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Stability Analysis of Neutral Type Time-Delay Positive Systems with Commensurate Delays
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress Control The International Federation of Toulouse, France,Federation July 9-14, 2017 The International of Automatic Automatic Control Available online at www.sciencedirect.com The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 3093–3098
Stability Analysis of Stability Analysis of Stability Analysis of Stability Analysis of Neutral Type Time-Delay Positive Systems Neutral Type Time-Delay Positive Systems Neutral Type Time-Delay Positive Systems Neutralwith TypeCommensurate Time-Delay Positive Systems Delays with Commensurate Delays with Commensurate Delays with Commensurate Delays ∗
Yoshio Ebihara ∗ Yoshio Ebihara ∗∗ Yoshio Yoshio Ebihara Ebihara ∗ ∗ Department of Electrical Engineering, Kyoto University, ∗ Department of Electrical Engineering, Kyoto University, Engineering, Kyoto University, ∗ Department of Electrical Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan Department of Electrical Engineering, Kyoto University, Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, (e-mail: [email protected]). Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan Japan (e-mail: [email protected]). (e-mail: (e-mail: [email protected]). [email protected]). Abstract: Recently, the author derived a necessary and sufficient condition for the asymptotic Abstract: the author derived aa necessary sufficient condition for the type asymptotic Abstract: Recently, the author and sufficient condition for asymptotic stability of Recently, neutral type positive systemsand (TDPSs), where the neutral TDPS Abstract: Recently, the time-delay author derived derived a necessary necessary and sufficient condition for the the type asymptotic stability of neutral type time-delay positive systems (TDPSs), where the neutral TDPS stability of neutral type time-delay positive systems (TDPSs), where the neutral type TDPS of interestofisneutral given by a feedback connection between (TDPSs), a finite-dimensional LTI positive system stability type time-delay positive systems where the neutral type TDPS of interest is given by a feedback connection between a finite-dimensional LTI positive system of interest is given by a feedback connection between a finite-dimensional LTI positive system and the pure delay. The goal of this paper is to extend this result to neutral type TDPSs of interest is given by a feedback connection between a finite-dimensional LTI positive system and the pure delay. The goal of thisTopaper is toweextend this result to neutral TDPSs and the pure delay. of is this to type TDPSs with multiple this end, first represent a neutral typetype TDPS with and the pure commensurate delay. The The goal goaldelays. of this thisTopaper paper is to toweextend extend this result result to neutral neutral type TDPSs with multiple commensurate delays. this end, first represent a neutral type TDPS with with multiple commensurate delays. To this end, we first represent a neutral type TDPS with multiple commensurate delays as a neutral type TDPS with a single delay, by augmenting input with multiple commensurate delays. To this end, we first represent a neutral type TDPSinput with multiple commensurate delays as a neutral type TDPS with a single delay, by augmenting multiple commensurate delays as a neutral type TDPS with a single delay, by augmenting input and output signals in the feedback connection. By this conversion, we can readily apply the multiple commensurate delays as a neutral type TDPS with a single delay, by augmenting input and output signals in the feedback connection. By this conversion, we can readily apply the and output the feedback this conversion, we can apply the latest result signals alreadyin established for connection. single delay By neutral type TDPSs. As thereadily main result, we and output signals in the feedback connection. By this conversion, we can readily apply the latest result already established for single delay neutral type TDPSs. As the main result, we latest result already established for single delay neutral type TDPSs. As the main result, we show that a neutral type TDPS with commensurate delays is stable if and only if its delaylatest resulta already established for single delay neutral typeisTDPSs. Asand theonly mainif result, we show that neutral type TDPS with commensurate delays stable if its delayshow that a neutral type TDPS with commensurate delays is stable if and only if its delayfree counterpart is stable and an additional “admissibility” condition is satisfied. This result in show that a neutral typeand TDPS with commensurate delayscondition is stableisifsatisfied. and onlyThis if its delayfree counterpart is stable an additional “admissibility” result in free counterpart stable an additional “admissibility” condition is satisfied. This result in particular impliesis that theand stability is irrelevant of the length of delays. free counterpart is stable and an additional “admissibility” condition is satisfied. This result in particular implies that the stability is irrelevant of the length of delays. particular implies that the stability is irrelevant of the length of delays. particular implies that the stability is irrelevant of the length of delays. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: neutral type time-delay positive systems, commensurate delays, asymptotic stability Keywords: neutral type type time-delay positive positive systems, commensurate commensurate delays, asymptotic asymptotic stability Keywords: Keywords: neutral neutral type time-delay time-delay positive systems, systems, commensurate delays, delays, asymptotic stability stability 1. INTRODUCTION To deal with stability analysis of neutral type TDPSs, in 1. INTRODUCTION INTRODUCTION To deal deal with stability analysis of neutral neutral type TDPSs, in 1. To of in [Ebihara et al.stability [2017a]],analysis we focused on thetype the TDPSs, time-delay 1. INTRODUCTION To deal with with stability analysis of neutral type TDPSs, in [Ebihara et al. [2017a]], we focused on the the time-delay [Ebihara et al. [2017a]], we focused on the the time-delay feedback system (TDFS)we form shownon in the Fig.the 1 totime-delay represent [Ebihara et al. [2017a]], focused feedback system (TDFS) form shown in Fig. 1 to represent This paper is concerned with asymptotic stability analy- feedback system (TDFS) shown in to type TDSs (that form are not necessarily In feedback system (TDFS) form shown in Fig. Fig. 11 positive). to represent represent Thisof paper paper is concerned concerned with asymptotic asymptotic stability analy- neutral neutral type TDSs (that are not not necessarily positive). In This is with stability analysis neutral type time-delay positive systems (TDPSs) neutral type TDSs (that are necessarily positive). In Fig. 1, G is an FDLTI system described by x(t) ˙ = Ax(t) + This paper is concerned with asymptotic stability analyneutral type TDSs (that are not necessarily positive). In sis of of commensurate neutral type type time-delay time-delay positive systems (TDPSs) Fig. 1, G is an FDLTI system described by x(t) ˙ = Ax(t) + sis neutral positive systems (TDPSs) with delays. In retrospect, the theory of Fig. 1, G is an FDLTI system described by x(t) ˙ = Ax(t) + Bw(t) and z(t) = Cx(t)+Dw(t), while H is the pure delay sis of neutral type time-delay positive systems (TDPSs) Fig. 1, and G isz(t) an FDLTI system described byisx(t) ˙the pure = Ax(t) + with commensurate commensuratelinear delays. In retrospect, retrospect, the theory theory of Bw(t) = Cx(t)+Dw(t), while H delay with delays. In the of finite-dimensional time-invariant positive systems Bw(t) and z(t) = Cx(t)+Dw(t), while H is the pure delay delay length h Cx(t)+Dw(t), > 0 and thus w(t) =Hz(t − h) (t ≥ h). with commensuratelinear delays. In retrospect, the theory of of Bw(t) and z(t) = while is the pure delay finite-dimensional time-invariant positive systems of delay length h > 0 and thus w(t) = z(t − h) (t ≥ h). finite-dimensional linear time-invariant positive systems (FDLTIPSs) is deeply in the theory of nonnegaof length > thus = z(t h). particular, theh thew(t) direct finite-dimensional linearrooted time-invariant positive systems In of delay delay length h existence > 00 and and of thus = feedthrough z(t − − h) h) (t (t ≥ ≥term h). (FDLTIPSs) is[Berman deeply rooted in the the theory theory of Horn nonnegaIn particular, the existence of thew(t) direct feedthrough term (FDLTIPSs) deeply rooted in nonnegative matrices is and Plemmons [1979],of and In particular, the existence of the direct feedthrough term D implies that the corresponding TDS is of neutral type. (FDLTIPSs) is deeply rooted in the theory of nonnegaIn particular, the existence of the direct feedthrough term tive matrices [Berman and Plemmons [1979], Horn and D implies implies that the corresponding corresponding TDSallows is of of neutral type. tive matrices [Berman and [1979], and Johnson [1991]], and celebrated Perron-Frobenius theorem that the TDS is type. This TDFS representation naturally us to adopt tive matrices [Berman and Plemmons Plemmons [1979], Horn Horn and D D implies that the corresponding TDSallows is of neutral neutral type. Johnson [1991]], and[1991]] celebrated Perron-Frobenius theorem This TDFS representation naturally us ato tosolution adopt Johnson [1991]], and celebrated Perron-Frobenius theorem [Horn and Johnson has played a central role. Very This TDFS representation naturally allows us adopt the continuous concatenated solution (CCS) as Johnson [1991]], and[1991]] celebrated Perron-Frobenius theorem This TDFS representation naturally allows us to adopt [Horn and Johnson has played a central role. Very the continuous concatenated solution (CCS) as a solution [Horn Johnson [1991]] has aa central role. Very recently, system theory has gained renewed continuous concatenated solution (CCS) aa solution which is explicitly defined by Hagiwara andas [Horn and andpositive Johnson [1991]] has played played central role.interVery the the continuous concatenated solution (CCS) as Kobayashi solution recently, positive system theory has gained renewed interwhich is explicitly defined by Hagiwara and Kobayashi recently, positive system gained renewed interest from the viewpoint oftheory convexhas optimization, and excelwhich is explicitly defined by Hagiwara and Kobayashi [2011]. On the basis of this firm set up, in [Ebihara et al. recently, positive system theory has gained renewed interwhich is explicitly defined by Hagiwara and Kobayashi est from the viewpoint of convex optimization, and excel[2011]. On Onwethe the basis of this this firm firm set up, up, definition in [Ebihara [Ebiharaforet et al. al. est from the viewpoint of convex optimization, and excellent papers have been published along this direction, see, [2011]. basis of set in [2017a]], first introduced a proper est from the viewpoint of convex optimization, and excel[2011]. On the basis of this firm set up, in [Ebihara al. lent papers papers have been published published along this direction, direction, see, [2017a]], we first introduced a proper definition foretthe the lent have been along this see, e.g., those by Rantzer [2015, 2016], Gurvits et al. [2007], [2017a]], we first introduced a proper definition for the positivity of TDSs given by TDFS form. This definition lent papers have been published along this direction, see, [2017a]], we first introduced a proper definition for the e.g., those those by Rantzer [2015,Shorten 2016], Gurvits Gurvits