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Event-triggered Control for Nonlinear Systems withCenter Manifolds

Akshit Saradagi1, Vijay Muralidharan2, Arun D. Mahindrakar3, Senior Member, IEEE and PavankumarTallapragada4, Member, IEEE

Abstract—In this work, we consider the problem of event-triggered implementation of control laws designed for the localstabilization of nonlinear systems with center manifolds. Wepropose event-triggering conditions which are derived from alocal input-to-state stability characterization of such systems.The triggering conditions ensure local ultimate boundedness ofthe trajectories and the existence of a uniform positive lowerbound for the inter-event times. The ultimate bound can bemade arbitrarily small, but by allowing for smaller inter-eventtimes. Under certain assumptions on the controller structure,local asymptotic stability of the origin is also guaranteed. Twosets of triggering conditions are proposed, that cater to the caseswhere the exact center manifold and only an approximationof the center manifold is computable. The closed-loop systemexhibits some desirable properties when the exact knowledgeof the center manifold is employed in checking the triggeringconditions. Three illustrative examples that explore differentscenarios are presented and the applicability of the proposedmethods is demonstrated. The third example concerns the event-triggered implementation of a position stabilizing controller forthe open-loop unstable Mobile Inverted Pendulum (MIP) robot.

Index Terms—Event-triggered control, Input-to-state stability,Center manifold theory, Mobile Inverted Pendulum Robot

I. INTRODUCTION

Our view of what constitutes as resource in the implemen-tation of control laws has evolved over the years. In manyoptimal control formulations, the control effort or controlenergy captures the sensing, computation and actuation costsand is traded with closed-loop performance. The focus onlarge-scale multi-agent systems in the recent years has revealedthat, resources such as CPU time and wireless channel band-width cannot be overlooked, as they are shared and costly[1], [2]. In response, various resource-aware implementationshave emerged, that look to utilize the resources judiciously,while guaranteeing pre-specified performance. Event-triggeredcontrol [3]–[5] is one such resource-aware technique whichhas gained popularity in the recent years and presents analternative to time-triggered control.

In event-triggered control, the control loop is closed whencertain events occur in the system and not periodically asin time-triggered control. As the event instants are implicitlydefined and the inter-event times are aperiodic, non-existence

1Akshit and 3Arun are with the Department of Electrical Engineering,Indian Institute of Technology Madras, Chennai, India (emails: akshit-saradagi@gmail.com, arun dm@iitm.ac.in)

2Vijay is with the Department of Electrical Engineering, Indian Institute ofTechnology Palakkad, Palakkad, India (email: vijay@iitpkd.ac.in)

4Pavankumar Tallapragada is with the Department of Electrical Engineer-ing, Indian Institute of Science, Bengaluru, India (email: pavant@iisc.ac.in)

of Zeno behaviour (existence of a uniform positive lowerbound for the inter-event times) must be ensured. Many event-triggered implementations require continuous monitoring ofthe states to detect the occurrence of an event. As this isnot feasible in digital implementations, solutions are beingexplored where the occurrence of an event needs to bechecked only periodically [6], [7]. Often, controllers are de-signed first and then event-triggering conditions are designed.This sequential design procedure is called emulation-basedapproach. Recently, control and implementation co-design isbeing investigated, where both are designed parallelly [8] [9].

Although research in event-triggered control has investi-gated a wide variety of settings, as surveyed in [5] and [10],the case of event-triggered control of nonlinear systems withcenter manifolds has not been looked at so far. Center manifoldanalysis is a crucial design and analysis tool for nonlinearsystems with degenerate equilibria and is widely used inthe areas of control theory, bifurcation theory and multi-scale modelling, among many others [11], [12]. We realisedthe need for new research in this direction when analysingthe applicability of existing results for the Mobile InvertedPendulum (MIP) robot [13]. Control of a tethered satellitesystem is another example where a controller is designedin the presence of a center manifold [14]. In this work, wepresent our solution to the problem of event-triggered controlof nonlinear systems with center manifolds.

The work presented here builds on the preliminary findingsfrom our recent work [15], where an explicit Lyapunov-basedcharacterisation of local input-to-state stability (LISS) for non-linear systems with center manifolds was derived, assuminga general controller structure proposed in [16]. This LISScharacterization is used in the present work to design event-triggering conditions. In [15], we also identified hurdles inthe way of designing event-triggered implementation, namely,requirement of the exact knowledge of the center manifold inchecking the triggering conditions and the non-applicability ofthe existing ISS-based results for nonlinear systems [3], [17]for a large class of systems under consideration.

For the general class of nonlinear systems, two approachesto the design of event-triggered control are popular in liter-ature. The approach assuming input-to-state stability of theclosed-loop system with respect to measurement errors [3],[17]–[20], guarantees asymptotic stability of the origin. How-ever, the non-existence of Zeno behaviour can be ensured onlyunder the assumption that the composition of the comparisonfunctions involved in the ISS characterization is Lipschitzcontinuous over compact sets. This assumption is not satisfied

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for a large subset of the systems under consideration. Underthe assumption of only asymptotic tracking, the work [21]proposes event-triggered tracking (which can be adapted forstabilization) for nonlinear systems and guarantees Zeno-free uniform ultimate boundedness of the tracking error. Forthe systems under consideration, the Lyapunov analysis isessentially local and the results from [21] are not directlyapplicable.

The main contributions of this work are the following. Inthis work, event-triggered implementation of control laws de-signed for local stabilization of nonlinear systems with centermanifolds is being investigated for the first time. The pro-posed methods ensure Zeno-free local ultimate boundednessof the trajectories, with an ultimate bound that can be madearbitrarily small. Under some assumptions on the controllerstructure, Zeno-free asymptotic stability of the origin is alsoensured. For a subclass of the systems under consideration,the LISS characterization meets the sufficient conditions from[3] and [17] and triggering conditions proposed in these workscan be employed. For systems which do not fall in this class,which include many practical systems, such as the MIP robot,these triggering conditions cannot be used. The triggeringconditions presented in this work cater to this class of systemsand this forms our main contribution. As in [21], for a largeclass of systems under consideration, the triggering conditionspresented in this work ensure ultimate boundedness of thetrajectories, to an ultimate bound that can be made arbitrarilysmall. However, under some assumptions on the controllerstructure, Zeno-free asymptotic stability of the origin is alsoensured. The proposed triggering conditions cater to both thecases where the exact and the approximate knowledge ofthe center manifold is available. Simulation results for threeillustrative examples are presented, that explore the differentscenarios being considered in this work. The third exampleconcerns the event-triggered position stabilization of an open-loop unstable MIP robot.

