# On the Asymptotic Hyperstability of Linear Time-Invariant Continuous-Time Systems under a Class of Controllers Satisfying Discrete-Time Popov’s Inequality

Authors: M. De la Sen
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper is concerned with the property of asymptotic hyperstability of a continuous-time linear system under a class of continuous-time nonlinear and perhaps time-varying feedback controllers belonging to a certain class with two main characteristics; namely, (a) it satisfies discrete-type Popov’s inequality at sampling instants and (b) the control law within the intersample period is generated based on its value at sampling instants being modulated by two design weighting auxiliary functions. The closed-loop continuous-time system is proved to be asymptotically hyperstable, under some explicit conditions on such weighting functions, provided that the discrete feed-forward transfer function is strictly positive real.


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## Body

## 1. Introduction

Continuous-time and discrete-time positive systems have been studied in detail in recent years [1–10]. If the state and output possess the positivity property under nonnegative initial conditions and controls, the positivity is said to be internal or, simply, the system is positive. If the output possesses such a property, the system is said to be externally positive. Thus, positive systems are intrinsically interesting to describe some problems like Markov chains, queuing problems, certain distillation columns and biological, and other physical compartmental problems where populations or concentrations cannot be negative [2, 3]. It is well known that time-invariant dynamic linear systems which are externally positive, while having positive real or strictly positive real transfer matrices, are, in addition, hyperstable or asymptotically hyperstable, i.e., globally Lyapunov stable for any nonlinear and/or time-varying feedback device satisfying Popov’s type inequality for all time, [11, 12]. The converse assertion is not generically true in the sense that a hyperstable linear system, then characterized by a positive real transfer function, is not necessarily an externally positive one. In particular, hyperstable linear systems can have a positive instantaneous input-output power and a positive input-output energy measure for all time. Thus, they have identical signs of the input and output for all time instants while they are not externally positive if those signals are not everywhere nonnegative. Thus, external positivity of a SISO system is not implied by the positive realness of its transfer function. The property of asymptotic hyperstability generalizes that of absolute stability [13–15] which generalizes the most basic concept of stability of dynamic systems. See, for instance [2, 3, 11, 13–26], and references therein. It is well known that closed-loop hyperstability is, by nature, a powerful version of closed-loop stability since it refers to the stability of a hyperstable linear feed-forward plant (in the sense of the positive realness of the associated transfer matrix) under a wide class of feedback controllers applied. The above important properties make very attractive potential research issues for kind of more complex dynamic systems with applied projection including those lying in the class of continuous/digital hybrid systems. On the other hand, the class of hybrid systems consisting of continuous-time and discrete-time (or digital) systems are of an increasing interest since many existing industrial installations combine both kinds of systems. An elementary well-known case is when a discrete-time controller is used for a continuous-time plant. Another case is related to teleoperation systems where certain variables evolve in a discrete-time or digital fashion. A background literature and related relevant results are given in [1, 7, 11, 17, 18, 27–30] and some of the references therein.Some conditions have been given in the background literature to guarantee the positive realness of discrete transfer functions from related conditions on their continuous-time counterparts which are then maintained under discretization. See, for instance [31]. The results are useful to guarantee the hyperstability of discrete-time systems under the hyperstability of the continuous-time ones when the class of feedback discrete-time controllers satisfies Popov’s type inequality. However, it turns out that positive realness of a continuous-time system does not guarantee the positive realness of its discretized transfer function by a zero-order hold for any sampling period and, conversely, the strong positive realness of a discretized transfer function does not guarantee in some limited cases the strong positive realness of its continuous-time counterpart, [31–33]. This paper is devoted to a certain kind of inverse problem stated in the following terms. Given is an asymptotically hyperstable discrete-time system of sampling period T such that its feed- forward transfer function is strictly positive real and the nonlinear time-varying feedback controller belongs to a class ΦdT satisfying certain discrete Popov’s type inequality. The elementary question that arises is how the asymptotic hyperstability of the continuous-time system is guaranteed for a certain modified class of controllers such that its member at sampling instants is in the class ΦdT? To answer this question, an intersample controller based on the class ΦdT at sampling instants is designed which corrects the control law along the intersample period via the concourse of two modulating continuous-time functions which have to satisfy certain reasonable constraints. The paper is organized as follows. Firstly, a notation and terminology subsection is allocated in the subsequent section. Section 3 presents the linear and time-invariant system of the feed-forward part of the closed-loop system together with its analytic output expressions some auxiliary results of positivity and boundedness for all the sampling instants of an input-output energy measure. The continuous-time control input is proposed to be generated within each intersample period from the last sampling instant defining the starting point of the current intersample period and some design auxiliary functions which modulate the control signal along such an intersample period. Some of the formulas presented are derived in detail in the appendix. On the other hand, some auxiliary preparatory lemmas concerning the input-output energy measure are also formulated which are of usefulness for the main result. Section 4 is devoted to obtain the main asymptotic hyperstability result which is proved based on some given and previously proved auxiliary preparatory lemmas. Basically, it is proved that the continuous-time closed-loop system is asymptotically hyperstable provided that (a) the discretized feed-forward transfer function is strictly positive real, (b) the discretized system at sampling instants is asymptotically hyperstable for all nonlinear and time-varying feedback controllers which satisfy certain Popov’s type inequality at sampling instants, and (c) some extra additional conditions on smallness and boundedness are fulfilled by the abovementioned auxiliary modulated functions which define the controls along the intersample periods. Such conditions make any continuous-time controller belonging to the appropriate class to satisfy continuous-type Popov’s inequality for a class of controllers provided that the former one at sampling instants is fulfilled by the discretized controllers. Some examples are discussed in Section 5. Finally, conclusions end the paper.


