1 On Event Triggered Tracking for Nonlinear Systems Pavankumar Tallapragada and Nikhil Chopra Abstract In this paper we study an event based control algorithm for trajectory tracking in nonlinear systems. The desired trajectory is modelled as the solution of a reference system with an exogenous input and it is assumed that the desired trajectory and the exogenous input to the reference system are uniformly bounded. Given a continuous-time control law that guarantees global uniform asymptotic tracking of the desired trajectory, our algorithm provides an event based controller that not only guarantees uniform ultimate boundedness of the tracking error, but also ensures non-accumulation of inter-execution times. In the case that the derivative of the exogenous input to the reference system is also uniformly bounded, an arbitrarily small ultimate bound can be designed. If the exogenous input to the reference system is piecewise continuous and not differentiable everywhere then the achievable ultimate bound is constrained and the result is local, though with a known region of attraction. The main ideas in the paper are illustrated through simulations of trajectory tracking by a nonlinear system. 3 1 0 I. INTRODUCTION 2 Traditional computer based control systems rely on periodic sampling of the sensors and computation/execution of the n control. The reason for the popularity of this paradigm is a well developed theory and the ease of analysis of such systems. a However, such control algorithms may be very inefficient from a computational perspective as the period for sampling and J control execution is determined by a worst case analysis and the rate of control execution is independent of the system’s state. 5 On the other hand, in event based control systems, timing of control execution is not necessarily periodic and can be state ] dependent. Thus, event based control is useful in systematically designing controllers that make better use of computational Y and communication resources in a wide variety of applications such as embedded control systems and decentralized systems S (a representative list of references includes [1]–[5]). . s Whiletherehavebeensomeeffortsinthepasttostudyeventbasedcontrolsystems[6]–[8],theirsystematicdesignfortasks c such as stabilization has been undertaken only recently [1], [9]–[12]. Of these, [1] has significantly influenced the proposed [ controller in this paper. In [1], an event-triggering algorithm was proposed that ensures global asymptotic stability as well as 1 a lower bound on the inter-execution times of the control law for general nonlinear systems that are rendered Input-to-State v Stable (ISS) with respect to measurement errors by a continuous time controller. 5 In this paper, we investigate an event triggered control algorithm for trajectory tracking. Tracking a time varying trajectory 7 or even a set-point is of tremendous practical importance in many control applications. In these applications, the goal is to 8 0 make the state of the system follow a reference or desired trajectory, which is usually specified as an exogenous input to the . system. In this paper, the reference trajectory is generated by a reference system. To the best of our knowledge, the majority 1 0 of the previous works in the event-triggered control literature assumed a state feedback control strategy with no exogenous 3 input, some exceptions being [9]–[15], where unknown disturbances appear as exogenous inputs. However, in this paper, we 1 consider exogenous inputs that are available to the controller through measurements, namely the reference trajectory and the : v input to the reference system. i The main contribution ofthis paperis thedesign