Time-delay Robustness Analysis for Systems with Negative Degree of Homogeneity*

10th IFAC Symposium on Nonlinear Control Systems 10th IFAC Symposium on Nonlinear Control Systems August 23-25, 2016. Monterey, California, USA 10th IFAC IFAC Symposium on Nonlinear Nonlinear Control Systems 10th Symposium on Control Systems August 23-25, 2016. Monterey, California, USA Available online at www.sciencedirect.com August 23-25, 2016. Monterey, California, USA August 23-25, 2016. Monterey, California, USA

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Time-delay Robustness Analysis for Time-delay Robustness Analysis for Time-delay Robustness Analysis for Time-delay Robustness Analysis for Systems with Negative Degree of Systems with Negative Degree of Systems with Negative Degree of Systems with NegativeDegree of Homogeneity Homogeneity Homogeneity  Homogeneity

Konstantin Zimenko ∗∗ , Denis Efimov ∗,∗∗,∗∗∗ ∗,∗∗,∗∗∗ , Konstantin Zimenko ,, Denis Efimov ,, , ∗,∗∗,∗∗∗ ∗,∗∗,∗∗∗ ∗ ∗∗,∗∗∗ ∗ Konstantin Zimenko Denis Efimov Andrey Polyakov , Wilfrid Perruquetti ∗,∗∗,∗∗∗ , Denis Efimov ∗,∗∗,∗∗∗ ∗∗,∗∗∗ Konstantin Zimenko , Andrey Polyakov , Wilfrid Perruquetti ∗,∗∗,∗∗∗ ∗∗,∗∗∗ Andrey Polyakov Polyakov ∗,∗∗,∗∗∗ ,, Wilfrid Wilfrid Perruquetti Perruquetti ∗∗,∗∗∗ ,,, Andrey ∗ ∗ Department of Control Systems and Informatics, ITMO University, Department of Control Systems and Informatics, ITMO University, ∗ ∗ Department of Systems and ITMO University, 49 Kronverkskiy av., 197101 Petersburg, Russia Department of Control Control SystemsSaint and Informatics, Informatics, ITMO (e-mail: University, 49 Kronverkskiy av., 197101 Saint Petersburg, Russia (e-mail: 49 Kronverkskiy av., 197101 Saint Petersburg, Russia (e-mail: [email protected]). 49 Kronverkskiy av., 197101 Saint Petersburg, Russia (e-mail: [email protected]). ∗∗ [email protected]). Parc Scientifique de la Haute Borne, 40 ∗∗ Non-A team, INRIA-LNE, [email protected]). ∗∗ Non-A team, INRIA-LNE, Parc Scientifique de la Haute Borne, 40 ∗∗ Non-A team, INRIA-LNE, Parc de Borne, Halley, 59650 Villeneuve d’Ascq, France (e-mail: Non-Aav. team, INRIA-LNE, Parc Scientifique Scientifique de la la Haute Haute Borne, 40 40 av. Halley, 59650 Villeneuve d’Ascq, France (e-mail: av. Halley, 59650 Villeneuve d’Ascq, France (e-mail: [email protected], [email protected], [email protected], Halley, 59650 Villeneuve d’Ascq, France (e-mail: [email protected], [email protected], [email protected], [email protected]). [email protected], [email protected], [email protected]). ∗∗∗ [email protected]). 9189), Ecole Centrale de Lille, BP 48, Cit´e ∗∗∗ CRIStAL (UMR-CNRS [email protected]). CRIStAL (UMR-CNRS 9189), Ecole Centrale de Lille, BP 48, Cit´ee ∗∗∗ ∗∗∗ CRIStAL (UMR-CNRS 9189), Ecole Lille, 59651 Villeneuve-d’Ascq, France. CRIStALScientifique, (UMR-CNRS 9189), Ecole Centrale Centrale de de Lille, BP BP 48, 48, Cit´ Cit´e Scientifique, 59651 Villeneuve-d’Ascq, France. Scientifique, Scientifique, 59651 59651 Villeneuve-d’Ascq, Villeneuve-d’Ascq, France. France. Abstract: Time-delay robustness analysis for homogeneous systems with negative degree is Abstract: analysis homogeneous systems with negative degree is Abstract: Time-delay robustness analysis for homogeneous systems with negative degree is presented inTime-delay the paper. Itrobustness is established that iffor a system is homogeneous and Abstract: Time-delay robustness analysis for homogeneous systems with withnegative negativedegree degree is presented in the paper. It is established that if a system is homogeneous with negative degree and presented in the paper. It is established that if a system is homogeneous with negative degree and asymptotically stable at its origin in the absence of delays, then stability property is preserved presented in thestable paper.atItits is established that if a system is homogeneous withproperty negativeisdegree and asymptotically origin in the absence of delays, then stability preserved asymptotically at in absence delays, then stability is with respect to astable compact setorigin containing originof any delay It isproperty shown that obtained asymptotically stable at its its origin in the thethe absence offor delays, thenvalue. stability property is preserved preserved with respect to a compact set containing the origin for any delay value. It is shown that obtained with respect to a compact set containing the origin for any delay value. It is shown that obtained results can also be used for locally homogeneous systems. A simulation example is included to with respect to a compact set containing the origin for any delay value. It is shown that obtained results can also be used for locally homogeneous systems. A simulation example is included to results can also be used for locally homogeneous systems. A simulation example is included support theoretical results. results can also be used for locally homogeneous systems. A simulation example is included to to support theoretical results. support theoretical results. support theoretical results. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Homogeneous system, time-delay system, robustness. Keywords: Homogeneous system, system, time-delay system, system, robustness. Keywords: Keywords: Homogeneous Homogeneous system, time-delay time-delay system, robustness. robustness. 1. INTRODUCTION linear system is that its local behavior is the same as 1. INTRODUCTION INTRODUCTION linear system that its local behavior is theassame as 1. linear system is that behavior is as the global oneis and Rosier (2001), in the 1. INTRODUCTION linear system is Bacciotti that its its local local behavior is the theassame same as the global one Bacciotti and Rosier (2001), in the the global one Bacciotti and Rosier (2001), as in the case. In addition, the homogeneous stable/unstable The time-delay dynamical systems (whose models are rep- linear the global one Bacciotti and Rosier (2001), as in the linear case. In addition, the homogeneous stable/unstable The time-delay time-delay dynamical systems (whose (whose models are repreplinear case. In the stable/unstable admit homogeneous Lyapunov/Chetaev functions The dynamical systems models are resented by functional differential equations) appear in systems linear case. In addition, addition, the homogeneous