The organization of this work is as follows. We begin withnotations and preliminaries in Section II. In Section III, werecollect important results from center manifold theory, setupthe problem and summarize the findings from our work in[15] on LISS of nonlinear systems with center manifolds.This LISS characterization is used in designing event-triggeredcontrol in Section IV, for systems with exactly computablecenter manifolds. In Section V, possibility of designing trig-gering conditions that do not require the exact knowledge ofthe center manifold is explored. In section VI, we presenttriggering conditions utilizing the approximate knowledge ofthe center manifold and draw a comparison with the casewhere the exact knowledge of the center manifold is assumed.In subsection VI-C, using the triggering conditions presentedin section VI, event-triggered implementation of a positionstabilizing controller for an MIP robot is presented, followingwhich concluding remarks are made in section VII.

II. NOTATIONS AND PRELIMINARIES

We denote by R, the set of real numbers and by R+ theset of non-negative real numbers. Given two vectors y and

w, we denote by (y;w) the concatenation of the two vectors[y> w>]>. We denote by | · |, the absolute value of a realnumber and by || · ||, the Euclidean norm of a vector or theinduced 2-norm of a matrix, depending on the argument. Ann × n identity matrix is denoted by In. Given a matrix A ∈Rn×n, A � 0 denotes that A is a symmetric positive definitematrix. A continuous function α : [0, a) → [0,∞) is said tobe a class-K function if α(0) = 0 and it is strictly increasing.The notations f(x) = O(||x||p) and f(x) ∈ O(||x||p), p ∈ Rdenote that |f(x)| ≤ c1||x||p for all x such that ||x|| < ε, forsome c1, ε > 0.

Definition 1 (Local input-to-state stability (LISS) [22]). Thesystem x = f(x, d), x ∈ Rn and d ∈ Rm with f being locallyLipschitz and f(0, 0) = 0, is said to be locally input-to-statestable in the domain Dx ⊂ Rn with respect to input d inthe domain Dd ⊂ Rm, if there exists a Lipschitz continuousfunction V : Dx → R+ and class-K functions α1, α2, α3 andβ such that

α1(||x||) ≤ V (x) ≤ α2(||x||)

and

ζ>f(x, d) ≤ −α3(||x||) + β(||d||)

hold for all x ∈ Dx, d ∈ Dd and ζ ∈ ∂V (x), thesubdifferential [23] of V at x. The function V satisfying theabove conditions is called an LISS Lyapunov function.

III. PROBLEM SETUP

In this section, we introduce the systems under consideration- nonlinear systems with center manifolds, review the controldesign techniques for such systems and setup the problemof designing event-triggered control with the most generalcontroller structure.

Consider the nonlinear dynamical system

x = f(x, u), x(t0) = x0 (1)

where x ∈ Rn, u ∈ Rm and f : Rn × Rm → Rn is atwice continuously differentiable function with f(0, 0) = 0.The Taylor series expansion of the function f about x = 0and u = 0 yields

x = Ax+Bu+ f(x, u) (2)

where A ∈ Rn×n, B ∈ Rn×m and f(x, u) constitutes thehigher-order terms and satisfies

f(0, 0) = 0,∂f

∂x(0, 0) = 0 and

∂f

∂u(0, 0) = 0. (3)

In this work, we focus on nonlinear systems whose linearizedmodels have uncontrollable modes on the imaginary axis.For such systems, there exists a linear transformation x =T (y; z), T ∈ Rn×n such that system (2) is transformed intothe following form

y = A1y + g1(y, z, u)

z = A2z +B2u+ g2(y, z, u)(4)

where A1 ∈ Rk×k, A2 ∈ R(n−k)×(n−k), B2 ∈ R(n−k)×m,g1, g2 are the nonlinearities, the real parts of the eigenvalues

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of A1 are zero and the pair (A2, B2) is controllable. Thefunctions g1(y, z, u) and g2(y, z, u) satisfy

gi(0, 0, 0) = 0,∂gi∂y

(0, 0, 0) = 0,

∂gi∂z

(0, 0, 0) = 0,∂gi∂u

(0, 0, 0) = 0, for i = 1, 2.

(5)

This is easily inferred from the properties of function f in (3).Suppose we design a controller K(y, z) for system (4) and

wish to assess the local stability of the origin (y, z) = 0 ofthe closed-loop system. The control u cannot influence theeigenvalues of the matrix A1, even though the poles of A2 canbe arbitrarily placed. The indirect method of Lyapunov throughlinearization cannot be used and results from center manifoldtheory have to be invoked. This theory provides a model-reduction technique to determine the stability of the closed-loop system by assessing the stability of a reduced system,which governs the dynamics on an invariant manifold calledthe center manifold. If the dynamics on the center manifold isstable (unstable), then the overall system is stable (unstable).In [16], [24], [25], systematic control design procedures werepresented for nonlinear systems with center manifolds. In thefollowing subsections, we review important results from centermanifold theory and the control design approach from [16].

A. Choice of a controller

The origin of the z-subsystem in (4) can be locallyasymptotically stabilized by a controller u = K11z,K11 ∈Rm×(n−k), under the assumptions of stabilizability of the pair(A2, B2). The feedback matrix K11 indirectly determines boththe center manifold and the dynamics on the center manifold.Stabilizability, while limiting the freedom in specifying theperformance of the z-subsystem, also limits the indirect influ-ence of the matrix K11 on the centre manifold structure and thestability of the center manifold. This justifies the assumptionof controllability of the pair (A2, B2) rather than stabilizabilityof the pair (A2, B2).

When the overall system cannot be stabilized by justu = K11z, feedback of the form u = K11z + K12y must beemployed with K12 chosen to stabilize the dynamics on thecenter manifold. This may also be inadequate, following whichan additive pseudo-control Kn(y) must be introduced to yieldu = K11z + K12y + Kn(y), as in [16]. The pseudo-controlKn : Rk → Rm is a continuously differentiable nonlinearfunction of y and is chosen to stabilize the dynamics on thecenter manifold. In the rest of the work, we use the controlstructure u = K(y; z) = K11z + K12y + Kn(y), as this isthe most general form of the controller that can be used inthe stabilization of nonlinear systems with center manifolds.Denoting A2 +B2K11 by AK , we arrive at

y = A1y + g1(y, z,K(y; z))

z = AKz +B2K12y +B2Kn(y) + g2(y, z,K(y; z)).(6)

B. Background from Center Manifold Theory

At this juncture, we recall some important definitions andresults from center manifold theory. For the next few results

from center manifold theory to hold, the cross-coupling linearterm, B2K12y between the y and z subsystems in (6) mustbe eliminated. The cross-coupling term is eliminated using thechange of variables v , z−Ey, where the matrix E is foundby solving the equation

AKE − EA1 +B2K12 = 0. (7)

As the sum of any eigenvalue of A1 and any eigenvalue ofAK is non-zero, we use from [25], the result which states thatthere exists a unique matrix E ∈ R(n−k)×k such that (7) issatisfied. Using this change of variables and the notations

g1(y, v+Ey,K(y; v + Ey)) = g1(y, v + Ey,K(y; v + Ey))

g2(y, v+Ey,K(y; v + Ey)) = B2Kn(y) + g2(y, v + Ey,

K(y; v + Ey))− Eg1(y, v + Ey,K(y; v + Ey))

we arrive at

y = A1y + g1(y, v + Ey,K(y; v + Ey))

v = AKv + g2(y, v + Ey,K(y; v + Ey)).(8)

Again, it can be easily checked that g1 and g2 also satisfyconditions (5).