## 2. Notation

(i)R+=R0+∪0, Z+=Z0+∪0, R0+=r∈R:r≥0, and Z0+=z∈Z:z≥0.(ii)cl⁡R=-∞,+∞ is the closure of the real field, that is, the set of real numbers together with the ± infinity points.(iii) The continuous and discrete-time argumentst∈R0+ and k∈Z0+ are denoted with parenthesis and brackets, that is, t and k, respectively.(iv) IfT>0 is the sampling period then ut=uk+u~t;∀t∈kT,k+1T;∀t∈kT,k+1T;∀k∈Z0+ is the continuous-time input, where the piecewise-continuous intersample incremental input is u~t=ut-uk with uk=ukT and u~k=u~kT=0;∀t∈kT,k+1T;∀k∈Z0+. The auxiliary purely discrete input is udk, such that udk=uk;∀k∈Z0+ if u~t=0;∀t∈R0+.(v)yht;∀t∈R0+ and yhk=yhkT;∀k∈Z0+ are the forced and homogeneous (unforced) output of the system, respectively.(vi)ydhk=ydhkT and ydfk=ydfkT;∀k∈Z0+ are the forced and homogeneous output of the system, respectively, when the input is piecewise-constant with constant value in-between each two consecutive sampling instants, i.e., when the input is generated from a zero- order-hold (ZOH) device.(vii) The Fourier transform of any real vector functionft is denoted by f^iω if it exists;∀t,ω∈R, where i=-1 is the imaginary complex unit.(viii) For the purpose of formally using Fourier transforms for the subsequent developments, we define continuous-time and discrete-time truncated signals from their untruncated counterparts on the respective continuous-time and discrete-time intervals0,t and 0,kT as follows:(1)vtτ=vτif0≤τ≤t0ifτ>t0ifτ<0;∀t,τ∈R0+,vkj=vjif0≤j≤k0ifj>k0ifj<0;∀j,k∈Z0+.(ix) Lets and z=e-Ts the Laplace and z-transforms (for sampling period T) complex arguments. The set SPR is the set of strictly positive real continuous g^s (respectively, discrete) transfer functions g^z; i.e., they are (strictly) stable, i.e., with poles in Re⁡s<0 (respectively, z<1) which satisfy Re⁡g^s>0 for all Re⁡s≥0 (respectively, Re⁡g^z>0 for all z≥1). These conditions imply that Re⁡g^iω>0 , respectively, Re⁡g^eiω>0;∀ω∈R, [13, 31, 34]. The strictly positive continuous and discrete real sets are not distinguished at the level of notation since they are easy to identify them according to context.(x) The so-called setSSPR is the set of (continuous or discrete depending on context) strongly positive real transfer functions which are those in SPR whose real part is strictly positive also as s→∞. It can be pointed out that Szëgo-Kalman-Popov Lemma (also so-called Discrete Positive Real Lemma) is a discrete version of the Yakubovich-Kalman-Popov Lemma (also so-called Positive Real Lemma) which relates positive realness of transfer functions to associated state-space realization properties states that discrete transfer function in SPR are of relative degree (namely, pole-zero excess) equal to zero so that they are also in SSPR and both sets are equivalent. This equivalence concern between the discrete sets SPR and SSPR does not apply to continuous transfer functions, [13, 31].