homogeneous stable/unstable The time-delay dynamical systems (whose models are repsystems admit homogeneous Lyapunov/Chetaev functions resented by functional differential equations) appear in systems admit homogeneous Lyapunov/Chetaev functions Zubov (1958), Rosier (1992), Efimov and Perruquetti resented by functional differential equations) appear in many areas of science and technology, like e.g. systems systems admit homogeneous Lyapunov/Chetaev functions resented by of functional differential equations) appear in Zubov (1958), Rosier (1992), Efimov and Perruquetti many areas science and technology, like e.g. systems Zubov (1958), Rosier (1992), Efimov and Perruquetti (2010), Efimov et al. (2014a). Since the subclass of nonlinmany areas of science and technology, like e.g. systems biology and networked/distributed systems Chiasson and Zubov (1958), Rosier (1992), Efimov and Perruquetti many areas of science and technology, like e.g. systems (2010), Efimov et al. (2014a). Since the subclass of nonlinbiology and and networked/distributed systems Chiasson and (2010), Efimov et al. (2014a). Since the subclass of ear systems, having a global behavior, is rather small, the biology networked/distributed systems Chiasson and Loiseau (2007), Erneux (2009). Then influence of delays on (2010), Efimov et al.a(2014a). Since theissubclass of nonlinnonlinbiology and networked/distributed systems Chiasson and ear systems, having global behavior, rather small, the Loiseau (2007), Erneux (2009). Then influence of delays on ear systems, having a global behavior, is rather small, the concept of local homogeneity has been introduced Zubov Loiseau (2007), Erneux (2009). Then influence of delays on the system stability and performance is critical for many ear systems, having a global behavior, is rather small, the Loiseau (2007), Erneux (2009). Then influence of delays on concept of local homogeneity has been introduced Zubov the system stability and performance is critical for many concept of local homogeneity has been introduced Zubov (1958), Andrieu et al. (2008), Efimov and Perruquetti the system and performance critical many natural andstability human-developed systemsis Gu et al.for (2003), concept of local homogeneity has been introduced Zubov the system stability and performance is critical for many Andrieu etto al. (2008), and Perruquetti natural and human-developed human-developed systems Gu et al. al. (2003), (1958), (1958), Andrieu (2008), Efimov and Perruquetti Attempts apply and Efimov extend the natural and et (2003), Hale (1977), Kolmanovsky andsystems Nosov Gu (1986). Synthesis (1958), Andrieu et etto al. al. (2008), Efimov and homogeneity Perruquetti natural and human-developed systems Gu et al. (2003), (2010). (2010). Attempts apply and extend the homogeneity Hale (1977), Kolmanovsky and Nosov (1986). Synthesis (2010). Attempts to apply and extend the homogeneity theory for nonlinear functional differential equations, in Hale (1977), Kolmanovsky and Nosov (1986). Synthesis of control and estimation algorithms, which are robust (2010). Attempts to apply and extend the homogeneity Hale (1977), Kolmanovsky and Nosov (1986). Synthesis theory for nonlinear functional differential equations, in of control control and estimation algorithms, whichdelays, are robust robust theory for nonlinear functional differential equations, in order to simplify their stability analysis and design, have of and estimation algorithms, which are with respect to uncertain and time-varying is an theory for nonlinear functional differential equations, in of control and estimation algorithms, whichdelays, are robust order to simplify their stability analysis and design, have with respect to uncertain and time-varying is an order to simplify their stability analysis and design, have been performed in Efimov and Perruquetti (2011), Efimov with respect uncertain and time-varying an important andto quickly developing topic of the delays, modernis conorder to simplify their stability analysis and design, have with respect to uncertain and time-varying delays, is an in Efimov Perruquetti (2011), Efimov important and quickly developing topic of the the modern con- been been performed in and Perruquetti (2011), al.performed (2014b), Efimov et and al. (2016). Applications of the important quickly developing topic of control theoryand Fridman (2014). Despite of variety of methods been performed in Efimov Efimov and Perruquetti (2011), Efimov Efimov important and quickly developing topic of the modern modern con- et et al. (2014b), Efimov et al. (2016). Applications of the trol theory Fridman (2014). Despite of variety of methods et al. (2014b), Efimov et al. (2016). Applications of the conventional homogeneity framework for analysis of trol theory Fridman (2014). Despite of variety of methods solving this problem, most of them deal with linear timeet al. (2014b), Efimov et framework al. (2016). for Applications oftimethe trol theory Fridman (2014). Despite of variety of methods conventional homogeneity analysis of timesolving this problem, most of them deal with linear timeconventional homogeneity framework for analysis of timedelay systems (considering delay as a kind of perturbation, solving this problem, most of them deal with linear timedelay models, which is originated by complexity of stability conventional homogeneity framework for analysis of timesolving this problem, most of them deal with linear time- delay systems (considering delay as a kind of perturbation, delay models, which is originated by complexity of stability delay systems (considering delay as aa kind of instance) been carried out earlier in Aleksandelay models, which is complexity of analysis for the nonlinear case by (design of a Lyapunovdelay systemshave (considering delay aseven kind of perturbation, perturbation, delay models, which is originated originated complexity of stability stability for for instance) have been carried out even earlier in Aleksananalysis forfunctional the nonlinear case by (design of aa LyapunovLyapunovfor instance) have been carried out even earlier in Aleksandrov and Zhabko (2012), Asl and Ulsoy (2000), analysis for the nonlinear case (design of Krasovskii or a Lyapunov-Razumikhin function for instance) have (2012), been