Definition 2 (Center Manifold). For the dynamical system (8),a manifold v = h(y) is called a center manifold, if

h(0) = 0 and∂h

∂y(0) = 0 (9)

and v(0) = h(y(0)) implies v(t) = h(y(t)), ∀ t ∈ [0,∞).

The center manifold is an invariant manifold of system (8).The center manifold h(y) satisfying (9) is found by solvingthe partial differential equation

0 =AKh(y) + g2(y, h(y) + Ey,K(y;h(y) + Ey))

− ∂h(y)

∂y(A1y + g1(y, h(y) + Ey,K(y;h(y) + Ey))).

(10)

Theorem 1 (Existence of a center manifold [26]). Considerthe system (8). If g1(y, v) and g2(y, v) are twice continuouslydifferentiable and satisfy conditions in (5), with all eigenvaluesof A1 having zero real parts and all eigenvalues of AK havingnegative real parts, then there exists a constant δ > 0 anda continuously differentiable function h(y), defined for all||y|| ≤ δ, such that v = h(y) is a center manifold for thesystem (8).

System (8) satisfies the hypothesis of Theorem 1 for theexistence of a k-dimensional center manifold v = h(y) andthe dynamics on the center manifold is governed by

y = A1y + g1(y, h(y) + Ey,K(y;h(y) + Ey)) (11)

which is referred to as the reduced system. We next recall aresult, known popularly as the Reduction Theorem.

Theorem 2 (Reduction Theorem [26]). Under the assumptionsof Theorem 1, if the origin y = 0 of the reduced system (11)is locally asymptotically stable (unstable), then the origin ofthe full system (8) is locally asymptotically stable (unstable).

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C. Event-based control and measurement errors

In event-triggered implementation, the control is updated atdiscrete instants ti, i ≥ 0, called the event times. Between twoevents, in the interval [ti, ti+1), the control is held constant tou = K(y(ti); v(ti) +Ey(ti)) and measurement and actuationerrors are introduced to account for the sample and hold natureof the implementation. With the measurement error ey(t) ,y(ti) − y(t) and ev(t) , v(ti) − v(t), the control can berewritten as u = K(y + ey; v + ev + E(y + ey)). Under thisimplementation, the closed-loop system is

y =A1y + g1(y, v + Ey, u)

v =AKv +B2K11ev +B2(K12 + EK11)ey

+ g2(y, v + Ey, u).

(12)

For further analysis of the system, we introduce a new coordi-nate w = v−h(y) (the need for which is explained in sectionV). With control u = K(y + ey, w + h(y) + ev +E(y + ey))and K1 = [K12 +EK11 K11] and e = (ey, ev), the dynamics(12) in the (y;w) coordinates is found to be

y = A1y + g1(y, w + h(y) + Ey, u)

w = AK(w + h(y)) +B2K1e+ g2(y, w + h(y) + Ey, u)

− ∂h(y)

∂y(A1y + g1(y, w + h(y) + Ey, u). (13)

Subtracting (10) from (13) and using the following notations

N1(y, w, e) ,g1(y, w + h(y) + Ey,K(y + ey;w + h(y)

+ ev + E(y + ey)))

− g1(y, h(y) + Ey,K(y;h(y) + Ey))

N2(y, w, e) ,g2(y, w + h(y) + Ey,K(y + ey;w + h(y)

+ ev + E(y + ey)))− g2(y, h(y) + Ey,

K(y;h(y) + Ey))− ∂h(y)

∂y(A1y + g1(y, w

+ h(y) + Ey,K(w + h(y) + e))− (A1y

+ g1(y, h(y) + Ey,K(y;h(y) + Ey))))

we obtain

y =A1y + g1(y, h(y) + Ey,K(y;h(y) + Ey))

+N1(y, w, e)

w =AKw +B2K1e+N2(y, w, e).

(14)

By substitution, we see that for i = 1, 2

Ni(y, 0, 0) = 0,∂Ni∂w

(0, 0, 0) = 0 and∂Ni∂e

(0, 0, 0) = 0.

(15)

When Ni satisfy conditions (15), there exists a constant δyw >0 such that, in the set

S = {(y;w) | ||(y;w)|| < δyw} (16)

we have for i = 1, 2,

||Ni|| ≤ ki||(w; e)|| ≤ ki(||w||+ ||e||) (17)

where the constants ki > 0 can be made arbitrarily smallby decreasing δyw. Note that the stability properties of system(14) are the same as that of system (1) in view of the sequence

of smooth transformations w = v − h(y), v = z − Ey andx = T (y; z) relating the two systems.

In this work, our design of event-triggered implementationuses the ISS based approach proposed in [3] and generalized in[17]. Verifying that the closed-loop system is ISS with respectto measurement errors forms the first step of this approach.This led us to the question : Does a controller that locallyasymptotically stabilizes (8) also render (14) locally input-to-state stable with respect to measurement errors? Although theresult [27, Lemma 1] answers this question in the affirmative,this qualitative result is useful for event-triggered control onlywhen the class-K function α3 and β that characterize localinput-to-state stability in Definition 1 are known. Motivated bythe unavailability of such explicit characterization of input-to-state stability in terms of comparison functions for nonlinearsystems with center-manifolds and with the intent of designingevent-triggered control, we investigated the local ISS propertyof the systems under consideration in our preliminary work[15]. In the next subsection, we recall an important result fromthis work.

D. Local input-to-state stability of nonlinear systems withcenter manifolds

The Proposition presented in this subsection establishes thata controller that locally asymptotically stabilizes system (8),renders (14) LISS with respect to measurement errors. InDefinition 1 of local input-to-state stability, it is only requiredthat V be a Lipschitz continuous function. It is well knowthat a system is input-to-state stable if and only if thereexists an ISS Lyapunov function [28], which is a continuouslydifferentiable function. However, a continuously differentiablefunction is not necessary to establish ISS [29]. In the followingproposition from [15], LISS of system (14) is shown byemploying a nonsmooth Lyapunov function.

Proposition 1 ( [15]). Under the assumptions of Theorem1, if the origin y = 0 of the reduced system (11) is locallyasymptotically stable, then the overall system (14) is locallyinput-to-state stable with respect to the error e = (ey; ev).

