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Robust Event-Triggered Control Subject to External Disturbance*
Proceedings of the 20th World Congress Proceedings of the 20th World The International Federation of Automatic The International Federation of Congress Automatic Control Control Proceedings of the 20th9-14, World The International Federation of Congress Automatic Control Toulouse, France, July 2017 Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 7899–7904 Robust Event-Triggered Control Subject Robust Robust Event-Triggered Event-Triggered Control Control Subject Subject Robust Event-Triggered Control Subject External Disturbance External Disturbance External Disturbance External Disturbance ∗ ∗ ∗∗
to to to to
Pengpeng Pengpeng Zhang Zhang ∗ Tengfei Tengfei Liu Liu ∗ Zhong-Ping Zhong-Ping Jiang Jiang ∗∗ Pengpeng Zhang ∗∗ Tengfei Liu ∗∗ Zhong-Ping Jiang ∗∗ ∗∗ ∗ Pengpeng Zhang Tengfei Liu Zhong-Ping Jiang ∗ State Key Laboratory of Synthetical Automation for Process ∗ State Key Laboratory of Synthetical Automation for Process State Northeastern Key Laboratory of Synthetical Automation for Process Industries, University, Shenyang, 110004, China (e-mail: ∗ Industries, University, Shenyang, 110004,for China (e-mail: State Northeastern Key Laboratory of Synthetical Automation Process Industries, Northeastern University, Shenyang, 110004, China (e-mail: [email protected]; [email protected]). [email protected]; [email protected]). Industries, Northeastern University, Shenyang, 110004, China (e-mail: ∗∗ [email protected]; [email protected]). ∗∗ Tandon School of Engineering, New York University, 5 MetroTech of Engineering, New York University, 5 MetroTech ∗∗ Tandon School [email protected]; [email protected]). Tandon School of Engineering, New York University, 5 MetroTech Brooklyn, NY (e-mail: [email protected]) ∗∗ Center, Center,School Brooklyn, NY 11201, 11201, USA USA York (e-mail: [email protected]) Tandon of Engineering, University, 5 MetroTech Center, Brooklyn, NY 11201, New USA (e-mail: [email protected]) Center, Brooklyn, NY 11201, USA (e-mail: [email protected]) Abstract: Abstract: This This paper paper studies studies the the event-triggered event-triggered control control problem problem for for systems systems subject subject to to both both Abstract: This paper and studies the event-triggered control problem for systems subject to both dynamic uncertainties external disturbances. To overcome infinitely fast sampling caused dynamic uncertainties and external disturbances. To overcome infinitely fast sampling caused Abstract: This paper studies the event-triggered control problem for systems subject to both dynamic uncertainties and external disturbances. To overcome infinitely fastuses sampling caused by a event-triggering mechanism is which not the by the the disturbance, disturbance, a new new external event-triggering mechanism is proposed, proposed, which uses not only only the dynamic uncertainties disturbances. Tothe overcome infinitely fastuses sampling caused by the disturbance, a and newbut event-triggering mechanism is proposed, which not only the measurable system state also an estimate of unmeasurable state and the external measurable system state but also an estimate of the unmeasurable state and the external by the disturbance, a newbut event-triggering mechanism is to proposed, which uses not the measurable system state alsointervals an estimate of the unmeasurable state and by thea only external disturbance. The can bounded positive disturbance. system The inter-sampling inter-sampling intervals can be beof proved proved to be be lower lower state bounded by a external positive measurable state but also an estimate the unmeasurable and the disturbance. The inter-sampling intervals can be proved to be lower bounded by a positive constant, is independent the of state constant, which which is inter-sampling independent on onintervals the magnitudes magnitudes of the the unmeasurable unmeasurable state and andbythe the external external disturbance. Theis can be proved to be lower bounded positive constant, which independent ondesign, the magnitudes of the event-triggered unmeasurable state andcan theabeexternal disturbance. With the proposed the closed-loop system inputdisturbance. With the proposed design, the closed-loop event-triggered system can be inputconstant, which is independent on the magnitudes of the unmeasurable state and the external disturbance. With with the proposed design, the closed-loop event-triggered system can be inputto-state stabilized the external disturbance as the input. Refined tools of input-to-state to-state stabilized with the external disturbance as the input. Refined tools of input-to-state disturbance. With with the proposed design, the closed-loop event-triggered system be inputto-state stabilized the external disturbance as the in input. Refined tools of can input-to-state stability theorem are solving the stability (ISS) (ISS) and and the the small-gain small-gain theorem are employed employed in solving the problem. problem. to-state the external disturbance as the in input. Refined tools of input-to-state stability stabilized (ISS) and with the small-gain theorem are employed solving the problem. stability (ISS)(International and the small-gain are employed in solving the problem. © 2017, IFAC Federationtheorem of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Event-Triggered Event-Triggered Control, Control, Input-to-State Input-to-State Stabilization, Stabilization, External External Disturbances, Disturbances, Keywords: Event-Triggered Control, Input-to-State Stabilization, External Disturbances, Dynamic Dynamic Uncertainty. Uncertainty. Keywords: Event-Triggered Control, Input-to-State Stabilization, External Disturbances, Dynamic Uncertainty. Dynamic Uncertainty. 1. terize 1. INTRODUCTION INTRODUCTION terize the the robustness robustness with with respect respect to to sampling sampling errors, errors, and and 1. INTRODUCTION terize the robustness with respectaccording to sampling errors, and the threshold signal is designed to the margin the threshold signal is designed according to the margin 1. INTRODUCTION terize the robustness with respect to sampling errors, and the threshold signal is designed according to the margin robustness, for (asymptotic) stability of closed-loop Event-triggered of robustness, for (asymptotic) stability of the the closed-loop Event-triggered control control has has attracted attracted a a lot lot of of attention attention of the threshold signal is designed according to the margin of robustness, for (asymptotic) stability of the closed-loop Event-triggered control has attracted a lot ofthe attention system Tabuada (2007); (2015). from Compared periodevent-triggered system Tabuadastability (2007); Girard Girard (2015). For For from the the controls controls community. community. Compareda with with the period- event-triggered robustness, for (asymptotic) of the control, closed-loop Event-triggered has the attracted lot ofthe attention event-triggered system Tabuada (2007); Girard (2015). For from the controlscontrol community. Compared with period- of practical implementation of event-triggered inic time-triggered sampling, event-triggered sampling practical implementation of event-triggered control, inic time-triggered sampling, the event-triggered sampling event-triggered systemshould Tabuada (2007); Girard (2015). For from the controls community. Compared with thesampling periodpractical implementation of be event-triggered control, inic time-triggered sampling, the event-triggered finitely fast sampling avoided, i.e., the sampling time instants are determined by an event generator that finitely fast sampling should be avoided, i.e., the sampling time instants are determined by an event generator that practical implementation of event-triggered control, inic time-triggered sampling, the event-triggered sampling finitely fast sampling should be avoided, i.e., the sampling time instants are determined by an event generator that intervals are are required required to to be be lower lower bounded bounded by by some some positive positive depends on system state, into account the depends on the the system state,byto toantake take into account that the intervals finitely fast sampling should be avoided, i.e., the sampling time instants are determined event generator are required to be lower bounded by some positive depends on the during system the state, to takeintervals. into account constant; see e.g., in (2010). One system behavior behavior sampling Early the pa- intervals constant; seerequired e.g., the thetodiscussions discussions in Lemmon Lemmon (2010). One system during the sampling intervals. Early paintervals are be lower bounded byZeno some positive depends on the system˚ state, to take into account the constant; see e.g., the discussions in Lemmon (2010). One ˚ system behavior during the sampling intervals. Early paspecial case of infinitely fast sampling is the behavior, pers, e.g., A rz´ e n (1999); A str¨ o m and Bernhardsson (1999), ˚ ˚ special case of infinitely fast sampling is the Zeno behavior, pers, e.g., A rz´ e n (1999); A str¨ o m and Bernhardsson (1999), constant; see e.g., the discussions insampling Lemmon (2010). One system behavior during ˚ the sampling intervals. Early pa- special ˚ case ofan infinitely fast sampling is the Zeno behavior, pers, e.g., Arz´ enadvantages (1999); A str¨ om and Bernhardsson (1999), i.e., there is infinite number of time instants have shown the of event-triggered control over there isofaninfinitely infinite fast number of sampling timebehavior, instants have shown the advantages ofom event-triggered control over i.e., ˚ ˚ special case sampling is the Zeno pers, e.g., A rz´ e n (1999); A str¨ and Bernhardsson (1999), i.e., there is an infinite number of sampling time instants have shown the advantages of event-triggered control over which converge to value; see et periodic time-triggered control. converge to aa finite finite value; see e.g., e.g., Goebel Goebel et al. al. periodic time-triggered control. i.e., there isthe anZeno infinite number ofhybrid sampling time In instants have shown the advantages of event-triggered control over which which converge to abehavior finite value; see e.g., Goebel et al. periodic time-triggered control. (2009) for of systems. most (2009) for the Zeno behavior of hybrid systems. In most converge to abehavior finite value; see e.g., Goebel al. In decade, effort periodic time-triggered control. (2009) for the results, Zeno of hybrid systems. In et most In the the past past decade, great great effort has has been been spent spent in in dede- which of the existing the event-triggered control problem the existing results, the event-triggered control problem In the past decade, great effort hastheory, been see, spent in Ande- of (2009) for the Zeno behavior of hybrid systems. In most veloping the event-triggered control e.g., of the existing results, the event-triggered control problem veloping the event-triggered control theory, see, e.g., Antransformed into of appropriate In the past great effort hastheory, been see, spent in Ande- is is the transformed into the thetheproblem problem of choosing choosing appropriate veloping the decade, event-triggered control e.g., of existing results, event-triggered control problem ta (2010); and (2009); is transformed into the problem of fast choosing appropriate ta and and Tabuada Tabuada (2010); Gawthrop Gawthrop and Wang Wang (2009); threshold signals to avoid infinitely sampling. veloping the event-triggered control theory, see, e.g., Anthreshold signals to avoid infinitely fast sampling. ta and Tabuada (2010); Gawthrop and Wang (2009); is transformed into the problem of choosing appropriate Heemels et al. (2012); Heemels and Donkers (2013); Lunze Heemels et al. (2012); Heemels and Donkers (2013); Lunze threshold signals to avoid infinitely fast sampling. ta and Tabuada (2010); Gawthrop andMarchand Wang Heemels et al. (2012); Heemels and Donkers (2013);(2009); Lunze unmeasurable disturbances, threshold signals of to avoid infinitely external fast sampling. and (2010); Tabuada (2007); et In the the presence presence of unmeasurable external disturbances, and Lehmann Lehmann (2010);Heemels Tabuadaand (2007); Marchand et al. al. In Heemels et al. (2012); Donkers (2013); Lunze In the presence of to unmeasurable external disturbances, and Lehmann (2010); Tabuada (2007); Marchand etconal. an intuitive design avoid infinitely fast sampling is (2013) and the references therein. In event-triggered an intuitive design to avoid infinitely fast sampling is to to (2013) and the (2010); references therein.