et al. al. [2007], [2007], positivity of TDSs given by TDFS form. This definition e.g., Rantzer [2015, 2016], et Mason andby Shorten [2007], et al. [2009], of by This definition led us to the soundgiven conclusion that form. a TDS is positive if e.g., those Rantzer [2015,Shorten 2016], Gurvits et al. Tanaka [2007], positivity positivity of TDSs TDSs given by TDFS TDFS form. This definition Mason andby Shorten [2007], et[2012], al. [2009], [2009], Tanaka led us us to ifthe the sound conclusion that a TDS TDS isthe positive if Mason and Shorten [2007], Shorten et al. Tanaka and Langbort [2011], Blanchini et al. Briat [2013], led to sound conclusion that a is positive if and only the FDLTI system G is positive (in sense of Mason and Shorten [2007], Shorten et al. [2009], Tanaka led us to ifthe sound conclusion that a TDS isthe positive if and Langbort Langbort [2011], Blanchini et al. al. [2012], [2012], Briat [2013], and only the FDLTI system G is positive (in sense of and [2011], Blanchini et Briat [2013], Najson [2013]. We also emphasize that the study on and only if the FDLTI system G is positive (in the sense of FDSs). Then, as the main result, we showed(inthat asense TDPS and Langbort [2011], Blanchini et al. [2012], Briat [2013], and only if the FDLTI system G is positive the of and Najson [2013]. We also emphasize that the study on FDSs). Then, as the main result, we showed that a TDPS and Najson [2013]. also that study on consensus of multi-agent positive systems a FDSs). Then, the main we in TDFS formas asymptotically stable if andthat onlyaa ifTDPS D is and Najsonproblems [2013]. We We also emphasize emphasize that the the studyis on FDSs). Then, asis main result, result, we showed showed consensus problems of multi-agent positive systems isby in TDFS TDFS form is the asymptotically stable if and andthat only ifTDPS D is −1 consensus problems of positive aaa in promising direction, andmulti-agent this issue is treatedsystems activelyis form is asymptotically stable if only if D C is Hurwitz stable. Schur stable and A + B(I − D) consensus problems of multi-agent positive systems is −1 TDFS form is A asymptotically stable if and only if D is is promising direction, and this this issue is treated treated actively On by in −1 C is Hurwitz stable. Schur stable and + B(I − D) promising direction, and issue is actively by Valcher and Misra [2014] and Ebihara et al. [2017b]. stable and A + B(I − D) −1 C is Hurwitz stable. promising direction, and this issue is treated actively On by Schur C isthis Hurwitz stable. Schur stable and A + B(I − D) Valcher and Misra [2014] and Ebihara et al. [2017b]. The goal of this paper is to extend latest result to Valcher and Misra [2014] al. [2017b]. the other hand, study onand the Ebihara analysiset of The goal of this paper is to extend this latest result to Valcher and Misra [2014] Ebihara etand al. synthesis [2017b]. On On the other other hand, study onand the analysis and synthesis of neutral The goal of paper is extend this to TDPSs with multiple delays the hand, study on the analysis and synthesis of goaltype of this this paper is to to extendcommensurate this latest latest result result to TDPSs has also been active, and fruitful results have been neutral type TDPSs with multiple commensurate delays the other hand, study on the analysis and synthesis of The TDPSs has also been active, and fruitful results have been neutral type TDPSs with multiple commensurate delays (see Fig. 2). In Fig. 2, the system G is an FDLTIPS TDPSs has also been active, and fruitful results have been type TDPSs with multiple commensurate delays obtained, e.g., by Haddad Chellaboina [2004], Ait neutral (see Fig. 2). In Fig. 2, the system G is an FDLTIPS TDPSs has also been active, and fruitful results have been obtained, e.g., by and Haddad and Lam Chellaboina [2004], Ait (see Fig. Fig. 2, is an the as before, andIn nonnegative matrixG i represents obtained, by Haddad and Chellaboina Ait (see Fig. 2). 2). Inthe Fig. 2, the the system system GΩ is an FDLTIPS FDLTIPS Rami et al.e.g., [2013], Shen and [2014]. In[2004], particular, the as before, and the nonnegative matrix Ω obtained, e.g., by and Haddad and Lam Chellaboina [2004], Ait feedback i represents Rami et al. [2013], Shen and [2014]. In particular, represents the as before, and the nonnegative matrix Ω connection between z and w over which retari Rami et al. [2013], and Shen and Lam [2014]. In particular, represents the as before, and the nonnegative matrix Ω Haddad and Chellaboina [2004] showed a prominent result i L which retarfeedback connection between z and w over Rami et al. [2013], and Shen and Lam [2014]. In particular, Haddad and Chellaboina [2004] showed a prominent result feedback connection between z and w over which retarΩ z(t − ih), dation of length ih occurs, i.e., w(t) = L i Haddad and Chellaboina [2004] showed a prominent result i=1 feedback connection between z and w over which retarverifying that a retarded type TDPS is stable if and only L Ωi z(t ih), dation of length ih occurs, i.e., w(t) = Haddad and [2004] showed prominent result L i=1To verifying thatChellaboina a retarded type TDPS TDPS is astable stable if is and only where − ih), dation of length ih occurs, i.e., w(t) 0 stands the base length. deal− i=1 Ω verifying that type is and only Ωii z(t z(t −with ih), dation h of> length ih for occurs, i.e., delay w(t) = = if its delay-free finite-dimensional stable. i=1To h > 00 stands for the base delay length. deal with verifying that aa retarded retarded type TDPScounterpart is stable if if is and only where if its delay-free finite-dimensional counterpart stable. where h > stands for the base delay length. To deal with such a neutral type TDPSs with multiple commensurate if its delay-free finite-dimensional counterpart is stable. where h > 0 stands for the base delay length. To deal with However, to the best of the author’s knowledge, existing such a neutral type TDPSs with multiple commensurate if its delay-free finite-dimensional counterpart is stable. However, to the best best of the author’s author’s knowledge, existing delays, such aa neutral type multiple in this paper, we first with represent it as commensurate a neutral type However, the the knowledge, existing neutral type TDPSs TDPSs with multiple commensurate studies onto TDPSs are of restricted to retarded type TDPSs, delays, in this paper, we first represent it as a neutral type However, to the best of the author’s knowledge, existing such studies on TDPSs are restricted to retarded type TDPSs, delays, in this paper, we first represent it as type TDPS with a single delay by augmenting input and output studies on TDPSs are restricted retarded type in this paper,delay we first represent it input as aa neutral neutral type and those neutral TDPSsto surprisingly scarce. delays, TDPS with a single by augmenting and output studies on for TDPSs are type restricted toare retarded type TDPSs, TDPSs, and those for neutral type TDPSs are surprisingly scarce. TDPS with a single delay by augmenting input and output signals in the feedback connection. By this conversion, and those for neutral type TDPSs are surprisingly scarce. TDPS with a single delay connection. by augmenting input and output This is probably due totype theTDPSs fact that the definition of solu- signals in the feedback By this conversion, and those for neutral are surprisingly scarce. This is isforprobably probably due toTDSs the fact fact that the the definition of solusolu- we signals in this can readily apply the connection. latest resultBy already established This the that definition of in the the feedback feedback connection. By this conversion, conversion, tions neutral due typeto is rather difficult irrespective we can readily apply the latest result already established This isforprobably due toTDSs the fact that the definition of solu- signals tions neutral type is rather difficult irrespective we can readily apply the latest result already established for single delay neutral type TDPSs. As the main result, tions for neutral type TDSs is rather difficult irrespective we can readily apply thetype latest result As already established of positivity [Bellman and Cooke [1963]]. for single delay neutral TDPSs. the main result, tions for neutral type TDSs is rather difficult irrespective of positivity [Bellman and Cooke [1963]]. for single delay neutral type TDPSs. As the main of for single delay neutral type TDPSs. As the main result, result, of positivity positivity [Bellman [Bellman and and Cooke Cooke [1963]]. [1963]]. Copyright © 2017, 2017 IFAC 3148Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 3148 Copyright ©under 2017 responsibility IFAC 3148Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 3148 10.1016/j.ifacol.2017.08.681
Proceedings of the 20th IFAC World Congress 3094 Yoshio Ebihara / IFAC PapersOnLine 50-1 (2017) 3093–3098 Toulouse, France, July 9-14, 2017
w
z
G
H
Fig. 1. Time-Delay Systems in Time-Delay Feedback System Form (H(s) = Ie−sh ). w
G
z
2.1 Delay-Differential Equations (DDEs)
H{Ωi } [L]
Fig. 2. Neutral Type TDPS with Multiple Commensurate L [L] Ωi e−sih ). Delays (H{Ωi } (s) = i=1
we show that a neutral type TDPS with commensurate delays is stable if and only if DΩ is Schur stable and A + BΩ(I − DΩ)−1 C is Hurwitz stable where Ω := L i=1 Ωi . This result in particular implies that the stability is irrelevant of the length of delays. The former condition (i.e., the Schur stability of the nonnegative matrix DΩ) has already appeared in our preceding study on the analysis of (delay-free) interconnected positive systems [Ebihara et al. [2017b]]. By following the terminology introduced there, this condition is referred to as admissibility in this paper. Using this terminology, we can restate our main result in the way that a neutral type TDPS with commensurate delays is stable if and only if its delay-free finite-dimensional counterpart is admissible and stable. We use the following notation. We denote by R, R+ , and R++ the set of real, nonnegative, and strictly positive numbers, respectively. For given two matrices A and B of the same size, we write A > B (A ≥ B) if Aij > Bij (Aij ≥ Bij ) holds for all (i, j), where Aij stands for the (i, j)-entry of A. We define Rn+ := {x ∈ Rn : x ≥ 0} and Rn++ := {x ∈ Rn : x > 0}. We also define Rn×m + and Rn×m with obvious modifications. The set of Metzler ++ (Hurwitz stable) matrices of size n is denoted by Mn (Hn ). The set of natural numbers is denoted by N. For x ∈ Rn , n we denote its 1-norm by �x�, i.e., �x� := i=1 |xi |. In relation to the definition of CCS and positivity for TDSs in TDFS form, we introduce the following function spaces: m := {f : f (θ) ∈ Rm , f is continuous C[0,h) over [0, h)} , Khm := m Kh+
m f ∈ C[0,h) :
:= {f ∈
Khm
Therefore, we start our discussion from TDSs in DDE form, and review the result by Hagiwara and Kobayashi [2011] for the definition of solution and the conversion from DDE form to TDFS form. We note that the contents of Sections 2 and 3 are in part borrowed from the full paper version of [Ebihara et al. [2017a]]. Since this reference paper is not yet publically available, we decided to review the results in [Ebihara et al. [2017a]] as quickly as possible.
lim f (θ) exists ,
θ→h−0
: f (θ) ≥ 0 (∀θ ∈ [0, h))} .
The L1 [0, h) norm for f ∈ Khm h �f (θ)�dθ.
(1) is defined by �f � :=
0
2. REPRESENTATION OF TIME-DELAY SYSTEMS Even though we concentrate our attention on the analysis of TDSs in TDFS form, the denomination “neutral” is historically given for TDSs in delay-differential equation (DDE) form [Bellman and Cooke [1963], Hale [1977]].