carried outUlsoy even earlier inBokharaie Aleksananalysis forfunctional the nonlinear case (design of a Lyapunovdrov and Zhabko Asl and (2000), Bokharaie Krasovskii or a Lyapunov-Razumikhin function drov and Zhabko (2012), Asl and Ulsoy (2000), Bokharaie et al. (2010), Diblik (1997), Mazenc et al. (2001). Krasovskii functional or a Lyapunov-Razumikhin function is a complex problem), and that constructive and compudrov and Zhabko (2012), Asl and Ulsoy (2000), Bokharaie Krasovskii functional or a Lyapunov-Razumikhin function et al. (2010), Diblik (1997), Mazenc et al. (2001). is aa complex complex problem), and that that constructive and compucompuet al. al. (2010), (2010), Diblik (1997), (1997), Mazenc Mazenc et et al. al. (2001). (2001). is problem), and constructive and tationally tractable conditions exist for linear systems only et is a complex problem), and that constructive and compuEfimov et Diblik al. (2016) it has been shown that homogetationally tractable conditions exist for linear systems only In tationally tractable conditions exist for linear systems only Richard (2003). In Efimov et al. (2016) it has been shown that homogetationally tractable conditions exist for linear systems only In Efimov et al. (2016) it has been shown that homogeneous time-delay systems have certain stability Richard (2003). In Efimov et al. (2016) ithave has certain been shown thatrobustness homogeRichard (2003). neous time-delay systems stability robustness Richard (2003). neous time-delay systems have certain stability robustness with respect to delays, e.g. if certain they are globally asympThe theory of homogeneous dynamical systems has been neous time-delay systems have stability robustness with respect to delays, e.g. if they are globally asympThe theory of homogeneous dynamical systems has been with respect to delays, if they globally asymptotically stable for somee.g. delay, thenare they preserve this The theory of homogeneous dynamical systems has been established and well explored for ordinary differential with respect to delays, e.g. if they are globally asympThe theory of homogeneous dynamical systems has been totically stable for some delay, then they preserve this established and well explored for ordinary differential totically stable for some delay, then they preserve this property independently of delay (IOD). For the case of established and well explored for ordinary differential equations Bacciotti and Rosier (2001), Bhat and Bernstein totically stable for someofdelay, then they preserve this established and well explored for ordinary differential property independently delay (IOD). For the case of equations Bacciotti and Rosier (2001), Bhat and Bernstein property independently of delay (IOD). For the case nonnegative degree it has been proven that global asympequations Bacciotti and Rosier (2001), Bhat and Bernstein (2005), Kawski (1991), Zubov (1958), Qian et al. (2015) property independently ofbeen delay (IOD). For the asympcase of of equations Bacciotti and Rosier (2001), Bhat and Bernstein nonnegative degree it has proven that global (2005), Kawski (1991), Zubov (1958), Qian et al. (2015) nonnegative degree it has been proven that global asymptotic stabilitydegree in theit delay-free case implies local asymp(2005), Kawski Zubov (1958), et (2015) and differential inclusions Bernuau et Qian al. (2014), has been proven that global (2005), Kawski (1991), (1991), Zubov (1958), Qian et al. al. Levant (2015) nonnegative totic stability in the delay-free case implies local asympand differential inclusions Bernuau et al. (2014), Levant stability in delay-free asympforthe a sufficiently smallimplies delay local (similarly to and differential inclusions Bernuau et (2014), Levant (2005). Linear systems form a subclass homogeneous totic stability in the delay-free case case implies local asympand differential inclusions Bernuau et al. al.of Levant totic totic stability for aaa sufficiently small delay (similarly to (2005). Linear systems systems form a subclass subclass of(2014), homogeneous totic stability for sufficiently small delay (similarly to linear systems, i.e. case of zero degree). The main goal (2005). Linear form a of homogeneous ones, moreover the main feature of a homogeneous nontotic stability for aa sufficiently small delayThe (similarly to (2005). Linear systems form a subclass of homogeneous linear systems, i.e. case of zero degree). main goal ones, moreover the main feature of a homogeneous nonlinear systems, i.e. a case of zero degree). The main goal of this work is to develop that result for the case of negative ones, moreover the main feature of a homogeneous nonsystems, i.e. a casethat of zero degree). The ofmain goal  ones, moreover the maininfeature of aGovernment homogeneous non- linear This work was supported part by the of Russian of this work is to develop result for the case negative  This work was supported in part by the Government of Russian of this to that result for case negative degree, suchis extension not trivial proof  of this work work is an to develop develop thatis result for the the since case of ofthe negative Federation (Grant 074-U01)inand the Ministry of Education and This was part by the of degree, such an extension is not trivial since the proof  This work work(Grant was supported supported inand partthe by Ministry the Government Government of Russian Russian degree, such an extension is not trivial since the proof of Efimov et al. (2016) was heavily based on Lipschitz Federation 074-U01) of Education and degree, such an extension is heavily not trivial since the proof Science of Russian Federation (Project 14.Z50.31.0031). Federation (Grant 074-U01) and the Ministry of Education and of Efimov et al. (2016) was based on Lipschitz Federation (Grant 074-U01) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031). of Efimov et al. (2016) was heavily based on Lipschitz of Efimov et al. (2016) was heavily based on Lipschitz Science of Russian Federation (Project 14.Z50.31.0031). Science of Russian Federation (Project 14.Z50.31.0031). Copyright © 2016, 2016 IFAC 558Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 558 Copyright © 2016 IFAC 558 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 558Control. 10.1016/j.ifacol.2016.10.222