(2007); In event-triggered conIn the presence of unmeasurable external disturbances, and Lehmann Tabuada Marchand et al. intuitive design tothreshold avoid infinitely fastpaying sampling is to (2013) and the signals references therein. Inemployed event-triggered con- an introduce a constant signal, by the price trol, threshold are usually to generate introduce a constant threshold signal, by paying the price trol, threshold signals are usually employed to generate an intuitive design to avoid infinitely fastpaying sampling is to (2013) and of the signals references Inemployed event-triggered conintroduce a constant threshold signal, by the price trol, threshold aretherein. usually to generate convergence. The conthe sampling. In earlier e.g., of losing losing asymptotic asymptotic convergence. The event-triggered event-triggered conthe events events of data data sampling. In the the earlier results, results, e.g., of introduce a constant threshold signal, by paying the price trol, threshold signals are usually employed to generate of losing asymptotic convergence. The event-triggered conthe events of data sampling. In the earlier results, e.g., trol problem of linear systems pHeemels et al. Henningsson et constant trol problem of continuous-time continuous-time systems in in the the pHeemels et of al. (2008); (2008); Henningsson et al. al. (2008), (2008), constant losing asymptotic convergence.islinear The event-triggered conthe events sampling. In the results, e.g., of trol problem of continuous-time linear systems in the pHeemels etsignals al. data (2008); Henningsson et earlier al. (2008), constant resence of external disturbances discussed in Lunze and threshold are used, while in the recent results, resence of external disturbances is discussed in Lunze and threshold signals are used, while in the recent results, trol problem of continuous-time linear systems in the pHeemels et al. (2008); Henningsson et al. (2008), constant resence of external disturbances is discussed in Lunze and threshold signals are used, while(2009); in theWang recentand results, Lehmann (2010) in which the disturbance behaviour of e.g., Dimarogonas and Johansson Lem(2010) indisturbances which the disturbance behaviour of e.g., Dimarogonas and used, Johansson (2009); Wang and Lem- Lehmann resence of external is discussed in Lunze and threshold signals are while in the recent results, Lehmann (2010) in which the disturbance behaviour of e.g., Dimarogonas and Johansson (2009); Wang and Lema feedback loop is the main concern of the paper. Howmon (2009, 2008), the threshold signals are designed to a feedback loop is the main concern of the paper. Howmon (2009, 2008), the threshold signals are designed to Lehmann (2010) in which the disturbance behaviour of e.g., Dimarogonas and Johansson (2009); Wang and Lema feedback loop is the main concern of the paper. Howmon (2009, 2008), the threshold signals are designed to ever, in and Lehmann (2010), interdepend on state and the historical data. ever, in Lunze Lunze and Lehmann (2010), ofthe thethesmallest smallest interdepend on real-time real-time state and even even the are historical data. a feedback loop is the main concern paper. Howmon (2009, 2008), the threshold signals designed to in Lunze and to Lehmann (2010), theofsmallest interdepend on real-time anddesign even methods the historical sampling is the bound disturbances. Also, refined refined classicalstate control have data. been ever, sampling is related related to the upper upper bound ofsmallest disturbances. Also, classical control design methods have been ever, in Lunze and Lehmann (2010), the interdepend onfor real-time state anddesign even methods the historical sampling is related to thestrategy upper bound of disturbances. Also, refined classical control have data. been A self-triggered sampling for a class of nonlinear employed event-triggered control. For example, the self-triggered sampling strategy for a class of nonlinear employed for classical event-triggered control.methods For example, the A sampling is related to the upper bound of disturbances. Also, refined control design have been A self-triggered sampling strategy for a class of nonlinear employed for event-triggered control. Forused example, the systems subject to disturbances is in concept stability (ISS) to subject sampling to external external disturbances is proposed proposed in concept of of input-to-state input-to-state stability (ISS) is is used to characcharacA self-triggered strategy for a bound class ofofnonlinear employed event-triggered control. example, the systems systems subject to external disturbances is proposed in concept of for input-to-state stability (ISS) For is used to characLiu and Jiang (2015) in which the upper external Liu and Jiang (2015) in whichdisturbances the upper bound of external This work was supported in part by NSF grant ECCS-1501044, in systems subject to external is proposed in concept of input-to-state stability (ISS) is used to characThis work was supported in part by NSF grant ECCS-1501044, in Liu and Jiang (2015) in which the upper bound of external disturbances is assumed to be known a priori. This disturbances is(2015) assumed to be the known a priori. part by NSFC grants 61374042, 61522305, 61633007 and 61533007, in was supported in part by NSF grant ECCS-1501044, Liu and Jiang in which upper bound of external part by work NSFC grants 61374042, 61522305, 61633007 and 61533007, in This disturbances is assumed to be known a priori. was supported in part by NSF grant ECCS-1501044, in part by work NSFC grants 61374042, 61522305, 61633007 and 61533007, in the Research Funds the Universities The main of is disturbances is assumed to paper be known priori.the part by NSFC the Fundamental Fundamental Research Funds for for the Central Central Universities The main purpose purpose of this this paper is to toa refine refine the existing existing part by grants 61374042, 61522305, 61633007 and 61533007, in part byGrants the Fundamental Research Funds for the Central Universities under N130108001 and N140805001 in China, and in part by The main purpose of this paper control is to refine the existing under Grants N130108001Research and N140805001 in China, and in part by results of robust event-triggered for systems with part by the Fundamental Funds for the Central Universities results of robust event-triggered control for systems with under Grants N130108001 and N140805001 in China, andofinComplex part by State Key Laboratory of Intelligent Control and Decision The main purpose of this paper isexternal to refine the existing results of robust event-triggered control for systems with State Key Laboratory of Intelligent Control and Decision of Complex both dynamic uncertainties and disturbances. underKey Grants N130108001 and N140805001 in China, andofinComplex part by both dynamic uncertainties and external disturbances. State Laboratory of Intelligent Control and Decision Systems at BIT. results of robust event-triggered control for disturbances. systems with Systems atLaboratory BIT. both dynamic uncertainties and external This paper does not assume a known upper bound of State Key of Intelligent Control and Decision of Complex Systems at BIT. author: Corresponding Tengfei Liu This paper doesuncertainties not assume aand known upper disturbances. bound of the the both dynamic external Corresponding author: Tengfei Liu This paper does not assume a known upper bound of the Systems at BIT. author: Tengfei Liu Corresponding This paper does not assume a known upper bound of the Corresponding author: Tengfei Liu
Copyright © 2017 8175 Copyright 2017 IFAC IFAC 8175Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Copyright ©under 2017 responsibility IFAC 8175Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 8175 10.1016/j.ifacol.2017.08.752
Proceedings of the 20th IFAC World Congress 7900 Pengpeng Zhang et al. / IFAC PapersOnLine 50-1 (2017) 7899–7904 Toulouse, France, July 9-14, 2017
external disturbance. To avoid infinitely fast sampling, a new event-triggering mechanism is proposed, which uses not only the measurable system state but also an estimate of the unmeasurable state and the external disturbance. Interestingly, it is proved that the event-triggered sampling intervals are lower bounded by a positive constant, which is independent of both the dynamic uncertainty and the external disturbance. ISS of the closed-loop eventtriggered system is also proved by considering the external disturbance as the input. In the design, refined tools of ISS Sontag and Wang (1996); Sontag (2006) and small-gain theorem Jiang et al. (1994); Liu et al. (2014) are used. Moreover, as a by-product of the main result, this paper proposes a small-gain result for a new class of interconnected systems induced by event-triggered control. The rest of the paper is organized as follows. Section 2 gives the problem formulation of event-triggered control for continuous-time systems subject to both dynamic uncertainties and external disturbances. In Section 3, we propose a new event-triggering mechanism, and prove that the resulted closed-loop event-triggered system is ISS with the external disturbance as the input. An example with numerical simulation is employed to demonstrate the implementation and the effectiveness of the proposed design. Section 5 contains some concluding remarks. To make the paper self-contained, some notations are given here. We use Z+ to denote the set of all nonnegative integers. For any vector x ∈ Rn , |x| represents its Euclidean norm. For any function ψ : Z+ → Rn , mean ψ(τ ) a≤τ ≤b b 1 represents b−a a ψ(τ ) dτ , where b > a.
According to the definition of sampling error w in (4), the closed-loop event-triggered system can be represented as an interconnected system, as shown in Figure 1. δ(t) x(t) x(t)
Define w(t) = x(ti ) − x(t), t ∈ [ti , ti+1 ), i ∈ S as the sampling error, and rewrite u(t) = v(x(t) + w(t)).
(4) (5)
Then, the controlled x-subsystem can be represented by x(t) ˙ = f (x(t), z(t), v(x(t) + w(t)), δ(t)) =: g(x(t), z(t), w(t), δ(t)). (6)
z(t)
x(t) ˙ = g(x(t), z(t), w(t), δ(t)) S
x(ti ) +
δ(t) w(t) −
Fig. 1. The block diagram of the closed-loop eventtriggered system composed of (1) and (6), where S represents the event-triggered sampler. In this paper, we study the case in which the system dynamics g is globally Lipschitz. δ Assumption 1. There exist constants Lxg , Lzg , Lw g , Lg ≥ 0 such that δ (7) |g(x, z, w, δ)| ≤ Lxg |x| + Lzg |z| + Lw g |w| + Lg |δ| holds for all x, z, w, δ. Also, it is assumed that the dynamic uncertainty, i.e., the z-subsystem, is ISS with x and δ as the inputs, and there exists a v such that the nominal system, i.e., the x-subsystem, is ISS with z, w and δ as the inputs. Assumption 2. System (1) is ISS with x and δ as the inputs, and admits an ISS-Lyapunov function Vz : Rnz → R+ satisfying the following conditions: (1) There exist constants αz , αz > 0 such that
2. PROBLEM FORMULATION
We consider an event-triggered control system in the following form: z(t) ˙ = h(z(t), x(t), δ(t)) (1) x(t) ˙ = f (x(t), z(t), u(t), δ(t)) (2) u(t) = v(x(ti )), t ∈ [ti , ti+1 ), i ∈ S ⊆ Z+ (3) n where x ∈ R is the measurable system state, z ∈ Rnz is the unmeasurable portion of the state, u ∈ Rm is the control input, δ ∈ Rnδ represents the external disturbance, h : Rnz × Rn × Rnδ → Rnz and f : Rn × Rnz × Rm × Rnδ → Rn represent system dynamics, v : Rn → Rm represents the feedback control law based on eventtriggered data-sampling. In this paper, the z-subsystem is considered as dynamic uncertainty, while the x-subsystem is called the nominal system. The time sequence {ti }i∈S is determined online by an appropriately designed eventtriggering mechanism with S = {0, 1, 2, · · · } ⊆ Z+ being the set of the indices of the sampling times and t0 = 0.
z(t) ˙ = h(z(t), x(t), δ(t))
αz |z|2 ≤ Vz (z) ≤ αz |z|2 ,
∀z;
(8)
(2) There exist constants αz , kzx , kzδ > 0 such that ∇Vz (z)h(z, x, δ) ≤
(9) − αz Vz (z) + max{kzx |x|2 , kzδ |δ|2 }, ∀z, x, δ. Assumption 3. System (6) is ISS with z, w and δ as inputs, and admits an ISS-Lyapunov function Vx : Rn → R+ satisfying the following conditions: (1) There exist constants αx , αx > 0 such that αx |x|2 ≤ Vx (x) ≤ αx |x|2 ,
∀x;
(10)
(2) There exist constants αx , kxz , kxw , kxδ > 0 such that ∇Vx (x)g(x, z, w, δ) ≤ −αx Vx (x) (11) + max{kxz |z|2 , kxw |w|2 , kxδ |δ|2 }, ∀x, z, w, δ. Our objective is to develop a new event-triggering mechanism, which fully takes into consideration the external disturbance and the dynamic uncertainty such that the following two objectives are achieved: Objective 1: There exists a constant t∆ > 0, independent of the unmeasurable state z and the external disturbance δ, such that ti+1 − ti ≥ t∆ (12) holds for all i, i + 1 ∈ S. Objective 2: The closed-loop event-triggered system composed of (1)-(3) is ISS with respect to the external disturbance δ.