In the literature, LTITDSs represented by the following DDE are studied extensively [Bellman and Cooke [1963], Hale [1977]]. q(t) ˙ = Jq(t) + K q(t ˙ − h) + Lq(t − h), J, K, L ∈ Rn×n . (2) Here, h > 0 stands for the delay length. TDSs represented by the DDE (2) with K = 0 are historically referred to as retarded type TDSs, while TDSs represented by the DDE (2) with K �= 0 are referred to as neutral type TDSs [Bellman and Cooke [1963], Hale [1977]]. The “solution” of (2) is determined under the initial condition q(t) = φ(t) (t ∈ [−h, 0)),
q(0) = ξ
(3)
where φ(t) is usually assumed to be (continuous and) continuously differentiable on the closed interval [−h, 0]. However, it is rather difficult to define the concept of solutions due to the appearance of indifferentiability (or even discontinuity) especially in the neutral case. This issue is deeply studied by Hagiwara and Kobayashi [2011], and proper definitions of solutions are given. Among them, in this paper, we adopt the continuous concatenated solution (CCS) defined there. 2.2 Continuous Concatenated Solutions (CCSs) Let us introduce the definition of CCSs for the DDE (2) equipped with the initial condition (3). Definition 1. (Hagiwara and Kobayashi [2011]). Suppose φ(t) in (3) is bounded, continuously differentiable on [−h, 0), and has the limit lim φ(t). Then, q(t) (t ≥ −h) t→0−0
is said to be a continuous concatenated solution (CCS) of the DDE (2) under the initial condition (3) if (i) it is continuous for t ≥ 0 and (ii) it is differentiable and satisfies (2) for t ≥ 0 except possibly for time instants t = kh (k ∈ N). As the definition says, a CCS is continuous over t ≥ 0 but not necessarily differentiable at t = kh (k ∈ N). On the other hand, a stronger solution that is differentiable over t ≥ −h is referred to as a regular solution in [Hagiwara and Kobayashi [2011]]. However, for the existence of such a regular solution, it is obviously necessary that φ(0) = ξ ˙ ˙ and φ(−0) = Jφ(0) + K φ(−h) + Lφ(−h). The latter requirement is rather stringent, and hence it is reasonable to introduce somehow relaxed solutions. Among them the CCS is believed to be a natural one since it possesses continuity property that is essentially required in describing the behavior of physical (real-world) systems.
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3. REVIEW OF OUR PRECEDING RESULTS
2.3 Time-Delay Feedback Systems (TDFSs) In the community of control theory, it is common to describe LTITDSs in time-delay feedback system (TDFS) form shown in Fig. 1. In Fig. 1, G stands for an FDLTI system described by x(t) ˙ = Ax(t) + Bw(t), G: z(t) = Cx(t) + Dw(t), (4) A ∈ Rn×n , B ∈ Rn×m , C ∈ Rm×n , D ∈ Rm×m . On the other hand, H is the pure delay of constant delay length h > 0 and thus H : w(t) = z(t − h). (5) In the following we denote by G H the TDS in Fig. 1. Note that the behavior of TDS G H can be determined by the initial condition given for z(t) (t ∈ [−h, 0)), x(0). (6) Once we describe a given TDS in TDFS form, we can apply fully matured control-oriented techniques such as (scaled) small-gain theorem for its stability analysis. Moreover, recently, a more advanced and rigorous monodromy operator approach has been build for the stability analysis of TDSs in TDFS form, see Kim et al. [2015] and references cited there in. This is the motivation of [Hagiwara and Kobayashi [2011]] to seek for a conversion technique from DDE form to TDFS form so that the monodromy operator approach can also be applied to TDSs in DDE form. More precisely, the issue in [Hagiwara and Kobayashi [2011]] is how to determine the FDLTI system G (or say, the matrices A, B, C, and D) and the initial condition (6) from a given DDE (or say, the matrices J, K, and L) and a given initial condition (3) so that the (continuous concatenated) solution of the DDE can be represented as a signal in TDS G H. The results in Hagiwara and Kobayashi [2011] that are relevant to this issue are quickly reviewed in the next subsection. 2.4 Conversion from DDE to TDFS
This section is devoted to a quick review of [Ebihara et al. [2017a]]. Before getting into the details, we begin with the very basic definition and related results for FDLTIPSs. Definition 3. (Farina and Rinaldi [2000]). The FDLTI system (4) is said to be positive if its state and output are both nonnegative for any nonnegative initial state and nonnegative input. Proposition 4. (Farina and Rinaldi [2000]). The FDLTI system (4) is positive if and only if , C ∈ Rm×n , D ∈ Rm×m . A ∈ Mn×n , B ∈ Rn×m + + +
(10)
The first question in [Ebihara et al. [2017a]] is to seek for proper extensions of these definition and condition to LTITDSs in TDFS form. The counterparts are as follows. Definition 5. (Ebihara et al. [2017a]). A TDS in TDFS form G H constructed from (4) and (5) is said to be positive if x(t) ≥ 0 and w(t) ≥ 0 (∀t ≥ 0) hold for any x(0) ∈ Rn+ and z−h ∈ Kh+ . Lemma 6. (Ebihara et al. [2017a]). A TDS in TDFS form G H constructed from (4) and (5) is positive in the sense of Definition 5 if and only if FDLTI system G is positive, i.e., (10) holds. Even though Lemma 6 has been proved in [Ebihara et al. [2017a]], in the following we review the proof for the sufficiency part rather in detail. We believe that this is of great benefit for readers to see concretely what kind of continuous solution (which is nothing but CCS) we employ in this study. To see the sufficiency proof, suppose x(0) ∈ Rn+ and z−h ∈ Kh+ are given. Then, for the first time interval [0, h), note that w0 (θ) = z−h (θ) (θ ∈ [0, h)) and hence w0 ∈ Kh+ . Then, from the variation of constant formula, we see that t At x(t) = e x(0) + eA(t−τ ) Bw(τ )dτ (0 ≤ t < h) (11) 0
The next proposition shows that the CCS of the DDE (2) can always be represented by the state x of the FDLTI system G in the TDS G H. In the following we write wt = wt (θ) = w(t + θ) (θ ∈ [0, h)), (7) zt = zt (θ) = z(t + θ) (θ ∈ [0, h)). Proposition 2. (Hagiwara and Kobayashi [2011]). Suppose φ(t) in (3) is bounded, continuously differentiable on [−h, 0), and has the limit lim φ(t). Then, the DDE (2) t→0−0
has a unique CCS q(t), and it coincides, over t ≥ 0, with x(t) resulting from G H with A, B, C, and D given by A = J, B = I, C = L + KJ, D = K (8) and with the initial condition ˙ (θ ∈ [0, h)), x(0) = ξ. (9) z−h (θ) = K φ(θ−h)+Lφ(θ−h) As clearly shown in (8), the direct-feedthrough term D of the FDLTI system G coincides with the coefficient K in the DDE form. Since we are mainly interested in the neutral type TDSs and hence K �= 0 in (2), we justifiably focus on TDSs given by TDFS form with D �= 0 in the following.
holds and hence x(t) is uniquely determined and continuous over t ∈ [0, h). It is also true that x(t) ≥ 0 (t ∈ [0, h)) since G is positive and w0 ∈ Kh+ . Moreover, since w0 ∈ m (or more precisely since limt→h−0 w(t) exists from Kh+ m the definition of Kh+ ), we see that limt→h−0 x(t) exists as well, and from the continuity requirement on x we can let x(h) := limt→h−0 x(t)(∈ Rn+ ). On the other hand, from z(t) = Cx(t) + Dw(t) (0 ≤ t < h) (12) and again from the positivity of G, we see that the m important property z0 ∈ Kh+ holds. To summarize, for the next time interval t ∈ [h, 2h), we know that x(h) ∈ Rn+ is determined and wh has exactly the same property m since wh (θ) = z0 (θ) (θ ∈ [0, h) and with w0 ∈ Kh+ m z0 ∈ Kh+ . Therefore by repeating the same arguments, we see that continuous solution x(t) exists over t ∈ [0, 2h), and by repeating the same arguments recursively (or say, by concatenating the solutions determined over [kh, (k + 1)h) repeatedly), we can conclude that continuous solution x(t) uniquely exists over t ≥ 0 and in particular x(t) ≥ 0 (∀t ≥ 0). We note that x(t) thus constructed might not be differentiable for time instants t = kh (k ∈ N). On the other hand, it is also clear that w(t) thus constructed is unique and satisfies w(t) ≥ 0 (∀t ≥ 0), although it might
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not be continuous for time instants t = kh (k ∈ N). We thus completed the proof for the sufficiency. On the basis of these preliminary results, in [Ebihara et al. [2017a]], we analyzed stability of TDPS G H. The definition of asymptotic stability of TDS G H (that is not necessarily positive at this stage) is as follows. Definition 7. (Ebihara et al. [2017a]). A TDS in TDFS form G H constructed from (4) and (5) is said to be asymptotically stable if �xt � + �wt � → 0 (t → ∞) for any x0 ∈ Rn and w0 ∈ Kh . Then, as a main result of [Ebihara et al. [2017a]], we derived the next theorem. In the following, “asymptotic stability” is abbreviated as “stability” just for brevity. Lemma 8. (Ebihara et al. [2017a]). A TDPS in TDFS form G H constructed from FDLTIPS G given by (4) and (10) and the pure delay H given by (5) is stable if and only if D ∈ Rm×m is Schur stable and Acl = A + B(I − + D)−1 C ∈ Mn×n is Hurwitz stable. An immediate fact that follows from Lemma 8 is that the stability of G H is independent of the delay length h > 0. It is also true that, for the retarded type TDPS, i.e., if D = 0, the stability condition in Lemma 8 reduces to the well-known condition A − BC ∈ Hn×n that is shown by Haddad and Chellaboina [2004]. In other words, Lemma 8 includes this well-known result as a special case. In the next section we extend Lemma 8 to TDPSs with multiple commensurate delays. 4. STABILITY ANALYSIS OF TDPS WITH MULTIPLE COMMENSURATE DELAYS 4.1 TDSs with Multiple Commensurate Delays Let us consider the TDS shown in Fig. 2. In Fig. 2, G is an FDLTI system given by (4). On the other hand, Ωi ∈ Rm×m (i = 1, · · · , L) represents the interconnection between z and w over which retardation of length ih occurs, i.e., L w(t) = Ωi z(t − ih), (13) i=1
where h > 0 stands for the base delay length. In the [L] following we denote by G H{Ωi } the TDS in Fig. 2. Even though our main focus is on the stability analysis of TDPSs with multiple commensurate delays shown in Fig. 2, it must be too hasty to directly move onto this issue since at present we do not really know the existence [L] and uniqueness of the “solution” of G H{Ωi } and proper definition of positivity neither. This is the reason why we start our discussion in this subsection with G that is not necessarily positive and Ωi (i = 1, · · · , L) that are not necessarily nonnegative.
