝜏=_{𝑡𝑖∈Im(𝑡)}^𝑡𝑖+1_𝑡+𝑖𝑦_𝑡(𝜏)𝑢(𝜏)𝑑𝜏+^𝑡_{𝑡+𝑝(𝑡)}𝑦_𝑡(𝜏)𝑢(𝜏)𝑑𝜏+_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗=12𝜋^∞_−∞𝑌_𝑡(𝑖𝜔)𝑈_𝑡(−𝑖𝜔)𝑑𝜔+_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗=12𝜋^∞_−∞Re𝐺^∗(𝑖𝜔)𝑈_𝑡(𝑖𝜔)²+1𝑑𝜔2𝜋^∞_−∞Re𝐺(𝑖𝜔)𝑈_𝑡(𝑖𝜔)²𝑑𝜔+_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗≥12𝜋min_{𝜔∈𝐑0+}Re𝐺^∗(𝑖𝜔)+min_{𝜔∈𝐑0+}𝐺()×Re𝑖𝜔_∞−∞𝑈_𝑡(𝑖𝜔)²𝑑𝜔+_{𝑡𝑗(𝑡)∈Im}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗=𝑑−𝐾^∞_−∞𝑢²_𝑡(𝜏)𝑑𝜏=𝑑−𝐾^𝑡₀𝑢²(𝜏)𝑑𝜏+_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗,∀𝑡∈𝐑₊. (A.1) In the same way, and since,𝐸(𝑡⁺)=𝐸(𝑡)+∫^𝑡+_{𝑡𝑝(𝑡)}𝑦_𝑡(𝜏)𝑢(𝜏)𝑑𝜏, one has 𝛾²≥𝐸𝑡⁺≥𝑑−𝐾^∞_−∞𝑢²_𝑡(𝜏)𝐾×𝑑𝜏=𝑑−_𝑡0𝑢²(𝜏)𝑑𝜏+_{𝑡𝑗∈Im(𝑡+)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗,∀𝑡∈𝐑₊ (A.2)

and the proof is complete.

B. Proof of Lemma 5

The finite uniform upper-bound of (12) follows from Lemma 2. It is needed to guarantee a positive lower-bound for all nonzero time so that 𝛾²≥𝑑−𝐾^𝑡₀𝑢²(𝜏)𝑑𝜏+_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗>0,∀𝑡∈𝐑₊ (B.1) to hold with|∑_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔(𝑡_𝑗)𝑦(𝑡_𝑗)|<∞. This implies that −∞<(𝑑−𝐾)∫^𝑡₀𝑢²(𝜏)𝑑𝜏<∞and −∞<∑_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔(𝑡_𝑗)𝑦(𝑡_𝑗)<∞ while it is avoided that (12) can hold withlim_𝑡→∞(𝑑−𝐾)∫^𝑡₀𝑢²(𝜏)𝑑𝜏=−lim_𝑡→∞∑_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔(𝑡_𝑗)𝑦(𝑡_𝑗)=+∞. Thus, it suffices for any given strictly positive bounded function0<𝑚(𝑡)<𝛾²  forall𝑡∈𝐑₊to guarantee (B.1) in the form 𝛾²≥𝑑−𝐾^𝑡₀𝑢²(𝜏)𝑑𝜏+_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗≥𝑚(𝑡)>0,∀𝑡∈𝐑₊, (B.2) or, equivalently, 𝛾²−𝑑−𝐾^𝑡₀𝑢²(𝜏)𝑑𝜏≥_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗≥𝑚(𝑡)−𝑑−𝐾^𝑡₀𝑢²(𝜏)𝑑𝜏, (B.3) so that (12) holds ifcardIm_∞≤𝜒₀if and only if for each 𝑡_𝑘∈⋃_{𝑡∈𝐑0+}Im(𝑡).

Consider 𝛾²−𝑑−𝐾^𝑡𝑘₀𝑢²(𝜏)𝑑𝜏−_{𝑡𝑗∈Im(𝑡𝑘)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗≥𝐾_𝑘𝑔𝑡_𝑘𝑦𝑡_𝑘≥𝑚𝑡_𝑘−𝐾×𝑑−_𝑡𝑘0𝑢²(𝜏)𝑑𝜏−_{𝑡𝑗∈Im(𝑡𝑘)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗, (B.4) which holds if (10) holds for any𝑡_𝑘∈⋃_{𝑡∈𝐑0+}Im(𝑡), such that𝑦(𝑡_𝑘)≠0, which also implies that (11) holds ifcardIm_∞<𝜒₀.