## 3. The Continuous-Time and Discrete-Time Linear Time-Invariant Systems and Some Auxiliary Results

Consider annth-order linear and time-invariant dynamic system in state-space description:(2)x˙t=Axt+but;x0=x0∈Rn,yt=cTxt+dut,where xt∈Rn, ut∈R, and yt∈R are the state, piecewise-continuous control input, and output and A∈Rn×n, b∈Rn, c∈Rn, and d∈R are the matrix of dynamics, control vector, output vector, and input-output interconnection gain. The auxiliary purely discrete output and the continuous-time and sampled output are calculated from (2) via the impulse responses gt and gdk which become(3)gt-τ=cTeAt-τb+dδt-τ;∀τ,t≥τ∈R0+gdk-j=∫jTkTgkT-τdτ=∫0k-jTgk-jT-τdτ=∫jTkTcTeAkT-τb+dδkT-τdτ=∫0k-jTcTeAk-jT-τb+dδk-jT-τdτ;∀j≤k,k∈Z0+and gt-τ=0;∀τ,t<τ∈R0+, where δt-τ is the Dirac distribution, u-t=0;∀t∈R+and gdk-j=0;∀j>k,k∈Z0+ with gd0=d. The initial input and output conditions are uj=yj=0 for any integer j<0 and yt=u-t=u~-t=0;∀t∈R+ in causal dynamic systems subject to initial conditions x0=x0. Let the control input be(4)ut=uk+u~t=udk+u~dt;∀t∈kT,k+1T;∀k∈Z0+.Then, the auxiliary purely discrete output becomes under piecewise-constant input with eventual finite jumps at sampling instants:(5)ydk=∑j=-∞∞∫jTj+1TgkT-τdτudkj+ydhk;∀k∈Z0+.The continuous-time output becomes(6)yt=∑j=-∞∞gdk-jukj+∫kTtgt-τdτuk+∫-∞∞gt-τu~tτdτ+yht;∀t∈kT,k+1T;∀k∈Z0+while the sampled real output becomes(7)yk=ykT=∑j=-∞∞gdk-jukj+∫-∞∞gkT-τu~kTτdτ+yhk;∀k∈Z0+.Expressions (5)–(7) are derived in the appendix.Remark 1.
Note that the difference between the sampled continuous-time sequenceyk and the purely discrete output sequence ydk is that the first one is the exact value of the output at sampling instants including the effects of the intersample input ripple while the second one is the discretized output at sampling instants in the presence of a zero-order hold. As a result, note that yk=ydk for any given k∈Z+ if and only if ∫0kTgkT-τu~τdτ=0. In particular, yk=ydk if u~t=0;∀t∈R0+. Causality implies that (6)-(7) can be rewritten as(8)yt=∑j=0∞gdk-jukj+∫kTtgt-τdτuk+∫0∞gt-τu~tτdτ+yht;∀t∈kT,k+1T;∀k∈Z0+,yk=ykT=∑j=0∞gdk-jukj+∫0∞gkT-τu~kTτdτ+yhk;∀t∈kT,k+1T;∀k∈Z0+.The following result is direct.Lemma 2.
Any given piecewise-continuous control input can be decomposed into purely discrete-time control plus incremental intersample period ones as follows:(9)ut=uk+u~t=udk+u~dt=1+λtuk+σt;∀t∈kT,k+1T;∀k∈Z0+through piecewise-continuous functions λ,σ:∪k∈Z0+kT,k+1T→R defined by (10)λt=u~tukifuk≠00ifuk=0,σt=0ifuk≠0u~tifuk=0.The purely auxiliary input-output and true ones energy measures on the discrete-time interval0,kT as well as the continuous-time ones on 0,t;∀t∈kT,k+1T and ∀k∈Z0+ are, respectively, from (5)–(7) and (9)-(10), given by(11)Edk=∑n=0kTydnudn=∑n=0k∑l=0nT∫lTnTgnT-τdτudludn=∑n=0k-1∑l=0nTgdnn-ludnludk-1n+Tgd0ud2k;∀k∈Z0+(12)Ek=∫0kTyτuτdτ=∑j=0k-1∑i=0j∫jTj+1T∫iTi+1Tgττ-θ1+λjTθuji+σθ1+λk-1Tτuk-1j+στdθdτ+Tgd0u2k;∀k∈Z0+(13)Et=∫0tyτuτdτ=Ek+∫kTt∫0τgτ-θ1+λθuk+σθ1+λτuk+στdθdτ=Ek+∫kTt∫0τgττ-θ1+λτθutk+στθ1+λτutk+σττdθdτ;∀t∈kT,k+1T;∀k∈Z0+Remark 3.
Since the control input is piecewise-continuous, the truncated inpututτ for τ∈0,t has Fourier transforms for any t∈R0+ so that the corresponding truncated output ytτ for τ∈0,t and also the energy measure have also Fourier transforms for any t∈R0+. This follows from the fact that these signals are calculated via truncated functions which are then zero at infinity so that they are square-integrable and their sequences of sampled values of sampling period T are square-summable as a result.The subsequent two simple auxiliary lemmas will be then used in the next section to establish the main result.Lemma 4.
Assume that there exist constantsγd0,γd>γd0∈R+ such that γd0≤Edk≤γd;∀k>k0∈Z+ and some k0∈Z0+. Then, there exist γ0,γ>γ0∈R+ such that γ0≤Et≤γ;∀t∈k0+1T,∞ if(14)γ0-γd≤E~dkT,t=Ek-Edk+E~kT,kT+τ=Et-Edk=E~dkT,t≤γ-γd0;∀k>k0∈Z+;∀τ∈0,T,where(15a)Et=Ek+E~kT,t=Edk+E~dkT,t;∀t∈kT,k+1T(15b)E~kT,kT+τ=∫kTt∫0τgτ-θ1+λθuk+σθ1+λτuk+στdθ;∀τ∈0,T(15c)E~dkT,kT+τ=Ek-Edk+E~kT,t;∀k>k0∈Z+;∀τ∈0,T.Proof.
Note from (13), (15a)–(15c), and (11)-(12) that(16)E~dkT,t=Et-Edk=Ek-Edk+E~kT,t=∑j=0k-1∑i=0j-1∫jTj+1T∫iTi+1Tgτ-θ1+λjTθuj-1i+σθ1+λk-1Tτuk-1j+στdθdτ-∑n=0k-1∑l=0nT∫0n-lTgn-lT-τdτulun+∫kTt∫0τgτ-θ1+λθuk+σθ1+λτuk+στdθdτ+∑j=0kTyjuj-ydjudj;∀t∈kT,k+1T,∀k∈Z0+ so that (14) implies that γ0≤Et≤γ;∀t∈k0+1T,∞ provided that γd0≤Edk≤γd;∀k>k0∈Z+ for k0∈Z0+.Lemma 5.
Assume thatφx,t is subadditive for all x∈R, i.e., φx+y,t≤φx,t+φy,t; ∀x,y∈R,∀t∈R0+. Then, φyt,t≤φyk,k+φyt-yk,t,kT, and as a result u~t≥φyt-yk,t,kT and φyk,k≥-φ-yk,k;∀t∈kT,k+1T,∀k∈Z0+.Ifφx,t is superadditive for all x∈R, i.e., φx+y,t≥φx,t+φy,t;∀x,y∈R,∀t∈R0+, then, φyt,t≥φyk,kT+φyt-yk,t,kT, as a result u~t≤φyt-yk,t,kT and φyk≤-φ-yk;∀t∈kT,k+1T,∀k∈Z0+.Proof.
From the first identity in (9), if φx,t is subadditive then(17)-φyt=ut=uk+u~t≥-φyk+yt-yk≥-φyk-φyt-yk=-φyk-φyt+-yk≥-φyk-φyt-φ-yk;∀t∈kT,k+1T,∀k∈Z0+so that(18)φyt≤φyk+φyt-yk;φyk≥-φ-yk;∀t∈kT,k+1T,∀k∈Z0+.If φx,t is superadditive then(19)-φyt≤-φyk+yt-yk≤-φyk-φyt-yk=-φyk-φyt+-yk≤-φyk-φyt-φ-yk;∀t∈kT,k+1T,∀k∈Z0+so that(20)φyt≥φyk+φyt-yk,φyk≤-φ-yk;∀t∈kT,k+1T,∀k∈Z0+.