ofevent-triggeredcontrollers fortrajectory trackingin nonlinearsystems, X whichisaspecialcaseofnonlinearsystemswithexogenousinputs.Itisassumedthatthereferencetrajectoryandtheexogenous r input to the reference system are uniformly bounded. Given a nonlinear system and a continuous-time controller that ensures a global uniform asymptotic tracking of the desired trajectory, the proposed algorithm provides an event based controller that guarantees uniform ultimate boundedness of the tracking error and ensures that the inter-execution times of the controller are boundedawayfromzero.Inthespecialcasethatthederivativeoftheexogenousinputtothereferencesystemisalsouniformly bounded, an arbitrarily small ultimate bound for the tracking error can be designed. In this paper, unlike in the event-triggered control literature, the continuous-time control law is assumed to render the closed loop system asymptotically stable rather than ISS with respect to measurement errors. Although on compact sets the latter condition can be arrived at from the former, our choice allows a direct and clear procedure for designing an event-triggering condition with time-varying components that resultsinfewercontrollerexecutions.Apreliminaryversionoftheresultsinthispaperhavebeenpublishedin[16].Theresults therein have been expanded here. The rest of the paper is organized as follows. In Section II we set up the problem and introduce the notation used in the paper.Subsequently,inSectionIII,themajorassumptionsarestatedandtheeventtriggeringconditionisintroduced.Themain analytical results are presented in Section IV. The theoretical results in the paper are illustrated through numerical simulations of a second order nonlinear system in Section V. Finally, the results are summarized in Section VI. ThisworkwaspartiallysupportedbyMintaMartinFundandbytheNationalScienceFoundationthroughgrantnumber0931661. PavankumarTallapragadaiswiththeDepartmentofMechanicalEngineering,UniversityofMaryland,CollegePark,20742MD,USApavant@umd.edu N.ChopraiswiththeDepartmentofMechanicalEngineeringandTheInstituteforSystemsResearch,UniversityofMaryland,CollegePark,20742MD, USAnchopra@umd.edu 2 II. PROBLEMSTATEMENTANDNOTATION Consider a nonlinear system of the form x˙ =f(x,u), x∈Rn, u∈Rm (1) which has to track a reference trajectory defined implicitly by the dynamical system x˙ =f (x ,v), x ∈Rn, v ∈Rq (2) d r d d wheretheexternalsignalv andtheinitialconditionofthesignalx determinethespecificreferencetrajectory.Letthetracking d error be defined as x˜(cid:44)x−x . In general, a controller for tracking a reference trajectory depends on both the tracking error d as well as the reference trajectory. Hence, we assume that the control signal is of the form u=γ(ξ), where ξ (cid:44)[x˜;x ;v] (3) d where the notation [a ;a ;a ] denotes the column vector formed by the concatenation of the vectors a , a and a . Conse- 1 2 3 1 2 3 quently, the closed loop system that describes the tracking error is given as x˜˙ =f(x˜+x ,γ(ξ))−x˙ . (4) d d Now,consideracontrollerthatupdatesthecontrolonlyintermittentlyandnotcontinuouslyintime.Lett fori=0,1,2,... i be the time instants at which the control is computed and updated. Then, the tracking error evolves as x˜˙ =f(cid:0)x˜+x ,γ(ξ(t ))(cid:1)−x˙ , for t∈[t ,t ), i∈{0,1,2,...}. (5) d i d i i+1 The above dynamical system can also be viewed as a continuously updated control system, albeit with an error in the measurement of the state and the exogenous input. By defining the measurement error as     e x˜(t )−x˜ x˜ i e(cid:44)exd(cid:44)ξ(ti)−ξ (cid:44)xd(ti)−xd, for t∈[ti,ti+1), i∈{0,1,2,...