C. Proof of Theorem 9

(i) IfIm_∞=∅and 𝑑>𝐾, then also 𝑑>0as a result, and one has from (B.1) in the proof of Lemma 5 that for any nonzero piecewise continuous control input satisfying Popovian integral inequality the following: 𝛾²≥𝐸(𝑡)≥𝑑−𝐾^𝑡₀𝑢²(𝜏)𝑑𝜏>0,∀𝑡∈𝐑₊,⟹𝛾²𝑑−𝐾≥^𝑡₀𝑢²(𝜏)𝑑𝜏>0,∀𝑡∈𝐑₀₊,⟹lim_{𝜏→𝑡,𝑡→∞}𝑢_𝑡(𝜏)=lim_𝑡→∞𝑢(𝑡)=0. (C.1) The property  𝛾²≥𝐸(𝑡)>0forall𝑡∈𝐑₀₊with𝐸(0)=0follows from (B.1) in view of (A.1). Since𝑢(𝑡)is piecewise continuous then it is also bounded for all time. Thus, since the matrix𝐴is strictly stable,𝑥(𝑡)and𝑦(𝑡)are bounded for all time and also lim_𝑡→∞𝑥(𝑡)=0, and  lim_𝑡→∞𝑦(𝑡)=0for any𝑥₀∈𝐑^𝑛with the state and output being also uniformly bounded sincemin_{𝜔∈𝐑0+}Re(𝐺^∗(𝑖𝜔)+𝐺(𝑖𝜔))≥𝑑−𝐾>0implies thatRe(𝐺^∗(𝑠)+𝐺(𝑠))>0for allRe𝑠≥0 so that(𝐺^∗(𝑠)+𝐺(𝑠))∈SPR and then is strictly stable.

(ii) In this case,Im_∞=∅. One has from the given hypotheses that(1)Re(𝐺^∗(𝑖𝜔)+𝐺(𝑖𝜔))≥Re𝐺^∗(𝑖𝜔)+Re𝐺(𝑖𝜔)>0forall𝜔∈𝐑₀₊, (2)lim_𝜔→±∞𝐺(𝑖𝜔)=lim_𝜔→±∞Re𝐺(i𝜔)=lim_𝜔→±∞Re(𝐺^∗(𝑖𝜔)+𝐺(𝑖𝜔))=𝑑=0. Direct calculations similar to those performed in the proof of Lemma 2 yield 𝛾²(𝑡)≥1≥𝐸𝑀(𝑊)2𝜋^∞_−∞𝑈_𝑡(𝑖𝜔)²𝑑𝜔+2𝜋𝜀_𝑡(𝑊)=1𝑀(𝑊)2𝜋^𝑡₀𝑢²(𝜏)𝑑𝜏+𝜀_𝑡(𝑊)≥1×2𝜋min_{[]𝜔∈−𝑊,𝑊}Re𝐺(𝑖𝜔)^∞_−∞𝑈_𝑡(𝑖𝜔)²𝑑𝜔+min_{[)𝜔∈𝑊,∞}Re𝐺(𝑖𝜔)^∞_𝑊𝑈_𝑡(𝑖𝜔)²𝑑𝜔,∀𝑡∈𝐑₀₊, (C.2) where𝑀(𝑊)=min_{𝜔∈[−𝑊,𝑊]}Re𝐺(𝑖𝜔) and 𝜀_𝑡(𝑊)=(1/2𝜋)min_{𝜔∈[𝑊,∞)}Re𝐺(𝑖𝜔)∫^∞_𝑊|𝑈_𝑡(𝑖𝜔)|²𝑑𝜔forall𝑡∈𝐑₀₊ are everywhere continuous, nonnegative, and strictly decreasing so thatlim_𝑊→∞𝑀(𝑊)=lim_𝑊→∞𝜀_𝑡(𝑊)=0; for all 𝑡∈𝐑₀₊. Thus, for a sufficiently large finite𝑊₀∈𝐑⁺and for all 𝑡∈𝐑₀₊, (𝛾²−𝜀_𝑡(𝑊))/𝑀(𝑊) is finite and positive for all 𝑊≥𝑊₀ so that 𝛾∞>2𝜋²−𝜀_𝑡(𝑊)𝑀(𝑊)≥^𝑡₀𝑢²(𝜏)𝑑𝜏⟹lim_𝑡→∞𝑢(𝑡)=0. (C.3) The remaining of the proof follows in the same way as that of property (i).