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3. ROBUST EVENT-TRIGGERED CONTROL
Then, direct calculation yields:
The main contribution of this paper lies in a new eventtriggering mechanism. Significantly different from most of the published results, an estimation of the unmeasurable state and the external disturbance is introduced, so that the threshold signal depends not only on the system state but also on the magnitudes of the unmeasurable state and the external disturbance. In this way, the technical challenge caused by the dynamic uncertainty and the external disturbance is solvable at the same time. Specifically, the new event-triggering mechanism is proposed in the form of ti+1 = min{t ≥ti + t∆ : x δ |w(t)| = max{kw |x(t)|, kw ζ(t)}},
x kw
(13)
δ kw
> 0 and > 0 are constants to be where t∆ > 0, designed later, ζ(t) represents an estimate of the influence of the unmeasurable state and the external disturbance. 3.1 An Example as Motivation Before presenting the main result, we employ a simple example to show how an estimation of the external disturbance can be made to guarantee the positive lower bound the inter-sampling intervals. Consider a first-order linear system in the form of x(t) ˙ = x(t) + u(t) + δ(t) (14) where x ∈ R is the state, u ∈ R is the control input, and δ ∈ R represents external disturbances. The event-triggered control law is designed as u(t) = −2x(ti ) for ti ≤ t < ti+1 , i ∈ S. By using (4), (14) and (15), we have x(t) ˙ = −x(t) − 2w(t) + δ(t).
(15)
(16)
Also, according to the definition of sampling error w, we have (17) w(t) ˙ = x(t) ˙ = −w(t) − x(ti ) + δ(t) for ti ≤ t < ti+1 , i ∈ S. Note that w(ti ) = x(ti )−x(ti ) = 0. For ti ≤ t < ti+1 with i ∈ S, the solution of system (17) with initial condition w(ti ) = 0 is t w(t) = (e−(t−ti ) − 1)x(ti ) + e−(t−υ) δ(υ) dυ, (18) ti
and thus,
|x(t) − x(ti )| ≤ (1 − e−(t−ti ) )|x(ti )| +
t ti
7901
|δ(υ)| dυ (19)
holds for ti ≤ t ≤ ti+1 , i ∈ S. Here, (19) holds for t = ti+1 because of the continuity of both sides of the inequality. t Clearly, the term ti |δ(υ)| dυ represents the influence of the external disturbance δ. For t ∈ (ti , ti+1 ], a lower bound t of ti |δ(υ)| dυ can be calculated as
η(t) = max{|x(t) − x(ti )| − (1 − e−(t−ti ) )|x(ti )|, 0}. (20) It can be observed that η(t) is left-continuous at the discontinuous points.
|x(t) − x(ti )| ≤ (1 − e−(t−ti ) )|x(ti )| + η(t) for ti < t ≤ ti+1 , i ∈ S.
(21)
To represent the “average” influence of δ(t), define 0, for t = t0 = 0; ζ(t) = η(t) , for t ∈ (ti , ti+1 ]. t − ti
(22)
With an estimation in form of (22), it can be guaranteed that |x(t) − x(ti )| ≤ (1 − e−(t−ti ) )|x(ti )| + (t − ti )ζ(t)
≤ 2 max{t − ti , 1 − e−(t−ti ) } max{|x(ti )|, ζ(t)} for ti < t ≤ ti+1 , i ∈ S.
(23)
On the other hand, the event trigger (13) guarantees that x δ |x(t)|, kw ζ(t)} (24) |x(t) − x(ti )| ≤ max{kw holds for ti < t ≤ ti+1 , i ∈ S. With (Liu and Jiang, 2015, Lemma A.1), property (24) of the event trigger holds if x x δ /(1 + kw ), kw max {|x(ti )|, ζ(t)} |x(t) − x(ti )| ≤ min kw x x δ ≤ max kw /(1 + kw )|x(ti )|, kw ζ(t) (25) holds for ti < t ≤ ti+1 , i ∈ S.
Define
t∆ = inf{t > ti : 2 max{t − ti , 1 − e−(t−ti ) } x x δ } − ti = min kw /(1 + kw ), kw (26) for t ≥ ti , i ∈ S. That is, a positive lower bound of the inter-sampling intervals is guaranteed. Also, it can be directly proved that x δ |x(t)|, kw ζ(t) (27) |w(t)| ≤ max kw for all t ∈ [ti , ti + t∆ ], i ∈ S.
Moreover, in the following discussion, it is proved that the convergence of δ leads to the convergence of ζ, to satisfy the necessary condition for input-to-state stabilization. Figure 2 shows the structure of the estimation-based eventtriggering mechanism. u(t)
u(t) = −2x(ti )
x(ti ) S
δ(t)
x(t) ˙ = x(t) + u(t) + δ(t)
ζ(t)
E
x(t)
Fig. 2. The block diagram of the closed-loop eventtriggered system (14)-(15), where E represents the estimation (22) of the external disturbance. 3.2 ISS of Event-Triggered Controlled x-Subsystem In this subsection, we focus on the event-triggered control of the x-subsystem. In accordance with the design in Subsection 3.1, an estimation of the external disturbance δ and the unmeasurable state z is defined as 1 ζ(t) = (t − ti )Lzδ g E(t, ti ) max {|x(t) − x(ti )| − Lg (E(t, ti ) − 1)|x(ti )|, 0} (28)
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δ z for all t ∈ [ti + t∆ , ti+1 ], i ∈ S, where Lzδ g = 2 max{Lg , Lg }, w (Lx +L )(t−t ) x w x i E(t, ti ) = e g g , Lg = Lg /(Lg + Lg ) and t∆ is a positive constant to be determined later.
Theorem 4 presents the main result on ISS of the eventtriggered controlled x-subsystem. Theorem 4. Consider the closed-loop event-triggered system (6) with event-triggering mechanism (13) and (28). x Under Assumptions 1 and 3, by choosing constant kw >0 satisfying x 2 ) kxw (kw < 1, (29) (1 − σ1 ) αx αx and constant t∆ as t∆ = {t∆ : ϕ(t∆ ) − εmin = 0} (30) xw
with 0 < σ1 < 1, ϕ(t∆ ) = 2 max{Lg eLg t∆ − Lxw x g t∆ t }, Lxw = Lx + Lw , ε Lg , Lzδ min = min{kw /(1 + ∆ g g g e g x δ δ kw ), kw } and kw is to be designed later, Objective 1 is achieved, and the event-triggered controlled x-subsystem (6) is ISS with δ and z as the inputs. Proof. In this proof, we first find a positive t∆ , and then prove the ISS of the event-triggered controlled xsubsystem (6).
By the proof of Theorem 4, the event-triggered mechanism (13) and (28) satisfying (29) and (30) can achieve input-tostate stabilization of the closed-loop system (6). However, it cannot be directly clarified that the stability of the whole system composed of (1) and (6). 3.3 ISS of the Closed-Loop Event-Triggered System The x-subsystem and the z-subsystem are feedback interconnected with each other. In this subsection, a condition for ISS of the closed-loop event-triggered system is given by using the small-gain idea. Theorem 5. Consider the (z, x)-system in (1) and (6). Under Assumptions 1, 2 and 3, if the event-triggering mechanism composed of (13) and (28) satisfies (29) and (30), and χxz · χzx < 1, (35) (36) σχz < 1 where χxz = kxz /((1 − σ2 )σ1 αx αz ), χzx = kzx /((1 − δ 2 σ2 )αz αx ), χz = 2.3kxw (kw ) /((1 − σ2 )σ1 αx αz ), 0 < σ2 < 1 and σ satisfies 1 > χxz , σ > χzx , (37) σ then, ISS property with respect to external disturbances for the whole system composed of (1)-(3) is achieved.
(a) The Existence of t∆ > 0 As discussion in Subsection Proof. From (8) and (34), we can get the gain margin 3.1, one can find a t∆ satisfying formulation of the ISS-Lyapunov functions Vx : x xw xw kw δ , kw . Vx (x(t)) ≥ max{χxz Vz (z(t)), χz mean Vz (z(τ )), 2 max{Lg eLg t∆ − Lg , eLg t∆ Lδg t∆ } = min x ti ≤τ ≤t 1 + kw δ χ ¯x mean |δ(τ )|2 , χδx |δ(t)|2 } ti ≤τ ≤t In particular, t∆ is independent on the external disturbance δ and the unmeasurable state z. Please see Zhang ⇒ ∇Vx (x(t))g(x(t), z(t), w(t), δ(t)) ≤ −αx∗ Vx (x(t)) (38) et al. (2017) for more detailed proofs. for t ∈ (ti , ti+1 ], i ∈ S, where constants αx∗ , χxz , χz , χ ¯δx , χδx are chosen as αx∗ = σ1 σ2 αx , χxz = kxz /((1 − σ2 )σ1 αx αz ), (B) ISS The event-triggering mechanism (13) can guarχz = k¯xz /((1 − σ2 )σ1 αx αz ), χ ¯δx = k¯xδ /((1 − σ2 )σ1 αx ) and antee that δ δ χx = kx /((1 − σ2 )σ1 αx ). x δ |x(t) − x(ti )| ≤ max kw |x(t)|, kw ζ(t) (31) Similarly, under Assumption 2 and 3, we have for t ∈ (ti , ti+1 ], i ∈ S. Vz (z(t)) ≥ max{χzx Vx (x(t)), χδz |δ(t)|2 } (39) With the satisfaction of (29) and (31), (11) implies ⇒ ∇Vz (z)h(z(t), x(t), δ(t)) ≤ −αz∗ Vz (z(t)) ∇Vx (x(t))g(x(t), z(t), w(t), δ(t)) ≤ −σ1 αx Vx (x(t)) for all t ≥ 0, where constants α∗ , χ , χδ are chosen as δ 2 2 + max{kxz |z(t)|2 , kxw (kw ) ζ (t), kxδ |δ(t)|2 } for t ∈ (ti , ti+1 ], i ∈ S.
From (Zhang et al., 2017, Section 1), we have t max{|z(τ )|, |δ(τ )|} dτ ζ(t) ≤ ti t − ti ≤ max{1.5 mean |z(τ )|, 3 mean |δ(τ )|}
(32)
(33)
z
zx
x
αz∗ = σ2 αz , χzx = kzx /((1 − σ2 )αz αx ) and χδz = kzδ /((1 − σ2 )αz ).
An ISS-Lyapunov function candidate for the interconnected system composed of (1) and (6) is defined as V (x, z) = max{σVx (x), Vz (z)} (40) where σ is defined in (37).
For convenience of notations, we denote V (x(t), z(t)) =: V (t) in the following discussion. According to a smallfor all t ∈ (ti , ti+1 ], i ∈ S. gain result for a class of interconnected systems induced Substituting (33) into the right-hand side of (32) yields by event-triggered control (Zhang et al., 2017, Appendix), by using conditions (35)-(37), we have ∇Vx (x(t))g(x(t), z(t), w(t), δ(t)) ≤ −σ1 αx Vx (x(t)) z 2 ¯z 2 ¯δ 2 δ 2 V (t) ≥ max{σχz mean V (τ ), χ ¯δ mean |δ(τ )|2 , χδ |δ(t)|2 } + max{kx |z(t)| , kx mean |z(τ )| , kx mean |δ(τ )| , kx |δ(t)| } t ≤τ ≤t t ≤τ ≤t ti ≤τ ≤t
ti ≤τ ≤t
ti ≤τ ≤t
ti ≤τ ≤t
i
i
(34)
δ 2 for all t ∈ (ti , ti+1 ], i ∈ S, where k¯xz = 2.3 kxw (kw ) and δ w δ 2 ¯ kx = 9 kx (kw ) . Property (34) means (delayed) ISS of the closed-loop event-triggered system (6) with respect to δ and z. This ends the proof of Theorem 4.