## 4. The Main Result

Definition 6 (Popov’s inequality).
A continuous-time feedback nonlinear and eventually time-varying continuous-time controllerut=-φyt,t and ∀t∈R0+, where φ:R0+×R0+→R0+ is in the class Φ, denoted as φ∈Φ, is said to satisfy Popov’s type integral inequality if, for some γ0∈R+, one has(21)∫0tyτφyτ,τdτ≥-γ0;∀t∈R0+Definition 7 (discrete-time Popov’s inequality).
A discrete-time feedback nonlinear and eventually time-varying discrete-time controllerut=uk=ukT=-φykT,kT, ∀t∈kT,k+1T, and ∀k∈Z0+, where φ:R0+×Z0+→R0+ is in the class ΦdT of sampling period T∈R+, denoted as φ∈ΦdT, is said to satisfy a Popov’s type discrete-time inequality if, for some γd∈R+, one has(22)∑j=0kykφk=∑j=0kykTφkT≥-γdT;∀k∈Z0+I. D. Landau refers to controllers satisfying a Popov’s type inequality (21) (or, respectively, (22)) as hyperstable controllers of class Φ (or, respectively, of class ΦdT), [13], bearing in mind that if any controller of such a class is coupled to a linear time-invariant forward system of transfer function being continuous-time (or, respectively, discrete-time) strictly positive real then the overall closed-loop system is asymptotically hyperstable, namely, globally asymptotically stable for any arbitrary controller belonging to the respective class. It can be pointed out that the above comments referred to discrete-time systems and, in particular Definition 7, are also applicable to digital systems, in the same way as they are applicable to discrete-time systems, i.e., those which are fully described in the discrete domain without having specific links to a discretization process on a certain continuous-time system.It can be pointed out that Definition7 can be also established for a digital system not being related to the time discretization of a continuous-time system in terms of Popov’s inequality of the following form: (23)∑j=0kykφk≥-γda;∀k∈Z0+for some constant γda∈R+ for any digital controller of class Φd. In this context the sampling period either can be nonrelevant—then nonmade notational explicit in the class Φd—or can even have nonsense since the inequality is applied to a certain discrete sequence.The following result obtains useful expressions for the discretized input-output energy measure under some conditions concerning the subadditivity constraints on the controllers of classΦdT. The result is also a preparatory one for the next Lemma 9 which addresses the positivity and boundedness of the input-output energy measure at sampling instants.Lemma 8.
The following properties are fulfilled:
(i) The subsequent formulas hold:(24)yk=∑j=0∞∫jTj+1T1+λkj+ξkjgk-jTdτuj+∫0∞gkT-τσkTτdτ+yhk;∀k∈Z0+(25)Ek=∑l=0kylul=∑l=0k∑j=0l∫jTj+1T1+λlj+ξljgl-jTdτ+∫0lTglT-τσlTτdτuj+yhlul=∑l=0∞yklukl=∑l=0∞∑j=0l∫jTj+1T1+λlj+ξljgl-jTdτ+∫0∞glT-τσlTτdτuj+yhlul;∀k∈Z0+,where(26)ξkj=∫jTj+1TλjT+τgk-jT-τ-λjgk-jTdτ;∀j≤kk∈Z0+.
(ii) Assume thatφ∈ΦdT and that φx,kT is subadditive for all x∈R, i.e., φx+y,kT≤φx,kT+φy,kT;∀x,y∈R,∀k∈Z0+. Then, the subsequent relations are true if the sequence εk is defined by εk=λk+ξk;∀k∈Z0+:(27)ydhk+∫0∞glT-τσkTτdτ≤∑j=0∞∫jTj+1Tεkjgk-jTdτφkyj,j+∑j=0∞∫jTj+1Tgk-jTdτφkydj,j+φk-ydj,j;∀k∈Z0+(28)Ek≤∑l=0∞∑j=0l∫jTj+1T1+εkgl-jTdτujul+∑l=0∞∑j=0l∫jTj+1Tgk-jTdτφlydj,j+φl-ydj,jφlydl,j+φl-ydl,l-ydhk-∫0∞glT-τσkTτdτ;∀k∈Z0+Proof.
One has(29)yk=∫0kTgkT-τuτdτ+yhk=∑j=0k∫0kTgk-jTdτuj+∫0kTgt-τu~τdτ+yhk=∑j=0k∫jTj+1Tgk-jTdτuj+∑j=0k-1∫jTj+1TgkT-τλτuj+στdτ+yhk=∑j=0k-1∫jTj+1Tgk-jTdτuj+gd0uk+∑j=0k-1∫jTj+1TgkT-τλτuj+στdτ+yhk=∑j=0k-1∫jTj+1Tgk-jT+λjT+τgk-jT-τdτuj+gd0uk+∑j=0k-1∫jTj+1TgkT-τστdτ+yhk=∑j=0k-1∫jTj+1T1+λjTgk-jTdτuj+∑j=0k-1∫jTj+1TλjT+τgk-jT-τ-λjTgk-jTdτuj+gd0uk+∫0kTgkT-τστdτ+yhk;∀k∈Z0+Then, since λkk=σkk=0;∀k∈Z0+ then ξkk=0;∀k∈Z0+; it follows that(30)yk=∑j=0k∫jTj+1T1+λkj+ξkjgk-jTdτuj+∫0kTglT-τσkTτdτ+yhk;∀k∈Z0+which yields (24) and leads to (25). Now, define the sequence εk by εk=λk+ξk and ξkj=∫jTj+1TλjT+τgk-jT-τ-λjgk-jTdτ;∀k∈Z0+ and note that(31)
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+and then(32)-yk-ydk-yhk-∫0∞glT-τσkTτdτ≤∑j=0∞∫jTj+1Tgk-jTdτφkyj-ydj,j+∑j=0∞∫jTj+1Tεkjgk-jTdτφkyj,j≤∑j=0∞∫jTj+1Tgk-jTdτφkyj,j+φk-ydj,j+∑j=0∞∫jTj+1Tεkjgk-jTdτφkyj,j≤-∑j=0∞∫jTj+1Tgk-jTdτukk-φk-ydj,j-∑j=0∞∫jTj+1Tεkjgk-jTdτukk=-∑j=0∞∫jTj+1T1+εkjgk-jTdτukk+∑j=0∞∫jTj+1Tgk-jTdτφk-ydj,j;∀k∈Z0+and one gets (27), since yhk=ydhk;∀k∈Z0+, from the following relation:(33)ydhk+∫0∞glT-τσkTτdτ≤-∑j=0∞∫jTj+1Tεkjgk-jTdτukk+∑j=0∞∫jTj+1Tgk-jTdτφkydj,j+φk-ydj,j;∀k∈Z0+Finally, (28) follows from (25) and (27).The subsequent result establishes lower and upper-bounds for the discretized input-output energy measures based on the formulas obtained in Lemma8 