D. Proof of Theorem 10

It follows directly as the proof of Theorem 9 by using Lemma 5 and the guidelines for the proof of Lemma 2 with the following modifications. The control is bounded except at the impulsive time instants while the state and output are uniformly bounded while possessing discontinuities at the impulsive time instants. Furthermore, if cardIm_∞<𝜒₀, then it is possible to take an initial state after some sufficiently large finite time, which are bounded for any bounded𝑥₀, to conclude that the control, input, and output converge asymptotically to zero as in the impulsive-free case of Theorem 9. To prove that the control, state and output converge to zero as time tends to infinity if cardIm_∞=𝜒₀  with 𝐾_𝑗→0 as 𝑗→∞, note that 𝑥𝑡⁺_𝑗−𝑥𝑡_𝑗=𝑏𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗𝑦𝑡⟶0,_+𝑗−𝑦𝑡_𝑗=𝑐^𝑇𝑏𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗⟶0, (D.1) as 𝑗→∞,𝑢(𝑡)→0since𝑢(𝑡⁺_𝑗)−𝑢(𝑡_𝑗)=𝐾_𝑗𝛿(0)𝑔(𝑡_𝑗)𝑦(𝑡_𝑗)→0 as 𝑗→∞. Then, 𝑥(𝑡)→0, 𝑦(𝑡)→0 as 𝑡→∞ since 𝐴 is strictly stable, and 𝑥𝑡⁺_𝑗+1=𝐼+𝑏𝐾_𝑗𝑔𝑡_𝑗𝑐^𝑇𝑥𝑡_𝑗+1=𝐼+𝑏𝐾_𝑗𝑔𝑡_𝑗𝑐^𝑇×𝑒^{𝐴(𝑡𝑗+1−𝑡𝑗)}𝑥𝑡⁺_𝑗+𝑏𝐾_𝑗𝛿(0)𝑔𝑡_𝑗𝑦𝑡_𝑗=𝐼+𝑏𝐾_𝑗𝑔𝑡_𝑗𝑐^𝑇×𝑒^{𝐴(𝑡𝑗+1−𝑡𝑗)}𝑥𝑡⁺_𝑗+𝑏𝐾_𝑗𝛿(0)𝑔𝑡_𝑗𝑐^𝑇𝑥𝑡_𝑗+𝑑𝑢𝑡_𝑗=𝐼+𝑏𝐾_𝑗𝑔𝑡_𝑗𝑐^𝑇×𝑒^{𝐴(𝑡𝑗+1−𝑡𝑗)}𝑥𝑡_𝑗+𝑏𝐾_𝑗𝑔𝑡_𝑗((0×𝑐1+𝛿))_𝑇𝑥𝑡_𝑗+𝑑𝑢𝑡_𝑗,𝑥𝑡⁺_𝑗+1−𝑥𝑡_𝑗=𝐼+𝑏𝐾_𝑗𝑔𝑡_𝑗𝑐^𝑇𝑒^{𝐴(𝑡𝑗+1−𝑡𝑗)}−𝐼𝑥𝑡_𝑗+𝐼+𝑏𝐾_𝑗𝑔𝑡_𝑗𝑐^𝑇𝑏𝐾_𝑗𝛿(0)𝑔𝑡_𝑗𝑦𝑡_𝑗+𝐼+𝑏𝐾_𝑗𝑔𝑡_𝑗𝑐^𝑇×𝑒^{𝐴(𝑡𝑗+1−𝑡𝑗)}𝑥𝑡_𝑗𝑏𝐾_𝑗𝑔𝑡_𝑗×𝑐^𝑇𝑥𝑡_𝑗+𝑑𝑢𝑡_𝑗, (D.2) and, since𝐾_𝑗→0 and 𝑥(𝑡⁺_𝑗)−𝑥(𝑡_𝑗)=𝑏𝐾_𝑗𝑔(𝑡_𝑗)𝑦(𝑡_𝑗)→0 as 𝑗→∞, one concludes that lim_𝑗→∞𝑥𝑡⁺_𝑗+1−𝑥𝑡_𝑗−𝐼+𝑏𝐾_𝑗𝑔𝑡_𝑗𝑐^𝑇𝑒^{𝐴(𝑡𝑗+1−𝑡𝑗)}−𝐼𝑥𝑡_𝑗=−lim_𝑗→∞𝐼+𝑏𝐾_𝑗𝑔𝑡_𝑗𝑐^𝑇𝑒^{𝐴(𝑡𝑗+1−𝑡𝑗)}−𝐼𝑥𝑡_𝑗=lim_𝑗→∞𝐼−𝑒^{𝐴(𝑡𝑗+1−𝑡𝑗)}𝑥𝑡_𝑗=lim_𝑗→∞𝑥𝑡_𝑗−𝑥𝑡_𝑗+1=0. (D.3) Then,lim_{𝑡𝑗→∞}[𝑥(𝑡_𝑗)−𝑥(𝑡⁺_𝑗)]=lim_{𝑡𝑗→∞}[𝑥(𝑡_𝑗+1)−𝑥(𝑡⁺_𝑗)]=0, so that there is a limit lim_{𝑡𝑗→∞}𝑥(𝑡⁺_𝑗)=lim_{𝑡𝑗→∞}𝑥(𝑡⁺_𝑗)=𝑥^∗. Then lim_{𝑡𝑗→∞}(𝐼−𝑒^{𝐴(𝑡𝑗+1−𝑡𝑗)})𝑥^∗≠0 or it does not exist, which would be contradiction, unless𝑥^∗=0. Then,𝑥^∗=lim_{𝑡𝑗→∞}𝑥(𝑡⁺_𝑗)=lim_{𝑡𝑗→∞}𝑥(𝑡_𝑗)=0 and, since 𝐴  is strictly stable, and 𝑢(𝑡)→0 as 𝑡→∞, 𝑥(𝑡)→0 on  [𝑡_𝑗,𝑡_𝑗+1)  as 𝑡_𝑗→∞. Since𝑥^∗=lim_{𝑡𝑗→∞}𝑥(𝑡⁺_𝑗)=lim_{𝑡𝑗→∞}𝑥(𝑡⁺_𝑗)=0 then𝑥(𝑡)→0on [𝑡_𝑗,𝑡_𝑗+1]as𝑡_𝑗→∞. As a result and since𝑡_𝑗→∞,𝑥(𝑡)→0as𝑡→∞. The same property follows for the output so that𝑦(𝑡)→0as𝑡→∞, and𝑢(𝑡_𝑗)→0as𝑡_𝑗→∞, since ifcardIm_∞=𝜒₀, |𝑡_𝑗+1−𝑡_𝑗|<∞and𝑢(𝑡_𝑗)→0as𝑡_𝑗→∞, thenlimsup_{𝑡𝑗→∞}∫^𝑡𝑗+1_𝑡+𝑗𝑢²(𝜏)𝑑𝜏<∞so that𝑢(𝑡)→0as𝑡→∞since𝑢(𝑡)is almost everywhere piecewise continuous on its definition domain. It also follows from (12) and (C.3) that 𝑑−𝐾^𝑡₀𝑢²(𝜏)𝛾𝑑𝜏<∞,∞>2𝜋²−𝜀_𝑡(𝑊)𝛿(𝑊)^𝑡₀𝑢²(𝜏)𝑑𝜏_{𝑡𝑗∈Im(𝑡)},𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗<∞,∀𝑡∈𝐑₊ (D.4) so that (11) in Lemma 5 holds, in fact, if cardIm_∞≤𝜒₀and 𝑑≥𝐾≥0. The boundedness of the state and the output for all time follow directly from that of the input and the strict stability of the feed-forward system. The boundedness properties of the impulsive contributions (16) follow from Lemma 2 and (8)-(9), for both properties (i)-(ii).

