Asymptotic Hyperstability and Input–Output Energy Positivity of a Single-Input Single-Output System Which Incorporates a Memoryless Non-Linear Device in the Feed-Forward Loop

Citation: De la Sen, M. Asymptotic Hyperstability and Input–Output Energy Positivity of a Single-Input Single-Output System Which Incorporates a Memoryless Non-Linear Device in the Feed-Forward Loop. Mathematics 2022,10, 2051. https://doi.org/ 10.3390/math10122051 Academic Editors: Lijun Pei and Youming Lei Received: 24 April 2022 Accepted: 12 June 2022 Published: 13 June 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Asymptotic Hyperstability and Input–Output Energy Positivity of a Single-Input Single-Output System Which Incorporates a Memoryless Non-Linear Device in the Feed-Forward Loop Manuel De la Sen Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country (UPV/EHU), 48940 Leioa, Bizkaia, Spain; [email protected] Abstract: This paper visualizes the role of hyperstable controllers in the closed-loop asymptotic stability of a single-input single-output system subject to any nonlinear and eventually time-varying controller within the hyperstable class. The feed-forward controlled loop (or controlled plant) contains a strongly strictly positive real transfer function in parallel with a non-linear and memory-free device. The properties of positivity and boundedness of the input–output energy are examined based on the “ad hoc” use of the Rayleigh energy theorem on the truncated relevant signals for finite time intervals. The cases of minimal and non-minimal state-space realizations of the linear part are characterized from a global asymptotic stability (asymptotic hyperstability) point of view. Some related extended results are obtained for the case when the linear part is both positive real and externally positive and for the case of incorporation of other linear components which are stable but not necessarily positive real. Keywords: positive realness; hyperstability; asymptotic hyperstability; passivity; sample and hold devices MSC: 93D10; 93D15; 93D20; 34D05 1. Introduction Positive realness is a very relevant property of linear systems. A positive real transfer function has non-negative real part on the closed complex right half-plane. It has a relative degree (that is, a pole-zero excess) of 0, +1, or − 1 [ 1 – 8 ]. Several ways and methods of designing such transfer functions in circuitry synthesis problems are given in [ 3 – 7 ]. Their design in the context of recursive parameter adaptation is focused on in [ 8 ]. In [ 9 ], the global asymptotic stability property is studied for a composite system with an asymptotically hyperstable subsystem. A consequence of positive realness of transfer functions is that the frequency response hodograph is confined within the first and four complex quadrants so that the maximum absolute phase of the frequency response is not larger than π/ 2. On the other hand, if the system is state-space realizable, then its transfer function is proper, that is, with not less than zero poles, so that its relative degree is either 0 (i.e., the transfer function is biproper, that is, it is proper with a proper inverse) or 1. Another property of such transfer functions is that they are stable, including the critical case, and so they are non-necessarily strictly stable, but eventual critical poles, if any, have to be single and with non-negative associated residuals. Furthermore, the inverses of positive real transfer functions are also positive real. On the other hand, the input–output energy of the systems described by positive real transfer functions is non-negative for all times. In this way, the positive realness of a transfer is associated with input–output energy dissipation for all times of the corresponding dynamic system. It has to be pointed out that positive realness does not directly imply the joint non-negativity of the input and the output through time, which Mathematics 2022,10, 2051. https://doi.org/10.3390/math10122051 https://www.mdpi.com/journal/mathematics Mathematics 2022,10, 2051 2 of 20 is the so-called external positivity property, which also implies that the non-negativity for all times of both the input–output power and the input–output energy. A particular subclass of that set of positive real transfer functions is that of the so-called strictly positive real transfer functions which are strictly stable, that is, without poles at the imaginary and whose real parts are strictly positive at the open right half-plane. Positive real transfer functions are very common in the description of classical circuitry involving tandems of resistors, capacitors and inductances. On the other hand, the class of non-linear and eventually time-varying hyperstable controllers is defined by the set of controllers which satisfy a so-called Popov’s