and strict positive realness of the discrete-time transfer function of the feed-forward linear and time-invariant system. The result invokes the boundedness and the smallness of the parameters which define the calculation of the intersample control input from their preceding values at sampling instants. Note that the control deviation in the intersample periods related to the sampling time instants has to be sufficiently moderate enough so that the hyperstability property is kept from the discrete-time system to the continuous-time one. In particular, the positivity and boundedness of the energy measure at the sampling instants is guaranteed if supt∈R0+⁡σt is small enough and ΦdT is subadditive.Lemma 9.
Assume thatg^d∈SPR and that φ∈ΦdT.
Then, the following properties hold:
(i) The input-output energy system measure of the auxiliary discretized system satisfies(34)0<γ0da≤T2πminω∈R0+⁡Re⁡g^de-iωT∑k=0∞ud2k≤Edk=∑j=0kTydkudk≤γda;∀k>k0∈Z+for some γ0da,γda>γ0da∈R+ and for any nonidentically zero controls and zero initial conditions, i.e., for any forced solution.
(ii) Assume, in addition, that the classΦdT consists of subadditive functions. Assume also that supt∈R0+⁡σt is sufficiently small, infk∈Z0+⁡εk>-1 and supk∈Z0+⁡εk<∞, where εk=λk+ξk;∀k∈Z0+, and that, furthermore,(35)supj,l≤j∈Z0+⁡φlydj,j+φl-ydj,j≤φ-<+∞;∀φ∈ΦdTThen, the input-output energy measure of the auxiliary discretized system satisfies(36)0<γ0d≤T2πminω∈R0+⁡Re⁡g^de-iωT∑k=0∞u2k≤Ek=∑j=0kTykuk≤γd;∀k>k0∈Z+for some γ0d,γd>γ0d∈R+ and for any nonidentically zero controls and zero initial conditions.Proof.
The upper-bounding constraint of (34) follows since(37)Edk=∑j=0kydkφydk,kT=∑j=0kydkTφydkT,kT≥-γdaT;∀k∈Z0+for some γda∈R+ from the first assumption of the theorem and udk=-φydk,kT; ∀k∈Z0+, since Popov’s inequality (22) is fulfilled by any given discrete-time feedback controller φ∈ΦdT generating a feedback control udk=-φydk,kT;∀k∈Z0+. On the other hand, the initial conditions can be neglected since the assumption g^d∈SPR, implying that the discrete-time transfer function is (strictly) stable which makes their values irrelevant for purposes of stability analysis. Then, the discrete frequency response being applicable to piecewise-constant inputs in-between any two consecutive sampling instants is given by (38)g^de-inω=∑n=0∞gdne-inTω=∑n=-∞∞gdne-inTωsince the subsequence g-n=0; ∀n∈Z+ and the transfer frequency response of a sampling and zero-order-hold operator Z of period T defined by ZT,vt=vkT=vk for any given v:R0+→R and all t∈kT,k+1T, k∈Z0+ is given by ZT,iω=1-e-iTω/iω [31, 35]. It turns out that ZT,iω=1-e-iTω/iω<1;∀ω≠0∈R and ZT,i0=1. Thus, the auxiliary discretized system satisfies the subsequent equivalence relation in the frequency domain to the discrete-time relation (11). It turns out that Fourier transforms exist in the impulse responses and truncated control and auxiliary functions of (11)–(13) since the truncated functions are square-integrable in cl⁡R=-∞,+∞. By using the discrete Parseval’s theorem under zero initial conditions one has(39)Edk=T2π∑n=-∞∞∫-ππgdne-inTωu^dkiωu^dk-iωdω=T2π∑n=-∞∞∫-ππgdne-inTωu^dkiω2dω=T2π∫-ππg^de-inωu^dkiω2dω=12π∫-∞∞1-e-iTωiωg^iωu^dkiω2dω;∀k∈Z0+The lower-bounding constraint of (37) follows by using (39) and the discrete Parseval’s theorem for the equivalence between the input-output energy measures from the frequency domain to the discrete-time domain(40)Edk≥T2πminω∈R0+⁡Re⁡g^de-iωT∫-ππu^dkiω2dω=T2πminω∈R0+⁡Re⁡g^de-iωT∑k=0∞ud2k≥γ0d;∀k∈Z0+from the assumption g^d∈SPR; i.e., minω∈R0+⁡Re⁡g^de-iωT≥γ0da>0;∀k0,kk≥k0∈Z+ since(a)
strictly positive real discrete transfer functions have zero relative degree (i.e., an identical number of poles and zeros) from the Discrete Positive Real Lemma (Szëgo-Kalman-Popov Lemma) [13, 31, 34], so that the real parts of their frequency hodographs in the argument e-iωT are positively lower-bounded;(b)
Im⁡g^de-iωT=-Im⁡g^deiωT;∀ω∈R0+ so that their integrals in the argument ω on frequency intervals -kπ,kπ for any k∈Z+ are null.
Property (i) has been proved. To prove Property (ii), note from (28) [Lemma 8(ii)] that(41)0<Ek≤1+supk∈Z0+⁡εkγda+∑l=0∞∑j=0l∫jTj+1Tgk-jTdτφlydj,j+φl-ydj,jφlydl,j+φl-ydl,l-ydhk-∫0∞glT-τσkTτdτ≤γd=Kγda;∀k∈Z+for some K∈R+, provided that supt∈R0+⁡σt is sufficiently small, infk∈Z0+⁡εk>-1, and supk∈Z0+⁡εk<∞, where εk=λk+ξk;∀k∈Z0+ since gt is bounded and converges exponentially to zero since gd∈SPR and then convergent (that is, strictly stable in the discrete context), ydhk→0, and it is a bounded sequence.The next result, which is a preparatory result to then establish the main asymptotic hyperstability result, addresses the relevant property that the input-output measureEt of the continuous-time system is positively lower-bounded and finitely upper-bounded for all time under the conditions of Lemmas 4 and 9.Lemma 10.