⇒ ∇V (t)F (x(t), z(t), w(t), δ(t)) ≤ −αV (t)
(41) (42)
for t ∈ (ti , ti+1 ], i ∈ S, where χ ¯δ = σ χ ¯δx , χδ = max{σχδx , χδz }, F (x, z, w, δ) = [hT (x, z, δ), g T (x, z, w, δ)]T and α = min{ 12 αx∗ , 12 αz∗ }.
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Since the right-hand side of (41) contains V , the implication (41)-(42) does not directly imply the ISS of the closed-loop event-triggered system. Define (43) V ∗ (x(t), z(t)) = max{V (t), σχz mean V (τ )}
composed of (1)-(3) with respect to external disturbances is achieved. This ends the proof of Theorem 5.
for t ∈ (ti , ti+1 ], i ∈ S, where V (x, z) satisfies condition (40). We denote V ∗ (x(t), z(t)) =: V ∗ (t) in the following discussion. To calculate the derivative of V ∗ (x, z) along the trajectories of interconnected system composed of (1) and (6). We consider the cases of V (t) ≥ σχz mean V (τ ) and
In this section, we employ an example to illustrate the effectiveness of the obtained results in this paper. We consider the system z(t) x(t) + 0.4δ(t) (50) z(t) ˙ = −z(t) + 0.1 1 + z 2 (t) x(t) ˙ = x(t) + 0.4 sin(x(t))z(t) + u(t) + δ(t) (51) where x ∈ R is the measurable system state, z ∈ R is the unmeasured portion of the state, u ∈ R is the control input, δ ∈ R represents the external disturbance.
ti ≤τ ≤t
ti ≤τ ≤t
V (t) < σχz mean V (τ ) for t ∈ (ti , ti+1 ], i ∈ S, respectively. ti ≤τ ≤t
Case 1: V (t) ≥ σχz mean V (τ ). In this case, V ∗ (x, z) = ti ≤τ ≤t
V (x, z). With the condition (42) is satisfied, we have (44) ∇V ∗ (t)F (x(t), z(t), w(t), δ(t)) ≤ −αV ∗ (t) ∗ 2 2 ¯δ mean |δ(τ )| , χδ |δ(t)| } for t ∈ whenever V (t) ≥ max{χ ti ≤τ ≤t
(ti , ti+1 ], i ∈ S.
Case 2: V (t) < σχz mean V (τ ). In this case, we have for ti ≤τ ≤t
t ∈ (ti , ti+1 ], i ∈ S
∇V ∗ (t)F (x(t), z(t), w(t), δ(t)) ≤
σχz − 1 ∗ V (t). t − ti
(45)
By combining the two cases, it follows that, for t ∈ (ti , ti+1 ], i ∈ S ¯δ mean |δ(τ )|2 , χδ |δ(t)|2 } V ∗ (t) ≥ max{χ ∗
ti ≤τ ≤t
⇒ ∇V (t)F (x(t), z(t), w(t), δ(t)) ≤ −α∗ (t)V ∗ (t),
(46)
z where α∗ (t) = min{α, 1−σχ t−ti }. We can find an arbitrary constant > 0 such that 1 − σχz }. (47) α∗ (t) ≥ min{α, t+ In the case of V ∗ (x(t), z(t)) ≥ χδ max |δ(τ )|2 , we consider
ti ≤τ ≤t
the cases of α > (1 − σχz )/(t + ) for t > 0 and α < (1 − σχz )/, respectively.
Case 1: α > (1 − σχz )/(t + ) for t > 0. We have (t + )(σχz −1) =: β1 (V ∗ (0), t) (σχz −1) for t > 0, where β1 is a KL function. V ∗ (t) ≤ V ∗ (0)
(48)
Case 2: α < (1 − σχz )/. Then, there exists a constant T satisfying (1 − σχz )/α − such that α ≤ (1 − σχz )/(t + ) for 0 < t ≤ T and α > (1 − σχz )/(t + ) for t > T . Then, We have ∗ −αt =: β2 (V ∗ (0), t), 0 < t ≤ T ; V (0)e V ∗ (t) ≤ (t + )(σχz −1 ) V ∗ (0)e−αT =: β3 (V ∗ (0), t), t > T, (T + )(σχz −1) where β2 , β3 are KL function.
4. AN ILLUSTRATIVE EXAMPLE
The control law is chosen as u(t) = −2x(ti ) for ti ≤ t < ti+1 , i ∈ S.
Then, it can be directly proved that |g(x, z, w, δ)| ≤ |x| + 0.4|z| + 2|w| + |δ| holds for all x, z, w, δ.
(52)
(53)
To verify the satisfaction of Assumption 2 and 3, we define Vx (x) = 12 x2 and Vz (z) = 12 z 2 . It holds that ∇Vx (x)g(x, z, w, δ) (54) ≤ −0.5Vx (x) + max{0.24|z|2 , 14.4|w|2 , 18|δ|2 } for all x, z, w, δ, and ∇Vz (z)h(z, x, δ) ≤ −Vz (z) + max{0.03|x|2 , 0.48|δ|2 } (55) for all z, x, δ. By using conditions (29), (30) and (36), we choose conx δ , t∆ , kw as follows stants kw x δ = 0.117, t∆ = 0.0207, kw = 0.089. (56) kw Then, we have ti+1 = min{t ≥ ti + 0.0207 : (57) |w(t)| = max{0.117|x(t)|, 0.089ζ(t)}} where ζ(t) satisfies 1 ζ(t) = (t − ti )2e3(t−ti ) × max |x(t) − x(ti )| − 1/3(e3(t−ti ) − 1)|x(ti )|, 0 (58)
for all t ∈ [ti + 0.0207, ti+1 ], i ∈ S.
The simulation result with initial states x(0) = −1, z(0) = 1 and external disturbances δ(t) = (cos(7t) + sin(5t) + sin(9t)+sin(11t))/6 is shown in Figs 3, 4 and 5. Compared with the existing event-triggering mechanism in the form of (59), the event-triggering mechanism (13) designed in this paper can avoid infinitely fast sampling, see Fig.5 (59) ti+1 = min{t ≥ ti : |w(t)| = 0.117|x(t)|}.
By combining the two cases, it follows that 5. CONCLUSIONS (49) V ∗ (0) ≤ β(V ∗ (0), t) for 0 ≤ t ≤ T ∗ with T ∗ = min{t : V ∗ (t) = This paper has proposed a new event-triggering mechamax{χ ¯δ mean |δ(τ )|2 , χδ |δ(t)|2 }, where β = max{β1 , β2 , β3 }. nism for systems subject to both dynamic uncertainties ti ≤τ ≤t Moreover, one can find a β satisfying (49) whenev- and external disturbances. Significantly different from the er V ∗ (t) ≥ max{χ ¯δ mean |δ(τ )|2 , χδ |δ(t)|2 } for t ∈ existing designs, an estimation of the unmeasurable state and the external disturbance is introduced, to avoid inti ≤τ ≤t (ti , ti+1 ], i ∈ S. Then, ISS property of the whole system finitely fast sampling. Interestingly, it is proved that the 8179
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1 x(t) x(ti) z(t)
x(t),x(ti) and z(t)
0.5
0
−0.5
−1
0
2
4
6
8
10
time
Fig. 3. State trajectories of the numerical example. 2
control input
1.5 1 0.5 0 −0.5
0
2
4
6
8
10
time
Fig. 4. Control input of the example. 0
ti+1−ti
10
−2
10
Event−triggering mechanism (57) t∆ −4
10
0
2
4
6
8
10
ti 0
10
ti+1−ti
Event−triggering mechanism (59) −2
10
−4
10
0
2
4
6
8
10
ti
Fig. 5. The sequence of times of event-triggered sampling with event-triggering mechanism (57) and (59). inter-sampling intervals are lower bounded by a positive constant, which is independent on the magnitudes of the unmeasurable state and the external disturbance. With the proposed design, the closed-loop event-triggered system has also been guaranteed to be ISS with the external disturbance as the input. Folowing this line of research, more general systems (e.g., systems formulated by inputto-output stability) and event-triggered control design problems would be of interest to study in the future. REFERENCES Anta, A. and Tabuada, P. (2010). To sample or not to sample: Self-triggered control for nonlinear systems. IEEE Transactions on Automatic Control, 55, 2030– 2042. ˚ Arz´en, K.E. (1999). A simple event-based PID controller. In Proceedings of the IFAC World Congress, 423–428. ˚ Astr¨ om, K.J. and Bernhardsson, B. (1999). Comparison of periodic and event based sampling for first order stochastic systems. In Proceedings of the IFAC World Congress. Beijing. Dimarogonas, D.V. and Johansson, K.H. (2009). Eventtriggered control for multi-agent systems. In Proceedings of the 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, 7131–7136. Gawthrop, P.J. and Wang, L. (2009). Event-driven intermittent control. International Journal of Control, 82,
2235–2248. Girard, A. (2015). Dynamic triggering mechanisms for event-triggered control. IEEE Transactions on Automatic Control, 60, 1992–1997. Goebel, R., Sanfelice, R.G., and Teel, A.R. (2009). Hybrid dynamical systems: Robust stability and control for systems that combine continuous-time and discrete-time dynamics. IEEE Control Systems Magazine, 4, 28–93. Heemels, W.P.M.H. and Donkers, M.C.F. (2013). Modelbased periodic event-triggered control for linear systems. Automatica, 49, 698–711. Heemels, W.P.M.H., Johansson, K.H., and Tabuada, P. (2012). An introduction to event-triggered and selftriggered control. In Proceedings of the 51st IEEE Conference on Decision and Control, 3270–3285. Heemels, W.P.M.H., Sandee, J.H., and Van Den Bosch, P.P.J. (2008). Analysis of event-driven controllers for linear systems. International Journal of Control, 81, 571–590. Henningsson, T., Johannesson, E., and Cervin, A. (2008). Sporadic event-based control of first-order linear stochastic systems. Automatica, 44, 2890–2895. Jiang, Z.P., Teel, A.R., and Praly, L. (1994). Small-gain theorem for ISS systems and applications. Mathematics of Control, Signals and Systems, 7, 95–120. Lemmon, M.D. (2010). Event-triggered feedback in control, estimation, and optimization. Berlin: Springer-Verlag. Liu, T. and Jiang, Z.P. (2015). A small-gain approach to robust event-triggered control of nonlinear systems. IEEE Transactions on Automatic Control, 60, 2072– 2085. Liu, T., Jiang, Z.P., and Hill, D.J. (2014). Nonlinear Control of Dynamic Networks. Boca Raton: CRC Press. Lunze, J. and Lehmann, D. (2010). A state-feedback approach to event-based control. Automatica, 46, 211– 215. Marchand, N., Durand, S., and Castellanos, J.F.G. (2013). A general formula for event-based stabilization of nonlinear systems. IEEE Transactions on Automatic Control, 58, 1332–1337. Sontag, E.D. (2006). Input to state stability: Basic concepts and results. In: P. Nistri, G. Stefani. (Eds.), Nonlinear and Optimal Control Theory. Berlin: Springer-Verlag. Sontag, E.D. and Wang, Y. (1996). New characterizations of input-to-state stability. IEEE Transactions on Automatic Control, 41, 1283–1294. Tabuada, P. (2007). Event-triggered real-time scheduling of stabilizing control tasks. IEEE Transactions on Automatic Control, 52, 1680–1685. Wang, X. and Lemmon, M.D. (2008). Event design in event-triggered feedback control systems. In Proceedings of the 47th IEEE Conference on Decision and Control, 2105–2110. Wang, X. and Lemmon, M.D. (2009). Event-Triggering in Distributed Networked Systems with Data Dropouts and Delays, Hybrid Systems: Computation and Control. Heidelberg: Springer Berlin. Zhang, P., Liu, T., and Jiang, Z.P. (2017). Robust event-triggered control subject to external disturbance. Technical report. http://faculty.neu.edu.cn/tfliu/Tech17IFAC.pdf.