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continuity property of the system, but this property is not satisfied for a negative degree. It will be shown that in the case of negative degree, if the system is globally asymptotically stable in the delay-free case, then for any delay it is globally asymptotically stable with respect to a compact set containing the origin. The outline of this paper is as follows. The preliminary definitions and the homogeneity for time-delay systems are given in Section 2. The main result is presented in Section 3. An example is considered in Section 4.

547

(b) asymptotically stable at the origin into Ω if it is stable into Ω and limt→+∞ |x(t, x0 )| = 0 for any x0 ∈ Ω;

(c) finite-time stable at the origin into Ω if it is stable into Ω and for any x0 ∈ Ω there exists 0 ≤ T x0 < +∞ such that x(t, x0 ) = 0 for all t ≥ T x0 . The functional T0 (x0 ) = inf{T x0 ≥ 0 : x(t, x0 ) = 0 ∀t ≥ T x0 } is called the settling time of the system (1).

If Ω = C[−τ,0] , then the corresponding properties are called global stability/asymptotic stability/finite-time stability. 2.2 Homogeneity

2. PRELIMINARIES Consider an autonomous functional differential equation of retarded type Kolmanovsky and Nosov (1986): dx(t)/dt = f (xt ), t ≥ 0, (1) n where x ∈ R and xt ∈ C[−τ,0] is the state function, xt (s) = x(t + s), −τ ≤ s ≤ 0 (we denote by C[−τ,0] the Banach space of continuous functions φ : [−τ, 0] → Rn with the uniform norm ||φ|| = sup−τ ≤ς≤0 |φ(ς)|, where | · | is the standard Euclidean norm); f : C[−τ,0] → Rn ensures existence and uniqueness of solutions in forward time, f (0) = 0. The representation (1) includes pointwise or distributed time-delay systems. We assume that solutions of the system (1) satisfy the initial functional condition x0 ∈ C[−τ,0] for which the system (1) has a unique solution x(t, x0 ) and xt,x0 (s) = x(t + s, x0 ) for −τ ≤ s ≤ 0, which is defined on some finite time interval [−τ, T ) (we will use the notation x(t) to reference x(t, x0 ) if the origin of x0 is clear from the context). The upper right-hand Dini derivative of a locally Lipschitz continuous functional V : C[−τ,0] → R+ along the system (1) solutions is defined as follows for any φ ∈ C[−τ,0] : 1 D+ V (φ) = lim sup [V (φh ) − V (φ)], h + h→0 where φh ∈ C[−τ,0] for 0 < h < τ is given by  φ(θ + h), θ ∈ [−τ, −h) φh = φ(0) + f (φ)(θ + h), θ ∈ [−h, 0]. For a locally Lipschitz continuous function V : Rn → R+ the upper directional Dini derivative is defined as follows: V [xt (0)+hf (xt )]−V [xt (0)] D+V [xt (0)]f (xt ) = lim sup . h h→0+ A continuous function σ : R+ → R+ belongs to class K if it is strictly increasing and σ (0) = 0; it belongs to class K∞ if it is also radially unbounded. A continuous function β : R+ × R+ → R+ belongs to class KL if β(·, r) ∈ K and β(r, ·) is decreasing to zero for any fixed r ∈ R+ . The symbol 1, m is used to denote a sequence of integers 1, ..., m. 2.1 Stability definitions