The proof involved the construction of a nonsmooth ISSLyapunov function V : Rn → R+

V (y, w) = V1(y) +√w>Pw. (18)

By the converse Lyapunov theorem, the local asymptoticstability of the equilibrium point of the center manifolddynamics implies the existence of a continuously differentiableLyapunov function V1 : Rk → R+ and class-K functionsα4, α5 such that

V1 =∂V1

∂y(A1y + g1(y, h(y) + Ey,K(y;h(y) + Ey)))

≤ −α4(||y||),∣∣∣∣∣∣∣∣∂V1

∂y

∣∣∣∣∣∣∣∣ ≤ α5(||y||) ≤ kv

for some kv > 0, in a neighbourhood of the origin. Also,since the matrix AK is Hurwitz, for every Q � 0 there existsa unique P � 0 such that A>KP + PAK = −Q. Taking the

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time derivative of V along the trajectories of the system (14),it was shown that

ζ>(y; w) ≤ −α4(||y||)− (1− sf )λmin(Q)

2√λmax(P )

||w||︸ ︷︷ ︸−αD(||(y;w)||)

+

(kvk1 + k2

λmax(P )√λmin(P )

+||PB2K1||√λmin(P )

)||e||︸ ︷︷ ︸

βG(||e||)

+ γ(||w||)(19)

where ζ ∈ ∂V (x) and sf ∈ (0, 1). The constantδyw defining the set S in (16) is chosen such thatthe constants k1 and k2 in (17) lead to γ(||w||) =(kvk1 + k2

λmax(P )√λmin(P )

− sfλmin(Q)

2√λmax(P )

)||w|| ≤ 0. The func-

tions αD(||(y;w)||) and βG(||e||) are class-K functions of||(y;w)|| and ||e|| respectively. From Definition 1, (19) impliesthat the origin of system (14) is locally input-to-state stablewith respect to the error e.

Proposition 1 generalizes the Reduction Theorem (Theorem2), as in the absence of the error e, local asymptotic stabilityof the overall system is recovered. As mentioned before, thestability properties of system (14) are the same as that ofsystem (1), in view of the sequence of smooth transformationsrelating the two systems.

Remark 1. The work in [11] consolidated earlier work oncenter manifold theory and presented proofs of Theorems 1and 2 using the contraction mapping principle. In [26], aLyapunov based proof of the Reduction Theorem was presentedfor the first time using a nonsmooth Lyapunov function. Thestructure of the ISS Lyapunov functions chosen in our workis inspired by the nonsmooth Lyapunov function in [26].In [16], where design of control laws in the presence ofcenter manifolds is presented, also uses the same nonsmoothLyapunov function.

In the rest of this work, for ease of notation, we denote by V ,the derivative of V when it exists and ζ>(y; w), ζ ∈ ∂V , whenthe derivative does not exist. When we arrive at inequalities ofthe form V ≤ fr(y, w), it is made sure that such inequalitieshold in both the sets where the function is differentiable andnon-differentiable.

IV. EVENT-TRIGGERED CONTROL

In this section, we use the LISS characterization of nonlin-ear systems with center manifolds presented in the previoussection, to propose event-triggered control implementations.We begin by assessing the applicability of methods that alreadyexist in literature. The LISS characterization

V ≤ −αD(||(y;w)||) + βG(||e||) (20)

where, αD and βG are class-K functions was derived in theprevious section. In [3] and [17], a relative threshold basedevent-triggered control was proposed, where the events aretriggered and the control is updated when the condition

βG(||e||) ≥ σαD(||(y;w)||), σ ∈ (0, 1) (21)

is satisfied at event times ti, i ≥ 0. The input u(t) under thisimplementation evolves as

u(t) =

{K(y(ti); z(ti)) if βG(||e||) < σαD(||(y;w)||)K(y(t); z(t)) if βG(||e||) ≥ σαD(||(y;w)||).

(22)Note that the state x of system (1) is measured, from which(y; z) is obtained through a linear transformation and thecontrol K(y; z) (design of which has been discussed in SectionIII-A) is computed. From (20) and (22), we have

V ≤ −(1− σ)αD(||(y;w)||) < 0, ∀ (y;w) 6= 0

and local asymptotic stability of the origin of system (14) isguaranteed. Although the thresholding condition (21) is easilyand directly derived from the LISS characterization, we seetwo major hurdles on closer observation.

1) The triggering rule (21) can be accurately checked onlywhen w = v−h(y) can be exactly computed. The centermanifold h(y) is determined by solving (10) and thereare systems for which h(y) can be exactly computed(one such system is considered in Example 1). However,for most systems, only an approximation of h(y) can befound.

2) The non-existence of Zeno behaviour must be ensured.This involves showing that the inter-event times ti+1−tiare lower bounded by a positive constant for all i ≥0. Sufficient conditions that rule-out Zeno behaviourhave been proposed in [3], [17] and these conditionsrequire that the comparison functions αD and βG in(20) are such that α−1

D ◦ βG is Lipschitz continuousover compact sets. This assumption on the comparisonfunctions has since been made in [18]–[20], amongmany others. In our case, this assumption holds onlywhen α4 ∈ O(||y||p), p ≤ 1. When α4 ∈ O(||y||p),p > 1, the sufficient conditions from [3], [17] are notsatisfied and no conclusion can be drawn regarding theexistence or non-existence of Zeno behaviour under theimplementation (22).

In this section, we look to overcome these hurdles for the sys-tems under consideration. To begin with, we focus on systemsfor which the center manifold can be exactly computed andlook to overcome the second difficulty by proposing triggeringconditions which are different from (21). Then, we turn ourattention on systems for which the center manifold can onlybe approximately computed and propose modified triggeringconditions.

We begin by considering a class of nonlinear systems withcenter manifolds for which, in the Lyapunov characterization(20), α4 ∈ O(||y||p), p ≤ 1. As mentioned before, for suchsystems, the comparison functions in the LISS characterization(20) satisfy the sufficient conditions from [3], [17] and thetriggering condition (21) can be used for Zeno-free event-triggered implementation. However, the function βG in (20) isdependent on the constants kv , k1 and k2, which characterizethe local region where (20) holds and must be known tocheck the triggering condition (21). The triggering conditionsproposed in this work possess a simpler relative thresholding

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structure and are easier to check than (21). In the followingProposition, we present the triggering condition

||e|| ≥σ||(y;w)||,

0 < σ ≤ λmin(Q)

4||PBK1||

√λmin(P )√λmax(P )

(23)

which is computationally a simpler condition to check than(21).

Proposition 2. Consider the system (8). Under the assump-tions of Theorem 1, if the origin y = 0 of the reduced system(11) is locally asymptotically stable and there exists a LISSLyapunov function such that in (20), α4 ∈ O(||y||p), p ≤ 1,then the origin of the overall system (14) is locally asymptoti-cally stable under the event-triggered implementation with thetriggering condition (23). Moreover, the inter-execution times(ti+1 − ti) are lower bounded by a positive constant for alli ≥ 0.