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IFAC PapersOnLine 50-1 (2017) 7899–7904 Robust Event-Triggered Control Subject Robust Robust Event-Triggered Event-Triggered Control Control Subject Subject Robust Event-Triggered Control Subject External Disturbance External Disturbance External Disturbance External Disturbance ∗ ∗ ∗∗
to to to to
Pengpeng Pengpeng Zhang Zhang ∗ Tengfei Tengfei Liu Liu ∗ Zhong-Ping Zhong-Ping Jiang Jiang ∗∗ Pengpeng Zhang ∗∗ Tengfei Liu ∗∗ Zhong-Ping Jiang ∗∗ ∗∗ ∗ Pengpeng Zhang Tengfei Liu Zhong-Ping Jiang ∗ State Key Laboratory of Synthetical Automation for Process ∗ State Key Laboratory of Synthetical Automation for Process State Northeastern Key Laboratory of Synthetical Automation for Process Industries, University, Shenyang, 110004, China (e-mail: ∗ Industries, University, Shenyang, 110004,for China (e-mail: State Northeastern Key Laboratory of Synthetical Automation Process Industries, Northeastern University, Shenyang, 110004, China (e-mail: [email protected]; [email protected]). [email protected]; [email protected]). Industries, Northeastern University, Shenyang, 110004, China (e-mail: ∗∗ [email protected]; [email protected]). ∗∗ Tandon School of Engineering, New York University, 5 MetroTech of Engineering, New York University, 5 MetroTech ∗∗ Tandon School [email protected]; [email protected]). Tandon School of Engineering, New York University, 5 MetroTech Brooklyn, NY (e-mail: [email protected]) ∗∗ Center, Center,School Brooklyn, NY 11201, 11201, USA USA York (e-mail: [email protected]) Tandon of Engineering, University, 5 MetroTech Center, Brooklyn, NY 11201, New USA (e-mail: [email protected]) Center, Brooklyn, NY 11201, USA (e-mail: [email protected]) Abstract: Abstract: This This paper paper studies studies the the event-triggered event-triggered control control problem problem for for systems systems subject subject to to both both Abstract: This paper and studies the event-triggered control problem for systems subject to both dynamic uncertainties external disturbances. To overcome infinitely fast sampling caused dynamic uncertainties and external disturbances. To overcome infinitely fast sampling caused Abstract: This paper studies the event-triggered control problem for systems subject to both dynamic uncertainties and external disturbances. To overcome infinitely fastuses sampling caused by a event-triggering mechanism is which not the by the the disturbance, disturbance, a new new external event-triggering mechanism is proposed, proposed, which uses not only only the dynamic uncertainties disturbances. Tothe overcome infinitely fastuses sampling caused by the disturbance, a and newbut event-triggering mechanism is proposed, which not only the measurable system state also an estimate of unmeasurable state and the external measurable system state but also an estimate of the unmeasurable state and the external by the disturbance, a newbut event-triggering mechanism is to proposed, which uses not the measurable system state alsointervals an estimate of the unmeasurable state and by thea only external disturbance. The can bounded positive disturbance. system The inter-sampling inter-sampling intervals can be beof proved proved to be be lower lower state bounded by a external positive measurable state but also an estimate the unmeasurable and the disturbance. The inter-sampling intervals can be proved to be lower bounded by a positive constant, is independent the of state constant, which which is inter-sampling independent on onintervals the magnitudes magnitudes of the the unmeasurable unmeasurable state and andbythe the external external disturbance. Theis can be proved to be lower bounded positive constant, which independent ondesign, the magnitudes of the event-triggered unmeasurable state andcan theabeexternal disturbance. With the proposed the closed-loop system inputdisturbance. With the proposed design, the closed-loop event-triggered system can be inputconstant, which is independent on the magnitudes of the unmeasurable state and the external disturbance. With with the proposed design, the closed-loop event-triggered system can be inputto-state stabilized the external disturbance as the input. Refined tools of input-to-state to-state stabilized with the external disturbance as the input. Refined tools of input-to-state disturbance. With with the proposed design, the closed-loop event-triggered system be inputto-state stabilized the external disturbance as the in input. Refined tools of can input-to-state stability theorem are solving the stability (ISS) (ISS) and and the the small-gain small-gain theorem are employed employed in solving the problem. problem. to-state the external disturbance as the in input. Refined tools of input-to-state stability stabilized (ISS) and with the small-gain theorem are employed solving the problem. stability (ISS)(International and the small-gain are employed in solving the problem. © 2017, IFAC Federationtheorem of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Event-Triggered Event-Triggered Control, Control, Input-to-State Input-to-State Stabilization, Stabilization, External External Disturbances, Disturbances, Keywords: Event-Triggered Control, Input-to-State Stabilization, External Disturbances, Dynamic Dynamic Uncertainty. Uncertainty. Keywords: Event-Triggered Control, Input-to-State Stabilization, External Disturbances, Dynamic Uncertainty. Dynamic Uncertainty. 1. terize 1. INTRODUCTION INTRODUCTION terize the the robustness robustness with with respect respect to to sampling sampling errors, errors, and and 1. INTRODUCTION terize the robustness with respectaccording to sampling errors, and the threshold signal is designed to the margin the threshold signal is designed according to the margin 1. INTRODUCTION terize the robustness with respect to sampling errors, and the threshold signal is designed according to the margin robustness, for (asymptotic) stability of closed-loop Event-triggered of robustness, for (asymptotic) stability of the the closed-loop Event-triggered control control has has attracted attracted a a lot lot of of attention attention of the threshold signal is designed according to the margin of robustness, for (asymptotic) stability of the closed-loop Event-triggered control has attracted a lot ofthe attention system Tabuada (2007); (2015). from Compared periodevent-triggered system Tabuadastability (2007); Girard Girard (2015). For For from the the controls controls community. community. Compareda with with the period- event-triggered robustness, for (asymptotic) of the control, closed-loop Event-triggered has the attracted lot ofthe attention event-triggered system Tabuada (2007); Girard (2015). For from the controlscontrol community. Compared with period- of practical implementation of event-triggered inic time-triggered sampling, event-triggered sampling practical implementation of event-triggered control, inic time-triggered sampling, the event-triggered sampling event-triggered systemshould Tabuada (2007); Girard (2015). For from the controls community. Compared with thesampling periodpractical implementation of be event-triggered control, inic time-triggered sampling, the event-triggered finitely fast sampling avoided, i.e., the sampling time instants are determined by an event generator that finitely fast sampling should be avoided, i.e., the sampling time instants are determined by an event generator that practical implementation of event-triggered control, inic time-triggered sampling, the event-triggered sampling finitely fast sampling should be avoided, i.e., the sampling time instants are determined by an event generator that intervals are are required required to to be be lower lower bounded bounded by by some some positive positive depends on system state, into account the depends on the the system state,byto toantake take into account that the intervals finitely fast sampling should be avoided, i.e., the sampling time instants are determined event generator are required to be lower bounded by some positive depends on the during system the state, to takeintervals. into account constant; see e.g., in (2010). One system behavior behavior sampling Early the pa- intervals constant; seerequired e.g., the thetodiscussions discussions in Lemmon Lemmon (2010). One system during the sampling intervals. Early paintervals are be lower bounded byZeno some positive depends on the system˚ state, to take into account the constant; see e.g., the discussions in Lemmon (2010). One ˚ system behavior during the sampling intervals. Early paspecial case of infinitely fast sampling is the behavior, pers, e.g., A rz´ e n (1999); A str¨ o m and Bernhardsson (1999), ˚ ˚ special case of infinitely fast sampling is the Zeno behavior, pers, e.g., A rz´ e n (1999); A str¨ o m and Bernhardsson (1999), constant; see e.g., the discussions insampling Lemmon (2010). One system behavior during ˚ the sampling intervals. Early pa- special ˚ case ofan infinitely fast sampling is the Zeno behavior, pers, e.g., Arz´ enadvantages (1999); A str¨ om and Bernhardsson (1999), i.e., there is infinite number of time instants have shown the of event-triggered control over there isofaninfinitely infinite fast number of sampling timebehavior, instants have shown the advantages ofom event-triggered control over i.e., ˚ ˚ special case sampling is the Zeno pers, e.g., A rz´ e n (1999); A str¨ and Bernhardsson (1999), i.e., there is an infinite number of sampling time instants have shown the advantages of event-triggered control over which converge to value; see et periodic time-triggered control. converge to aa finite finite value; see e.g., e.g., Goebel Goebel et al. al. periodic time-triggered control. i.e., there isthe anZeno infinite number ofhybrid sampling time In instants have shown the advantages of event-triggered control over which which converge to abehavior finite value; see e.g., Goebel et al. periodic time-triggered control. (2009) for of systems. most (2009) for the Zeno behavior of hybrid systems. In most converge to abehavior finite value; see e.g., Goebel al. In decade, effort periodic time-triggered control. (2009) for the results, Zeno of hybrid systems. In et most In the the past past decade, great great effort has has been been spent spent in in dede- which of the existing the event-triggered control problem the existing results, the event-triggered control problem In the past decade, great effort hastheory, been see, spent in Ande- of (2009) for the Zeno behavior of hybrid systems. In most veloping the event-triggered control e.g., of the existing results, the event-triggered control problem veloping the event-triggered control theory, see, e.g., Antransformed into of appropriate In the past great effort hastheory, been see, spent in Ande- is is the transformed into the thetheproblem problem of choosing choosing appropriate veloping the decade, event-triggered control e.g., of existing results, event-triggered control problem ta (2010); and (2009); is transformed into the problem of fast choosing appropriate ta and and Tabuada Tabuada (2010); Gawthrop Gawthrop and Wang Wang (2009); threshold signals to avoid infinitely sampling. veloping the event-triggered control theory, see, e.g., Anthreshold signals to avoid infinitely fast sampling. ta and Tabuada (2010); Gawthrop and Wang (2009); is transformed into the problem of choosing appropriate Heemels et al. (2012); Heemels and Donkers (2013); Lunze Heemels et al. (2012); Heemels and Donkers (2013); Lunze threshold signals to avoid infinitely fast sampling. ta and Tabuada (2010); Gawthrop andMarchand Wang Heemels et al. (2012); Heemels and Donkers (2013);(2009); Lunze unmeasurable disturbances, threshold signals of to avoid infinitely external fast sampling. and (2010); Tabuada (2007); et In the the presence presence of unmeasurable external disturbances, and Lehmann Lehmann (2010);Heemels Tabuadaand (2007); Marchand et al. al. In Heemels et al. (2012); Donkers (2013); Lunze In the presence of to unmeasurable external disturbances, and Lehmann (2010); Tabuada (2007); Marchand etconal. an intuitive design avoid infinitely fast sampling is (2013) and the references therein. In event-triggered an intuitive design to avoid infinitely fast sampling is to to (2013) and the (2010); references therein.