type input– output integral inequality (referred to as Popov’s hyperstability condition of the whole class of controllers) [ 9 – 20 ]. In particular, the use of theory in different adaptive control problems is widely developed in [ 14 , 15 ] and some of the references therein. Its usefulness in qualitative behaviours of dynamic systems and in neural networks are focused on in [ 16 , 17 ], while the hyperstability in the discrete-time context is addressed in [ 18 ] for linear time-varying systems. The case of impulsive controls in hyperstability problems is focused on in [ 19 ]. On the other hand, an important stability property of the obtained closed-loop system is that a positive real transfer function under any controller belonging to the hyperstable class of controllers is “hyperstable”. What this means is that it is globally stable in the large (that is, for any given finite initial condition) in the Lyapunov’s sense. If the feed-forward transfer function is strictly positive real, then the closed-loop system is “globally hyperstable”, that is, globally asymptotically stable in the large. It can be pointed out that Popov’s hyperstability condition on the controller is also satisfied for more elementary static non-linear controllers invoked in the context of absolute stability (like the well-known Lur’e absolute stability problem within a Lur’e’s sector, Popov’s absolute stability criterion within a Popov’s sector, etc.) [ 20 – 22 ]. The concept of hyperstability is closely related to the more general one of dissipativity, or its particular version of passivity, through the above-mentioned positivity/boundedness properties of the input–output energy [ 23 – 27 ]. On the other hand, a variety of applications in different designs in the fields of mechanics, electric machinery, circuit synthesis, model reference adaptive control, and delayed systems has been performed. See, for instance, refs. [ 28 – 35 ] and some references therein for more details. Positive realness has been widely applied and linked to the hyperstability concept in adaptive control designs by taking advantage of the large universe of useful controllers, which allows a large flexibility in the design of the adaptive laws, the input/output filters to be used, and the family of free-design parameters of the estimation algorithm being compatible with the closed-loop stabilization. For similar reasons, they have been very popular for the synthesis of a wide set of regulators in electrical machinery problems. Basically, the hyperstability condition of the feedback part obtained under appropriate transformations and equivalence manipulations of the involved equations is used to get the adaptive law, which ensures the global stability of the whole scheme [ 14 , 15 , 33 , 34 ]. In [ 36 ], a double-convection system exhibiting chaotic behaviour with three nonlinearities is discussed, and its stability and dissipativity properties and their equilibria are investigated. On the other hand, in [ 37 ], a chaotic dissipative attractor with two quadratic nonlinearities, which possesses three unstable equilibrium points, is investigated. Its realization through an electronic circuit is also described. On the other hand, it is well-known that discretetime models are widely invoked in practical applications because of their flexibility for the design of appropriate controllers which do not need to pick up information for all times, but only at certain sampling instants, even if the controlled system is totally or, part of, a continuous-time nature. This fact allows for simplification of the whole design, and, in general, closed-loop stabilization is achievable anyway, as are the basic needed design performances [ 38 – 42 ]. In this context, this paper also gives some further ideas about hyperstability designs when the continuous-time control input to the continuous-time controlled plant is generated by sample and hold devices, which pick up input-registered values at previous sampling instants, which are used to generate the continuous-time input. Mathematics 2022,10, 2051 3 of 20 The main objective of this paper is to visualize the role of the class of hyperstable controllers in the closed-loop asymptotic stability of a single-input single-output system subject to negative feedback generated by, in general, a nonlinear and eventually timevarying controller. The controller is any element within a class which satisfies a Popov’s type time-integral inequality. The feed-forward loop consists of a strongly strictly positive real transfer function operating in parallel with, in general, a non-linear, memory-free device. The incorporation of such a device in the whole configuration, while guaranteeing the asymptotic hyperstability of the closed-loop system, is the main contribution of this work. Because of the