Proof. Consider the LISS characterization (19) of system (14).

V ≤− α4(||y||)− (1− sf )λmin(Q)

2√λmax(P )

||w||

+

(kvk1 + k2

λmax(P )√λmin(P )

− sfλmin(Q)

2√λmax(P )

)||w||

+

(kvk1 + k2

λmax(P )√λmin(P )

+||PB2K1||√λmin(P )

)||e||.

As α4 ∈ O(||y||p), p ≤ 1, there exists a neighbourhood of theorigin in which α4(||y||) ≥ l1||y|| or −α4(||y||) ≤ −l1||y||,for some l1 > 0. Using the notation mp1 = ||PB2K1||√

λmin(P )and

mp1 =

(kvk1 + k2

λmax(P )√λmin(P )

+ ||PB2K1||√λmin(P )

)and with the

event-triggering mechanism ensuring ||e|| ≤ σ||(y;w)|| ≤σ(||y||+ ||w||), we arrive at

V ≤ −(l1 − mp1σ)||y|| −

(λmin(Q)

4√λmax(P )

−mp1σ

)||w||

+

((1 + σ)(kvk1 +

k2λmax(P )√λmin(P )

)− λmin(Q)

4√λmax(P )

)||w||.

(24)

When σ is chosen as in (23) and δyw defining the set S in(16) is chosen such that the constants k1 and k2 satisfy(

(1 + σ)

(kvk1 + k2

λmax(P )√λmin(P )

)− λmin(Q)

4√λmax(P )

)≤ 0

(l1 − mp1σ) > 0,

we have

V ≤ −(l1 − mp1σ)||y|| − (λmin(Q)

4√λmax(P )

−mp1σ)||w|| < 0,

which implies that local asymptotic stability of the origin(y;w) = 0 of system (14) is guaranteed. The existence of apositive lower bound for the inter-execution times (ti+1 − ti)for all i ≥ 0 can be established as in the proof presentedin [3, Theorem III.1], where the evolution of the fraction

||e(t)||||(y(t);w(t))|| is analysed to find a lower bound τ for thetime taken by the fraction to reach σ from zero. For thetriggering condition (23), τ = σ/(L(1 + σ)) > 0, whereL = min(L1, L2) and the constants L1 and L2 are suchthat ||y|| ≤ L1||(y;w; e)|| ≤ L1||(y;w)|| + L1||e|| and||w|| ≤ L2||(y;w; e)|| ≤ L2||(y;w)|| + L2||e||. Therefore,the inter-execution times are uniformly lower bounded by apositive constant. �

Remark 2. In showing the non-existence of Zeno behaviourfor the triggering condition (21), the proof of [3, TheoremIII.1] analyses an equivalent triggering condition (σαD)−1 ◦βG(||e||) ≥ ||x||. Under the assumption that the function(σαD)−1 ◦ βG is Lipschitz continuous, we have (σαD)−1 ◦βG(||e||) ≤ Pl||e|| ≤ ||x||, where Pl is the Lipschitz constant.The time taken by ||e||/||x|| to reach 1/Pl from zero serves asan under-approximation for the time taken by α−1

D ◦ βG(||e||)to reach ||x||. As the triggering condition in our case is chosento be ||e|| ≥ σ||x|| (with x = (y;w)), the proof of non-existence of Zeno behaviour begins directly with the analysisof the evolution of the fraction ||e||/||(y;w)||.

Next, we consider a class of systems for which, α4 ∈O(||y||p), p > 1 in the LISS characterization (20). Thiscase arises when the function g1 in the reduced system (11)has a polynomial approximation in a neighbourhood of theorigin, that is, ||g1|| ≤ k5||y||p, p > 1 in a neighbourhoodof the origin and V1 in (18) is chosen as ||y|| or ||y||2. Forsimplicity, in the rest of this work, we use g1 to denoteg1(y, h(y) + Ey,K(y;h(y) + Ey)).

Assumption 1. For the system (8), the matrix A1 = 0. In thedynamics of the reduced system (11), the function g1 is suchthat ||g1|| ≤ k5||y||p and y>g1 ≤ −k6||y||p+1, with p > 1for some k5, k6 > 0 in a neighbourhood of the origin. Thecontroller u = K(y; z) = K11z+K12y+Kn(y) = K1(y; v)+Kn(y) considered in subsection III-A is such that Kn(y) = 0and the matrix K1 = [K12 + EK11 K11] = [0 K11].

As the focus of this work is on stabilization of the origin, wedo not focus on systems for which the matrix A1 has purelyimaginary eigenvalues, leading to periodic orbits. Therefore,the matrix A1 is assumed to be a zero matrix. Models andcontrollers of the two examples and the Mobile InvertedPendulum robot considered in this work satisfy the conditionsof Assumption 1. In the case of Examples 1 and 2, thecontroller u = K11z suffices to stabilize the dynamics on thecenter manifold and K12 and E are zero. In the case of theMIP robot, K12, E and K11 are such that K12 + EK11 = 0.The utility of the particular structure of the matrix K1 is inshowing both asymptotic stability of the origin and the non-existence on Zeno behaviour. This restriction on the matrixK1 will be relaxed in Proposition 4.

The proof of the proposition presented next employs anon-smooth Lyapunov function. Specifically, a sum of twofunctions of the form

√x>Px. We now recall a result that

presents the subdifferential of the function√x>Px.

Lemma 1 ( [23, page 181]). Let P ∈ Rn1×n1 be a symmetricpositive definite matrix and P = M>DM be its eigen-

7

decomposition, where M ∈ Rn1×n1 is an orthonormal matrixand D ∈ Rn1×n1 is a diagonal matrix. The subdifferentialof the function fP =

√x>Px : Rn1 → R+ at x = 0 is

∂fP (0) = {g ∈ Rn1 : ||g>MD−12 || ≤ 1}.

The above result will be used in evaluating the subdiffer-ential of a nonsmooth Lyapunov function in the proof of thefollowing proposition.

Proposition 3. Consider the system (8). Under the assump-tions of Theorem 1, if the origin y = 0 of the reduced system(11) is locally asymptotically stable and the conditions inAssumption 1 are satisfied, then the origin of the overall system(14) is locally asymptotically stable under the event-triggeringcondition

||ev|| ≥ σ(||w||+ ||y||(p+1))

0 < σ ≤ (1− sf )λmin(Q)

2||PBK1||

√λmin(P )√λmax(P )

(25)

where sf ∈ (0, 1). Moreover, the inter-execution times (ti+1−ti) are lower bounded by a positive constant for all i ≥ 0.