} (6) e v(t )−v v i the system in (5) can be rewritten as x˜˙ =(cid:2)f(x˜+x ,γ(ξ))−x˙ (cid:3)+(cid:2)f(x˜+x ,γ(ξ+e))−f(x˜+x ,γ(ξ))(cid:3) (7) d d d d where we have expressed the above system as a perturbed version of the dynamical system (4). Note that e is discontinuous at t=t , for each i, because e(t )=ξ(t )−ξ(t )=0 while e(t−)(cid:44)lime(t)=lim(ξ(t )−ξ(t)). i i i i i i−1 t↑ti t↑ti In time-triggered or periodic control systems, t −t =T for all i∈{0,1,2,...}, where T >0 is a constant sampling i+1 i s s time.Ontheotherhand,inanevent-triggeredsystemthetimeinstantst ingeneralarenotuniformlyspaced,andaredetermined i dynamically by an event-triggering condition. The objective of this paper is to develop an event based controller for tracking a trajectory within a desired ultimate bound. To this end, we assume that when the control is updated continuously in time, the state x tracks the desired trajectory asymptotically, that is, there exists γ such that system (4) satisfies x˜→0 as t→∞. In the next section, we specify the main assumptions of the paper and develop an event-triggering condition for tracking a given trajectory within a desired bound. III. EVENT-TRIGGERINGCONDITIONFOREMULATIONBASEDTRAJECTORYTRACKINGCONTROL There are two main requirements for an event based trajectory tracking controller. It needs to (i) guarantee that the tracking error is at least uniformly ultimately bounded, and (ii) ensure that there is no accumulation of execution times. In this section, an event-triggering condition that satisfies both of these requirements is developed. We begin by formally stating the main assumptions of this paper. (A1) Suppose f(0,γ(0)) − f (0,0) = 0 and that there exists a C1 Lyapunov function for the dynamical system in (4), r V :Rn →R, such that for all admissible x and v, d α ((cid:107)x˜(cid:107))≤V(x˜)≤α ((cid:107)x˜(cid:107)) 1 2 ∂V (cid:2) (cid:3) f(x˜+x ,γ(ξ))−f (x ,v) ≤−α ((cid:107)x˜(cid:107)) ∂x˜ d r d 3 where α (.), α (.), and α (.) are class K functions1. 1 2 3 ∞ (A2) The functions f, γ and f are Lipschitz on compact sets. r (A3) For all time t≥0, (cid:107)[x ;v](cid:107)≤d for some d≥0 and v is piecewise continuous. d (A4) For all time t≥0, v is differentiable and (cid:107)v˙(cid:107)≤c for some c≥0. 1Acontinuousfunctionα:[0,∞)→[0,∞)issaidtobelongtotheclassK∞ ifitisstrictlyincreasing,α(0)=0andα(r)→∞asr→∞[17]. 3 The notation (cid:107).(cid:107) denotes the Euclidean norm of a vector. In the sequel, it is also used to denote the induced Euclidean norm of a matrix. Note that the meaning of ‘admissible x and v’ in (A1) differs in each of our main results, where in each case d it is specified precisely. At this stage, it is enough to know that (A3) is satisfied in each case. Now, consider the following family of compact sets: S(R)={ξ :V(x˜)≤α (R), (cid:107)[x ;v](cid:107)≤d}, δS(R)={ξ :V(x˜)=α (R), (cid:107)[x ;v](cid:107)≤d} (8) 2 d 2 d Note that for each R≥0, the sets S(R) and δS(R) include all the admissible reference signals, x and v. For each set S(R) d there exists, by assumption (A2), a vector L(R)∈R2n+q, with each of its components greater than zero such that (cid:107)f(x˜+x ,γ(ξ+e))−f(x˜+x ,γ(ξ))(cid:107)≤L(R)T|e|≤(cid:107)L(R)(cid:107)(cid:107)e(cid:107), ∀ξ, (ξ+e)∈S(R) (9) d d where |e| denotes the vector of the absolute values of the components of e. Without loss of generality, it may be assumed that each component of L(R) is a non-decreasing function of R. In the sequel, we use the notation S , δS and L to denote i i i S((cid:107)x˜(t )(cid:107)), δS((cid:107)x˜(t )(cid:107)) and L((cid:107)x˜(t )(cid:107)), respectively. Next, we define a continuous function, β(.), that satisfies i i i (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)∂V(w)(cid:12)(cid:12) β(R)≥ max (cid:12)(cid:12) (cid:12)(cid:12), ∀R≥0 (10) (cid:107)w(cid:107)≤R(cid:12)(cid:12) ∂w (cid:12)(cid:12) We now