m by the set Definition 9. For given L ∈ N, let us define Kh,L m of function f : [0, Lh) → R that satisfies the following two conditions for all j = 1, · · · , L: (i) f is continuous over [(j − 1)h, jh), and (ii) lim f (t) exists. In addition, we t→jh−0
m m ⊂ Kh,L by define Kh,L+ m m : f (θ) ≥ 0 (∀θ ∈ [0, Lh)) . Kh,L+ := f ∈ Kh,L
In the following we investigate the behavior of the signals [L] x and w in G H{Ωi } under the initial condition z−Lh,Lh ∈ m where zt,Lh = zt (θ) = z(t + θ) (θ ∈ [0, Lh)). Kh,L 4.2 Conversion to TDS with a Single Delay [L]
To grasp the behavior of G H{Ωi } , in this section, we convert it to a TDS with a single delay of the form F HLm . Here, F is an FDLTI system constructed from G and Ωi (i = 1, · · · , L). On the other hand, HLm is the pure delay of delay length h and of Lm-input and Lm-output signals, i.e., HLm (s) = ILm e−sh . This conversion can be done by an elementary discussion using block-diagram [L] representation of G H{Ωi } . Lemma 10. For given FDLTI system G represented by (4) and interconnection matrices Ωi ∈ Rm×m (i = 1, · · · , L), define the Lm-input and Lm-output FDLTI system F by x(t) + B w(t), x ˙ (t) = A F : (14) x w(t) z(t) = C (t) + D
where input and output signals are partitioned as in w T = [ u1 (t)T · · · uL (t)T ]T ∈ RLm , ui (t) ∈ Rm , (15) zT = [ y1 (t)T · · · yL (t)T ]T ∈ RLm , yi (t) ∈ Rm and = A, BΩ BΩ · · · · · · BΩ , A C B = [ DΩ1 DΩ2 · · · · · · DΩL ] 1 2 L 0 ··· ··· 0 0 Im . .. . (16) = .. , D = 0 Im . . . C . . .. .. .. .. .. . . .. . . . 0 0 · · · 0 Im 0 Then, the following three assertions hold. (t) (t ≥ 0) for the initial (i) F HLm has a unique CCS x n condition x (0) = ξ ∈ R and z−h = ζ ∈ KhLm . [L] (ii) G H{Ωi } has a unique continuous solution x(t) (t ≥ 0) for the initial condition x(0) = ξ ∈ Rn and m . z−Lh,Lh = ζ ∈ Kh,L (iii) If we let ζ0 ζh ∈ KhLm , .. (17) ξ = ξ ∈ Rn , ζ = . ζ(L−1)h
then the continuous solution x(t) and z(t) (t ≥ 0) of [L] (t) G H{Ωi } coincides respectively with the CCS x and y1 (t) (t ≥ 0) of F HLm . Here ζt = ζt (θ) = ζ(t + θ)(θ ∈ [0, h)).
[L]
The behavior of TDS G H{Ωi } can be determined by the initial condition given for z(t) (t ∈ [−Lh, 0)) and x(0). In relation to this initial condition, we introduce the following function spaces that are conformable to the discussions in Sections 2 and 3.
Proof of Lemma 10: The first assertion (i) readily follows from the preceding results on TDSs with a single delay which are reviewed in Sections 2 and 3. Since CCS
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x (t) (t ≥ 0) of F HLm is unique (as just proved in (i)), we see that (ii) is verified by proving (iii). We first note that, in (17), ζ ∈ KhLm surely holds from the assumption m ζ ∈ Kh,L . To prove the assertion of (iii) concretely using block-diagram, let us consider the case where L = 3. Then, [3] the dynamics of G HΩ can be represented by the blockdiagram shown in Fig. 3. If we name the signals as in Fig. 3, it is very clear that z and w defined by (15) satisfy w(t) = z (t − h) (t ≥ h). It is also elementary to see that the relationship between the input w and the output z can be represented by (14) and (16). It follows that F HLm [L] is nothing but an alternative representation of G H{Ωi } obtained by extracting internal signals and including them to input and output signals in the feedback connection. In particular, it is very clear from Fig. 3 that y1 in F HLm [L] represents the output z of G H{Ωi } . Therefore, it remains to prove that the initial condition x(0) = ξ ∈ Rn and [L] m for G H{Ωi } corresponds to the z−Lh,Lh = ζ ∈ Kh,L initial condition x (0) = ξ ∈ Rn and z −h = ζ ∈ KhLm for F HLm where ζ is given by (17). However, this is again obvious from the signal flow shown in Fig. 3 implying that
y1 (t) y2 (t) z (t) = ... = yL (t)
y1 (t) y1 (t − h) .. .
y1 (t − (L − 1)h)
=
z(t) z(t − h) .. .
z(t − (L − 1)h)
.
This clearly shows that the initial function ζ must be given by (17).
In this lemma, we (mainly) consider the conversion from [L] G H{Ωi } to F HLm . Conversely, if we are given F HLm with F being (14) and (16) and the initial condition stated in (i), we can reproduce its CCS x (t) (t ≥ 0) by the [L] continuous solution x(t) (t ≥ 0) of G H{Ωi } if we let the ζ(i−1)h = (eT ⊗ initial condition stated in (ii) as in ξ = ξ, i Im )ζ ∈ Khm (i = 1, · · · , L) where ei stands for the ith standard basis of the L-dimensional Euclidean space. [L] Namely, G H{Ωi } and F HLm is identical, and hence it is quite natural and reasonable to proceed the analysis of [L] G H{Ωi } by working on F HLm .
On the basis of the basic spirit stated in detail in the preceding subsection, we first introduce the definition of [L] positivity and stability of G H{Ωi } .
+
+
+
Ω3 u3
z
G
+
u2 y3
[L]
Regarding the positivity of G H{Ωi } we just defined, it is straightforward to see that the next preliminary result holds. [L] Lemma 12. A TDS G H{Ωi } with multiple commensurate delays that is constructed from (4) and (13) is positive if G is positive, i.e., (10) holds, and Ωi (i = 1, · · · , L) are all nonnegative, i.e., Ωi ∈ Rm×m (i = 1, · · · , L). (18) + From now on we assume that G is positive and Ωi (i = [L] 1, · · · , L) are all nonnegative to ensure that G H{Ωi } is [L]
positive. Then, regarding the stability of TDPS G H{Ωi } , it is again straightforward to see that the next preliminary result holds. [L] Lemma 13. A TDPS G H{Ωi } with multiple commensurate delays that is constructed from (4), (10), (13), and ∈ RLm×Lm is Schur stable (18) is stable if and only if D + −1 n×n and Acl = A + B(ILm − D) C ∈ M is Hurwitz ∈ Rn×Lm , C ∈ RLm×n , and ∈ Mn×n , B stable where A + + ∈ RLm×Lm D are given by (16). +
Even though Lemma 13 is a direct consequence of Lemma 8, what is interesting here is that we can drastically simplify the condition in Lemma 13 by taking special cl = A + B(I Lm − ∈ RLm×Lm and A structure of D + −1 C ∈ Mn×n into consideration. The next two lemmas D) are the key to achieve such simplification. ∈ RLm×Lm given by (16) is Lemma 14. The matrix D + Schur stable if and only if DΩ ∈ Rm×m is Schur stable + where L Ωi ∈ Rm×m . (19) Ω := + i=1
∈ Rn×Lm ∈ RLm×n ∈M ,B ,C , and Lemma 15. For A + + Lm×Lm D ∈ R+ given by (16), suppose D is Schur stable. Then, we have −1 C = A + BΩ(Im − DΩ)−1 C ∈ Mn×n + B(I Lm − D) A (20) where Ω ∈ Rm×m is defined by (19). + In view of these two lemmas, we arrive at the next theorem that is the main result of this paper. [L]
Ω2 Hm
[L]
Definition 11. A TDS G H{Ωi } with multiple commensurate delays that is constructed from (4) and (13) is said to be positive if F HLm is positive in the sense of Definition 5, where F is given by (14) and (16). Similarly, [L] G H{Ωi } is said to be stable if F HLm is stable in the sense of Definition 7.
n×n
4.3 Stability Analysis of TDPSs with Multiple Commensurate Delays
w
3097
Ω1 Hm [L]
u1 y2
Fig. 3. Block-diagram of G H{Ωi } for L = 3.