For any ri > 0, i = 1, n and λ > 0, define the dilation matrix Λr (λ) = diag{λri }ni=1 and the vector of weights r = [r1 , ..., rn ]T . For any ri > 0, i = 1, n and x ∈ Rn the homogeneous norm can be defined as follows 1/ρ  n  ρ/ri |xi | , ρ ≥ max ri . |x|r = i=1

1≤i≤n

For all x ∈ R , its Euclidean norm |x| is related with the homogeneous one: σ r (|x|r ) ≤ |x| ≤ σ ¯r (|x|r ) for some σ r , σ ¯r ∈ K∞ . The homogeneous norm has an important property that is |Λr (λ)x|r = λ|x|r for all x ∈ Rn . Define Sr = {x ∈ Rn : |x|r = 1}. n

For any ri > 0, i = 1, n and φ ∈ C[−τ,0] the homogeneous norm can be defined as follows  n 1/ρ  ρ/ri ||φi || , ρ ≥ max ri . ||φ||r = i=1

1≤i≤n

There exist two functions ρr , ρ¯r ∈ K∞ such that for all φ ∈ C[−τ,0] Efimov et al. (2014b): (2) ρr (||φ||r ) ≤ ||φ|| ≤ ρ¯r (||φ||r ).

The homogeneous norm in the Banach space has the same important property that ||Λr (λ)φ||r = λ||φ||r for all φ ∈ C[−τ,0] . In C[−τ,0] the corresponding unit sphere Sr = {φ ∈ C[−τ,0] : ||φ||r = 1}. Define Bρτ = {φ ∈ C[−τ,0] : ||φ||r ≤ ρ} as a closed ball of radius ρ > 0 in C[−τ,0] . Definition 2 Efimov and Perruquetti (2011) The function g : C[−τ,0] → R is called r-homogeneous (ri > 0, i = 1, n), if for any φ ∈ C[−τ,0] the relation g(Λr (λ)φ) = λd g(φ) holds for some d ∈ R and all λ > 0.

The function f : C[−τ,0] → Rn is called r-homogeneous (ri > 0, i = 1, n), if for any φ ∈ C[−τ,0] the relation f (Λr (λ)φ) = λd Λr (λ)f (φ) holds for some d ≥ − min1≤i≤n ri and all λ > 0.

In both cases, the constant d is called the degree of homogeneity.

Let Ω be a neighborhood of the origin in C[−τ,0] . Definition 1 Moulay et al. (2008) The system (1) is said to be (a) stable at the origin into Ω if there is σ ∈ K such that for any x0 ∈ Ω, |x(t, x0 )| ≤ σ(||x0 ||) for all t ≥ 0; 559

The introduced notion of weighted homogeneity in C[−τ,0] is reduced to the standard one in Rn if τ → 0. Lemma 1. Let f : C[−τ,0] → Rn be locally bounded and r-homogeneous with degree d, then there exist k > 0 such that

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i ||f (x)||r ≤ k max ||x||1+d/r r

∀x ∈ C[−τ,0] .

1≤i≤n

Proof. Take x ∈ C[−τ,0] , then there exists ξ ∈ Sr such that x = Λr (λ)ξ for λ = ||x||r . By definition of homogeneity we obtain: ||f (x)||r = ||f (Λr (λ)ξ)||r ≤ max λ1+d/ri ||f (ξ)||r 1≤i≤n

i ≤ k max ||x||1+d/r r

1≤i≤n

for k = supξ∈Sr ||f (ξ)||r . Note that d > − min1≤i≤n ri for a continuous f , then 1 + d/ri > 0.  This lemma implies that if a continuous f : C[−τ,0] → Rn is r-homogeneous with degree d, then it admits a kind of H¨older continuity at the origin (i.e. λ < 1) for the homogeneous norm of exponent 1+d/ max1≤i≤n ri if d ≥ 0 or 1 + d/ min1≤i≤n ri if d < 0, and taking into account the properties of the functions ρr , ρ¯r a similar conclusion holds for the conventional norm also. Lemma 2. Let f : C[−τ,0] → Rn be r-homogeneous with degree d and uniformly continuous in Bρτ 1 for some ρ > 0, then for any η > 0 there exists k > 0 such that 1+ rd

||f (x) − f (z)||r≤ max{k max ||x − z||r 1≤i≤n

i

, η} ∀x, z ∈ Bρτ .