intrinsic nature of the hyperstability concept, the global asymptotic stability in the large of the obtained closed-loop system is characterized for the whole class of controllers satisfying a Popov’s type integral inequality. Special attention is also paid to the properties of positivity and the uniform boundedness of the input–output energy of the feed-forward-loop for all times so that the controlled system has a dissipative nature. In particular, the minimum upper-bound of such an input–output energy is given for all times by the largest negative parameter, which bounds from below the time–integral Popov’s constraint that defines the class of hyperstable controllers. The main results are derived for the case that the state-space realization of the transfer function is minimal, that is, controllable and observable. There are also some further extensions of the main above results that dealt with the case of non-minimal realizations, which are stable, and for the case when the transfer function is weakly positive real, or, simply, (non-strictly) positive real. In this last case, the input–output energy is guaranteed to be non-negative and uniformly bounded for all times, and the closed-loop hyperstability is not asymptotic. Further related results are also obtained for the case when the linear part is both positive real and externally positive. In this case, the non-negative and boundedness properties of the input–output energy for all times are also fulfilled by the instantaneous input–output power, or passivity supply rate. The results are obtained for the mentioned devices being saturated and linear, while non-necessarily being proportional to the input, and nonlinear, including the constant, linear, and quadratic terms of the input. Some further results are also obtained for when the linear part of the system is a parallel connection of a strictly positive real transfer function with a strictly stable one, which has a sufficiently small resonance peak compared to the minimum (positive) value of the real part of its counterpart integrated with the mentioned linear tandem and connected in parallel. Further, some applications are developed for the case when the continuous-time input is generated from a very general sampling and hold device, which generates the current inter-sample input value, in general, from its two last previous sampled values. The paper is organized as follows: Section 2states the main results for a closed-loop system whose feed-forward loop is a linear system described by a strictly positive real transfer function operating in series with a bounded nonlinear operator on the input, and the feedback loop is any controller belonging to an hyperstable class defined by an integral-type, Popov’s-type hyperstability constraint. The closed-loop system is proven to be asymptotically hyperstable if the transfer function of the feed-forward loop is strongly strictly positive real. Other proven results are the integrability of the squared input and the squared output on the whole interval of time and the non-negativity and boundedness of the input–output integral energy. Some extensions are given in Section 3for: (a) weaker constraints related to weak strict positive realness on the transfer function; (b) the tandems of the strictly positive real transfer function with another strictly stable one which does not have, in general, positive realness properties; and (c) other alternative constraints on the cascaded nonlinear operator on the input combined with the above variants. In addition, in the case where the transfer function is only positive real but not strictly positive real, some further parallel conditions are obtained for the input and output to those in Section 2. In particular, some hyperstability conditions are proven if the transfer function is both externally positive and positive real. However, the asymptotic hyperstability property is not concluded, in general. Section 4develops some applications of the former theoretical Mathematics 2022,10, 2051 4 of 20 results to the case where the input is generated from a general sampling and hold device of speed correction, which generates the current inter-sample input value from the two last sampled values according to a correcting design coefficient, and, at the same time, it satisfies an “ad hoc” Popov’s-type hyperstability integral constraint. Finally, conclusions end the paper. Notation The following notation will be used through the manuscript: R0+=R+∪{0};R+={r∈R:r>0}, Z0+=Z+∪{0};Z+={z∈Z:z>0}, and C0+=C+∪{iR};C+={w∈C:Rew>0}, where R , Z , and C are the sets of real, integer, and complex numbers, respectively, the real set R can be extended, including the infinity points, to ¯ R=R∪{±∞} . In the same way, the extended ¯ R0+=R0+∪{+∞} , ¯ R+=R+∪{+∞} , and iR ={iω:ω∈R} are defined as the set of