(2007); In event-triggered conIn the presence of unmeasurable external disturbances, and Lehmann Tabuada Marchand et al. intuitive design tothreshold avoid infinitely fastpaying sampling is to (2013) and the signals references therein. Inemployed event-triggered con- an introduce a constant signal, by the price trol, threshold are usually to generate introduce a constant threshold signal, by paying the price trol, threshold signals are usually employed to generate an intuitive design to avoid infinitely fastpaying sampling is to (2013) and of the signals references Inemployed event-triggered conintroduce a constant threshold signal, by the price trol, threshold aretherein. usually to generate convergence. The conthe sampling. In earlier e.g., of losing losing asymptotic asymptotic convergence. The event-triggered event-triggered conthe events events of data data sampling. In the the earlier results, results, e.g., of introduce a constant threshold signal, by paying the price trol, threshold signals are usually employed to generate of losing asymptotic convergence. The event-triggered conthe events of data sampling. In the earlier results, e.g., trol problem of linear systems pHeemels et al. Henningsson et constant trol problem of continuous-time continuous-time systems in in the the pHeemels et of al. (2008); (2008); Henningsson et al. al. (2008), (2008), constant losing asymptotic convergence.islinear The event-triggered conthe events sampling. In the results, e.g., of trol problem of continuous-time linear systems in the pHeemels etsignals al. data (2008); Henningsson et earlier al. (2008), constant resence of external disturbances discussed in Lunze and threshold are used, while in the recent results, resence of external disturbances is discussed in Lunze and threshold signals are used, while in the recent results, trol problem of continuous-time linear systems in the pHeemels et al. (2008); Henningsson et al. (2008), constant resence of external disturbances is discussed in Lunze and threshold signals are used, while(2009); in theWang recentand results, Lehmann (2010) in which the disturbance behaviour of e.g., Dimarogonas and Johansson Lem(2010) indisturbances which the disturbance behaviour of e.g., Dimarogonas and used, Johansson (2009); Wang and Lem- Lehmann resence of external is discussed in Lunze and threshold signals are while in the recent results, Lehmann (2010) in which the disturbance behaviour of e.g., Dimarogonas and Johansson (2009); Wang and Lema feedback loop is the main concern of the paper. Howmon (2009, 2008), the threshold signals are designed to a feedback loop is the main concern of the paper. Howmon (2009, 2008), the threshold signals are designed to Lehmann (2010) in which the disturbance behaviour of e.g., Dimarogonas and Johansson (2009); Wang and Lema feedback loop is the main concern of the paper. Howmon (2009, 2008), the threshold signals are designed to ever, in and Lehmann (2010), interdepend on state and the historical data. ever, in Lunze Lunze and Lehmann (2010), ofthe thethesmallest smallest interdepend on real-time real-time state and even even the are historical data. a feedback loop is the main concern paper. Howmon (2009, 2008), the threshold signals designed to in Lunze and to Lehmann (2010), theofsmallest interdepend on real-time anddesign even methods the historical sampling is the bound disturbances. Also, refined refined classicalstate control have data. been ever, sampling is related related to the upper upper bound ofsmallest disturbances. Also, classical control design methods have been ever, in Lunze and Lehmann (2010), the interdepend onfor real-time state anddesign even methods the historical sampling is related to thestrategy upper bound of disturbances. Also, refined classical control have data. been A self-triggered sampling for a class of nonlinear employed event-triggered control. For example, the self-triggered sampling strategy for a class of nonlinear employed for classical event-triggered control.methods For example, the A sampling is related to the upper bound of disturbances. Also, refined control design have been A self-triggered sampling strategy for a class of nonlinear employed for event-triggered control. Forused example, the systems subject to disturbances is in concept stability (ISS) to subject sampling to external external disturbances is proposed proposed in concept of of input-to-state input-to-state stability (ISS) is is used to characcharacA self-triggered strategy for a bound class ofofnonlinear employed event-triggered control. example, the systems systems subject to external disturbances is proposed in concept of for input-to-state stability (ISS) For is used to characLiu and Jiang (2015) in which the upper external Liu and Jiang (2015) in whichdisturbances the upper bound of external This work was supported in part by NSF grant ECCS-1501044, in systems subject to external is proposed in concept of input-to-state stability (ISS) is used to characThis work was supported in part by NSF grant ECCS-1501044, in Liu and Jiang (2015) in which the upper bound of external disturbances is assumed to be known a priori. This disturbances is(2015) assumed to be the known a priori. part by NSFC grants 61374042, 61522305, 61633007 and 61533007, in was supported in part by NSF grant ECCS-1501044, Liu and Jiang in which upper bound of external part by work NSFC grants 61374042, 61522305, 61633007 and 61533007, in This disturbances is assumed to be known a priori. was supported in part by NSF grant ECCS-1501044, in part by work NSFC grants 61374042, 61522305, 61633007 and 61533007, in the Research Funds the Universities The main of is disturbances is assumed to paper be known priori.the part by NSFC the Fundamental Fundamental Research Funds for for the Central Central Universities The main purpose purpose of this this paper is to toa refine refine the existing existing part by grants 61374042, 61522305, 61633007 and 61533007, in part byGrants the Fundamental Research Funds for the Central Universities under N130108001 and N140805001 in China, and in part by The main purpose of this paper control is to refine the existing under Grants N130108001Research and N140805001 in China, and in part by results of robust event-triggered for systems with part by the Fundamental Funds for the Central Universities results of robust event-triggered control for systems with under Grants N130108001 and N140805001 in China, andofinComplex part by State Key Laboratory of Intelligent Control and Decision The main purpose of this paper isexternal to refine the existing results of robust event-triggered control for systems with State Key Laboratory of Intelligent Control and Decision of Complex both dynamic uncertainties and disturbances. underKey Grants N130108001 and N140805001 in China, andofinComplex part by both dynamic uncertainties and external disturbances. State Laboratory of Intelligent Control and Decision Systems at BIT. results of robust event-triggered control for disturbances. systems with Systems atLaboratory BIT. both dynamic uncertainties and external This paper does not assume a known upper bound of State Key of Intelligent Control and Decision of Complex Systems at BIT. author: Corresponding Tengfei Liu This paper doesuncertainties not assume aand known upper disturbances. bound of the the both dynamic external Corresponding author: Tengfei Liu This paper does not assume a known upper bound of the Systems at BIT. author: Tengfei Liu Corresponding This paper does not assume a known upper bound of the Corresponding author: Tengfei Liu
Copyright © 2017 8175 Copyright 2017 IFAC IFAC 8175Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Copyright ©under 2017 responsibility IFAC 8175Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 8175 10.1016/j.ifacol.2017.08.752
Proceedings of the 20th IFAC World Congress 7900 Pengpeng Zhang et al. / IFAC PapersOnLine 50-1 (2017) 7899–7904 Toulouse, France, July 9-14, 2017
external disturbance. To avoid infinitely fast sampling, a new event-triggering mechanism is proposed, which uses not only the measurable system state but also an estimate of the unmeasurable state and the external disturbance. Interestingly, it is proved that the event-triggered sampling intervals are lower bounded by a positive constant, which is independent of both the dynamic uncertainty and the external disturbance. ISS of the closed-loop eventtriggered system is also proved by considering the external disturbance as the input. In the design, refined tools of ISS Sontag and Wang (1996); Sontag (2006) and small-gain theorem Jiang et al. (1994); Liu et al. (2014) are used. Moreover, as a by-product of the main result, this paper proposes a small-gain result for a new class of interconnected systems induced by event-triggered control. The rest of the paper is organized as follows. Section 2 gives the problem formulation of event-triggered control for continuous-time systems subject to both dynamic uncertainties and external disturbances. In Section 3, we propose a new event-triggering mechanism, and prove that the resulted closed-loop event-triggered system is ISS with the external disturbance as the input. An example with numerical simulation is employed to demonstrate the implementation and the effectiveness of the proposed design. Section 5 contains some concluding remarks. To make the paper self-contained, some notations are given here. We use Z+ to denote the set of all nonnegative integers. For any vector x ∈ Rn , |x| represents its Euclidean norm. For any function ψ : Z+ → Rn , mean ψ(τ ) a≤τ ≤b b 1 represents b−a a ψ(τ ) dτ , where b > a.
According to the definition of sampling error w in (4), the closed-loop event-triggered system can be represented as an interconnected system, as shown in Figure 1. δ(t) x(t) x(t)
Define w(t) = x(ti ) − x(t), t ∈ [ti , ti+1 ), i ∈ S as the sampling error, and rewrite u(t) = v(x(t) + w(t)).
(4) (5)
Then, the controlled x-subsystem can be represented by x(t) ˙ = f (x(t), z(t), v(x(t) + w(t)), δ(t)) =: g(x(t), z(t), w(t), δ(t)). (6)
z(t)
x(t) ˙ = g(x(t), z(t), w(t), δ(t)) S
x(ti ) +
δ(t) w(t) −
Fig. 1. The block diagram of the closed-loop eventtriggered system composed of (1) and (6), where S represents the event-triggered sampler. In this paper, we study the case in which the system dynamics g is globally Lipschitz. δ Assumption 1. There exist constants Lxg , Lzg , Lw g , Lg ≥ 0 such that δ (7) |g(x, z, w, δ)| ≤ Lxg |x| + Lzg |z| + Lw g |w| + Lg |δ| holds for all x, z, w, δ. Also, it is assumed that the dynamic uncertainty, i.e., the z-subsystem, is ISS with x and δ as the inputs, and there exists a v such that the nominal system, i.e., the x-subsystem, is ISS with z, w and δ as the inputs. Assumption 2. System (1) is ISS with x and δ as the inputs, and admits an ISS-Lyapunov function Vz : Rnz → R+ satisfying the following conditions: (1) There exist constants αz , αz > 0 such that
2. PROBLEM FORMULATION
We consider an event-triggered control system in the following form: z(t) ˙ = h(z(t), x(t), δ(t)) (1) x(t) ˙ = f (x(t), z(t), u(t), δ(t)) (2) u(t) = v(x(ti )), t ∈ [ti , ti+1 ), i ∈ S ⊆ Z+ (3) n where x ∈ R is the measurable system state, z ∈ Rnz is the unmeasurable portion of the state, u ∈ Rm is the control input, δ ∈ Rnδ represents the external disturbance, h : Rnz × Rn × Rnδ → Rnz and f : Rn × Rnz × Rm × Rnδ → Rn represent system dynamics, v : Rn → Rm represents the feedback control law based on eventtriggered data-sampling. In this paper, the z-subsystem is considered as dynamic uncertainty, while the x-subsystem is called the nominal system. The time sequence {ti }i∈S is determined online by an appropriately designed eventtriggering mechanism with S = {0, 1, 2, · · · } ⊆ Z+ being the set of the indices of the sampling times and t0 = 0.
z(t) ˙ = h(z(t), x(t), δ(t))
αz |z|2 ≤ Vz (z) ≤ αz |z|2 ,
∀z;
(8)
(2) There exist constants αz , kzx , kzδ > 0 such that ∇Vz (z)h(z, x, δ) ≤
(9) − αz Vz (z) + max{kzx |x|2 , kzδ |δ|2 }, ∀z, x, δ. Assumption 3. System (6) is ISS with z, w and δ as inputs, and admits an ISS-Lyapunov function Vx : Rn → R+ satisfying the following conditions: (1) There exist constants αx , αx > 0 such that αx |x|2 ≤ Vx (x) ≤ αx |x|2 ,
∀x;
(10)
(2) There exist constants αx , kxz , kxw , kxδ > 0 such that ∇Vx (x)g(x, z, w, δ) ≤ −αx Vx (x) (11) + max{kxz |z|2 , kxw |w|2 , kxδ |δ|2 }, ∀x, z, w, δ. Our objective is to develop a new event-triggering mechanism, which fully takes into consideration the external disturbance and the dynamic uncertainty such that the following two objectives are achieved: Objective 1: There exists a constant t∆ > 0, independent of the unmeasurable state z and the external disturbance δ, such that ti+1 − ti ≥ t∆ (12) holds for all i, i + 1 ∈ S. Objective 2: The closed-loop event-triggered system composed of (1)-(3) is ISS with respect to the external disturbance δ.