Proof. Proof. Since f is uniformly continuous in Bρτ , then for η > 0 there is δη > 0 such that ||f (x) − f (z)||r < η for all x, z ∈ Bρτ with ||x − z||r < δη . Take e = x − z ∈ C[−τ,0] , then there exists ∈ Sr such that e = Λr (λ) with λ = ||e||r . For λ ≥ δη define z = Λr (λ)ζ for some ζ ∈ C[−τ,0] , then ||f (x) − f (z)||r = ||f (z + e) − f (z)||r = ||λd Λr (λ)[f (ζ + ) − f (ζ)]||r ≤ max λ1+d/ri ||f (ζ + )−f (ζ)||r . 1≤i≤n

Since ||z||r ≤ ρ, then ||ζ||r ≤ δη−1 ρ and

||f (ζ + ) − f (ζ)||r ≤ k for some k > 0 dependent on ρ and η for any x, z ∈ Bρτ with ||x − z||r ≥ δη . The claim follows by combining these both cases.  Corollary 3. Let f : C[−τ,0] → Rn be r-homogeneous with degree d < 0 and uniformly continuous in Bρτ for some ρ > 0, then for any η > 0 there exist k  > 0 such that ||f (x) − f (z)||r ≤ max{k  ||x − z||r , η} ∀x, z ∈ Bρτ . Proof. The result follows Lemma 2 noticing that for d < 0 we have 0 < 1 + d/ri < 1 for all 1 ≤ i ≤ n, then for any η > 0 and ρ > 0 there exists k˜ > 0 such that i ˜ − z||r , η } ∀x, z ∈ B τ . ≤ max{k||x max ||x − z||1+d/r r ρ 1≤i≤n k 

can be obtained from another solution under the dilation rescaling and a suitable time re-parametrization. A similar property holds for functional homogeneous systems: Proposition 4. Efimov et al. (2014b) Let x : R+ → Rn be a solution of the r-homogeneous system (1) with the degree d = 0 for an initial condition x0 ∈ C[−τ,0] . For any λ > 0 define y(t) = Λr (λ)x(t) for all t ≥ 0, then y(t) is also a solution of (1) with the initial condition y0 = Λr (λ)x0 . Proposition 5. Efimov et al. (2016) Let x(t, x0 ) be a solution of the r-homogeneous system (1) with the degree d = 0 for an initial condition x0 ∈ C[−τ,0] , τ ∈ (0, +∞). For any λ > 0 the functional differential equation dy(t)/dt = f (yt ), t ≥ 0 (3) with yt ∈ C[−λ−d τ,0] , has a solution y(t, y0 ) = Λr (λ)x(λd t, x0 ) 2 for all t ≥ 0 with the initial condition y0 ∈ C[−λ−d τ,0] , y0 (s) = Λr (λ)x0 (λd s) for s ∈ [−λ−d τ, 0]. The following results have also been obtained in Efimov et al. (2016): Lemma 6. Let the system (1) be r-homogeneous with degree d = 0 and globally asymptotically stable at the origin for some delay 0 < τ0 < +∞, then it is globally asymptotically stable at the origin IOD. Lemma 7. Efimov et al. (2016) Let the system (1) be rhomogeneous with degree d < 0 and asymptotically stable at the origin into the set Bρτ for some 0 < ρ < +∞ for any value of delay 0 ≤ τ ≤ τ0 with 0 < τ0 < +∞, then it is globally asymptotically stable at the origin IOD. Lemma 8. Efimov et al. (2016) Let the system (1) be rhomogeneous with degree d > 0 and the set Bρτ for some 0 < ρ < +∞ be uniformly globally asymptotically stable for any value of delay 0 ≤ τ ≤ τ0 < +∞ 3 , then (1) is globally asymptotically stable at the origin IOD. Lemma 9. Efimov et al. (2016) Let f (xτ ) = F [x(t), x(t − τ )] in (1) and the system (1) be r-homogeneous with degree d ≥ 0 and globally asymptotically stable at the origin for τ = 0, then for any ρ > 0 there is 0 < τ0 < +∞ such that (1) is asymptotically stable at the origin into Bρτ for any delay 0 ≤ τ ≤ τ0 . Thus, (1) is locally robustly stable with respect to a sufficiently small delay if it is r-homogeneous with a nonnegative degree and stable in the delay-free case. 2.3 Local homogeneity A disadvantage of the global homogeneity introduced so far is that such systems possess the same behavior ”globally” Efimov and Perruquetti (2011), Efimov et al. (2014b). Definition 3 Efimov and Perruquetti (2011) The function g : C[−τ,0] → R is called (r,λ0 ,g0 )-homogeneous (ri > 0, i = 1, n; λ0 ∈ R ∪ {+∞}; g0 : C[−τ,0] → R) if for any φ ∈ Sr the relation lim λ−d0 g(Λr (λ)φ) − g0 (φ) = 0 λ→λ0

An advantage of homogeneous systems described by ordinary differential equations is that any of its solutions Function f : C[−τ,0] → Rn is called uniformly continuous in Bρτ if for any ε > 0 there is a δ > 0 such that for all x, z ∈ Bρτ satisfying ||x − z||r < δ, the inequality ||f (x) − f (z)||r < ε holds (the homogeneous norm is used for simplicity of notation). 1