Hm y1
Theorem 16. A TDPS G H{Ωi } with multiple commensurate delays that is constructed from (4), (10), (13), and (18) is stable if and only if DΩ ∈ Rm×m is Schur stable + and Acl = A + BΩ(Im − DΩ)−1 C ∈ Mn×n is Hurwitz stable where Ω ∈ Rm×m is defined by (19). + Proof of Theorem 16: The assertion readily follows from Lemmas 13, 14, and 15. 3152
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To see the implication of this main result deeply, let us [L] consider the delay-free counterpart of G H{Ωi } shown in Fig. 4 (that is obtained by letting h = 0). In the following, we denote this FD system by G Ω. In [Ebihara et al. [2017b]], we dealt with the same system configuration with G Ω where G is constructed from FDLTI positive stands for the interconnection subsystems and Ω ∈ Rm×m + over subsystems. For such interconnected system G Ω, we presupposed the satisfaction of “admissibility,” which as in is nothing but the Schur stability of DΩ ∈ Rm×m + Theorem 16. Under the admissibility condition, we see that G Ω is well-posed and positive, and this drastically facilitated our positivity-based analysis [Ebihara et al. [2017b]]. Using this terminology, we can restate Theorem 16 in the way that a neutral type TDPS with commensurate delays is stable if and only if its delay-free finite-dimensional counterpart is admissible and stable. This result in particular implies that the stability is irrelevant of the length of delays. Moreover, it has been clarified that it is meaningless to analyze an FDLTI interconnected system G Ω violating the admissibility, since it is unstable under unavoidable communication delays in practice. Namely, our main result clearly and strongly justify the admissibility presupposition introduced in Ebihara et al. [2017b]. 5. CONCLUSION In this paper we dealt with stability analysis of neutral type time-delay positive systems (TDPSs) with commensurate delays. To deal with such a system, the key issues were the proper definition of solution and the conversion to a neutral type TDPS with a single delay. As a main result, we showed that a neutral type TDPS with commensurate delays is stable if and only if its delay-free finitedimensional counterpart is admissible and stable. Acknowledgements: This work was supported by JSPS KAKENHI Grant Number 25420436. REFERENCES Ait Rami, M., Jordan, A.J., and Schonlein, M. (2013). Estimation of linear positive systems with unknown timevarying delays. European Journal of Control, 19(3), 179– 187. Bellman, R. and Cooke, K.L. (1963). Differential Difference Equations. Academic Press, New York. Berman, A. and Plemmons, R.J. (1979). Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York. Blanchini, F., Colaneri, P., and Valcher, M.E. (2012). Co-positive Lyapunov functions for the stabilization of positive switched systems. IEEE Transactions on Automatic Control, 57(12), 3038–3050. Briat, C. (2013). Robust stability and stabilization of uncertain linear positive systems via integral linear conw
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[L]
Fig. 4. Delay-Free Counterpart of TDPS G H{Ωi } . 3153
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Stability Analysis of Stability Analysis of Stability Analysis of Stability Analysis of Neutral Type Time-Delay Positive Systems Neutral Type Time-Delay Positive Systems Neutral Type Time-Delay Positive Systems Neutralwith TypeCommensurate Time-Delay Positive Systems Delays with Commensurate Delays with Commensurate Delays with Commensurate Delays ∗
Yoshio Ebihara ∗ Yoshio Ebihara ∗∗ Yoshio Yoshio Ebihara Ebihara ∗ ∗ Department of Electrical Engineering, Kyoto University, ∗ Department of Electrical Engineering, Kyoto University, Engineering, Kyoto University, ∗ Department of Electrical Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan Department of Electrical Engineering, Kyoto University, Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, (e-mail: [email protected]). Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan Japan (e-mail: [email protected]). (e-mail: (e-mail: [email protected]). [email protected]). Abstract: Recently, the author derived a necessary and sufficient condition for the asymptotic Abstract: the author derived aa necessary sufficient condition for the type asymptotic Abstract: Recently, the author and sufficient condition for asymptotic stability of Recently, neutral type positive systemsand (TDPSs), where the neutral TDPS Abstract: Recently, the time-delay author derived derived a necessary necessary and sufficient condition for the the type asymptotic stability of neutral type time-delay positive systems (TDPSs), where the neutral TDPS stability of neutral type time-delay positive systems (TDPSs), where the neutral type TDPS of interestofisneutral given by a feedback connection between (TDPSs), a finite-dimensional LTI positive system stability type time-delay positive systems where the neutral type TDPS of interest is given by a feedback connection between a finite-dimensional LTI positive system of interest is given by a feedback connection between a finite-dimensional LTI positive system and the pure delay. The goal of this paper is to extend this result to neutral type TDPSs of interest is given by a feedback connection between a finite-dimensional LTI positive system and the pure delay. The goal of thisTopaper is toweextend this result to neutral TDPSs and the pure delay. of is this to type TDPSs with multiple this end, first represent a neutral typetype TDPS with and the pure commensurate delay. The The goal goaldelays. of this thisTopaper paper is to toweextend extend this result result to neutral neutral type TDPSs with multiple commensurate delays. this end, first represent a neutral type TDPS with with multiple commensurate delays. To this end, we first represent a neutral type TDPS with multiple commensurate delays as a neutral type TDPS with a single delay, by augmenting input with multiple commensurate delays. To this end, we first represent a neutral type TDPSinput with multiple commensurate delays as a neutral type TDPS with a single delay, by augmenting multiple commensurate delays as a neutral type TDPS with a single delay, by augmenting input and output signals in the feedback connection. By this conversion, we can readily apply the multiple commensurate delays as a neutral type TDPS with a single delay, by augmenting input and output signals in the feedback connection. By this conversion, we can readily apply the and output the feedback this conversion, we can apply the latest result signals alreadyin established for connection. single delay By neutral type TDPSs. As thereadily main result, we and output signals in the feedback connection. By this conversion, we can readily apply the latest result already established for single delay neutral type TDPSs. As the main result, we latest result already established for single delay neutral type TDPSs. As the main result, we show that a neutral type TDPS with commensurate delays is stable if and only if its delaylatest resulta already established for single delay neutral typeisTDPSs. Asand theonly mainif result, we show that neutral type TDPS with commensurate delays stable if its delayshow that a neutral type TDPS with commensurate delays is stable if and only if its delayfree counterpart is stable and an additional “admissibility” condition is satisfied. This result in show that a neutral typeand TDPS with commensurate delayscondition is stableisifsatisfied. and onlyThis if its delayfree counterpart is stable an additional “admissibility” result in free counterpart stable an additional “admissibility” condition is satisfied. This result in particular impliesis that theand stability is irrelevant of the length of delays. free counterpart is stable and an additional “admissibility” condition is satisfied. This result in particular implies that the stability is irrelevant of the length of delays. particular implies that the stability is irrelevant of the length of delays. particular implies that the stability is irrelevant of the length of delays. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: neutral type time-delay positive systems, commensurate delays, asymptotic stability Keywords: neutral type type time-delay positive positive systems, commensurate commensurate delays, asymptotic asymptotic stability Keywords: Keywords: neutral neutral type time-delay time-delay positive systems, systems, commensurate delays, delays, asymptotic stability stability 1. INTRODUCTION To deal with stability analysis of neutral type TDPSs, in 1. INTRODUCTION INTRODUCTION To deal deal with stability analysis of neutral neutral type TDPSs, in 1. To of in [Ebihara et al.stability [2017a]],analysis we focused on thetype the TDPSs, time-delay 1. INTRODUCTION To deal with with stability analysis of neutral type TDPSs, in [Ebihara et al. [2017a]], we focused on the the time-delay [Ebihara et al. [2017a]], we focused on the the time-delay feedback system (TDFS)we form shownon in the Fig.the 1 totime-delay represent [Ebihara et al. [2017a]], focused feedback system (TDFS) form shown in Fig. 1 to represent This paper is concerned with asymptotic stability analy- feedback system (TDFS) shown in to type TDSs (that form are not necessarily In feedback system (TDFS) form shown in Fig. Fig. 11 positive). to represent represent Thisof paper paper is concerned concerned with asymptotic asymptotic stability analy- neutral neutral type TDSs (that are not not necessarily positive). In This is with stability analysis neutral type time-delay positive systems (TDPSs) neutral type TDSs (that are necessarily positive). In Fig. 1, G is an FDLTI system described by x(t) ˙ = Ax(t) + This paper is concerned with asymptotic stability analyneutral type TDSs (that are not necessarily positive). In sis of of commensurate neutral type type time-delay time-delay positive systems (TDPSs) Fig. 1, G is an FDLTI system described by x(t) ˙ = Ax(t) + sis neutral positive systems (TDPSs) with delays. In retrospect, the theory of Fig. 1, G is an FDLTI system described by x(t) ˙ = Ax(t) + Bw(t) and z(t) = Cx(t)+Dw(t), while H is the pure delay sis of neutral type time-delay positive systems (TDPSs) Fig. 1, and G isz(t) an FDLTI system described byisx(t) ˙the pure = Ax(t) + with commensurate commensuratelinear delays. In retrospect, retrospect, the theory theory of Bw(t) = Cx(t)+Dw(t), while H delay with delays. In the of finite-dimensional time-invariant positive systems Bw(t) and z(t) = Cx(t)+Dw(t), while H is the pure delay delay length h Cx(t)+Dw(t), > 0 and thus w(t) =Hz(t − h) (t ≥ h). with commensuratelinear delays. In retrospect, the theory of of Bw(t) and z(t) = while is the pure delay finite-dimensional time-invariant positive systems of delay length h > 0 and thus w(t) = z(t − h) (t ≥ h). finite-dimensional linear time-invariant positive systems (FDLTIPSs) is deeply in the theory of nonnegaof length > thus = z(t h). particular, theh thew(t) direct finite-dimensional linearrooted time-invariant positive systems In of delay delay length h existence > 00 and and of thus = feedthrough z(t − − h) h) (t (t ≥ ≥term h). (FDLTIPSs) is[Berman deeply rooted in the the theory theory of Horn nonnegaIn particular, the existence of thew(t) direct feedthrough term (FDLTIPSs) deeply rooted in nonnegative matrices is and Plemmons [1979],of and In particular, the existence of the direct feedthrough term D implies that the corresponding TDS is of neutral type. (FDLTIPSs) is deeply rooted in the theory of nonnegaIn particular, the existence of the direct feedthrough term tive matrices [Berman and Plemmons [1979], Horn and D implies implies that the corresponding corresponding TDSallows is of of neutral type. tive matrices [Berman and [1979], and Johnson [1991]], and celebrated Perron-Frobenius theorem that the TDS is type. This TDFS representation naturally us to adopt tive matrices [Berman and Plemmons Plemmons [1979], Horn Horn and D D implies that the corresponding TDSallows is of neutral neutral type. Johnson [1991]], and[1991]] celebrated Perron-Frobenius theorem This TDFS representation naturally us ato tosolution adopt Johnson [1991]], and celebrated Perron-Frobenius theorem [Horn and Johnson has played a central role. Very This TDFS representation naturally allows us adopt the continuous concatenated solution (CCS) as Johnson [1991]], and[1991]] celebrated Perron-Frobenius theorem This TDFS representation naturally allows us to adopt [Horn and Johnson has played a central role. Very the continuous concatenated solution (CCS) as a solution [Horn Johnson [1991]] has aa central role. Very recently, system theory has gained renewed continuous concatenated solution (CCS) aa solution which is explicitly defined by Hagiwara andas [Horn and andpositive Johnson [1991]] has played played central role.interVery the the continuous concatenated solution (CCS) as Kobayashi solution recently, positive system theory has