Asymptotic hyperstability of dynamic systems with point delays

It is proved that a linear time-invariant system with internal point delays is asymptotically hyperstable independent of the delays if an associate delay-free system is asymptotically hyperstable and the delayed dynamics is sufficiently small.


I. INTRODUCTION
Global Lyapunov's stability (asymptotic stability) configurations consisting of linear time-invariant systems in the forward loop with arbitrary nonlinear (and, perhaps time -varying) devices satisfying Popov's-type timeintegral inequalities is the so-called hyperstability (asymptotic hyperstability) property, [1][2][3]. The importance of the topic relies on the fact that the stability property holds for all nonlinearity satisfying Popov's inequality for all time. In this brief, related results are obtained when the linear plant is subject to a finite number of bounded incommensurate delays (i.e. the delays are not necessarily an integer multiple of a real number) if its associated dynamics is sufficiently small. The importance of such systems is well-known in many physical systems including problems such as war /peace or population growth models, large scale systems as well as in many engineering applications as, for instance, those involving transmission and telecommunication problems, [4][5][6][7][8]. The study of stability / hyperstability properties for systems involving external (i.e. in the inputs or outputs) delays may be addressed by direct extensions from the analysis methods concerning delay-free systems by simply transforming the relevant signals in new ones influenced by delays, [2], [4], [5]. However, the related problems become much more involved for the case of internal (i.e. in the state) delays since the relevant dynamics possesses infinitely many modes. Different techniques have been used to deal with the stability of such systems including Lyapunov's theory and its extensions as well as frequency analysis methods, [4][5][6][7][8].
In this manuscript, the asymptotic hyperstability of continuous time-delay systems is focused on for systems including any finite number of incommensurate internal point delays of arbitrary sizes provided that the plant free of delayed dynamics satisfies a strict positive realness condition. The obtained results are independent of the sizes of the delays provided that the delayed dynamics is normalizing scalar parameter. Such results are also independent of the delayed dynamics being structured or not. The above issues are the main novelties related to previous results. The manuscript organization follows in brief the following guidelines. First, a model for the linear plant is fixed with a free parameter to then deal with the amount of tolerated delayed dynamics being compatible with the asymptotic hyperstability. Also the class of admissible feedback nonlinear/ time-varying devices satisfying a Popov's type integral inequality for all time is defined. Subsequently, a preliminary result -Lemma 1 -is given concerned with the sufficiency-type conditions guaranteeing strict positive realness of the transfer function for all sizes of delays and an upper-bound of the absolute value of the above mentioned parameter. Then, the main asymptotic hyperstability result -Theorem 1-is derived based on the intuitive physical idea that, if the transfer function is strictly positive real, then the input/output energy is strictly positive and bounded for all time what implies the asymptotic vanishing of the relevant signals in the closed loop system.

II. FORMULATION
Consider the open-loop linear and time-invariant singleinput single-output system with an arbitrary number r of incommensurate internal point delays h > 0 ( i= 1, r) given by and y(t) are , respectively, the nstate vector and the scalar input and output and A, A (i= ,r), b E R n, ce R n and d E R; 6 is a scalar parameter which allows the characterization of the size of the delayed dynamics for given A ()-matrices.
The initial condition of (1) is any piecewise continuous function P[C  (1) is where Z (s) denotes Laplace's transform of z(t) with d . 0 if G(s) is biproper and d=O if G(s) is strictly proper. The asymptotic hyperstability of (1) is investigated assuming that the delay-free system z(t) = A z(t) + bu(t) satisfies a strict positiveness condition. Note, in particular, that if the proper rational transfer function of such a delay-free system, namely, G 0(s) = G(s,0) = (cT(sI-A ) 1 b+d ) is strictly positive real (denoted by G 0 E SPR ) then Re G 0 (s) > 0 if Re s . 0 and, furthermore, all the poles of including possible zero/ pole cancellations (if any) are strictly stable (i.e. located in Re s < 0), [9][10][11][12]. Thus, in Ro =R+U {O}=tseR:Res>O}. It is wellknown that this delay-free system is asymptotically hyperstable under the same output equation as in (1) and any feedback controller (2) if G 0 E SPR. The subsequent result addresses the problem of guaranteeing sufficiently small. This fact will then allow to prove the asymptotic hyperstability of (1) for any feedback (2), [3], [11].   (A, b) is stabilizable. Note that if d=O then G 0 (s) and G (s) are strictly proper and then at most (non strictly) positive real.
(2) Lemma 1 may be used with q=O implying c = P b.
557 (5) such that P < 6 a if 6 < -1~-< I -1-6 c I Thus, direct calculations using (6)-(7) as well as the odd symmetry of the involved frequency response curves lead to (7) 2ReG(jCo)=G (j)+GT(_ jco)=2d +m T (jw)(jII-A ) lb-bT(jI+A T )m (-jwo) 2[ ( max (P ) + a min ( P) ) b 1l 2 + C 1 1 2] where T(y)= L T(>jI-A ) b with L such that L = L T L > 0. The last identity of (8) is greater than zero if (5) holds and the proof follows directly. Proof: Define truncated time signals z t( ) = z ( t )for all t E [ 0, t ] and z t ( ) = 0, otherwise), of Fourier transform Z t ( p) ) for any vector signal z (t), of Fourier transform Z (j co ), for any time t > 0 and any frequency co. From Parseval's theorem relating the inputoutput energy in the time and frequency domains, (1)-(2), G E SPR fromLemma I and Im G (j o) = -Im G ( -j co ) for all cow, one gets for all t > 0: O< Inf(ReG(ico))Ju u(')d <j G (jwd) U t(jco) dwc=J Y(t jCo)U2 (-jco)2daw These expressions dictate that the input/output product time-integral (a measure of the input/output energy) is monotonically non-decreasing with time since it is positive and strictly positive for all t > 0 since Re G (jco) > 0 so that u(t) is bounded piecewise continuous on R + and converges asymptotically to zero as t -* oo except (potentially) over a set of zero measure of bounded discontinuities. Thus, y(t) -O0 ast-* oo and is bounded on R + since G E R H .. Now, since the nonlinear feedback function ( (.) eq. 2, is locally continuous around the equilibrium, y (t) -* 0 implies u (t) -* 0 as t o so that the above mentioned set of potential discontinuities of u(t) is empty after some finite time. As a result, both u(t) and y(t) converge asymptotically to zero. Thus, from Theorem 1, the feedback configuration (1)-(2) is asymptotically hyperstable for any nonlinear device satisfying the constraint in (2). The associate transfer function possesses a strictly stable zero/pole cancellation at s = -f which has not been taken into account in the above calculations. This is reasonable when the transfer function numerator and denominator are not factored explicitly from the state-space description especially for high order systems. If such a cancellation is known and removed for a minimum state-space realization of (1) resulting in A=-a, A 1 = -a , b=k, c=1 then ada 6 = from Lemma 1 with 0 al (da + 2k (a + l) a ) P = 1/k , q=0. In this simple example, the calculations may also be performed from the real part of the transfer function once the cancellation, if known, is removed. In this case, this leads to d>0, 6 0 = which is the a1 weakest found constraint. However, obtaining factored transfer functions from a state-space realization is not easy for high-order systems in the presence of delays. This fact justifies the adequacy of the proposed method to practical problems. The author is grateful to MCYT and UPV /EHUfor their partial support of this work.