pure imaginary complex numbers, i=√−1 is the complex unit, ut is the truncation in the [0, t] of u:R→R , that is, ut(τ) = u(τ) if τ∈[0, t] and ut(τ) = 0 if τ∈(−∞, 0)∪(t,+∞) and (f∗h)(t) = R∞ −∞f(τ)g(t−τ)dτ , and ∀t∈R0+ is the convolution of f,h:R→R. If f,h:R→R, then (f∗h)(t) = Zt 0f(τ)g(t−τ)dτ=Z∞ −∞ft(τ)g(t−τ)dτ=Z∞ −∞f(τ)gt(t−τ)dτ=Z∞ −∞ft(τ)gt(t−τ)dτ =(ft∗h)(t) = (f∗ht)(t) = (ft∗ht)(t) =(ft∗h)t(t) = (f∗ht)t(t) = (ft∗ht)t(t) = (f∗h)t(t);∀t∈R0+, where ˆ g(s) and ˆ g(iω) are the Laplace and Fourier transforms of g:R0+→R , if they exist. The strictly positive real transfer functions (in the set SPR ), and, respectively, positive real transfer functions (in the set PR ) ˆ g(s) and s∈C are analytic in Re s ≥ 0 (respectively, in Re s > 0) and, if they are state-space realizable, then they have a relative degree (i.e., a pole-zero excess) of either unity or zero. The set SPR of the strictly positive real transfer functions is included in the set PR of (non-strict) positive real transfer functions, the first ones being strictly stable while those ones in the second set are required to be only stable. The set SSPR , a subset of SPR , is a set of strongly strictly positive real transfer functions of interest though the manuscript such that ˆ g∈SSPR if, and only if, Re ˆ g(s)> 0 for all Re s ≥ 0 and also for Re s →+∞ . In addition, the set of the so-called weakly strictly positive real transfer functions, WSPR [ 1 ], does not necessarily maintain the strict positive realness of Re s →+∞ and can be proper (that is, with the number of zeros not exceeding the number of poles), while not necessarily bi-proper (i.e., those being proper with a proper inverse, so with an identical number of poles and zeros). We note that the above sets possess the set inclusion properties SSPR ⊂SPR ⊂PR and WSPR ⊂SPR ⊂PR from more restrictive to less restrictive conditions. On the other hand, strictly positive real transfer functions are strictly stable, while positive real transfer functions can have single poles at the imaginary complex axis. The main of the above-mentioned sets of transfer functions for our central purposes in this paper is that of the strongly strictly positive real transfer functions SSPR, whose members have a strictly positive real part of the transfer function on the open right-halfplane. Such transfer functions are also bi-proper (that is, they have the same number of poles and zeros) and strictly stable (all poles are in Res < 0). We will refer to a strictly stable linear system as being one with all the poles of its transfer function in the open left-hand-side complex plane C0−={s∈C:Re s <0} , and we refer to a stable linear system as being one with poles in the closed left-handside complex plane. In the first case, the matrix of dynamics is a stability matrix whose Mathematics 2022,10, 2051 5 of 20 eigenvalues are such poles. In the second case, some eigenvalues can be allocated at the imaginary complex axis. We refer indistinctly to both of the above system stability concepts, as well to the respective transfer functions, as a strictly stable, or respectively stable, systems or transfer functions. 2. Problem Statement and Main Results It is well known that the Fourier transform ˆ g(iω)=F(g(t))=R∞ −∞g(t)e−iωtdt with ˆ g:iR→C of g:R0+→R exists if g is absolutely integrable on R and the Laplace transform ˆ g(s) = L(g(t))=R∞ 0g(t)e−(σ+iω)tdt is defined for a real σ≥σ0and some σ0∈R. For a given stable linear dynamic single-input (u(t)) single-output (y(t)) system of impulse response g(t) : ˆ g(s) = ˆ y(s)/ˆ u(s) is the so-called transfer function, which is the Laplace transform of g(t) , which equalizes the quotient of the Laplace transform of the output to the Laplace transform of the input under null initial conditions, and ˆ g(iω) is its so-called frequency response, which is the Fourier transform of g(t). Note that unstable linear systems can still be analysed though a Laplace transforms context, but not under a Fourier transform one. Remember also that ˆ g(s)∈SSPR if, and only if, Re ˆ g(s)> 0 for Re s ≥ 0. This property also implies that Re ˆ g(iω)> 0 and ∀ω∈ ¯ R (thus, Re ˆ g(iω)> 0, ∀ω∈R , and lim ω→±∞Re ˆ g(iω)> 0), and that ˆ g(s) is bi-proper (i.e., it has the same number of zeros and poles) and strictly stable, i.e., all its poles are in Re s <0. L∞ is the set of essentially bounded real functions on R and L2 is the set of squareintegrable functions on R . The functions considered in this paper are identically zero on the negative real semi-axis. Therefore, if essential boundedness and square-integrability, respectively, are proven to hold on R0+, then they are in L∞, respectively, in L2. For any given control u:R0+→R , the output of the controlled dynamic system P (or plant), under zero initial conditions, is: y(t) = (g∗u)(t) + W(ut);∀t∈R0+(1) where g:R0+→R is