E. Outline of Proof of Theorem 13

Most of the proof is close to that of Theorem 10. It has to be rearranged as the proof of the validity of (16). One has from (20), the modified property of the boundedness of the input-output energy measure (8) and resulting modified inequalities arising in the proof of Theorem 10 that 𝛾²≥𝑑−𝐾^𝑡₀𝑢²(𝜏)𝑑𝜏+^𝑡₀𝑦𝜂,𝜆,𝜏𝑢(𝜏)𝑑𝜏+_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗≥𝑚(𝑡)>0,∀𝑡∈𝐑₀₊,_𝑡0𝑦𝜂,𝜆,𝜏𝑢(𝜏)𝑑𝜏+_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗𝛾<∞∞>max2𝜋²−𝜀_𝑡(𝑊)𝛿(𝑊),𝐾≥𝑑−_𝑡0𝑢²(𝜏)𝑑𝜏;∀𝑡∈𝐑₀₊⟹^𝑡₀𝑢²(𝜏)𝑑𝜏<∞;∀𝑡∈𝐑₀₊∧lim_𝑡→∞𝑢(𝑡)=0. (E.1) Then,|∑_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔(𝑡_𝑗)𝑦(𝑡_𝑗)|forall𝑡∈𝐑₀₊ is bounded since sup_{𝑡∈𝐑0+}|𝑦(𝜂,𝜆,𝑡)| is bounded and _{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗−sup_{𝑡∈𝐑0+}𝑦𝜂,𝜆,𝑡^𝑡₀|𝑢(𝜏≤)|𝑑𝜏_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗−^𝑡₀𝑦𝑢(𝜏)≤𝜂,𝜆,𝜏𝑑𝜏_𝑡0𝑦𝜂,𝜆,𝜏𝑢(𝜏)𝑑𝜏+_{𝑡𝑗∈Im(𝑡)}𝐾_𝑗𝑔𝑡_𝑗𝑦𝑡_𝑗<∞,∀𝑡∈𝐑₊. (E.2) The rest of the proof follows “mutatis-mutandis” correspondingly to that of Theorem 10.

Acknowledgments

The authors are grateful to the Spanish Government for its support of this research through Grant DPI2012-30651 and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. They are also grateful to the University of Basque Country for its financial support through Grant UFI 2011/07. Finally, the authors are very grateful to the reviewers by their useful comments.