Ifλ- and σ- are sufficiently small then (42)γ0-γd≤Et-Edk≤1+λ-22π∫-∞∞g^iωu^kiω2dω+σ-2λ-sup0≤j≤k⁡uj2+σ-∫0kTgkT-τdτ≤γ-γd0;∀k>k0∈Z+;∀τ∈0,TThen, γ0≤Et≤γ; ∀t∈k0+1T,∞ for some k0∈Z0+ and some γ0,γ>γ0∈R+ subject to (43)+∞>γ≥γd0+1+λ-22π∫-∞∞g^iωu^kiω2dω+σ-2λ-sup0≤j≤k⁡uj2+σ-∫0kTgkT-τdτ0<γ0≤γd-λ-22π∫-∞∞g^iωu^kiω2dω-σ-2λ-sup0≤j≤k⁡uj2+σ-∫0kTgkT-τdτ-T2π∑n=-∞∞∫-ππgdne-inTωu^dkiω2dωprovided that γd0≤Edk≤γd;∀k>k0∈Z+.Proof.
One gets the following from direct calculations by using the continuous-time control laws (9)-(10) and (16) if λ-=supt∈R0+⁡λt and σ-=supt∈R0+⁡σt:(44)-λ-22π∫-∞∞g^iωu^kiω2dω-σ-2λ-sup0≤j≤k⁡uj2+σ-∫0kTgkT-τdτ≤Et-EdkT,t=12π∫-∞∞y^iωu^-iωdω-EdkT,t=12π∫-∞∞g^iωu^iω2dω-EdkT,t=∑j=0k∑i=0j∫jTj+1T∫iTi+1Tgτ-θλjTθuji+σθλk-1Tτuk-1j+στdθdτ+12π∫-∞∞g^iωu^kiω2dω-T2π∑n=-∞∞∫-ππgdne-inTωu^dkiω2dω≤1+λ-22π∫-∞∞g^iωu^kiω2dω-T2π∑n=-∞∞∫-ππgdne-inTωu^dkiω2dω+σ-2λ-sup0≤j≤k⁡uj2+σ-∫0kTgkT-τdτ;∀t∈kT,k+1T,∀k∈Z0+Now, the use of (44) in Lemma 4, (14), implies that Lemma 9(ii), (36), holds implying the existence of γ0,γ>γ0∈R+ such that γ0≤Et≤γ;∀t∈k0+1T,∞, provided that γd0≤Edk≤γd for γd0,γd>γd0∈R+ since γd0≤Edk≤γd;∀k>k0∈Z+ and some k0∈Z0+ if(45)γ0-γd≤-λ-22π∫-∞∞g^iωu^kiω2dω-σ-2λ-sup0≤j≤kuj2+σ-∫0kTgkT-τdτ≤Et-Edk≤1+λ-22π∫-∞∞g^iωu^kiω2dω+σ-2λ-sup0≤j≤kuj2+σ-∫0kTgkT-τdτ≤γ-γd0;∀k>k0∈Z+;∀τ∈0,Tif λ- and σ- are sufficiently small such that (43) holds leading to γ0>0 for any given positive γd0 and γd>γd0.The main first result follows.Theorem 11.
Consider the dynamic system (2) subject to a control:(46)ut=1+λtuk+σt;∀t∈R0+uk=-φyk,k;∀k∈Z0+,whereφk∈ΦdT.Assume that
(1)φyk,kT is in the class ΦdT (that is, φyk,kT∈ΦdT) defined by the discrete Popov’s type sum inequality ∑l=0kydlul≥-γd/T for some γd∈R; ∀k∈Z0+;
(2)g^d∈SPR; there exist constants γd0,γd>γd0∈R+ such that γd0≤Edk≤γd;∀k>k0∈Z+ and some k0∈Z0+;
(3)φx,kT is subadditive for all x∈R;∀k∈Z0+;
(4)λ-=supt∈R0+⁡λt and σ-=supt∈R0+⁡σt are sufficiently small;
(5)infk∈Z0+⁡εk>-1 and supk∈Z0+⁡εk<∞, where εk=λk+ξk;∀k∈Z0+ with (47)ξkj=∫jTj+1TλjT+τgk-jT-τ-λjgk-jTdτ;∀k∈Z0+Then, the following properties hold:
(i) The discrete closed-loop system is asymptotically hyperstable.
(ii)ut→0 as t→∞ and the control and output are bounded for all time.
(iii) Assume, furthermore, thatσt=φyt,t-1+λtφyk,kT, ∀t∈kT,k+1T, ∀k∈Z0+ut=-φyt,t, and ∀t∈R0+ and that the class Φ satisfies Popov’s type integral inequality for all time and results in the class ΦdkT at sampling instants. Then, the continuous- time closed-loop system is asymptotically hyperstable.Proof.
From Lemma9 via assumptions (1)–(5), constraints (36) hold so that the input- output energy measure associated with the forced solution is positively lower-bounded and finitely upper-bounded for all sampling instants t=kT; ∀k∈Z+. Since the discrete positive transfer function is strictly positive real then it is also strictly stable in the discrete context (i.e., convergent [29, 36]) so that the homogeneous response is irrelevant for stability analysis purposes. Thus, φk∈Φd, uk→0, and uk and yk are bounded for all time for any given bounded initial conditions. In addition, one has from Lemma 10 that (42) holds and the forced input-output energy measure is positively lower-bounded and finitely upper-bound for all t∈R+. Then, ut→0 as t→∞ and ut and yt are bounded for all time for any given bounded initial conditions. Properties [(i)-(ii)] have been proved. Property (iii) follows by equalizing ut=-φyt,t=1+λtuk+σt and φ∈Φ from the fact that -∫0tφyτ,τyτdτ=∫0tuτyτdτ≤γ0.Note that the use of Fourier transforms is useful to characterize the error of the input-output energy measure in the frequency domain along the intersample periods related to its values at the preceding sampling instants. This tool is addressed in Lemma10 which is used in the proof of Theorem 11. It allows to keep easily the characterization of the frequency response of the continuous-time transfer function to evaluate the input-output energy measure in the intersample period and to calculate the minimum value of the real part of such a transfer function to guarantee the positivity and boundedness of such an energy. Note that the characterization of the input-output energy measure of the feed-forward linear loop evaluated in the frequency domain, together with Popov’s inequalities of the class of hyperstable controllers, is relevant to the hyperstability proofs of the closed-loop system.