the impulse response of the linear part, which is the Laplace inverse transform of the transfer function ˆ g(s),W:R→Ris, in general, nonlinear, and u(t) = −v(t);∀t∈R0+(2) gives the control action under negative feedback of the hyperstable feedback controller K∈K (the class of hyperstable controllers) of input y(t) (that is, is, the output of the feed-forward controlled system) and output v:R0+→R , which is assumed to satisfy the subsequent Popov’s hyperstability input–output integral condition for some nonzero finite real constant γ0: Zt 0v(τ)y(τ)dτ≥ −γ2 0>−∞;∀t∈R0+(3) The definitions of hyperstability and asymptotic hyperstability in Popov’s sense follow below. See, for instance, refs. [9–15]. Definition 1. The controlled dynamic system P of an input–output relation defined by (1) is hyperstable if, for any control input satisfying the integral inequality (3), the zero-state solution x(t) for all t∈R0+ of any minimal state-space realization of n -th order of the linear part of (1) is globally stable in the large (that is, for any given finite initial condition x0∈Rn ) in the sense that the subsequent relation holds for some positive real constants δand K: kx(t)k ≤ K(kx(0)k+δ);∀t∈R0+ Mathematics 2022,10, 2051 6 of 20 Definition 2. The controlled dynamic system P of an input–output relation defined by (1) is asymptotically hyperstable if it is hyperstable in the sense of Definition 1 and, in addition, x(t)→0 as t →∞. The following features can be emphasized concerning the above hyperstability concepts : 1. The hyperstability (asymptotic hyperstability) property is a global Lyapunov stability (global Lyapunov asymptotic stability) property in the large (i.e., on the whole state space) for any control which satisfies (3), which defines a whole class of admissible controllers. Thus, it is not a global stability property for a particular control law, but it holds inherently for a whole class of controllers. The whole class of controllers can include linear and nonlinear static members, as well as time-varying ones, subject to the constraints of (3). If the feed-forward controlled plant is linear and time-invariant, it is well known that it has to be defined by a positive real (strictly positive real) transfer function in order to achieve the hypesrtability (asymptotic hyperstability) of the closed-loop system for the whole class of controllers satisfying the integral inequality of (3). 2. It has been common in the classical background literature to use the terminology that a closed-loop configuration (1)–(3) is hyperstable if both the feed-forward block (or controlled plant (1)) is hyperstable and the feedback loop (or the class of stabilizing controller (2) and (3)) is hyperstable as well. See, for instance [ 13 – 15 ] and some references therein. However, it can be pointed out that the class of stabilizing feedback controllers (in the hyperstability context) can include static members so that it can be preferable to refer to the hyperstability as a property of the plant under all controller members belonging to the hyperstable class of controllers. 3. Since hyperstability and asymptotic hyperstability are very wide classes of global Lyapunov’s stability, those properties can be characterized via Lyapunov function candidates. Exhaustive discussion on their associated Lyapunov function can be found in [ 13 – 15 ]. It can be pointed out as well that the hyperstability approach is not more general for stability characterization than the standard Lyapunov theory, but it allows for characterization of the stability for a whole class of controllers which satisfy and input–output integral constraint. This class contains eventually linear controllers, classes of static non-linear ones under a sector-type (Lur’e or Popov type) constraint, or eventually time-varying controllers. 4. In Definition 1, it is assumed that the state-space realization is minimal, that is, of minimum order of the state for the given transfer function, which implies that the transfer function has no zero-pole cancellation and the state-space realization is jointly controllable and observable. This constraint is not strictly necessary and some extensions under its removal will be given in Section 3. However, that minimality constraint helps to create an easy understanding of the property at a first glance since it becomes obvious that non-minimal realizations with eventual zero-pole unstable cancellations in the transfer function are not stable, and so they could never be hyperstable, either. 5. Some basic properties associated with hyperstability of a closed-loop configuration rely on the fact that the input–output energy of the feed-forward block is both nonnegative and uniformly bounded for all times. This is the main mathematical tool addressed in this research to obtain the given results. 