Proof. Consider the LISS Lyapunov function candidate V :Rn → R+

V = ||y||+√w>Pw. (26)

The function V is continuously differentiable everywhereexcept on the set Ne = {(y;w) ∈ Rn : w = 0 or y = 0}. Thefunctions α1(||(y;w)||) = min(1,

√λmin(P ))||(y;w)|| and

α2(||(y;w)||) =√

2 max(1,√λmax(P ))||(y;w)|| are class-K

functions satisfying

α1(||(y;w)||) ≤ V ((y;w)) ≤ α2(||(y;w)||).

Taking the time derivative of V along the trajectories of thesystem (14) on the set Rn \Ne, we get

V =y>y

||y||+

1

2√w>Pw

(w>Pw + w>Pw

).

By Assumption 1, A1 = 0, the function g1 is such that y>g1 ≤−k6||y||p+1, the control u = K11(v+ev), B2K1e = B2K11evin (14) and the functions N1 and N2 are functions of y, w andev . This leads us to

V ≤ −k6||y||p +y>N1(y, w, ev)

||y||

+1

2√w>Pw

((AKw +B2K11ev +N2(y, w, ev))

>Pw

+w>P (AKw +B2K11ev +N2(y, w, ev)))

≤ −k6||y||p + ||N1(y, w, ev)|| −w>Qw

2√w>Pw

+1√

w>Pw(w>PB2K11ev + w>PN2(y, w, ev))

≤ −k6||y||p −λmin(Q)

2√λmax(P )

||w||+ k1(||w||+ ||ev||)

+||PB2K11||√λmin(P )

||ev||+k2λmax(P )√λmin(P )

(||ev||+ ||w||).

With sf ∈ (0, 1),

V ≤ −k6||y||p − (1− sf )λmin(Q)

2√λmax(P )

||w||

+

(k1 + k2

λmax(P )√λmin(P )

− sfλmin(Q)

2√λmax(P )

)||w||

+

(k1 + k2

λmax(P )√λmin(P )

+||PB2K11||√λmin(P )

)||ev||.

(27)

Using the notation mp2 = ||PB2K11||√λmin(P )

and mp2 =(k1 + k2

λmax(P )√λmin(P )

+ ||PB2K11||√λmin(P )

)and with the relative

event-triggering rule ensuring ||ev|| ≤ σ(||w||+ ||y||p+1), wearrive at

V ≤− k6||y||p + mp2σ||y||p+1 − (1− sf )λmin(Q)

2√λmax(P )

||w||

+mp2σ||w||+

((1 + σ)

(k1 + k2

λmax(P )√λmin(P )

)

−sfλmin(Q)

2√λmax(P )

)||w||.

(28)With sy ∈ (0, 1)

V ≤− k6(1− sy)||y||p + (mp2σ||y||p+1 − k6sy||y||p)

− (1− sf )λmin(Q)

2√λmax(P )

||w||+mp2σ||w||

+

((1 + σ)

(k1 + k2

λmax(P )√λmin(P )

)

−sfλmin(Q)

2√λmax(P )

)||w||.

With σ chosen as in (25), in a small neighbourhood S1 ={(y, w) : ||(y;w)|| < δyw}, the constants k1, k2 from (17) canbe chosen such that(

(1 + σ)

(k1 + k2

λmax(P )√λmin(P )

)

−sfλmin(Q)

2√λmax(P )

)||w|| ≤ 0 (29)

and

(mp2σ||y||p+1 − k6sy||y||p) ≤ 0. (30)

For (y;w) ∈ S1, we have

V ≤ −k6(1− sy)||y||p − (1− sf )λmin(Q)

2√λmax(P )

||w||

= −ws((y;w)) < 0,

(31)

where ws((y;w)) is a positive definite function of (y;w).It can be verified that, on the set Ne where the Lyapunovfunction (26) is non-differentiable, the inequality (31) holds,that is, ζ>(y; w) ≤ −ws((y;w)), for all ζ ∈ ∂V and ∂Vis found through Lemma 1. Although (31) holds in the setS1, S1 is not positively invariant as it is not a sub-level set

8

of V (which is positively invariant). The largest, connectedsub-level set of V contained in S1 is Sv = {(y;w) ∈S1 | V ((y;w)) ≤ α1(δyw)}. By (31), asymptotic stability ofthe origin is guaranteed for all x(0) ∈ Sv .

Next, we prove the existence of a uniform positive lowerbound for the inter-event times, (ti+1−ti), when the system isinitialized in the positively invariant set Sv . The error ev(t) =v(ti)−v(t) = (w(ti)+h(y(ti)))−(w(t)+h(y(t))) = ew(t)+eh(t). Consider

d

dt

(||ev||

||w||+ ||y||p+1

)

=e>v ev

(||w||+ ||y||p+1) ||ev||−

(w>w||w|| + (p+1)||y||py>y

||y||

)||ev||

(||w||+ ||y||p+1)2

≤ ||ev||(||w||+ ||y||p+1)

+(||w||+ (p+ 1)||y||p||y||) ||ev||

(||w||+ ||y||p+1)2

≤ ||ew||+ ||eh||(||w||+ ||y||p+1)

+(||w||+ (p+ 1)||y||p||y||) ||ev||

(||w||+ ||y||p+1)2 .

(32)

From Assumption 1, ||g1|| ≤ k5||y||p for some k5 > 0.As ||h(y)|| ∈ O(||y||2) and ||∂h(y)

∂y || ∈ O(||y||), there existconstants k7 > 0 and k8 > 0 such that ||h(y)|| ≤ k7||y||2 and||∂h(y)

∂y || ≤ k8||y|| in a small neighbourhood of y = 0. Fromequations (14) and (17), we have ||y|| ≤ k5||y||p + k1||w||+k1||ev||, ||w|| ≤ (||Ac||+k2)||w||+(||B2K11||+k2)||ev|| and||eh|| = ||h(y)|| ≤ ||∂h∂y ||||y|| ≤ k8k5||y||p+1 +k8k1δyw||w||+δywk1k8||ev|| (||y|| ≤ δyw has been used as (y;w) ∈ S1).Consider the numerator of the first term in the right-hand sideof (32). Using the inequalities derived so far, we have

||ew||+ ||eh|| ≤ a1

(||w||+ ||y||p+1

)+ a2||ev|| (33)

where, a1 = max(||Ac|| + k2 + k8k1δyw, k8k5) and a2 =||B2K11|| + k2 + δywk1k8. For the numerator of the secondterm in (32), we have

||w||+ (p+ 1)||y||p||y||≤ a3

(||w||+ ||y||p+1

)||e||+ a4||ev||2

(34)

where a3 = max(||Ac|| + k2 + (p + 1)δpywk1, δ(p−1)yw k5) and

a4 = ||B2K11||+ k2 + (p+ 1)δpywk1. From (33) and (34), wehave

d

dt

(||ev||

||w||+ ||y||p+1

)≤ a1 +

((a2 + a3)||ev||||w||+ ||y||p+1

)+ a4

(||ev||

||w||+ ||y||p+1

)2

.