## 5. Examples

Example 12.
The positive realness of a discretized transfer function depends not only on the continuous-time parameters but on the sampling period as well. Therefore, the positive realness of the continuous-time system does not guarantee that of any discretized counterpart irrespective of the sampling period and conversely. For instance, the transfer functiong^s=1/s+a is strictly positive real for a>0 but it is not strongly strictly positive real since Re⁡g~iω→0 as ω→±∞. Its discretized version without the concourse of an ideal, that is instantaneous, sampler, and zero-order hold, becomes sg^dz=z/z-e-aT which is strongly positive real for T∈0,∞ and as the sampling period T→∞ since g^dz→1 as T→∞. However g^dz=z/z-1 as T→0. Since it has a critically stable pole in the discrete domain it is positive real although it is not strictly positive real.Example 13.
Consider the discrete transfer functiong^dz=z2+d1z+d2/z2-a1z-a2 with real numerator and denominator coefficients. Its strict positive realness requires as necessary conditions its convergence (i.e., strict stability in the discrete sense), so that its poles have to be within the complex open unit circle as well as the convergence of its inverse, so that, its zeros have also to lie inside the open unit circle and, furthermore, its relative degree (i.e., its pole-zero excess has to be zero). Note that the last condition is not needed for strict (nonstrong) positive realness of continuous transfer functions where their positive real part can converge to zero as s→∞. By using the homographic bilinear transformation z=1+s/1-s, which transforms the unit circle in the left-hand-side complex plane, to get the auxiliary continuous transfer function g^as=g^dz=1+s/1-s, it is found in [13] that the denominator and numerator polynomials of g^dz are subject to a2-a1<1, a1+a2<1, and a2>-1. Take the particular values a1=0.5 and a2=0.4 leading to the values of poles z1=0.5 and z2=0.4 and the coefficients of the numerator polynomial are fixed to d1=1.2, d2=0.3. Then, we fix a discrete transfer function g^dz=z2+1.2z+0.3/z2-0.5z-0.4 yielding minz=1⁡Re⁡g^dz≥0.09, so g^d∈SSPR. The decomposition of g^dz in simple fractions leads to g^dz=1+1.68/z-0.93+0.758/z+0.43. By using an ideal (i.e., with instantaneous sampling) sampler and zero-order-hold device of period T, of transfer function 1-e-Ts/s, on a continuous transfer function g^s to be specified to generate g^dz one has the identity Z1-e-Tsg^s/s=g^dz, where ZFs is an abbreviate usual informal notation to denote F∗s=Lf∗t, where L(·) and Z(·) denote the Laplace transform of (·), of argument s, and Z-transform of (·), of argument z=eTs, and f∗t=∑n=0∞fnTδt-nT denotes the pulsed function of the function ft with sampling period T. As a result, we get for a sampling period T=0.1 sec.:(48)g^s=sZ-1zz-11+1.68z-0.93+0.758z+0.43=1+0.0298s+0.726+15.92s+8.44=s2+25.086s+17.966172s2+9.166s+6.1274Note that g^∈SPR (and also g^∈SSPR since its relative degree is zero) is the continuous-time transfer function which generates g^dz under an ideal sampler and zero-order-hold device of period T=0.1 sec. so that minω∈cl⁡R⁡g^iω=minω∈R0+⁡g^iω>0. Thus, it is guaranteed that the continuous-time energy measure is associated with the forced output Et>0 for all t>0 (i.e., it is positively lower-bounded for all time) for any piecewise-continuous control irrespectively of the particular controller in operation. In order to guarantee that any such a control that converges asymptotically to zero it is necessary to finitely upper-bound such an energy for all time. The above concerns also guarantee that Ek=EkT≥γd0>0 for all integer k>0 under any piecewise-constant control uk=ukT with eventual jumps at sampling instants of any period T>0. For such a control, identical conclusion arises from the discrete g^dz which is also strongly strictly positive real. Assume that uk=-φyk,k and ∀k∈Z+ with φ∈Φd, where the class Φd is defined as the set of sequences φyk,k which satisfy Popov’s inequality T∑j=0kykφyk,k≥-γd;∀k∈Z+ for some finite γa∈R+ which is identical to (49)Ek=T∑j=0kykφyk,k≤γd<+∞;∀k∈Z+A valid choice to parameterize the discrete Popov’s inequality is the following one:(50)T∑j=0kykφyk,k=-TM∑j=0kνk+∑j=0kεk=-TM1-νk+11-ν-T∑j=0kεk≥-γd;∀k∈Z+for some real constants M>0, ν∈0,1 and some real summable sequence εk. Then, the class Φd is defined by any real sequence φyk,k being parameterized by any real constants M>0, ν∈0,1 and any real summable sequence εk such that(51)uk=-φyk,k=Mνk+εyk,kykifyk≠00ifyk=0;∀k∈Z0+ and γd≥max⁡TM/1-ν+ε-,γ- for some given γ-≥γd0, where ε-=Tmaxk∈Z+⁡∑j=0kεk. The continuous-time control is generated as(52)ut=1+λyt,tuyk,k+σyt,t;∀t∈R0+Theorem 11 guarantees that the continuous-time controller (53)ut=1+λtuk+σt;uk=-φyk,k;∀t∈kT,k+1T,∀k∈Z0+,where φk∈ΦdT, satisfies Popov’s type discrete inequality, which makes the feedback system globally asymptotically stable under the theorem hypothesis. Thus, the discrete-time closed-loop system is asymptotically hyperstable. In particular, under the additional assumptions in Theorem 11(iii), the closed-loop continuous-time system is asymptotically hyperstable.One of the referees has given a comment of interest on this example related to continuous versus discrete counterpart properties which has motivated us to give a further intuitive discussion on it. Note the following. In the continuous-time case, there are two kinds of strictly positive real transfer functions as it is well known, namely, the so-called strict positive real which can be zero (or, respectively, positive) at infinity frequency and which, a realizable, has relative degree one (or, respectively zero). In the first case, the direct input-output interconnection gaind is zero in a state-space representation (A,b,c,d=0) of the transfer function. The transfer functions of the second case are often referred to as the class of strong strict positive real transfer functions. This class is not a disjoint class of first above one of strict positive real transfer functions since it is included in it as a potential particular case. Note that the members of this strong class of strict positive real transfer functions have relative degree zero and a positive direct input-output interconnection gain d. Related to this concern, the Yakubovich-Kalman-Popov (refereed to the continuous-time