6. The main objective of this study is to extend the asymptotic hyperstability property to the presence of certain nonlinear devices in the feed-forward loop, which are allocated in a series tandem with the linear time-invariant part, and to characterize the strong type of strict positive realness of the linear time-invariant part, leading to the asymptotic hyperstability of the closed-loop configuration. Assumption 1. The control u:R0+→R is admitted to have “a priori”, any number of finite bounded discontinuities, and a finite number of impulsive discontinuities. Mathematics 2022,10, 2051 7 of 20 The following result holds for a feed-forward controlled dynamic system under the class of hyperstable controllers K . It is concerned with sufficiency-type conditions for the positivity and boundedness of the input–output energy and asymptotic vanishing conditions of the input and output of Punder certain stipulations on g:R0+→R. Theorem 1. The following properties hold for any controller K ∈K: (i) The input–output energy is bounded for all time, that is, E(t) = Zt 0y(τ)u(τ)dτ≤γ2 0<∞;∀t∈R0+ (ii) Assume that ˆ g(s)∈SSPR and that W(u) is bounded. Then, ess sup t∈R0+|u(t)|<∞ and lim in f t→∞|u(t)|≤ sup u∈R|W(u)| in f ω∈ ¯ R0+ Re ˆ g(iω). (iii) If, in addition to the conditions of Property (ii), W(ut(τ))≥λ(t,u(t))u(τ) , ∀τ∈[0 , t] , and ∀t∈R0+ for some λ:R0+×R→R , subject to in f t∈R0+ λ(t,u(t))>−in f ω∈ ¯ R0+ Re ˆ g(iω) , then u∈L∞∩L2 and |u| ∈ L1 as well, so that u(t)→0 as t→∞ , except, eventually, on an interval of zero measure, and E(t)∈0 , γ2 0 and ∀t∈R0+ , that is, the input–output energy is non-negative bounded for all time. If, furthermore, u(t) has support on some real interval of nonzero measure S , then E(t)∈0, γ2 0 and ∀t≥t1> 0, where (0, t1) is the first connected component of S , that is, the input–output energy is jointly positive and bounded on [t1,∞). (iv) If, in addition to the conditions of Properties (ii)–(iii), W( 0 ) = 0, then y(t) is bounded, ∀t∈R0+ for any given finite initial conditions, and y(t)→0 as t→∞ . Furthermore, y∈L2 so that both |u|,|y| ∈ L∞∩L1∩L2. Proof. Note that (3) combined with (2) proves Property (i). Now, note that by using Rayleigh (or Parseval’s) energy theorem [ 13 , 19 , 43 ], it follows that R∞ 0y(τ)ut(τ)dτ=1 2πR∞ −∞ˆ y(iω)ˆ ut(−iω)dω , so that, from the symmetry property of the Fourier transform, one gets: +∞>γ2 0≥E(t) = R∞ 0y(τ)ut(τ)dτ=R∞ 0[(g∗u)(τ) + W(uτ)]ut(τ)dτ =1 2πR∞ −∞ˆ y(iω)ˆ ut(−iω)dω=1 2πR∞ −∞ˆ g(iω)ˆ ut(iω)ˆ ut(−iω)dω =1 2πR∞ −∞ˆ g(iω)|ˆ ut(iω)|2dω+R∞ 0W(ut(τ))(ut(τ))dτ ≥1 2πin f ω∈ ¯ R0+ Re ˆ g(iω)R∞ −∞|ˆ ut(iω)|2dω+R∞ 0W(ut(τ))(ut(τ))dτ;∀t∈R0+, (4) since ˆ g∈SPR implies that Re ˆ g(iω)≥d=in f ω∈ ¯ R Re ˆ g(iω)> 0 and since the hodograph ˆ g(iω) again has the symmetry property Re ˆ g(iω)=Re ˆ g(−iω) , and Im ˆ g(iω)=−Im ˆ g(−iω) and ∀ω∈ ¯ R (due to the symmetry of the Fourier transform), one has to infer from the above inequality and the Rayleigh energy theorem that: Mathematics 2022,10, 2051 8 of 20 ∞>γ2 0≥E(t)≥in f ω∈ ¯ R0+ Re ˆ g(iω)R∞ 0u2 t(τ)dτ+R∞ 0W(ut(τ))(ut(τ))dτ =in f ω∈ ¯ R0+ Re ˆ g(iω)Rt 0u2(τ)dτ+Rt 0W(ut(τ))(u(τ))dτ ≥in f ω∈ ¯ R0+ Re ˆ g(iω)Rt 0u2(τ)dτ−sup u∈R|W(u)|Rt 0|u(τ)|dτ;∀t∈R0+ (5) It is proven by contradiction that u(t) is essentially bounded. If we assume that it is not essentially bounded, then there is a strictly increasing sequence {ti}∞ i=0(⊂R0+) such that Zti 0u2(τ)dτ/Zti 0|u(τ)|dτ≥Mi for some strictly increasing sequence {Mi}∞ 0(⊂R0+)→∞as i→∞. Then, γ2 0 Rti 0|u(τ)|dτ≥ in f ω∈R0+ Re ˆ g(iω)Rt 0u2(τ)dτ Rti 0|u(τ)|dτ−sup u∈R|W(u)|≥in f ω∈ ¯ R0+ Re ˆ g(iω)Mi−sup u∈R|W(u)| so that, since W(u)is bounded, ∞>lim in f t→∞  sup u∈R|W(u)|+γ2 0 Rti 0|u(τ)|dτ−Miin f ω∈ ¯ R0+ Re ˆ g(iω)  ≥0, (6) a contradiction since in f ω∈ ¯ R Re ˆ g(iω)> 0 and {Mi}∞ 0(⊂R0+)→∞ as i→∞ . Thus, ess sup t∈R0+|u(t)|<∞. It is now proven that lim in f t→∞|u(t)|≤ sup u∈R|W(u)| in f ω∈ ¯ R0+ Re ˆ g(iω) . Assume, on the contrary, that there is some |u|> sup u∈R|W(u)| in f ω∈ ¯ R0+ Re ˆ g(iω)>0 such that lim in f t→∞|u(t)|=|u|. Then, γ2 0≥lim in f t→∞  in f ω∈ ¯ R0+ Re ˆ g(iω)|u|−sup u∈R|W(u)|  Zt+θ t|u(τ)|dτand ∀θ∈R0+(7) so that one gets the subsequent contradiction: ∞>γ2 0≥lim θ→∞  lim in f t→∞  in f ω∈ ¯ R0+ Re ˆ g(iω)|u|−sup u∈R|W(u)|  Zt+θ t|u(τ)|dτ  =∞. Then, either lim in f t→∞|u(t)|≤ sup u∈R|W(u)| in f ω∈ ¯ R0+ Re ˆ g(iω) or lim in f t→∞|u(t)|> sup u∈R|W(u)| in f ω∈ ¯ R0+ Re ˆ g(iω) and u(t)→0 as t→∞ , except eventually on a time interval of zero measure, the second condition Mathematics 2022,10, 2051 9 of 20 being a contradiction itself. Therefore, lim in f t→∞|u(t)|≤ sup u∈R|W(u)| in f ω∈ ¯ R0+ Re ˆ g(iω) . Property (ii) has been proven. Now, also assume that W(ut(τ))≥λ(t,u(t))u(τ) , ∀τ∈[0 , t] , and ∀t∈R0+ , so that one obtains from (5) ∞>γ2 0≥E(t)≥in f ω∈ ¯ R0+ Re ˆ g(iω)R∞ 0u2 t(τ)dτ+R∞ 0λ(t,u(t))u(τ)ut(τ)dτ ≥  in