(35)Denoting ||ev||/(||w|| + ||y||p+1) by es, we have es ≤ a1 +(a2 + a3)es + a4e

2s. Using the Comparison Lemma [26], it

follows that es(t) ≤ φ(t), where φ(t) is the solution of φ =a1 + (a2 + a3)φ + a4φ

2, initialized at φ(0) = 0. When anevent occurs (when es rises from zero to meet σ), the controlis updated and es(t) is reset to zero. Let τ1 be the time taken byφ(t) to evolve from 0 to σ. As es(t) ≤ φ(t), the time taken byes(t) to reach σ is greater than τ1. By the comparison lemma,

es(t) ≤ φ(t) = b tan

(b

2(t+ c)

)− (a2 + a3)/(2a4).

where b =√

4a1a4 − (a2 + a3)2 and c = (2/b) tan−1((a2 +a3)/b). φ(τ1) = σ implies

τ1 =2

b

(tan−1

(2a4σ + (a2 + a3)

b

)− tan−1

(a2 + a3

b

))> 0.

(36)Thus, for all initializations (y(0);w(0)) ∈ Sv , there exists auniform positive lower bound for the inter-event times. �

Remark 3. A triggering condition of the form ||ev|| ≥σ(||w||+||y||(p+pe)), pe ≥ 1 can also be used to arrive at (31)to guarantee local asymptotic stability of the origin of (14).Proposition 3 presents the case with pe = 1, which leads to thelargest domain of attraction among this family of triggeringconditions, as (30) holds in the largest set when pe = 1. Aspe → ∞, the triggering condition tends to ||ev|| ≤ σ||w||and Zeno behaviour is observed in simulations using thistriggering condition.

Next, we recall an important qualitative property of thetrajectories of nonlinear systems with center manifolds, underthe assumptions of Theorem 1. This property, captured inthe following lemma, is used in analysing the asymptoticbehaviour of the inter-event times.

Lemma 2 (Exponential convergence of trajectories to thecenter manifold [11]). Consider the system (8) under theassumptions of Theorem 1. Let (y(t); v(t)) be a solution to (8)with ||(y(0); v(0))|| sufficiently small. Then, there exist positiveconstants C1 and µ such that

||v(t)− h(y(t))|| ≤ C1e−µt||v(0)− h(y(0))|| (37)

for all t ≥ 0.

Note that ||v(t) − h(y(t))|| = ||w(t)||, is the euclideandistance between the state at time t and the center manifold.Lemma 2 states that this distance decays exponentially to zero.The state y(t) on the other hand decays to the origin slowly(non-exponentially), governed by the dynamics (11) (if thedynamics (11) is asymptotically stable).

A. Asymptotic behaviour of the inter-event timesBy Lemma 2, we have

limw(t)→0

||ev||||w||+ ||y||p+1

exponentially−−−−−−−−−−→

||eh||||y||p+1

.

as ||ev|| = ||ew + eh||. The behaviour of the inter-eventtimes with the thresholding rule ||e|| ≥ σ(||w|| + ||y||(p+1))exponentially tends to the behaviour of the inter-event timeswith the thresholding rule ||eh|| ≥ σ||y||(p+1). Consider

d

dt

(||eh||||y||p+1

)=

−e>h h(y)

||eh|| ||y||p+1− (p+ 1) ||y||p (y>y)

(||y||p+1)2

||eh||||y||

≤||∂h(y)

∂y || ||y||||y||p+1

+(p+ 1) ||y||p ||y|| ||eh||

(||y||p+1)2.

Using the inequalities leading up to (33) and (34), but withw = 0, we arrive at

d

dt

(||eh||||y||p+1

)≤ b1 + (b2 + b3)

||eh||||y||p+1

+ b4

(||eh||||y||p+1

)2

9

where b1 = k8k5, b2 = δywk1k8, b3 = (p + 1)δp−1yw k5 and

b4 = (p+ 1)δpywk1. Following along the steps leading to (36),we arrive at an estimate τ2 with the same form as (36) butwith ai replaced by bi. The estimate for the uniform lowerbound on the inter-event times using bi is less conservative,as the inequalities leading up to (33) and (34) are now lessconservative due to Lemma 2.

B. Triggering rule guaranteeing local ultimate boundednessof the trajectories of the closed-loop system

In Assumption 1, the restriction on the matrix K1 wasneeded to ensure both asymptotic stability and non-existenceof Zeno behaviour. We now relax this assumption to considera more general class of systems and show that under theimplementation rule

u(t) =

{0 if t1 6= t0, ∀ t ∈ [t0, t1)

K(y(ti); z(ti)) if t ∈ [ti, ti+1), i ≥ 1

t1 = min{t ≥ t0 | (y(t);w(t)) ∈ Sv \ S2}ti+1 = min{t ≥ ti | ||e|| ≥ σ(||w||+ ||y||(p+1))

and (y(t);w(t)) ∈ Sv \ S2}, i ≥ 1

(38)

where

S2 = {(y;w) | ||(y;w)|| < α−12 α1(rs) = r1}, (39)

the trajectories of the closed-loop system are ultimatelybounded by a ball of desired radius rs. If (y(0);w(0)) ∈Sv \ S2, the first event instant t1 = t0 and the second casedefining u(t) is active for all t ≥ t0. If (y(0);w(0)) ∈ S2, thent1 6= t0 and the first case defining u(t) is active for t ∈ [t0, t1)before the second case takes over for all t ≥ t1.

The following Assumption and Proposition capture thisscenario. For simplicity, we use the notation Br to denote aball of radius r.

Assumption 2. For the system (8), the matrix A1 = 0. Thefunction g1 in the dynamics of the reduced system (11) is suchthat ||g1|| ≤ k5||y||p and y>g1 ≤ −k6||y||p+1, with p > 1 forsome k5, k6 > 0 in a neighbourhood of the origin.

Proposition 4. Consider the system (8). Under the assump-tions of Theorem 1, if the origin y = 0 of the reduced system(11) is locally asymptotically stable, the conditions in Assump-tion 2 are satisfied and Brs ⊂ Sv , then the trajectories of thesystem (14) are locally ultimately bounded by Brs , under theevent-triggered implementation (38), with σ chosen accordingto (25). Moreover, the inter-execution times (ti+1 − ti) arelower bounded by a positive constant for all i ≥ 0.

Proof. From (31), (39) and the implementation rule (38), weare lead to

V ≤ −ws((y;w)), ∀ ||(y;w)|| ≥ α−12 α1(rs).