domain) and the Szëgo-Kalman-Popov (referred to the discrete domain) lemmas can be invoked to compare positive realness of the transfer functions with their state-space associated properties (in fact, the properties are mutually equivalent from both sides). Thus, we see that, in the discrete case, the Szëgo-Kalman-Popov lemma needs, for its own coherency, that the input-output interconnection d-gain be nonzero since; otherwise, one of the relevant constraint in the lemma fails, [13]. So, strict positive realness in the discrete domain in this context is typically strong and then the relative degree of its transfer function is zero so that there is a direct interconnection positive gain from the input to the output. Since we assume that the discrete system is strictly positive real, it has a positive input-output interconnection from the input to the output and the minimum value of the real part of its frequency response is strictly positive even as the frequency tends to infinity. On the other hand, note that a zero-order hold “converts” any strictly proper (continuous-time) transfer function of any relative degree into a strictly discrete one of unity relative order while it “converts” a continuous one of zero relative order (i.e., biproper) into a discrete one still of zero-relative order and with the same input-output interconnection gain. See (48). Therefore, we deal in this example with a biproper strict positive real transfer function of positive input-output interconnection gain identical to that of its discrete-time counterpart got through a ZOH sampling and hold device.Example 14.
Consider that a controller of the formut=-φyt,t; ∀t∈R0+ is used for the continuous-time feed-forward transfer function (48) so that φ∈Φ and Φ is class in Example 13 satisfying an integral type Popov’s inequality (21) for all t≥t0∈R+ and some finite t0∈R+. That is, the control function belongs to a class satisfying continuous-time integral Popov’s inequality. Then, ut→0 as t→∞ and ut and yt are almost everywhere bounded for all time since g^∈SSPR, then being (strictly) stable with input-output energy measure being positively lower-bounded and also finitely upper-bounded for all t∈R+ from Popov’s-type integral inequality. Potential useful controllers are, for some given controller gain ku∈R+,(54)ut=0fort∈0,1kut2ytfort≥1(55)ut=kue-tytfort≥0(56)ut=0fort∈0,2kutln2⁡tytfort≥2(57)ut=0fort∈0,1kuettytfort≥1(58)ut=kut3e-tytfort≥0,where Popov’s inequality on 0,+∞ of (49)–(52) is ∫0∞ytφyt,tdt=-ku, that of (57) is ∫0∞ytφyt,tdt=-2ku/e, and that of (58) is ∫0∞ytφyt,tdt=-2ku/e. Since the respective integrands are monotonically decreasing then there is some t0∈R+ and some γ=γt0∈R+ such that ∫t0tytφyt,tdt≥-γ for all t≥t0∈R+. The obtained closed-loop systems are in all cases asymptotically hyperstable. The piecewise-constant controls with eventual discontinuities at sampling instants generated from any discrete-time controller (51) of class Φd of Example 13 lead also to asymptotic hyperstability of the discretized system since g^d∈SSPR.Example 15.
Consider the continuous-time open-loop system (2) under a real sampler of sampling period T and data picking-up nonzero duration ε∈0,T. The state and output trajectory solutions are(59)xkT+σ=eAσxkT+∫0min⁡σ,εeAσ-τbukT+τdτykT+σ=cTxkT+σ+dukT+σfor σ∈0,T;∀k∈Z0+. In particular,(60)xk+1T=eATxkT+∫0εeAT-τbukT+τdτ=eATxkT+∫0εeAT-τbdτuk+∫0εeAT-τbu~kT+τdτ=eATxkT+1-ρ∫0TeAT-τbdτuk+∫0εeAT-τbu~kT+τdτ;∀k∈Z0+,where ρ=∫εTeAT-τbdτ/∫0TeAT-τbdτ and u~kT+τ=ukT+τ-uk for τ∈0,T,∀k∈Z0+. Then, if ΦT=eAT and ΓT=∫0TeAT-τbdτ and provided that the control is continuous in the open intersample periods kT,kT+ε;∀k∈Z0+, one has by applying the mean value theorem(61)yk=cTekATx0+1-ρcTzI-ΦT-1ΓT+d1-ρuk+εcT∑j=0kek-jATeAT-ξjbu~jT+ξj=cTekATx0+cTzI-ΦT-1ΓT+d+dρ1-ρuk-ρcTzI-ΦT-1ΓT+d+dρ1-ρuk+εcT∑j=0k-1eAk-jT-ξjbu~jT+ξj=cTekATx0+∑j=0k1-ρgdk-juj+ρduk+εcT∑j=0k-1eAk-jT-ξjTbu~jT+ξj;∀k∈Z0+ for some real sequence ξk such that ξk∈kT,kT+ε, where g^dz is the discretized transfer function under a sampler and zero-order-hold device subject to instantaneous ideal sampling. Now, assume that the incremental control along intersample periods is generated from its preceding sampled value as u~jT+τ=λjT+εjT+τuj for any j∈Z0+ and τ∈0,T, where λjT+εjT+τ is a truncated function under the usual notation; one gets(62)yk=cTekATx0+∑j=0k1-ρgdk-juj+ρduk+εcT∑j=0k-1ek-jAT∫0TλjT+ετe-Aτbdτuj;∀k∈Z0+The above equations imply that if g^d∈SPR, then the forced output yfk=yk-cTekATx0;∀k∈Z0+ generates an associate input-output energy measure which is strictly positive for k∈Z0+ since ρ∈0,1 for real sampling of duration ε∈0,T. This is a necessary condition for asymptotic hyperstability under an appropriate class of controllers satisfying Popov’s inequality. However, the incremental input-output energy measure generated by the incremental forced output εcT∑j=0k-1ek-jAT∫0TλjT+ετe-Aτbdτuj is not guaranteed to be nonnegative in general. A sufficient condition for that is that limsupj→∞⁡esssupτ∈0,ε⁡λjT+ετ is sufficiently small related to the amount (1-ρ/ε)minω∈0,π⁡Re⁡g^deiω.


## 6. Conclusions

This paper has been devoted to obtain and to prove the asymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given and previously proved auxiliary preparatory lemmas. Basically, it is proved that the continuous-time closed-loop system is asymptotically hyperstable under the assumptions that the discretized feed-forward transfer function is strictly positive real and that the discretized system at sampling instants of constant sampling period is asymptotically hyperstable for all nonlinear and time-varying feedback controllers which satisfy certain Popov’s type inequality at sampling instants. It is assumed that the continuous-time controller belongs to a class defined in the following way: (a) its values at sampling instants are defined by a class of controllers which satisfy discrete Popov’s type inequality; (b) the intersample control is generated from its values at sampling instants with the use of two modulating auxiliary functions which are subject to some extra additional conditions on smallness and boundedness. Such conditions make the overall continuous-time controller to satisfy continuous-type Popov’s inequality for all time.



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*Source: 7861396-2018-06-04.xml*