gained renewed interwhich is explicitly defined by Hagiwara and Kobayashi recently, positive system gained renewed interest from the viewpoint oftheory convexhas optimization, and excelwhich is explicitly defined by Hagiwara and Kobayashi [2011]. On the basis of this firm set up, in [Ebihara et al. recently, positive system theory has gained renewed interwhich is explicitly defined by Hagiwara and Kobayashi est from the viewpoint of convex optimization, and excel[2011]. On Onwethe the basis of this this firm firm set up, up, definition in [Ebihara [Ebiharaforet et al. al. est from the viewpoint of convex optimization, and excellent papers have been published along this direction, see, [2011]. basis of set in [2017a]], first introduced a proper est from the viewpoint of convex optimization, and excel[2011]. On the basis of this firm set up, in [Ebihara al. lent papers papers have been published published along this direction, direction, see, [2017a]], we first introduced a proper definition foretthe the lent have been along this see, e.g., those by Rantzer [2015, 2016], Gurvits et al. [2007], [2017a]], we first introduced a proper definition for the positivity of TDSs given by TDFS form. This definition lent papers have been published along this direction, see, [2017a]], we first introduced a proper definition for the e.g., those those by Rantzer [2015,Shorten 2016], Gurvits Gurvits et al. al. [2007], [2007], positivity of TDSs given by TDFS form. This definition e.g., Rantzer [2015, 2016], et Mason andby Shorten [2007], et al. [2009], of by This definition led us to the soundgiven conclusion that form. a TDS is positive if e.g., those Rantzer [2015,Shorten 2016], Gurvits et al. Tanaka [2007], positivity positivity of TDSs TDSs given by TDFS TDFS form. This definition Mason andby Shorten [2007], et[2012], al. [2009], [2009], Tanaka led us us to ifthe the sound conclusion that a TDS TDS isthe positive if Mason and Shorten [2007], Shorten et al. Tanaka and Langbort [2011], Blanchini et al. Briat [2013], led to sound conclusion that a is positive if and only the FDLTI system G is positive (in sense of Mason and Shorten [2007], Shorten et al. [2009], Tanaka led us to ifthe sound conclusion that a TDS isthe positive if and Langbort Langbort [2011], Blanchini et al. al. [2012], [2012], Briat [2013], and only the FDLTI system G is positive (in sense of and [2011], Blanchini et Briat [2013], Najson [2013]. We also emphasize that the study on and only if the FDLTI system G is positive (in the sense of FDSs). Then, as the main result, we showed(inthat asense TDPS and Langbort [2011], Blanchini et al. [2012], Briat [2013], and only if the FDLTI system G is positive the of and Najson [2013]. We also emphasize that the study on FDSs). Then, as the main result, we showed that a TDPS and Najson [2013]. also that study on consensus of multi-agent positive systems a FDSs). Then, the main we in TDFS formas asymptotically stable if andthat onlyaa ifTDPS D is and Najsonproblems [2013]. We We also emphasize emphasize that the the studyis on FDSs). Then, asis main result, result, we showed showed consensus problems of multi-agent positive systems isby in TDFS TDFS form is the asymptotically stable if and andthat only ifTDPS D is −1 consensus problems of positive aaa in promising direction, andmulti-agent this issue is treatedsystems activelyis form is asymptotically stable if only if D C is Hurwitz stable. Schur stable and A + B(I − D) consensus problems of multi-agent positive systems is −1 TDFS form is A asymptotically stable if and only if D is is promising direction, and this this issue is treated treated actively On by in −1 C is Hurwitz stable. Schur stable and + B(I − D) promising direction, and issue is actively by Valcher and Misra [2014] and Ebihara et al. [2017b]. stable and A + B(I − D) −1 C is Hurwitz stable. promising direction, and this issue is treated actively On by Schur C isthis Hurwitz stable. Schur stable and A + B(I − D) Valcher and Misra [2014] and Ebihara et al. [2017b]. The goal of this paper is to extend latest result to Valcher and Misra [2014] al. [2017b]. the other hand, study onand the Ebihara analysiset of The goal of this paper is to extend this latest result to Valcher and Misra [2014] Ebihara etand al. synthesis [2017b]. On On the other other hand, study onand the analysis and synthesis of neutral The goal of paper is extend this to TDPSs with multiple delays the hand, study on the analysis and synthesis of goaltype of this this paper is to to extendcommensurate this latest latest result result to TDPSs has also been active, and fruitful results have been neutral type TDPSs with multiple commensurate delays the other hand, study on the analysis and synthesis of The TDPSs has also been active, and fruitful results have been neutral type TDPSs with multiple commensurate delays (see Fig. 2). In Fig. 2, the system G is an FDLTIPS TDPSs has also been active, and fruitful results have been type TDPSs with multiple commensurate delays obtained, e.g., by Haddad Chellaboina [2004], Ait neutral (see Fig. 2). In Fig. 2, the system G is an FDLTIPS TDPSs has also been active, and fruitful results have been obtained, e.g., by and Haddad and Lam Chellaboina [2004], Ait (see Fig. Fig. 2, is an the as before, andIn nonnegative matrixG i represents obtained, by Haddad and Chellaboina Ait (see Fig. 2). 2). Inthe Fig. 2, the the system system GΩ is an FDLTIPS FDLTIPS Rami et al.e.g., [2013], Shen and [2014]. In[2004], particular, the as before, and the nonnegative matrix Ω obtained, e.g., by and Haddad and Lam Chellaboina [2004], Ait feedback i represents Rami et al. [2013], Shen and [2014]. In particular, represents the as before, and the nonnegative matrix Ω connection between z and w over which retari Rami et al. [2013], and Shen and Lam [2014]. In particular, represents the as before, and the nonnegative matrix Ω Haddad and Chellaboina [2004] showed a prominent result i L which retarfeedback connection between z and w over Rami et al. [2013], and Shen and Lam [2014]. In particular, Haddad and Chellaboina [2004] showed a prominent result feedback connection between z and w over which retarΩ z(t − ih), dation of length ih occurs, i.e., w(t) = L i Haddad and Chellaboina [2004] showed a prominent result i=1 feedback connection between z and w over which retarverifying that a retarded type TDPS is stable if and only L Ωi z(t ih), dation of length ih occurs, i.e., w(t) = Haddad and [2004] showed prominent result L i=1To verifying thatChellaboina a retarded type TDPS TDPS is astable stable if is and only where − ih), dation of length ih occurs, i.e., w(t) 0 stands the base length. deal− i=1 Ω verifying that type is and only Ωii z(t z(t −with ih), dation h of> length ih for occurs, i.e., delay w(t) = = if its delay-free finite-dimensional stable. i=1To h > 00 stands for the base delay length. deal with verifying that aa retarded retarded type TDPScounterpart is stable if if is and only where if its delay-free finite-dimensional counterpart stable. where h > stands for the base delay length. To deal with such a neutral type TDPSs with multiple commensurate if its delay-free finite-dimensional counterpart is stable. where h > 0 stands for the base delay length. To deal with However, to the best of the author’s knowledge, existing such a neutral type TDPSs with multiple commensurate if its delay-free finite-dimensional counterpart is stable. However, to the best best of the author’s author’s knowledge, existing delays, such aa neutral type multiple in this paper, we first with represent it as commensurate a neutral type However, the the knowledge, existing neutral type TDPSs TDPSs with multiple commensurate studies onto TDPSs are of restricted to retarded type TDPSs, delays, in this paper, we first represent it as a neutral type However, to the best of the author’s knowledge, existing such studies on TDPSs are restricted to retarded type TDPSs, delays, in this paper, we first represent it as type TDPS with a single delay by augmenting input and output studies on TDPSs are restricted retarded type in this paper,delay we first represent it input as aa neutral neutral type and those neutral TDPSsto surprisingly scarce. delays, TDPS with a single by augmenting and output studies on for TDPSs are type restricted toare retarded type TDPSs, TDPSs, and those for neutral type TDPSs are surprisingly scarce. TDPS with a single delay by augmenting input and output signals in the feedback connection. By this conversion, and those for neutral type TDPSs are surprisingly scarce. TDPS with a single delay connection. by augmenting input and output This is probably due totype theTDPSs fact that the definition of solu- signals in the feedback By this conversion, and those for neutral are surprisingly scarce. This is isforprobably probably due toTDSs the fact fact that the the definition of solusolu- we signals in this can readily apply the connection. latest resultBy already established This the that definition of in the the feedback feedback connection. By this conversion, conversion, tions neutral due typeto is rather difficult irrespective we can readily apply the latest result already established This isforprobably due toTDSs the fact that the definition of solu- signals tions neutral type is rather difficult irrespective we can readily apply the latest result already established for single delay neutral type TDPSs. As the main result, tions for neutral type TDSs is rather difficult irrespective we can readily apply thetype latest result As already established of positivity [Bellman and Cooke [1963]]. for single delay neutral TDPSs. the main result, tions for neutral type TDSs is rather difficult irrespective of positivity [Bellman and Cooke [1963]]. for single delay neutral type TDPSs. As the main of for single delay neutral type TDPSs. As the main result, result, of positivity positivity [Bellman [Bellman and and Cooke Cooke [1963]]. [1963]]. Copyright © 2017, 2017 IFAC 3148Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 3148 Copyright ©under 2017 responsibility IFAC 3148Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 3148 10.1016/j.ifacol.2017.08.681
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w
z
G
H
Fig. 1. Time-Delay Systems in Time-Delay Feedback System Form (H(s) = Ie−sh ). w
G
z
2.1 Delay-Differential Equations (DDEs)
H{Ωi } [L]
Fig. 2. Neutral Type TDPS with Multiple Commensurate L [L] Ωi e−sih ). Delays (H{Ωi } (s) = i=1
we show that a neutral type TDPS with commensurate delays is stable if and only if DΩ is Schur stable and A + BΩ(I − DΩ)−1 C is Hurwitz stable where Ω := L i=1 Ωi . This result in particular implies that the stability is irrelevant of the length of delays. The former condition (i.e., the Schur stability of the nonnegative matrix DΩ) has already appeared in our preceding study on the analysis of (delay-free) interconnected positive systems [Ebihara et al. [2017b]]. By following the terminology introduced there, this condition is referred to as admissibility in this paper. Using this terminology, we can restate our main result in the way that a neutral type TDPS with commensurate delays is stable if and only if its delay-free finite-dimensional counterpart is admissible and stable. We use the following notation. We denote by R, R+ , and R++ the set of real, nonnegative, and strictly positive numbers, respectively. For given two matrices A and B of the same size, we write A > B (A ≥ B) if Aij > Bij (Aij ≥ Bij ) holds for all (i, j), where Aij stands for the (i, j)-entry of A. We define Rn+ := {x ∈ Rn : x ≥ 0} and Rn++ := {x ∈ Rn : x > 0}. We also define Rn×m + and Rn×m with obvious modifications. The set of Metzler ++ (Hurwitz stable) matrices of size n is denoted by Mn (Hn ). The set of natural numbers is denoted by N. For x ∈ Rn , n we denote its 1-norm by �x�, i.e., �x� := i=1 |xi |. In relation to the definition of CCS and positivity for TDSs in TDFS form, we introduce the following function spaces: m := {f : f (θ) ∈ Rm , f is continuous C[0,h) over [0, h)} , Khm := m Kh+
m f ∈ C[0,h) :
:= {f ∈
Khm
Therefore, we start our discussion from TDSs in DDE form, and review the result by Hagiwara and Kobayashi [2011] for the definition of solution and the conversion from DDE form to TDFS form. We note that the contents of Sections 2 and 3 are in part borrowed from the full paper version of [Ebihara et al. [2017a]]. Since this reference paper is not yet publically available, we decided to review the results in [Ebihara et al. [2017a]] as quickly as possible.
lim f (θ) exists ,
θ→h−0
: f (θ) ≥ 0 (∀θ ∈ [0, h))} .
The L1 [0, h) norm for f ∈ Khm h �f (θ)�dθ.
(1) is defined by �f � :=
0
2. REPRESENTATION OF TIME-DELAY SYSTEMS Even though we concentrate our attention on the analysis of TDSs in TDFS form, the denomination “neutral” is historically given for TDSs in delay-differential equation (DDE) form [Bellman and Cooke [1963], Hale [1977]].