derive the triggering condition that determines the time instants t at which the control is updated. i Consider the Lyapunov function, V(.), in assumption (A1) as a candidate Lyapunov function for the system (5). The time derivative of V(x˜), along the flow of the tracking error system, V˙ = (∂V/∂x˜)x˜˙, may be obtained through the measurement error interpretation, (7). V˙ =∂V (cid:2)f(x˜+x ,γ(ξ))−x˙ (cid:3)+ ∂V (cid:2)f(x˜+x ,γ(ξ+e))−f(x˜+x ,γ(ξ))(cid:3) ∂x˜ d d ∂x˜ d d ∂V (cid:2) (cid:3) ≤−α ((cid:107)x˜(cid:107))+ f(x˜+x ,γ(ξ+e))−f(x˜+x ,γ(ξ)) (11) 3 ∂x˜ d d ≤−α ((cid:107)x˜(cid:107))+β((cid:107)x˜(cid:107))L(R)T|e|, ∀ξ, (ξ+e)∈S(R) (12) 3 where(11)isobtainedfromassumption(A1),and(12)isthenobtainedfrom(8)-(10).Then,(12)suggestsatriggeringcondition. Consider the following triggering condition (for the sake of clarity, the complete system description including the state equation and the triggering condition are given). x˜˙ =f(cid:0)x˜+x ,γ(ξ(t ))(cid:1)−x˙ , for t∈[t ,t ), i∈{0,1,2,...} (13) d i d i i+1 t =min{t≥0:(cid:107)x˜(cid:107)≥r >0}, and 0 σα ((cid:107)x˜(cid:107)) t =min{t≥t :LT|ξ(t )−ξ|− 3 ≥0 and (cid:107)x˜(cid:107)≥r >0} (14) i+1 i i i β((cid:107)x˜(cid:107)) where 0 < σ < 1 and r is a design parameter that determines the ultimate bound of the tracking error. It is necessary to update the control only when (cid:107)x˜(cid:107)≥r, for some r >0, else it may result in the accumulation of control update times. Notice that each update instant t is defined implicitly with respect to t . Hence, the initial update instant t has been specified i+1 i 0 separately. As the proposed triggering condition does not allow the control to be updated whenever (cid:107)x˜(cid:107)0 (where V˙|t=t∗i denotes V˙ evaluated 4 aatcton=trat∗id)i.ctHioonw,ewveer,coanscξlu(dt∗ie)t∈hatδSnio⊂suSchi,t(∗12e)xiasntsda(n1d4)thiamtptlhyeVfi˙r|ts=ttc∗ila≤im−o(f1t−heσl)eαm3m((cid:107)ax˜i(st∗itr)u(cid:107)e).0 are any positive real numbers, the functions α and β are as defined in (A1) and (10), respectively. Since 2 1 3 α3 and β are continuous positive definite functions, ∆ss21 is well defined and positive for any given s2 ≥s1 >0. Now we present the first main result of the paper. It demonstrates, for a particular class of reference trajectories, that in the event-triggered system (13)-(14) the inter-execution times are uniformly bounded away from zero while the tracking error is uniformly ultimately bounded. Theorem 1: Consider the system (4). Suppose that assumptions (A1), (A2), (A3) and (A4) are satisfied. Then, for the event- triggered system (13)-(14), the tracking error, x˜, is uniformly ultimately bounded by a ball of radius r = α−1(α (r)), and 1 1 2 the inter-execution times (t −t ) for i∈{0,1,2,...} are uniformly bounded below by a positive constant that depends on i+1 i the bound of the initial tracking error. Proof: Uniform ultimate boundedness of the tracking error follows from Lemma 1. Only the existence of a positive lower bound for the inter-execution times remains to be shown. Note that for each i, (cid:107)e(t )(cid:107) = 0 and (cid:107)x˜(t )(cid:107) ≥ r. Hence, the i i triggeringcondition(14)impliesthattheith inter-updatetime,(t −t ),isatleastequaltothetimeittakes(cid:107)L (cid:107)(cid:107)e(cid:107)togrow i+1 i i from 0 to σα ((cid:107)x˜(cid:107))/β((cid:107)x˜(cid:107)). Recall from the proof of Lemma 1 that every solution, ξ, stays in the set S for all t∈[t ,t ), 3 0 0 i for each i. Thus, (cid:107)L (cid:107)≤(cid:107)L (cid:107) for each i. Notice that i 0 S ⊂{ξ :(cid:107)x˜(cid:107)≤µ , (cid:107)[x ;v](cid:107)≤d} (17) 0 0 d where µ = α−1(α ((cid:107)x˜(t )(cid:107))). Then, (16) implies t − t ≥ T, where T is the time it takes (cid:107)e(cid:107) to grow from 0 to 0 