f ω∈ ¯ R0+ Re ˆ g(iω)+in f t∈R0+ λ(t,u(t))  Z∞ 0u2 t(τ)dτ =  in f ω∈ ¯ R0+ Re ˆ g(iω)+in f t∈R0+ λ(t,u(t))  Zt 0u2(τ)dτ;∀t∈R0+, (8) and, if, furthermore, in f t∈R0+ λ(t,u(t))>−in f ω∈ ¯ R0+ Re ˆ g(iω) , then, in addition to the previously proved properties ess sup t∈R0+|u(t)|<∞ and u∈L2 , then u∈L∞∩L2 , so that u(t)→0 as t→∞ , except eventually on a time interval of zero measure, and it follows from (8) that E(t)∈0 , γ2 0 and ∀t∈R0+ ; that is the, input–output energy is non-negative bounded for all times. If, in addition, u(t) has support on some real interval of nonzero measure S , then E(t)∈0, γ2 0 and ∀t≥t1 , where (0, t1) is the first connected component of S . Property (iii) has been proven. Property (iv) follows since ˆ g(s)∈SSPR, thus: (1) the bi-proper and strictly stable system of the form ˆ g(s) = ˆ g1(s) + d with d=in f ω∈ ¯ R0+ Re ˆ g(iω)> 0, since ˆ g1(s)∈PR is the proper of unity relative order with Re ˆ g1(s)≥0 for Re s ≥0 and lim |s|→∞Re ˆ g1(s) = 0; (2) W(ut(τ))≥λ(t,u(t))u(τ),τ∈[0 , t];∀t∈R0+; and (3) in f t∈R0+ λ(t,u(t))>−d then sup t∈R0+|y(t)|<∞ and y(t)→0 as t→∞ for any given finite initial conditions since: (a) It has already been proven that ess sup t∈R0+|u(t)|<∞ and u(t)→0 as t→∞ (except eventually on a set of zero measure), and furthermore, (b) for any eventually non-zero initial conditions x( 0 ) = x0 , the solution of (1) has an extra additive function y0:R0+→R0+ , which is bounded and exponentially vanishing, since ˆ g(s)is strictly stable and g(t)also asymptotically vanishes, so that |y(t)|=|(g∗u)(t)|+|W(ut)|+|y0(t)|≤y<∞;∀t∈R0+(9) lim t→∞y(t) = lim t→∞(g∗u)(t) + W(0) + lim t→∞y0(t) = 0; ∀t∈R0+(10) since W(0) = 0. Since |u| ∈ L∞∩L1∩L2, then |y| ∈ L∞∩L1∩L2. The following result is a direct extension of Theorem 1 if there are two linear strictly stable systems in parallel connection in the feed-forward loop, with one of them having strict positive realness properties. Corollary 1. Assume that the output of the system is y(t) = ((g+ga)∗u)(t) + W(ut);∀t∈R0+(11) Mathematics 2022,10, 2051 16 of 20 (a) if λc(t)≡ 0 and ∀t∈R0+ then the sampling and hold device is a zero-order hold (Z0H) and u(kT +τ)=uk,∀τ∈[0 , T), and ∀k∈Z0+; (b) if λc(kT +τ)=τ , ∀τ∈[0 , T) , and ∀k∈Z0+ , then the sampling and hold device is a first-order-hold (FOH) and u(kT +τ)=u(kT)+τ T(u(kT)−u[(k−1)T]) , ∀τ∈[0 , T) , and ∀k∈Z0+; and (c) if λc:R0+→(0 , T) , then the sampling and hold device is a speed correction hold (SCH): u(kT +τ)=u(kT)+λc(kT +τ) T(u(kT)−u[(k−1)T]),∀τ∈[0 , T), and ∀k∈Z0+. In the case where λc(kT +τ)=kcτ , ∀k∈Z0+ , for some kc∈(0 , 1) , ∀τ∈[0 , T) , then the SCH is of constant slope kc , and SCH( kc ). If kc∈(0 , 1) , then the device is named as a partial speed correction hold (PSCH). We now discuss the hyperstable design for the more general SCH sampling and hold device. Assume that feedback of the hyperstable feedback controller K∈Kdr where Kdr has an input y(t) (that is, it is the output of the feed-forward controlled system) and output v(t)(=−u(t)):R0+→R , which is assumed to satisfy the following continuous/discrete Popov’s hyperstability input–output integral condition: Zt 0y(σ)v(σ)dτ=Zt 0(−u(σ))y(σ)dσ =− k−1 ∑ j=0Z(j+1)T jT u(σ)y(σ)dσ+Zτ 0u(kT +σ)y(kT +σ)dσ! =−γ2 0+Zk−1 j=0Z(j+1)T jT ε2(σ)dσ+Zτ 0 ε2(kT +σ)dσ≥ −γ2 0=>−∞ t=kT +τ,∀k∈Z0+,τ∈[0, T], (24) for some arbitrary square-integrable γ,ε:R0+→R fulfilling 0 ≤Rt 0ε2(σ)dσ<Rt 0γ2(σ)dσ< γ2 0<+∞ ; ∀t∈R0+ , with (y(kT)=0)⇒[(ε(kT)=0)∧u(kT)=0] and |ε(kT)|<|y(kT)| if y(kT)6= 0, ∀k∈Z0+ , and γ2 0=lim t→∞Rt 0γ2(τ)dτ for some finite nonzero real constant γ0 . Note that γ(t),ε(t)→0 as t→∞ . The purpose of the auxiliary functions γ,ε:R0+→R is to guarantee (24) under equalities for all t∈R0+ . Equation (24) holds for all t∈R0+ , subject to (23), if: λc(kT +τ) T(u(kT)−u[(k−1)T]) +u(kT)y(kT +τ)=γ2(kT +τ)−ε2(kT +τ);∀τ∈[0 , T),∀k∈Z0+, (25) which, at the sampling instants, i.e., τ=0, becomes: u(kT)y(kT)=γ2(kT)−ε2(kT);∀k∈Z0+, (26) which, when replaced in (25), yields: h1+λc(kT+τ) Tγ2(kT)−ε2(kT) y(kT)−λc(kT+τ) T γ2[(k−1)T]−ε2[(k−1)T] y[(k−1)T]iy(kT +τ) =γ2(kT +τ)−ε2(kT +τ);∀τ∈[0 , T),∀k∈Z0+, (27) which ensures the particular needed constraint at the sampling instants: λc(kT)=γ2(kT)−ε2(kT)y(kT)−γ2(kT)−ε2(kT)y(kT) (γ2(kT)−ε2(kT))y[(k−1)T]+(ε2[(k−1)T]−γ2[(k−1)T])y(kT) Ty[(k−1)T] y(kT)=0 and ∀k∈Z0+, Mathematics 2022,10, 2051 17 of 20 which guarantees u(t) = u(kT)=γ2(kT)−ε2(kT) y(kT) if t=kT and ∀k∈Z0+ , provided that y(kT)6=0 and u(kT)=0 if y(kT)=0. Thus, the hyperstable controller is synthesized as follows: (a) If t=kT (sampling instants), then u(kT)=γ2(kT)−ε2(kT) y(kT) with |ε(kT)|<|y(kT)| if y(kT)6=0 and u(kT)=0, with |ε(kT)|=|y(kT)|if y(kT)=0 and ∀k∈Z0+. (b) If t6=kT (inter-sampling instants): u(kT +τ)=u(kT)+λc(kT +τ) T(u(kT)−u[(k−1)T]) and ∀k∈Z0+, λc(kT +τ)=γ2(kT +τ)−ε2(kT +τ)y(kT)−γ2(kT)−ε2(kT)y(kT +τ) (γ2(kT)−ε2(kT))y[(k−1)T]+(ε2[(k−1)T]−γ2[(k−1)T])y(kT) Ty[(k−1)T] y(kT +τ), ∀τ∈(0 , T), and ∀k∈Z0+ |ε(kT +τ)|<|y(kT +τ)|;∀τ∈(0 , T),∀k∈Z0+. Note from (27) that as {u(kT)}∞ k=0→0 , lim k→∞(|y(kT +τ)|−|y(kT +τ)|)= 0; ∀τ∈[0 , T)and ∀k∈Z0+. The control law of the form (23), obtained from an SCH sampling and hold device, which is within the class of hyperstable controllers subject to the integral Popov’s-type constraint (24), satisfies Theorem if it is the feedback loop of a transfer function ˆ g(s)∈SSPR in parallel with a non-linear device W(u)under the given hypotheses in the theorem. Remark 4. It can be pointed out that the Popovian hyperstability constraints of (24) [ 10 – 12 ] are more general because of the more general inputs generated from their sampled values than the parallel constraints associated with discrete values of the inputs and outputs being of the form ∑j j=0y(kT)v(kT)≥ −γ2 0>−∞ ; ∀k∈Z0+ . A parallel result to Theorem 1 and its corollaries could be easily obtained by applying the Rayleigh theorem on the unit complex circle and using the Z-transform of the impulse response of the linear part. Basically, the integrals of (4) would be changed to the sums ˆ g(iω)→ˆ gD(z) for z=eiθ and θ∈[0 , 2π) (then |z|= 1), with ˆ gD(z) = Z(1−e−Ts)g ˆ(s) s where 1−e−Ts s is the transfer function of the ZOH. It turns out that d=in f ω∈ ¯ R0+ ˆ g(iω)=in f θ∈[0 ,2π) ˆ gDeiθ> 0since ˆ g(s)∈SSPR , which is then also bi-proper and strictly stable, that is, ˆ ga(z) and ˆ g(s) have an identical input–output interconnection gain, which is the quotient of the leading coefficients of both their numerator and denominator polynomials. However, the discretization approach addressed in this way has only information at the sampling instants, rather than for all time, and it is of interest only for the use of discretization under a ZOH and not for more general sampling and hold devices. Even in this case, note that the output is not fully addressed in the input–output energy formulas of Theorem 1 and the later corollaries since the output is not piece-wise constant. 5. Conclusions The paper has investigated the asymptotic hyperstability of a single-input singleoutput closed-loop control configuration whose feed-forward loop consists of a parallel connection of a strongly strictly positive real transfer function, together with (in general) a non-positive nonlinear operator which has to satisfy some discussed conditions. The feedback loop consists of, in general, a nonlinear and, perhaps, time-varying controller which satisfies a Popov-type integral inequality. The global asymptotic stability is proven to be “in the large”, that is, it is guaranteed for any given finite initial condition, and the asymptotic hyperstability property implies that the closed-loop asymptotic stability is guaranteed independently of the particular controller employed within the above class. Mathematics 2022,10, 2051 18 of 20 The property is addressed by proving, trough Parseval’s theorem, that the input–output energy of the feed-forward loop is always positive and bounded for all times. Extra sufficiency-type conditions to keep the asymptotic hyperstability property are obtained under the incorporation of an additional strictly stable linear and time-invariant system. In particular, and in order to keep the hyperstability properties of the whole closed-loop configuration, its frequency response resonance gain, which is sufficiently small, is related to the minimum value of the real part of the impulse response associated with the strongly strictly positive real transfer function. A case study is provided, which is concerned with the use of a fractional sampling and hold device to generate the continuous-time input from their sampled values at a constant sampling rate. Funding: This research was funded by the Spanish Government and the European Commission, grant number RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and by the Basque Government, grant number IT1207-19. The APC was funded by grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE). Acknowledgments: The author is grateful to the Spanish Government and the European Commission for its support through grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and to the Basque Government for its support through grant IT1207-19. He is also grateful to the Referees by their useful suggestions and comments. Conflicts of Interest: The author declares no conflict of interest. References 1. Hakimi-Moghaddam, M. A survey on positive real and strictly positive real scalar transfer functions. Int. J. Math. Comput. Sci. 2016,10, 57–63. 2. Fernandez-Anaya, G.; Flores-Godoy, J.-J.; Rodríguez-Palacios, A. Properties of strictly positive real functions: Products and compositions. Int. J. Syst. Sci. 2010,41, 457–466. 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