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3. ROBUST EVENT-TRIGGERED CONTROL
Then, direct calculation yields:
The main contribution of this paper lies in a new eventtriggering mechanism. Significantly different from most of the published results, an estimation of the unmeasurable state and the external disturbance is introduced, so that the threshold signal depends not only on the system state but also on the magnitudes of the unmeasurable state and the external disturbance. In this way, the technical challenge caused by the dynamic uncertainty and the external disturbance is solvable at the same time. Specifically, the new event-triggering mechanism is proposed in the form of ti+1 = min{t ≥ti + t∆ : x δ |w(t)| = max{kw |x(t)|, kw ζ(t)}},
x kw
(13)
δ kw
> 0 and > 0 are constants to be where t∆ > 0, designed later, ζ(t) represents an estimate of the influence of the unmeasurable state and the external disturbance. 3.1 An Example as Motivation Before presenting the main result, we employ a simple example to show how an estimation of the external disturbance can be made to guarantee the positive lower bound the inter-sampling intervals. Consider a first-order linear system in the form of x(t) ˙ = x(t) + u(t) + δ(t) (14) where x ∈ R is the state, u ∈ R is the control input, and δ ∈ R represents external disturbances. The event-triggered control law is designed as u(t) = −2x(ti ) for ti ≤ t < ti+1 , i ∈ S. By using (4), (14) and (15), we have x(t) ˙ = −x(t) − 2w(t) + δ(t).
(15)
(16)
Also, according to the definition of sampling error w, we have (17) w(t) ˙ = x(t) ˙ = −w(t) − x(ti ) + δ(t) for ti ≤ t < ti+1 , i ∈ S. Note that w(ti ) = x(ti )−x(ti ) = 0. For ti ≤ t < ti+1 with i ∈ S, the solution of system (17) with initial condition w(ti ) = 0 is t w(t) = (e−(t−ti ) − 1)x(ti ) + e−(t−υ) δ(υ) dυ, (18) ti
and thus,
|x(t) − x(ti )| ≤ (1 − e−(t−ti ) )|x(ti )| +
t ti
7901
|δ(υ)| dυ (19)
holds for ti ≤ t ≤ ti+1 , i ∈ S. Here, (19) holds for t = ti+1 because of the continuity of both sides of the inequality. t Clearly, the term ti |δ(υ)| dυ represents the influence of the external disturbance δ. For t ∈ (ti , ti+1 ], a lower bound t of ti |δ(υ)| dυ can be calculated as
η(t) = max{|x(t) − x(ti )| − (1 − e−(t−ti ) )|x(ti )|, 0}. (20) It can be observed that η(t) is left-continuous at the discontinuous points.
|x(t) − x(ti )| ≤ (1 − e−(t−ti ) )|x(ti )| + η(t) for ti < t ≤ ti+1 , i ∈ S.
(21)
To represent the “average” influence of δ(t), define 0, for t = t0 = 0; ζ(t) = η(t) , for t ∈ (ti , ti+1 ]. t − ti
(22)
With an estimation in form of (22), it can be guaranteed that |x(t) − x(ti )| ≤ (1 − e−(t−ti ) )|x(ti )| + (t − ti )ζ(t)
≤ 2 max{t − ti , 1 − e−(t−ti ) } max{|x(ti )|, ζ(t)} for ti < t ≤ ti+1 , i ∈ S.
(23)
On the other hand, the event trigger (13) guarantees that x δ |x(t)|, kw ζ(t)} (24) |x(t) − x(ti )| ≤ max{kw holds for ti < t ≤ ti+1 , i ∈ S. With (Liu and Jiang, 2015, Lemma A.1), property (24) of the event trigger holds if x x δ /(1 + kw ), kw max {|x(ti )|, ζ(t)} |x(t) − x(ti )| ≤ min kw x x δ ≤ max kw /(1 + kw )|x(ti )|, kw ζ(t) (25) holds for ti < t ≤ ti+1 , i ∈ S.
Define
t∆ = inf{t > ti : 2 max{t − ti , 1 − e−(t−ti ) } x x δ } − ti = min kw /(1 + kw ), kw (26) for t ≥ ti , i ∈ S. That is, a positive lower bound of the inter-sampling intervals is guaranteed. Also, it can be directly proved that x δ |x(t)|, kw ζ(t) (27) |w(t)| ≤ max kw for all t ∈ [ti , ti + t∆ ], i ∈ S.
Moreover, in the following discussion, it is proved that the convergence of δ leads to the convergence of ζ, to satisfy the necessary condition for input-to-state stabilization. Figure 2 shows the structure of the estimation-based eventtriggering mechanism. u(t)
u(t) = −2x(ti )
x(ti ) S
δ(t)
x(t) ˙ = x(t) + u(t) + δ(t)
ζ(t)
E
x(t)
Fig. 2. The block diagram of the closed-loop eventtriggered system (14)-(15), where E represents the estimation (22) of the external disturbance. 3.2 ISS of Event-Triggered Controlled x-Subsystem In this subsection, we focus on the event-triggered control of the x-subsystem. In accordance with the design in Subsection 3.1, an estimation of the external disturbance δ and the unmeasurable state z is defined as 1 ζ(t) = (t − ti )Lzδ g E(t, ti ) max {|x(t) − x(ti )| − Lg (E(t, ti ) − 1)|x(ti )|, 0} (28)
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δ z for all t ∈ [ti + t∆ , ti+1 ], i ∈ S, where Lzδ g = 2 max{Lg , Lg }, w (Lx +L )(t−t ) x w x i E(t, ti ) = e g g , Lg = Lg /(Lg + Lg ) and t∆ is a positive constant to be determined later.
Theorem 4 presents the main result on ISS of the eventtriggered controlled x-subsystem. Theorem 4. Consider the closed-loop event-triggered system (6) with event-triggering mechanism (13) and (28). x Under Assumptions 1 and 3, by choosing constant kw >0 satisfying x 2 ) kxw (kw < 1, (29) (1 − σ1 ) αx αx and constant t∆ as t∆ = {t∆ : ϕ(t∆ ) − εmin = 0} (30) xw
with 0 < σ1 < 1, ϕ(t∆ ) = 2 max{Lg eLg t∆ − Lxw x g t∆ t }, Lxw = Lx + Lw , ε Lg , Lzδ min = min{kw /(1 + ∆ g g g e g x δ δ kw ), kw } and kw is to be designed later, Objective 1 is achieved, and the event-triggered controlled x-subsystem (6) is ISS with δ and z as the inputs. Proof. In this proof, we first find a positive t∆ , and then prove the ISS of the event-triggered controlled xsubsystem (6).
By the proof of Theorem 4, the event-triggered mechanism (13) and (28) satisfying (29) and (30) can achieve input-tostate stabilization of the closed-loop system (6). However, it cannot be directly clarified that the stability of the whole system composed of (1) and (6). 3.3 ISS of the Closed-Loop Event-Triggered System The x-subsystem and the z-subsystem are feedback interconnected with each other. In this subsection, a condition for ISS of the closed-loop event-triggered system is given by using the small-gain idea. Theorem 5. Consider the (z, x)-system in (1) and (6). Under Assumptions 1, 2 and 3, if the event-triggering mechanism composed of (13) and (28) satisfies (29) and (30), and χxz · χzx < 1, (35) (36) σχz < 1 where χxz = kxz /((1 − σ2 )σ1 αx αz ), χzx = kzx /((1 − δ 2 σ2 )αz αx ), χz = 2.3kxw (kw ) /((1 − σ2 )σ1 αx αz ), 0 < σ2 < 1 and σ satisfies 1 > χxz , σ > χzx , (37) σ then, ISS property with respect to external disturbances for the whole system composed of (1)-(3) is achieved.
(a) The Existence of t∆ > 0 As discussion in Subsection Proof. From (8) and (34), we can get the gain margin 3.1, one can find a t∆ satisfying formulation of the ISS-Lyapunov functions Vx : x xw xw kw δ , kw . Vx (x(t)) ≥ max{χxz Vz (z(t)), χz mean Vz (z(τ )), 2 max{Lg eLg t∆ − Lg , eLg t∆ Lδg t∆ } = min x ti ≤τ ≤t 1 + kw δ χ ¯x mean |δ(τ )|2 , χδx |δ(t)|2 } ti ≤τ ≤t In particular, t∆ is independent on the external disturbance δ and the unmeasurable state z. Please see Zhang ⇒ ∇Vx (x(t))g(x(t), z(t), w(t), δ(t)) ≤ −αx∗ Vx (x(t)) (38) et al. (2017) for more detailed proofs. for t ∈ (ti , ti+1 ], i ∈ S, where constants αx∗ , χxz , χz , χ ¯δx , χδx are chosen as αx∗ = σ1 σ2 αx , χxz = kxz /((1 − σ2 )σ1 αx αz ), (B) ISS The event-triggering mechanism (13) can guarχz = k¯xz /((1 − σ2 )σ1 αx αz ), χ ¯δx = k¯xδ /((1 − σ2 )σ1 αx ) and antee that δ δ χx = kx /((1 − σ2 )σ1 αx ). x δ |x(t) − x(ti )| ≤ max kw |x(t)|, kw ζ(t) (31) Similarly, under Assumption 2 and 3, we have for t ∈ (ti , ti+1 ], i ∈ S. Vz (z(t)) ≥ max{χzx Vx (x(t)), χδz |δ(t)|2 } (39) With the satisfaction of (29) and (31), (11) implies ⇒ ∇Vz (z)h(z(t), x(t), δ(t)) ≤ −αz∗ Vz (z(t)) ∇Vx (x(t))g(x(t), z(t), w(t), δ(t)) ≤ −σ1 αx Vx (x(t)) for all t ≥ 0, where constants α∗ , χ , χδ are chosen as δ 2 2 + max{kxz |z(t)|2 , kxw (kw ) ζ (t), kxδ |δ(t)|2 } for t ∈ (ti , ti+1 ], i ∈ S.
From (Zhang et al., 2017, Section 1), we have t max{|z(τ )|, |δ(τ )|} dτ ζ(t) ≤ ti t − ti ≤ max{1.5 mean |z(τ )|, 3 mean |δ(τ )|}
(32)
(33)
z
zx
x
αz∗ = σ2 αz , χzx = kzx /((1 − σ2 )αz αx ) and χδz = kzδ /((1 − σ2 )αz ).
An ISS-Lyapunov function candidate for the interconnected system composed of (1) and (6) is defined as V (x, z) = max{σVx (x), Vz (z)} (40) where σ is defined in (37).
For convenience of notations, we denote V (x(t), z(t)) =: V (t) in the following discussion. According to a smallfor all t ∈ (ti , ti+1 ], i ∈ S. gain result for a class of interconnected systems induced Substituting (33) into the right-hand side of (32) yields by event-triggered control (Zhang et al., 2017, Appendix), by using conditions (35)-(37), we have ∇Vx (x(t))g(x(t), z(t), w(t), δ(t)) ≤ −σ1 αx Vx (x(t)) z 2 ¯z 2 ¯δ 2 δ 2 V (t) ≥ max{σχz mean V (τ ), χ ¯δ mean |δ(τ )|2 , χδ |δ(t)|2 } + max{kx |z(t)| , kx mean |z(τ )| , kx mean |δ(τ )| , kx |δ(t)| } t ≤τ ≤t t ≤τ ≤t ti ≤τ ≤t
ti ≤τ ≤t
ti ≤τ ≤t
ti ≤τ ≤t
i
i
(34)
δ 2 for all t ∈ (ti , ti+1 ], i ∈ S, where k¯xz = 2.3 kxw (kw ) and δ w δ 2 ¯ kx = 9 kx (kw ) . Property (34) means (delayed) ISS of the closed-loop event-triggered system (6) with respect to δ and z. This ends the proof of Theorem 4.
⇒ ∇V (t)F (x(t), z(t), w(t), δ(t)) ≤ −αV (t)
(41) (42)
for t ∈ (ti , ti+1 ], i ∈ S, where χ ¯δ = σ χ ¯δx , χδ = max{σχδx , χδz }, F (x, z, w, δ) = [hT (x, z, δ), g T (x, z, w, δ)]T and α = min{ 12 αx∗ , 12 αz∗ }.