1 2 0 i+1 i ∆µr0/(cid:107)L0(cid:107). If we show that T >0, then the proof is complete. From (7), and the triangle inequality property, we observe that (cid:107)x˜˙(cid:107)≤(cid:107)f(x˜+x ,γ(ξ))−x˙ (cid:107)+(cid:107)f(x˜+x ,γ(ξ+e))−f(x˜+x ,γ(ξ))(cid:107) (18) d d d d From (9), the second term is bounded by LT|e|≤(cid:107)L (cid:107)(cid:107)e(cid:107) on the set S . Since, according to (A1), f(0,γ(0))−f (0,0)=0, 0 0 0 r (A2) then implies that there exist Lipschitz constants P ≥0 and P ≥0 such that 1 2 (cid:107)x˜˙(cid:107)≤P (cid:107)x˜(cid:107)+P (cid:107)[x ;v](cid:107)+LT|e| 1 2 d 0 ≤P µ +P d+(cid:107)L (cid:107)(cid:107)e(cid:107) 1 0 2 0 where the second inequality is obtained from (17). Assumptions (A2)-(A3) imply that there exists a constant P ≥0 such that 3 (cid:107)x˙ (cid:107)≤ P d and (A4) implies (cid:107)v˙(cid:107)≤ c. Then, by letting P = P µ +(P +P )d and from the definition e˙ = −[x˜˙;x˙ ;v˙] it d 3 0 1 0 2 3 d follows that d(cid:107)e(cid:107) ≤(cid:107)e˙(cid:107)≤(cid:107)L (cid:107)(cid:107)e(cid:107)+P +c (19) dt 0 0 Note that for (cid:107)e(cid:107)=0, the first inequality holds for all the directional derivatives of (cid:107)e(cid:107). Then, according to the Comparison Lemma [17] P +c (cid:107)e(cid:107)≤ 0 (e(cid:107)L0(cid:107)(t−ti)−1), for t≥t . (20) (cid:107)L (cid:107) i 0 Thus, the inter-execution times are uniformly lower bounded by T, which satisfies 1 (cid:18) ∆µ0 (cid:19) T ≥ log 1+ r . (21) (cid:107)L (cid:107) P +c 0 0 5 As (cid:107)L0(cid:107) is finite and ∆µr0 >0, we conclude that the inter-execution times have a uniform lower bound, T, that is greater than zero. In the next result, the conditions on the reference trajectory are relaxed by no longer requiring it to satisfy assumption (A4). Instead, to ensure the absence of Zeno behavior, a new assumption is made - that d , the uniform bound on (cid:107)v(cid:107), is no larger v than a quantity determined by ∆µr0 and L0. The new assumptions, in contrast to Theorem 1, lead to a constraint on the choice of the radius r in the triggering condition and ensure only local uniform ultimate boundedness of the trajectory tracking error. Let L(R)(cid:44)[Q(R);M(R)] and L (cid:44)[Q ;M ] where Q(R),Q ∈R2n and M(R),M ∈Rq. Now, the second main result is i i i i i presented. Theorem 2: Consider the system defined by (4). Suppose that the assumptions (A1), (A2) and (A3) hold. Also, for some R0 ≥r suppose that ∆µr0−2dv(cid:107)M(R0)(cid:107)>0, where µ0 =α1−1(α2(R0)), ∆µr0 is given by (16) and dv is the uniform bound on (cid:107)v(cid:107). If (cid:107)x˜(0)(cid:107) ≤ R , then in the event-triggered system (13)-(14), the tracking error, x˜, is uniformly ultimately bounded 0 by a ball of radius r =α−1(α (r)), and the inter-update times (t −t ) for i∈{0,1,2,...} are uniformly bounded below 1 1 2 i+1 i by a positive constant that depends on R . 0 Proof: The proof is very similar to that of Theorem 1, and hence only the essential steps are described here. According to Lemma 1 each solution, ξ, with (cid:107)x˜(0)(cid:107) ≤ R stays in the set S(R ). Hence, (cid:107)M (cid:107) ≤ (cid:107)M(R )(cid:107) and (cid:107)Q (cid:107) ≤ (cid:107)Q(R )(cid:107) 0 0 i 0 i 0 for each i. Since (cid:107)v(cid:107) is uniformly bounded by d it follows that for each i, MT|e | ≤ (cid:107)M (cid:107)(cid:107)e (cid:107) ≤ 2d (cid:107)M(R )(cid:107), where v i v i v v 0 e =v(t )−v and |e | denotes the component-wise absolute value of the vector e . The definitions of Q and M imply that v i v v i i LT|e|=QT|[e ;e ]|+MT|e |≤QT|[e ;e ]|+2d (cid:107)M(R )(cid:107). i i x˜ xd i v i x˜ xd v 0 Note that for each i, r ≤(cid:107)x˜(ti)(cid:107)≤µ0. Thus, the triggering condition in (14) implies that for each i, LTi−1|e(t−i )|≥∆µr0, or equivalently, QTi−1|[ex˜(t−i );exd(t−i )]| ≥ δ (cid:44) ∆µr0 −2dv(cid:107)M(R0)(cid:107) > 0, the last inequality being one of the assumptions. Hence, the inter-execution times t −t ≥T, where T is the time it takes (cid:107)[e ;e ](cid:107) to grow from 0 to δ/(cid:107)Q(R )(cid:107). If we i+1 i x˜ xd 0 show that T >0, then the proof is complete. d(cid:107)[e ;e ](cid:107) FollowingstepssimilartothoseintheproofofTheorem1,weknowthatthereexistsafiniteP ≥0suchthat x˜ xd ≤ 0 dt (cid:107)Q (cid:107)(cid:107)[e ;e ](cid:107)+P +2d (cid:107)M(R )(cid:107). Note that for (cid:107)[e ;e ](cid:107) = 0, the inequality holds for all the directional derivatives. 0 x˜ xd 0 v 0 x˜ xd Thus, the inter-execution times are uniformly lower bounded by T, which satisfies T ≥ 1 log(cid:18)1+ ∆µr0 −2dv (cid:19). (22) (cid:107)Q (cid:107) P +2d (cid:107)M(R )(cid:107) 0 0 v 0 As (cid:107)Q (cid:107) is finite, we conclude that the inter-execution times have a lower bound, T, that is greater than zero. 0 Theorem 2 is somewhat conservative because only the uniform bound on (cid:107)v(cid:107) is utilized in determining the ultimate bound andthelowerboundontheinter-executiontimes.Amoreusefulresultisobtainedbyimposingonlyslightlystricterconstraints onv -thatjumpsinv areseparatedintimebyT >0,thatthemagnitudeofeachjumpisupperboundedbyaknownconstant v and that v is Lipschitz between jumps. This is expressed formally in the following assumption. (A5) There exist constants c ≥ 0, T ≥ 0 and J ≥ 0 such that for all t,s ≥ 0, the following holds: (cid:107)v(t) − v(s)(cid:107) ≤ v v (cid:108) (cid:109) c|t−s|+ |t−s| J , where (cid:100).(cid:101) is the ceiling function. Tv v Now, the final result is presented. Theorem 3: Considerthesystemdefinedby(4).Supposethattheassumptions(A1),(A2),(A3)and(A5)hold.Also,forsome R0 ≥r supposethat∆µr0−Jv(cid:107)M(R0)(cid:107)>0,whereµ0 =α1−1(α2(R0))and∆µr0 isgivenby(16).If(cid:107)x˜(0)(cid:107)≤R0,theninthe event-triggered system (13)-(14), the tracking error, x˜, is uniformly ultimately bounded by a ball of radius r =α−1(α (r)), 1 1 2 and the inter-update times (t −t ) for i ∈ {0,1,2,...} are uniformly bounded below by a positive constant that depends i+1 i on R . 0 (cid:108) (cid:109) Proof:Lete∗ (cid:44)[ex˜;exd;ev∗],whereev∗ (cid:44)c(t−ti)fort∈[ti,ti+1)andeachi.Then,by(A5),(cid:107)e(cid:107)≤(cid:107)e∗(cid:107)+ |tT−vs| Jv. Now, let T be the time it takes (cid:107)e∗(cid:107) to grow from zero to (∆µ−kJ (cid:107)M(R )(cid:107))/(cid:107)L (cid:107). Then, a lower bound on the inter- k r v 0 0 execution times is given by (cid:22) ∆µ (cid:23) max {min{kT ,T }}, N = r k∈{1,2,...,N} v k Jv(cid:107)M(R0)(cid:107) where(cid:98).(cid:99)denotesthefloorfunction.Notethat{kT }isanincreasingsequencewhile{T }isadecreasingsequence.Following v k the proof of Theorem 1, T is estimated as k 1 (cid:18) ∆µ−kJ (cid:107)M(R )(cid:107)(cid:19) T ≥ log 1+ r v 0 . (23) k (cid:107)L (cid:107) P +c 0 0 Thustheinter-executiontimesarelowerboundedbyapositiveconstant.Theultimateboundednessofthetrackingerrorfollows from Lemma 1. Remark 1: Noticefrom(9)thatinordertocomputeL =L((cid:107)x˜(t )(cid:107))itisnecessarytocomputethesetS =S((cid:107)x˜(t )(cid:107))orat i i i i leastasetofwhichS isasubset,suchasB (cid:44){ξ :(cid:107)x˜(cid:107)≤α−1(α ((cid:107)x˜(t (cid:107))),(cid:107)[x ;v](cid:107)≤d}.However,if(cid:107)x˜(t )(cid:107)≥(cid:107)x˜(t )(cid:107) i i 1 2 i d i i−1 then clearly some components of L may be greater than those of L . But from Lemma 1, we know that S ⊂ S for i i−1 i i−1 6 each i, so at time instant t instead of computing L based on B , we can let L = L . Following this rule, the sequence i i i i i−1 {L } can be chosen to be component-wise non-increasing. The triggering condition and the estimates of lower bounds on the i inter-update times depend critically on L and hence using a time-varying L lowers the overall average update rate. Computing L is in general a computationally costly task and it is not useful to update L continuously in time like α ((cid:107)x˜(cid:107)) and β((cid:107)x˜(cid:107)). 