560

If time is scaled t → λd t then the argument of f : C[−τ,0] → Rn in (1) is also scaled to f : C[−λ−d τ,0] → Rn in (3). 3 In this case for any 0 ≤ τ ≤ τ , any ε > 0 and κ ≥ 0 there is 0 ε ε 0 ≤ Tκ,τ < +∞ such that ||xt,x0 ||r ≤ ρ + ε for all t ≥ Tκ,τ for any x0 ∈ Bκτ , and |x(t, x0 )|r ≤ στ (||x0 ||r ) for all t ≥ 0 for some function στ ∈ K∞ for all x0 ∈ / Bρτ . 2

IFAC NOLCOS 2016 August 23-25, 2016. Monterey, California, Konstantin Zimenko et al. / IFAC-PapersOnLine 49-18 (2016) 546–551 USA

is satisfied (uniformly on Sr for λ0 ∈ {0, +∞}) for some d0 ∈ R.

The system (1) is called (r,λ0 ,f0 )-homogeneous (ri > 0, i = 1, n; λ0 ∈ R ∪ {+∞}; f0 : C[−τ,0] → Rn ) if for any φ ∈ Sr the relation lim λ−d0 Λ−1 r (λ)f (Λr (λ)φ) − f0 (φ) = 0 λ→λ0

is satisfied (uniformly on Sr for λ0 ∈ {0, +∞}) for some d0 ≥ − min1≤i≤n ri .

For a given λ0 , g0 and f0 are called approximating functions. For any 0 < λ0 < +∞ the following formulas give an example of r-homogeneous approximating functions g0 and f0 : 0 g0 (φ) = ||φ||dr λ−d g(Λr (λ0 )Λ−1 r (||φ||r )φ), d ≥ 0, 0 d −d0 −1 (λ f0 (φ)=||φ||r λ0 Λr(||φ||r )Λ−1 0 )f (Λr(λ0)Λr (||φ||r )φ), r d ≥ − min1≤i≤n ri . This property allows us to analyze local stability/instability of the system (1) on the basis of a simplified system (4) dy(t)/dt = f0 [yτ (t)], t ≥ 0, called the local approximating dynamics for (1). Theorem 10. Efimov and Perruquetti (2011) Let the system (1) be (r,λ0 ,f0 )-homogeneous for some ri > 0, i = 1, n, the function f0 be continuous and r-homogeneous with the degree d0 . Suppose there exists a locally Lipschitz continuous r-homogeneous Lyapunov-Razumikhin function V0 : Rn → R+ with the degree ν0 > max{0, −d0 }, α1 (|x|) ≤ V0 (x) ≤ α2 (|x|) for all x ∈ Rn and some α1 , α2 ∈ K∞ such that:

3. MAIN RESULTS In this work we propose the following extension of Lemma 9 for the case of negative degree. Lemma 11. Let f (xt ) = F [x(t), x(t − τ )] in (1) be uniformly continuous and the system (1) be r-homogeneous with degree d < 0 and globally asymptotically stable at the origin for τ = 0, then for any ε > 0 there is 0 < τ0 < +∞ such that (1) is globally asymptotically stable with respect to Bετ for any delay 0 ≤ τ ≤ τ0 4 . The proof is omitted by reason of the restriction on the number of pages. Nevertheless, it should be noted that the structure of the proof of Lemma 11 is similar to the proof of the Lemma 2 in Efimov et al. (2016) for homogeneous systems with positive degree d > 0. However, the proof of Efimov et al. (2016) was heavily based on Lipschitz continuity property of the system, but this property is not satisfied for a negative degree and proof is mainly based on application of Lemma 1 and Lemma2. Remark The conditions of Lemma 11 means that F [Λr (λ)x, Λr (λ)z] = λd Λr (λ)F [x, z] for any x, z ∈ Rn and λ ∈ (0, +∞). In addition, the point x = 0 for the system x˙ = F (x, x) (5) is globally asymptotically stable. Note that the system (5) is also homogeneous of degree d < 0 (F [Λr (λ)x, Λr (λ)x] = λd Λr (λ)F [x, x] for any x ∈ Rn and λ ∈ (0, +∞)), then according to Zubov (1958), Rosier (1992) there is a differentiable and r-homogeneous Lyapunov function V : Rn → R+ of degree v > −d such that a = − sup D+ V (ξ)F (ξ, ξ) > 0, ξ∈Sr    ∂V (ξ)    < +∞, 0 < b = sup  ∂ξ  |ξ|r ≤1 c1 = inf V (ξ), c2 = sup V (ξ),

(i) there exist functions α, γ ∈ K such that for all ϕ ∈ Sr max V0 [ϕ(θ)] < γ{V0 [ϕ(0)]} ⇒ θ∈[−τ,0]

D+ V0 [ϕ(0)]f0 (ϕ) ≤ −α(|ϕ(0)|); 

ξ∈Sr

ξ∈Sr

c1 |x|vr ≤ V (x) ≤ c2 |x|vr



(ii) there exists a function γ ∈ K such that λs < γ (λs) ≤ λγ(s) for all s, λ ∈ R+ \ {0}.