In the literature, LTITDSs represented by the following DDE are studied extensively [Bellman and Cooke [1963], Hale [1977]]. q(t) ˙ = Jq(t) + K q(t ˙ − h) + Lq(t − h), J, K, L ∈ Rn×n . (2) Here, h > 0 stands for the delay length. TDSs represented by the DDE (2) with K = 0 are historically referred to as retarded type TDSs, while TDSs represented by the DDE (2) with K �= 0 are referred to as neutral type TDSs [Bellman and Cooke [1963], Hale [1977]]. The “solution” of (2) is determined under the initial condition q(t) = φ(t) (t ∈ [−h, 0)),
q(0) = ξ
(3)
where φ(t) is usually assumed to be (continuous and) continuously differentiable on the closed interval [−h, 0]. However, it is rather difficult to define the concept of solutions due to the appearance of indifferentiability (or even discontinuity) especially in the neutral case. This issue is deeply studied by Hagiwara and Kobayashi [2011], and proper definitions of solutions are given. Among them, in this paper, we adopt the continuous concatenated solution (CCS) defined there. 2.2 Continuous Concatenated Solutions (CCSs) Let us introduce the definition of CCSs for the DDE (2) equipped with the initial condition (3). Definition 1. (Hagiwara and Kobayashi [2011]). Suppose φ(t) in (3) is bounded, continuously differentiable on [−h, 0), and has the limit lim φ(t). Then, q(t) (t ≥ −h) t→0−0
is said to be a continuous concatenated solution (CCS) of the DDE (2) under the initial condition (3) if (i) it is continuous for t ≥ 0 and (ii) it is differentiable and satisfies (2) for t ≥ 0 except possibly for time instants t = kh (k ∈ N). As the definition says, a CCS is continuous over t ≥ 0 but not necessarily differentiable at t = kh (k ∈ N). On the other hand, a stronger solution that is differentiable over t ≥ −h is referred to as a regular solution in [Hagiwara and Kobayashi [2011]]. However, for the existence of such a regular solution, it is obviously necessary that φ(0) = ξ ˙ ˙ and φ(−0) = Jφ(0) + K φ(−h) + Lφ(−h). The latter requirement is rather stringent, and hence it is reasonable to introduce somehow relaxed solutions. Among them the CCS is believed to be a natural one since it possesses continuity property that is essentially required in describing the behavior of physical (real-world) systems.
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3. REVIEW OF OUR PRECEDING RESULTS
2.3 Time-Delay Feedback Systems (TDFSs) In the community of control theory, it is common to describe LTITDSs in time-delay feedback system (TDFS) form shown in Fig. 1. In Fig. 1, G stands for an FDLTI system described by x(t) ˙ = Ax(t) + Bw(t), G: z(t) = Cx(t) + Dw(t), (4) A ∈ Rn×n , B ∈ Rn×m , C ∈ Rm×n , D ∈ Rm×m . On the other hand, H is the pure delay of constant delay length h > 0 and thus H : w(t) = z(t − h). (5) In the following we denote by G H the TDS in Fig. 1. Note that the behavior of TDS G H can be determined by the initial condition given for z(t) (t ∈ [−h, 0)), x(0). (6) Once we describe a given TDS in TDFS form, we can apply fully matured control-oriented techniques such as (scaled) small-gain theorem for its stability analysis. Moreover, recently, a more advanced and rigorous monodromy operator approach has been build for the stability analysis of TDSs in TDFS form, see Kim et al. [2015] and references cited there in. This is the motivation of [Hagiwara and Kobayashi [2011]] to seek for a conversion technique from DDE form to TDFS form so that the monodromy operator approach can also be applied to TDSs in DDE form. More precisely, the issue in [Hagiwara and Kobayashi [2011]] is how to determine the FDLTI system G (or say, the matrices A, B, C, and D) and the initial condition (6) from a given DDE (or say, the matrices J, K, and L) and a given initial condition (3) so that the (continuous concatenated) solution of the DDE can be represented as a signal in TDS G H. The results in Hagiwara and Kobayashi [2011] that are relevant to this issue are quickly reviewed in the next subsection. 2.4 Conversion from DDE to TDFS
This section is devoted to a quick review of [Ebihara et al. [2017a]]. Before getting into the details, we begin with the very basic definition and related results for FDLTIPSs. Definition 3. (Farina and Rinaldi [2000]). The FDLTI system (4) is said to be positive if its state and output are both nonnegative for any nonnegative initial state and nonnegative input. Proposition 4. (Farina and Rinaldi [2000]). The FDLTI system (4) is positive if and only if , C ∈ Rm×n , D ∈ Rm×m . A ∈ Mn×n , B ∈ Rn×m + + +
(10)
The first question in [Ebihara et al. [2017a]] is to seek for proper extensions of these definition and condition to LTITDSs in TDFS form. The counterparts are as follows. Definition 5. (Ebihara et al. [2017a]). A TDS in TDFS form G H constructed from (4) and (5) is said to be positive if x(t) ≥ 0 and w(t) ≥ 0 (∀t ≥ 0) hold for any x(0) ∈ Rn+ and z−h ∈ Kh+ . Lemma 6. (Ebihara et al. [2017a]). A TDS in TDFS form G H constructed from (4) and (5) is positive in the sense of Definition 5 if and only if FDLTI system G is positive, i.e., (10) holds. Even though Lemma 6 has been proved in [Ebihara et al. [2017a]], in the following we review the proof for the sufficiency part rather in detail. We believe that this is of great benefit for readers to see concretely what kind of continuous solution (which is nothing but CCS) we employ in this study. To see the sufficiency proof, suppose x(0) ∈ Rn+ and z−h ∈ Kh+ are given. Then, for the first time interval [0, h), note that w0 (θ) = z−h (θ) (θ ∈ [0, h)) and hence w0 ∈ Kh+ . Then, from the variation of constant formula, we see that t At x(t) = e x(0) + eA(t−τ ) Bw(τ )dτ (0 ≤ t < h) (11) 0
The next proposition shows that the CCS of the DDE (2) can always be represented by the state x of the FDLTI system G in the TDS G H. In the following we write wt = wt (θ) = w(t + θ) (θ ∈ [0, h)), (7) zt = zt (θ) = z(t + θ) (θ ∈ [0, h)). Proposition 2. (Hagiwara and Kobayashi [2011]). Suppose φ(t) in (3) is bounded, continuously differentiable on [−h, 0), and has the limit lim φ(t). Then, the DDE (2) t→0−0
has a unique CCS q(t), and it coincides, over t ≥ 0, with x(t) resulting from G H with A, B, C, and D given by A = J, B = I, C = L + KJ, D = K (8) and with the initial condition ˙ (θ ∈ [0, h)), x(0) = ξ. (9) z−h (θ) = K φ(θ−h)+Lφ(θ−h) As clearly shown in (8), the direct-feedthrough term D of the FDLTI system G coincides with the coefficient K in the DDE form. Since we are mainly interested in the neutral type TDSs and hence K �= 0 in (2), we justifiably focus on TDSs given by TDFS form with D �= 0 in the following.
holds and hence x(t) is uniquely determined and continuous over t ∈ [0, h). It is also true that x(t) ≥ 0 (t ∈ [0, h)) since G is positive and w0 ∈ Kh+ . Moreover, since w0 ∈ m (or more precisely since limt→h−0 w(t) exists from Kh+ m the definition of Kh+ ), we see that limt→h−0 x(t) exists as well, and from the continuity requirement on x we can let x(h) := limt→h−0 x(t)(∈ Rn+ ). On the other hand, from z(t) = Cx(t) + Dw(t) (0 ≤ t < h) (12) and again from the positivity of G, we see that the m important property z0 ∈ Kh+ holds. To summarize, for the next time interval t ∈ [h, 2h), we know that x(h) ∈ Rn+ is determined and wh has exactly the same property m since wh (θ) = z0 (θ) (θ ∈ [0, h) and with w0 ∈ Kh+ m z0 ∈ Kh+ . Therefore by repeating the same arguments, we see that continuous solution x(t) exists over t ∈ [0, 2h), and by repeating the same arguments recursively (or say, by concatenating the solutions determined over [kh, (k + 1)h) repeatedly), we can conclude that continuous solution x(t) uniquely exists over t ≥ 0 and in particular x(t) ≥ 0 (∀t ≥ 0). We note that x(t) thus constructed might not be differentiable for time instants t = kh (k ∈ N). On the other hand, it is also clear that w(t) thus constructed is unique and satisfies w(t) ≥ 0 (∀t ≥ 0), although it might
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not be continuous for time instants t = kh (k ∈ N). We thus completed the proof for the sufficiency. On the basis of these preliminary results, in [Ebihara et al. [2017a]], we analyzed stability of TDPS G H. The definition of asymptotic stability of TDS G H (that is not necessarily positive at this stage) is as follows. Definition 7. (Ebihara et al. [2017a]). A TDS in TDFS form G H constructed from (4) and (5) is said to be asymptotically stable if �xt � + �wt � → 0 (t → ∞) for any x0 ∈ Rn and w0 ∈ Kh . Then, as a main result of [Ebihara et al. [2017a]], we derived the next theorem. In the following, “asymptotic stability” is abbreviated as “stability” just for brevity. Lemma 8. (Ebihara et al. [2017a]). A TDPS in TDFS form G H constructed from FDLTIPS G given by (4) and (10) and the pure delay H given by (5) is stable if and only if D ∈ Rm×m is Schur stable and Acl = A + B(I − + D)−1 C ∈ Mn×n is Hurwitz stable. An immediate fact that follows from Lemma 8 is that the stability of G H is independent of the delay length h > 0. It is also true that, for the retarded type TDPS, i.e., if D = 0, the stability condition in Lemma 8 reduces to the well-known condition A − BC ∈ Hn×n that is shown by Haddad and Chellaboina [2004]. In other words, Lemma 8 includes this well-known result as a special case. In the next section we extend Lemma 8 to TDPSs with multiple commensurate delays. 4. STABILITY ANALYSIS OF TDPS WITH MULTIPLE COMMENSURATE DELAYS 4.1 TDSs with Multiple Commensurate Delays Let us consider the TDS shown in Fig. 2. In Fig. 2, G is an FDLTI system given by (4). On the other hand, Ωi ∈ Rm×m (i = 1, · · · , L) represents the interconnection between z and w over which retardation of length ih occurs, i.e., L w(t) = Ωi z(t − ih), (13) i=1
where h > 0 stands for the base delay length. In the [L] following we denote by G H{Ωi } the TDS in Fig. 2. Even though our main focus is on the stability analysis of TDPSs with multiple commensurate delays shown in Fig. 2, it must be too hasty to directly move onto this issue since at present we do not really know the existence [L] and uniqueness of the “solution” of G H{Ωi } and proper definition of positivity neither. This is the reason why we start our discussion in this subsection with G that is not necessarily positive and Ωi (i = 1, · · · , L) that are not necessarily nonnegative.