3 In the next section our theoretical results are illustrated through simulations of a second order nonlinear system. V. EXAMPLESANDSIMULATIONRESULTS The theoretical results developed in the previous sections are illustrated through simulations of the following second order nonlinear system. (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) x˙ 0 1 0 0 0 x˙ = 1 = x+ + u=Ax+ +Bu (24) x˙ 0 −1 −x3 1 −x3 2 1 1 The desired trajectory is a solution of the system [x˙ ;x˙ ]=[x ;v], where v is an exogenous input, which along with the d,1 d,2 d,2 initial conditions of the state of the reference system, x =[x ;x ], determines the specific trajectory. The control function d d,1 d,2 is chosen as γ(ξ)=Kx˜+v+(x˜ +x )3+x (25) 1 d,1 d,2 where K =[k ;k ]T is a 2×1 row vector such that A˜=(A+BK) is Hurwitz, and x˜=[x˜ ;x˜ ] is the tracking error. Then, 1 2 1 2 the closed-loop tracking error system with event-triggered control can be written, using the measurement error interpretation, as x˜˙ =x˜ 1 2 x˜˙ =−(x˜ +x )−(x˜ +x )3+γ(ξ+e)−v. (26) 2 2 d,2 1 d,1 Now, consider the quadratic Lyapunov function V =x˜TPx˜ where P is a positive definite matrix that satisfies the Lyapunov equation PA˜+A˜TP =−H, where H is a given positive definite matrix. The time derivative of V along the flow defined by (26) can be shown to satisfy V˙ ≤−x˜THx˜+2x˜TPB[γ(ξ+e)−γ(ξ)] ≤−σa(cid:107)x˜(cid:107)2+β((cid:107)x˜(cid:107))L(R)T|e|, ∀ξ,(ξ+e)∈S(R) (27) where a>0 is the minimum eigenvalue of H, β((cid:107)x˜(cid:107))=2(cid:107)PB(cid:107)(cid:107)x˜(cid:107) and L(R)=(cid:2)3(µ+d )2+|k |;|k |;3(µ+d )2;1;1(cid:3) (28) 1 1 2 1 where µ = α−1(α (R)) and d ≤ d is the uniform bound on x . If d is not known explicitly then d from assumption 1 2 1 d,1 1 (A3) may be used instead. Note that B has been absorbed in β rather than in L(R), as it should have been according to their definitions. This makes the β function point-wise lower. The vectors L were computed according to the procedure in Remark i 1. Finally, given a desired ultimate bound for the trajectory tracking error, the parameter r in the triggering condition can be designed. Next, we present simulation results for two cases corresponding to the two main classes of reference trajectories considered in this paper. Case I: The signals x , x , and v were chosen as sinusoidal signals with peak-to-peak amplitude 2. This was done by d,1 d,2 choosing[x (0),x (0);v(0)]=[π/3;1;0]andv˙ =−cos(t).Theinitialconditionoftheplantwas[x (0);x (0)]=[5;−1]. d,1 d,2 1 2 The parameter d was chosen as 2.5 while the actual uniform bounds on x and (cid:107)[x ;v](cid:107) were observed to be around 2 1 d,1 d and 2.28, respectively. The parameters in the controller were chosen as K =−[20;20]T, σ =0.95 and H was chosen as the identity matrix. According to Theorem 1, we chose r = 0.0154 in the triggering condition to achieve an ultimate bound of r =0.1 in the tracking error. 1 ThesimulationresultsareshowninFigure1a.TheFigureshowsthenormofthetrackingerror,theradiusr inthetriggering condition, the desired ultimate bound r and WT|e|, where W =(2(cid:107)PB(cid:107)L )/(σa). The figure demonstrates that the tracking 1 i i i error is ultimately bounded, and well below the desired bound. We recall that according to the triggering condition (14), the control is not updated when (cid:107)x˜(cid:107)

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