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7903
Since the right-hand side of (41) contains V , the implication (41)-(42) does not directly imply the ISS of the closed-loop event-triggered system. Define (43) V ∗ (x(t), z(t)) = max{V (t), σχz mean V (τ )}
composed of (1)-(3) with respect to external disturbances is achieved. This ends the proof of Theorem 5.
for t ∈ (ti , ti+1 ], i ∈ S, where V (x, z) satisfies condition (40). We denote V ∗ (x(t), z(t)) =: V ∗ (t) in the following discussion. To calculate the derivative of V ∗ (x, z) along the trajectories of interconnected system composed of (1) and (6). We consider the cases of V (t) ≥ σχz mean V (τ ) and
In this section, we employ an example to illustrate the effectiveness of the obtained results in this paper. We consider the system z(t) x(t) + 0.4δ(t) (50) z(t) ˙ = −z(t) + 0.1 1 + z 2 (t) x(t) ˙ = x(t) + 0.4 sin(x(t))z(t) + u(t) + δ(t) (51) where x ∈ R is the measurable system state, z ∈ R is the unmeasured portion of the state, u ∈ R is the control input, δ ∈ R represents the external disturbance.
ti ≤τ ≤t
ti ≤τ ≤t
V (t) < σχz mean V (τ ) for t ∈ (ti , ti+1 ], i ∈ S, respectively. ti ≤τ ≤t
Case 1: V (t) ≥ σχz mean V (τ ). In this case, V ∗ (x, z) = ti ≤τ ≤t
V (x, z). With the condition (42) is satisfied, we have (44) ∇V ∗ (t)F (x(t), z(t), w(t), δ(t)) ≤ −αV ∗ (t) ∗ 2 2 ¯δ mean |δ(τ )| , χδ |δ(t)| } for t ∈ whenever V (t) ≥ max{χ ti ≤τ ≤t
(ti , ti+1 ], i ∈ S.
Case 2: V (t) < σχz mean V (τ ). In this case, we have for ti ≤τ ≤t
t ∈ (ti , ti+1 ], i ∈ S
∇V ∗ (t)F (x(t), z(t), w(t), δ(t)) ≤
σχz − 1 ∗ V (t). t − ti
(45)
By combining the two cases, it follows that, for t ∈ (ti , ti+1 ], i ∈ S ¯δ mean |δ(τ )|2 , χδ |δ(t)|2 } V ∗ (t) ≥ max{χ ∗
ti ≤τ ≤t
⇒ ∇V (t)F (x(t), z(t), w(t), δ(t)) ≤ −α∗ (t)V ∗ (t),
(46)
z where α∗ (t) = min{α, 1−σχ t−ti }. We can find an arbitrary constant > 0 such that 1 − σχz }. (47) α∗ (t) ≥ min{α, t+ In the case of V ∗ (x(t), z(t)) ≥ χδ max |δ(τ )|2 , we consider
ti ≤τ ≤t
the cases of α > (1 − σχz )/(t + ) for t > 0 and α < (1 − σχz )/, respectively.
Case 1: α > (1 − σχz )/(t + ) for t > 0. We have (t + )(σχz −1) =: β1 (V ∗ (0), t) (σχz −1) for t > 0, where β1 is a KL function. V ∗ (t) ≤ V ∗ (0)
(48)
Case 2: α < (1 − σχz )/. Then, there exists a constant T satisfying (1 − σχz )/α − such that α ≤ (1 − σχz )/(t + ) for 0 < t ≤ T and α > (1 − σχz )/(t + ) for t > T . Then, We have ∗ −αt =: β2 (V ∗ (0), t), 0 < t ≤ T ; V (0)e V ∗ (t) ≤ (t + )(σχz −1 ) V ∗ (0)e−αT =: β3 (V ∗ (0), t), t > T, (T + )(σχz −1) where β2 , β3 are KL function.
4. AN ILLUSTRATIVE EXAMPLE
The control law is chosen as u(t) = −2x(ti ) for ti ≤ t < ti+1 , i ∈ S.
Then, it can be directly proved that |g(x, z, w, δ)| ≤ |x| + 0.4|z| + 2|w| + |δ| holds for all x, z, w, δ.
(52)
(53)
To verify the satisfaction of Assumption 2 and 3, we define Vx (x) = 12 x2 and Vz (z) = 12 z 2 . It holds that ∇Vx (x)g(x, z, w, δ) (54) ≤ −0.5Vx (x) + max{0.24|z|2 , 14.4|w|2 , 18|δ|2 } for all x, z, w, δ, and ∇Vz (z)h(z, x, δ) ≤ −Vz (z) + max{0.03|x|2 , 0.48|δ|2 } (55) for all z, x, δ. By using conditions (29), (30) and (36), we choose conx δ , t∆ , kw as follows stants kw x δ = 0.117, t∆ = 0.0207, kw = 0.089. (56) kw Then, we have ti+1 = min{t ≥ ti + 0.0207 : (57) |w(t)| = max{0.117|x(t)|, 0.089ζ(t)}} where ζ(t) satisfies 1 ζ(t) = (t − ti )2e3(t−ti ) × max |x(t) − x(ti )| − 1/3(e3(t−ti ) − 1)|x(ti )|, 0 (58)
for all t ∈ [ti + 0.0207, ti+1 ], i ∈ S.
The simulation result with initial states x(0) = −1, z(0) = 1 and external disturbances δ(t) = (cos(7t) + sin(5t) + sin(9t)+sin(11t))/6 is shown in Figs 3, 4 and 5. Compared with the existing event-triggering mechanism in the form of (59), the event-triggering mechanism (13) designed in this paper can avoid infinitely fast sampling, see Fig.5 (59) ti+1 = min{t ≥ ti : |w(t)| = 0.117|x(t)|}.
By combining the two cases, it follows that 5. CONCLUSIONS (49) V ∗ (0) ≤ β(V ∗ (0), t) for 0 ≤ t ≤ T ∗ with T ∗ = min{t : V ∗ (t) = This paper has proposed a new event-triggering mechamax{χ ¯δ mean |δ(τ )|2 , χδ |δ(t)|2 }, where β = max{β1 , β2 , β3 }. nism for systems subject to both dynamic uncertainties ti ≤τ ≤t Moreover, one can find a β satisfying (49) whenev- and external disturbances. Significantly different from the er V ∗ (t) ≥ max{χ ¯δ mean |δ(τ )|2 , χδ |δ(t)|2 } for t ∈ existing designs, an estimation of the unmeasurable state and the external disturbance is introduced, to avoid inti ≤τ ≤t (ti , ti+1 ], i ∈ S. Then, ISS property of the whole system finitely fast sampling. Interestingly, it is proved that the 8179
Proceedings of the 20th IFAC World Congress 7904 Pengpeng Zhang et al. / IFAC PapersOnLine 50-1 (2017) 7899–7904 Toulouse, France, July 9-14, 2017
1 x(t) x(ti) z(t)
x(t),x(ti) and z(t)
0.5
0
−0.5
−1
0
2
4
6
8
10
time
Fig. 3. State trajectories of the numerical example. 2
control input
1.5 1 0.5 0 −0.5
0
2
4
6
8
10
time
Fig. 4. Control input of the example. 0
ti+1−ti
10
−2
10
Event−triggering mechanism (57) t∆ −4
10
0
2
4
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8
10
ti 0
10
ti+1−ti
Event−triggering mechanism (59) −2
10
−4
10
0
2
4
6
8
10
ti
Fig. 5. The sequence of times of event-triggered sampling with event-triggering mechanism (57) and (59). inter-sampling intervals are lower bounded by a positive constant, which is independent on the magnitudes of the unmeasurable state and the external disturbance. With the proposed design, the closed-loop event-triggered system has also been guaranteed to be ISS with the external disturbance as the input. Folowing this line of research, more general systems (e.g., systems formulated by inputto-output stability) and event-triggered control design problems would be of interest to study in the future. REFERENCES Anta, A. and Tabuada, P. (2010). To sample or not to sample: Self-triggered control for nonlinear systems. IEEE Transactions on Automatic Control, 55, 2030– 2042. ˚ Arz´en, K.E. (1999). A simple event-based PID controller. In Proceedings of the IFAC World Congress, 423–428. ˚ Astr¨ om, K.J. and Bernhardsson, B. (1999). Comparison of periodic and event based sampling for first order stochastic systems. In Proceedings of the IFAC World Congress. Beijing. Dimarogonas, D.V. and Johansson, K.H. (2009). Eventtriggered control for multi-agent systems. In Proceedings of the 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, 7131–7136. Gawthrop, P.J. and Wang, L. (2009). Event-driven intermittent control. International Journal of Control, 82,
2235–2248. Girard, A. (2015). Dynamic triggering mechanisms for event-triggered control. IEEE Transactions on Automatic Control, 60, 1992–1997. Goebel, R., Sanfelice, R.G., and Teel, A.R. (2009). Hybrid dynamical systems: Robust stability and control for systems that combine continuous-time and discrete-time dynamics. IEEE Control Systems Magazine, 4, 28–93. Heemels, W.P.M.H. and Donkers, M.C.F. (2013). Modelbased periodic event-triggered control for linear systems. Automatica, 49, 698–711. Heemels, W.P.M.H., Johansson, K.H., and Tabuada, P. (2012). An introduction to event-triggered and selftriggered control. In Proceedings of the 51st IEEE Conference on Decision and Control, 3270–3285. Heemels, W.P.M.H., Sandee, J.H., and Van Den Bosch, P.P.J. (2008). Analysis of event-driven controllers for linear systems. International Journal of Control, 81, 571–590. Henningsson, T., Johannesson, E., and Cervin, A. (2008). Sporadic event-based control of first-order linear stochastic systems. Automatica, 44, 2890–2895. Jiang, Z.P., Teel, A.R., and Praly, L. (1994). Small-gain theorem for ISS systems and applications. Mathematics of Control, Signals and Systems, 7, 95–120. Lemmon, M.D. (2010). Event-triggered feedback in control, estimation, and optimization. Berlin: Springer-Verlag. Liu, T. and Jiang, Z.P. (2015). A small-gain approach to robust event-triggered control of nonlinear systems. IEEE Transactions on Automatic Control, 60, 2072– 2085. Liu, T., Jiang, Z.P., and Hill, D.J. (2014). Nonlinear Control of Dynamic Networks. Boca Raton: CRC Press. Lunze, J. and Lehmann, D. (2010). A state-feedback approach to event-based control. Automatica, 46, 211– 215. Marchand, N., Durand, S., and Castellanos, J.F.G. (2013). A general formula for event-based stabilization of nonlinear systems. IEEE Transactions on Automatic Control, 58, 1332–1337. Sontag, E.D. (2006). Input to state stability: Basic concepts and results. In: P. Nistri, G. Stefani. (Eds.), Nonlinear and Optimal Control Theory. Berlin: Springer-Verlag. Sontag, E.D. and Wang, Y. (1996). New characterizations of input-to-state stability. IEEE Transactions on Automatic Control, 41, 1283–1294. Tabuada, P. (2007). Event-triggered real-time scheduling of stabilizing control tasks. IEEE Transactions on Automatic Control, 52, 1680–1685. Wang, X. and Lemmon, M.D. (2008). Event design in event-triggered feedback control systems. In Proceedings of the 47th IEEE Conference on Decision and Control, 2105–2110. Wang, X. and Lemmon, M.D. (2009). Event-Triggering in Distributed Networked Systems with Data Dropouts and Delays, Hybrid Systems: Computation and Control. Heidelberg: Springer Berlin. Zhang, P., Liu, T., and Jiang, Z.P. (2017). Robust event-triggered control subject to external disturbance. Technical report. http://faculty.neu.edu.cn/tfliu/Tech17IFAC.pdf.
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