549

∀x ∈ Rn .

¯ ε such that the 1) if λ0 = 0, then there exists 0 < λ system (1) is locally asymptotically stable at the origin with the domain of attraction containing the set ¯ ε )}; X0 = {φ ∈ C[−τ,0] : ||φ|| ≤ α1−1 ◦ α2 ◦ ρ¯r (λ

Then the maximal possible time delay τ0 can be estimated as follows  −1   1 ρr (ρ) σ ¯r (aR−d−v − „) −1 , π , τ0 = min ιk (2ρr (ρ)) π2 (ε) b   1+d/ri , R = ¯r k max1≤i≤n [ρ−1 (s)] where ιk (s) = σ r

2) if λ0 = +∞, then there exists 0 < λε < +∞ such that the system (1) is globally asymptotically stable with respect to forward invariant set X∞ = {φ ∈ C[−τ,0] : ||φ|| ≤ α1−1 ◦ α2 ◦ ρr (λε )};

1+d/ri , π1 (τ ) = L max σ −1 r (M τ ) 1≤i≤n λ−1−d/ min1≤i≤n ri if λ ≥ 1 π2 (λ) = λ−1−d/ max1≤i≤n ri if λ < 1

¯ε < 3) if 0 < λ0 < +∞, then there exist 0 < λε ≤ λ0 ≤ λ +∞ such that the system (1) is asymptotically stable with respect to the forward invariant set X∞ with region of attraction X = {φ ∈ C[−τ,0] : α1−1 ◦ α2 ◦ ρr (λε ) < ||φ|| ¯ ε )} < α1−1 ◦ α2 ◦ ρ¯r (λ provided that the set X is connected and nonempty.

L > 0 : |F [ϕ(0), ϕ(−τ )] − F [ϕ(0), ϕ(0)]|r i ≤ max{L max |ϕ(0) − ϕ(−τ )|1+d/r , η}, r

Then (the functions ρ¯r and ρr have been defined in (2))

In Efimov et al. (2014b) analysis of the input-to-state stability property for the system (1) has been presented using the homogeneity theory. 561

1/v , γ > 1 : supθ∈[−τ,0] V [φ(θ)] < γV [φ(0)] n1/ρ (γc−1 1 c2 )

1≤i≤n

M=

sup |z|r

≤ρ ,|y|

r

≤ρ

|F [z, y]|,

  for some „ ∈ 0, aR−d−v .

ρ = ρ−1 [2ρr (ρ)] r

4 In this case for any 0 ≤ τ ≤ τ , any  > 0 and κ ≥ ε there is 0   0 ≤ Tκ,τ < +∞ such that ||xt,x0 ||r ≤ ε +  for all t ≥ Tκ,τ for any τ x0 ∈ Bκ , and |x(t, x0 )|r ≤ στ (||x0 ||r ) for all t ≥ 0 for some function στ ∈ K∞ for all x0 ∈ / Bετ .

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Fig. 1. Simulation results Using Theorem 10, results of Lemma 11 can be used for local analysis of stability for not necessary homogeneous systems. Theorem 12. Let system (1) be (r,+∞,f0 )-homogeneous with degree d0 < 0, f0 (xt ) = F0 [x(t), x(t − τ )] and the origin for the approximating system (4) be globally asymptotically stable for τ = 0. Then (1) has bounded trajectories IOD. Proof. The proof of Theorem 12 is a direct consequence of Theorem 10 and Lemma 11.  Roughly speaking the result of Theorem 12 says that if the approximating dynamics (4) has a negative degree and it is stable in the delay-free case, then the original system (1) has bounded trajectories for any delay. 4. NUMERICAL EXAMPLE Consider the following system  x˙ 1 = x2 − l1 x1 α , (6) x˙ 2 = −l2 x1 2α−1 , 1  β where l1 > 0, l2 > 0, α ∈ 2 , 1 and x = |x|β sign(x) for any real number β ≥ 0. The system (6) is homogeneous for r = [1, α]T with degree µ = α − 1 < 0.

Let us assume that the state x1 is available with delay 0 < τ < +∞. Then according to obtained results the system (6) is globally asymptotically stable with respect to Bετ for some ε > 0 dependent on the selected value of delay τ . The Fig. 1 shows the simulation results for α = 0.7, l1 = 1, l2 = 2 and two different values of delay τ . 5. CONCLUSIONS For time-delay systems with negative degree it has been shown that if for zero delay the system is globally asymptotically stable, then for any delay it is converging to some compact set around the origin, that is an interesting robustness property of nonlinear homogeneous systems with respect to delays, which is not observed in the linear case. Efficiency of the proposed approach is demonstrated on example. Design of new stabilization and estimation algorithms, which preserve boundedness of the system trajectories for any delays, is a direction of future research. REFERENCES Aleksandrov, A. and Zhabko, A. (2012). On the asymptotic stability of solutions of nonlinear systems with delay. Siberian Mathematical Journal, 53(3), 393–403. 562

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