m by the set Definition 9. For given L ∈ N, let us define Kh,L m of function f : [0, Lh) → R that satisfies the following two conditions for all j = 1, · · · , L: (i) f is continuous over [(j − 1)h, jh), and (ii) lim f (t) exists. In addition, we t→jh−0
m m ⊂ Kh,L by define Kh,L+ m m : f (θ) ≥ 0 (∀θ ∈ [0, Lh)) . Kh,L+ := f ∈ Kh,L
In the following we investigate the behavior of the signals [L] x and w in G H{Ωi } under the initial condition z−Lh,Lh ∈ m where zt,Lh = zt (θ) = z(t + θ) (θ ∈ [0, Lh)). Kh,L 4.2 Conversion to TDS with a Single Delay [L]
To grasp the behavior of G H{Ωi } , in this section, we convert it to a TDS with a single delay of the form F HLm . Here, F is an FDLTI system constructed from G and Ωi (i = 1, · · · , L). On the other hand, HLm is the pure delay of delay length h and of Lm-input and Lm-output signals, i.e., HLm (s) = ILm e−sh . This conversion can be done by an elementary discussion using block-diagram [L] representation of G H{Ωi } . Lemma 10. For given FDLTI system G represented by (4) and interconnection matrices Ωi ∈ Rm×m (i = 1, · · · , L), define the Lm-input and Lm-output FDLTI system F by x(t) + B w(t), x ˙ (t) = A F : (14) x w(t) z(t) = C (t) + D
where input and output signals are partitioned as in w T = [ u1 (t)T · · · uL (t)T ]T ∈ RLm , ui (t) ∈ Rm , (15) zT = [ y1 (t)T · · · yL (t)T ]T ∈ RLm , yi (t) ∈ Rm and = A, BΩ BΩ · · · · · · BΩ , A C B = [ DΩ1 DΩ2 · · · · · · DΩL ] 1 2 L 0 ··· ··· 0 0 Im . .. . (16) = .. , D = 0 Im . . . C . . .. .. .. .. .. . . .. . . . 0 0 · · · 0 Im 0 Then, the following three assertions hold. (t) (t ≥ 0) for the initial (i) F HLm has a unique CCS x n condition x (0) = ξ ∈ R and z−h = ζ ∈ KhLm . [L] (ii) G H{Ωi } has a unique continuous solution x(t) (t ≥ 0) for the initial condition x(0) = ξ ∈ Rn and m . z−Lh,Lh = ζ ∈ Kh,L (iii) If we let ζ0 ζh ∈ KhLm , .. (17) ξ = ξ ∈ Rn , ζ = . ζ(L−1)h
then the continuous solution x(t) and z(t) (t ≥ 0) of [L] (t) G H{Ωi } coincides respectively with the CCS x and y1 (t) (t ≥ 0) of F HLm . Here ζt = ζt (θ) = ζ(t + θ)(θ ∈ [0, h)).
[L]
The behavior of TDS G H{Ωi } can be determined by the initial condition given for z(t) (t ∈ [−Lh, 0)) and x(0). In relation to this initial condition, we introduce the following function spaces that are conformable to the discussions in Sections 2 and 3.
Proof of Lemma 10: The first assertion (i) readily follows from the preceding results on TDSs with a single delay which are reviewed in Sections 2 and 3. Since CCS
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x (t) (t ≥ 0) of F HLm is unique (as just proved in (i)), we see that (ii) is verified by proving (iii). We first note that, in (17), ζ ∈ KhLm surely holds from the assumption m ζ ∈ Kh,L . To prove the assertion of (iii) concretely using block-diagram, let us consider the case where L = 3. Then, [3] the dynamics of G HΩ can be represented by the blockdiagram shown in Fig. 3. If we name the signals as in Fig. 3, it is very clear that z and w defined by (15) satisfy w(t) = z (t − h) (t ≥ h). It is also elementary to see that the relationship between the input w and the output z can be represented by (14) and (16). It follows that F HLm [L] is nothing but an alternative representation of G H{Ωi } obtained by extracting internal signals and including them to input and output signals in the feedback connection. In particular, it is very clear from Fig. 3 that y1 in F HLm [L] represents the output z of G H{Ωi } . Therefore, it remains to prove that the initial condition x(0) = ξ ∈ Rn and [L] m for G H{Ωi } corresponds to the z−Lh,Lh = ζ ∈ Kh,L initial condition x (0) = ξ ∈ Rn and z −h = ζ ∈ KhLm for F HLm where ζ is given by (17). However, this is again obvious from the signal flow shown in Fig. 3 implying that
y1 (t) y2 (t) z (t) = ... = yL (t)
y1 (t) y1 (t − h) .. .
y1 (t − (L − 1)h)
=
z(t) z(t − h) .. .
z(t − (L − 1)h)
.
This clearly shows that the initial function ζ must be given by (17).
In this lemma, we (mainly) consider the conversion from [L] G H{Ωi } to F HLm . Conversely, if we are given F HLm with F being (14) and (16) and the initial condition stated in (i), we can reproduce its CCS x (t) (t ≥ 0) by the [L] continuous solution x(t) (t ≥ 0) of G H{Ωi } if we let the ζ(i−1)h = (eT ⊗ initial condition stated in (ii) as in ξ = ξ, i Im )ζ ∈ Khm (i = 1, · · · , L) where ei stands for the ith standard basis of the L-dimensional Euclidean space. [L] Namely, G H{Ωi } and F HLm is identical, and hence it is quite natural and reasonable to proceed the analysis of [L] G H{Ωi } by working on F HLm .
On the basis of the basic spirit stated in detail in the preceding subsection, we first introduce the definition of [L] positivity and stability of G H{Ωi } .
+
+
+
Ω3 u3
z
G
+
u2 y3
[L]
Regarding the positivity of G H{Ωi } we just defined, it is straightforward to see that the next preliminary result holds. [L] Lemma 12. A TDS G H{Ωi } with multiple commensurate delays that is constructed from (4) and (13) is positive if G is positive, i.e., (10) holds, and Ωi (i = 1, · · · , L) are all nonnegative, i.e., Ωi ∈ Rm×m (i = 1, · · · , L). (18) + From now on we assume that G is positive and Ωi (i = [L] 1, · · · , L) are all nonnegative to ensure that G H{Ωi } is [L]
positive. Then, regarding the stability of TDPS G H{Ωi } , it is again straightforward to see that the next preliminary result holds. [L] Lemma 13. A TDPS G H{Ωi } with multiple commensurate delays that is constructed from (4), (10), (13), and ∈ RLm×Lm is Schur stable (18) is stable if and only if D + −1 n×n and Acl = A + B(ILm − D) C ∈ M is Hurwitz ∈ Rn×Lm , C ∈ RLm×n , and ∈ Mn×n , B stable where A + + ∈ RLm×Lm D are given by (16). +
Even though Lemma 13 is a direct consequence of Lemma 8, what is interesting here is that we can drastically simplify the condition in Lemma 13 by taking special cl = A + B(I Lm − ∈ RLm×Lm and A structure of D + −1 C ∈ Mn×n into consideration. The next two lemmas D) are the key to achieve such simplification. ∈ RLm×Lm given by (16) is Lemma 14. The matrix D + Schur stable if and only if DΩ ∈ Rm×m is Schur stable + where L Ωi ∈ Rm×m . (19) Ω := + i=1
∈ Rn×Lm ∈ RLm×n ∈M ,B ,C , and Lemma 15. For A + + Lm×Lm D ∈ R+ given by (16), suppose D is Schur stable. Then, we have −1 C = A + BΩ(Im − DΩ)−1 C ∈ Mn×n + B(I Lm − D) A (20) where Ω ∈ Rm×m is defined by (19). + In view of these two lemmas, we arrive at the next theorem that is the main result of this paper. [L]
Ω2 Hm
[L]
Definition 11. A TDS G H{Ωi } with multiple commensurate delays that is constructed from (4) and (13) is said to be positive if F HLm is positive in the sense of Definition 5, where F is given by (14) and (16). Similarly, [L] G H{Ωi } is said to be stable if F HLm is stable in the sense of Definition 7.
n×n
4.3 Stability Analysis of TDPSs with Multiple Commensurate Delays
w
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Ω1 Hm [L]
u1 y2
Fig. 3. Block-diagram of G H{Ωi } for L = 3.
Hm y1
Theorem 16. A TDPS G H{Ωi } with multiple commensurate delays that is constructed from (4), (10), (13), and (18) is stable if and only if DΩ ∈ Rm×m is Schur stable + and Acl = A + BΩ(Im − DΩ)−1 C ∈ Mn×n is Hurwitz stable where Ω ∈ Rm×m is defined by (19). + Proof of Theorem 16: The assertion readily follows from Lemmas 13, 14, and 15. 3152
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To see the implication of this main result deeply, let us [L] consider the delay-free counterpart of G H{Ωi } shown in Fig. 4 (that is obtained by letting h = 0). In the following, we denote this FD system by G Ω. In [Ebihara et al. [2017b]], we dealt with the same system configuration with G Ω where G is constructed from FDLTI positive stands for the interconnection subsystems and Ω ∈ Rm×m + over subsystems. For such interconnected system G Ω, we presupposed the satisfaction of “admissibility,” which as in is nothing but the Schur stability of DΩ ∈ Rm×m + Theorem 16. Under the admissibility condition, we see that G Ω is well-posed and positive, and this drastically facilitated our positivity-based analysis [Ebihara et al. [2017b]]. Using this terminology, we can restate Theorem 16 in the way that a neutral type TDPS with commensurate delays is stable if and only if its delay-free finite-dimensional counterpart is admissible and stable. This result in particular implies that the stability is irrelevant of the length of delays. Moreover, it has been clarified that it is meaningless to analyze an FDLTI interconnected system G Ω violating the admissibility, since it is unstable under unavoidable communication delays in practice. Namely, our main result clearly and strongly justify the admissibility presupposition introduced in Ebihara et al. [2017b]. 5. CONCLUSION In this paper we dealt with stability analysis of neutral type time-delay positive systems (TDPSs) with commensurate delays. To deal with such a system, the key issues were the proper definition of solution and the conversion to a neutral type TDPS with a single delay. As a main result, we showed that a neutral type TDPS with commensurate delays is stable if and only if its delay-free finitedimensional counterpart is admissible and stable. Acknowledgements: This work was supported by JSPS KAKENHI Grant Number 25420436. REFERENCES Ait Rami, M., Jordan, A.J., and Schonlein, M. (2013). Estimation of linear positive systems with unknown timevarying delays. European Journal of Control, 19(3), 179– 187. Bellman, R. and Cooke, K.L. (1963). Differential Difference Equations. Academic Press, New York. Berman, A. and Plemmons, R.J. (1979). Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York. Blanchini, F., Colaneri, P., and Valcher, M.E. (2012). Co-positive Lyapunov functions for the stabilization of positive switched systems. IEEE Transactions on Automatic Control, 57(12), 3038–3050. Briat, C. (2013). Robust stability and stabilization of uncertain linear positive systems via integral linear conw
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[L]
Fig. 4. Delay-Free Counterpart of TDPS G H{Ωi } . 3153