By [26, Theorem 4.18], we conclude that the trajectories of theclosed-loop system (8)-(38) are ultimately bounded by Brs ⊂Sv and this ball is reached in finite time.

Next we show that the inter-event times (ti+1 − ti) areuniformly lower bounded by a positive constant for all

(y(t0);w(t0)) ∈ Sv . Under the implementation rule (38),events are triggered only in Sv \ S2 and the control is notupdated when the trajectory enters S2. When the system isinitialized in S2, the control is set to zero until the systemleaves S2. For each i, ||(y(ti);w(ti))|| ≥ r1 and e(ti) = 0.The next event occurs at ti+1, when ||e|| rises from zero andmeets σ(||w||+ ||y||p+1) in Sv \ S2. In Sv \ S2

σl , σ(r1 + rp+11 ) ≤ σ(||w||+ ||y||p+1)

≤ σ(δyw + δp+1yw ) , σu.

Consider the evolution of ||e|| along the trajectories of theevent-triggered closed-loop system, between two consecutiveevent instants

d||e||dt≤ ||e|| = || − (y; w)||.

The function f(x, u) in (1) is twice continuously differentiableand due to the sequence of smooth coordinate transformationsrelating x and (y;w), there exists a constant L1 in S1 suchthat ||(y; w)|| ≤ L1||(y;w; e)|| ≤ L1||(y;w)|| + L1||e|| ≤L1δyw + L1||e||. Using the notation δ , L1δyw,

d||e||dt≤ || − (y; w)|| ≤ L1δyw + L1||e|| = L1||e||+ δ.

Using the Comparison Lemma [26] with e(0) = 0, we have

||e(t)|| ≤ δ

L1(eL1t − 1). (40)

The time taken by the right-hand side of (40) to rise from 0to σl serves as a lower bound for the inter-event times. Thelower bound τ3 is found by solving

δ

L1(eL1τ3 − 1) = σl =⇒ τ3 =

1

L1ln

(1 +

σlL1

δ

)> 0.

(41)

Thus, we have shown that the inter-event times are uniformlylower bounded by τ3 > 0. �

The radius rs of the ultimate bound can be made arbitrarilysmall, but by allowing for small estimates of inter-event times,as τ3 in (41) is a function of σl which grows smaller as S2

becomes smaller.Next, we take up an example where the center manifold

is exactly computable and demonstrate the application of thetriggering condition presented in Proposition 3.

C. Example 1

Consider the system

y = −yvv = v + u+ y2 − 2v2.

(42)

The nonlinear function g1(y, v, u) = −yv is independent ofthe control input u. The control u determines the structureof the center manifold and its stability indirectly via the v-subsystem. With the controller u = −2v, the closed-loopsystem is

y = −yvv = −v + y2 − 2v2.

10

The center manifold of this system can be computed exactly[30]. By solving the PDE (10), the center manifold is foundto be v = y2. The dynamics on the center manifold is

y = g1(y, h(y)) = −y3. (43)

We have ||g1|| = |y|3 and y>g1 ≤ −|y|4 with k5, k6 = 1.Therefore, Assumption 1 is satisfied. The local asymptoticstability of y = 0 can be shown using the Lyapunov functionV1 = 1

2y2. With the change of variables w = v−y2, we obtain

y = −y(w + y2)

w = −w − 2w(w + y2).(44)

Introducing the measurement error ev = v(ti)−v(t), we obtain

y = −y3 − yww = −w − 2ev − w(w + y2).

(45)

Now N1 = −yw and N2 = −w(w + y2). The functions N1

and N2 satisfy conditions (15). As N1 and N2 are independentof u and therefore ev , the right-hand sides of the inequalities(17) are functions of only w. In the set S1 = {(y, w) | ||y|| ≤1√6

and ||w|| ≤ 16}, we have ||N1|| ≤ 1√

6||w|| and ||N2|| ≤

16 ||w|| with k1 = 1√

6and k2 = 1

6 . Using the LISS Lyapunov

function V = |y| +√w>w, with sf = 3

4 and sy = 12 and

the triggering condition ||ev|| ≥ σ(||w|| + ||y||4), σ = 1/16(according to (25)), we have

V ≤ −1

2|y|3 − 1

16||w||

and through Proposition 3, local asymptotic stability of theorigin of (45) is guaranteed with Zeno-free event-triggering.

In Figure 1, simulation results of the event-triggered closed-loop system are presented. Trajectories from three initialconditions are plotted in Figure 1a along with the centermanifold v = y2. The trajectories tend to the center manifoldexponentially and the evolution along the center manifold issignificantly slower in comparison. The mechanism of event-triggering is shown in Figure 1b, by plotting the evolution ofthe error ||ev|| and the threshold 1

16 (||w||+ ||y||4).The evolution of inter-event times for three initial conditions

are shown in Figure 1c. The inter-event times are lowerbounded by 30.3 ms. As h(y) has been determined exactly,the function g1(y, h(y) + Ey,K(y;h(y) + Ey)) is known inclosed-form and good estimates of the constants k5, k6, k7, k8

and the neighbourhoods in which they are valid can be found.However, the estimates of minimum inter-event times from(36) and (41) are conservative, as they depend on Lipschitzconstants and bounds on polynomial functions. To get a betterestimate, the event-triggered closed-loop system was simulatedfor ten initial conditions

(0.1 cos( 2πk

10 ), 0.1 sin( 2πk10 )), k =

0, 1, 2, . . . , 9 for 25 s. The trajectories practically convergeto the center manifold by 15 s. The mean inter-event timebefore 15 s is found to be 40.8 ms and between 15 s and25 s is found to be 33 ms. The minimum inter-event timein these simulations (MIETs) was found to be 30.3 ms andthe average inter-event time was found to be 36.3 ms. Toassess the performance of event-triggered control with respectto time-triggered control, we choose MIETs as the sampling

0 0.05 0.1 0.15

0

0.5

1

1.5

210

-1

(a) Phase portrait of the event-triggered closed-loop system

0 5 10 15 20 25

0

5

1010

-3

2.6 2.8 3

0

0.5

110

-3

(b) Evolution of the error and thethreshold

0 5 10 15 20 25

3

4

5

6

7

10-2

(c) Inter-event times

0 5 10 15 20 25

0.6

0.8

1

1.2

1.4

1.6

10-1

20 23 25

6

6.05

6.1

10-2

(d) Performance of time-triggeredand event-triggered control

Fig. 1: Simulation results for event-triggered implementationof a controller designed for system (42), with the triggeringrule ||ev|| ≥ σ(||w||+ y4) for σ = 1

16 .

time for time-triggered control. From Figure 1d, we see thatthe performances of time-triggered and event-triggered controlare a close match. However, the number of control updates ismuch higher in time-triggered control, thus making a case forthe use of event-triggered control.

The triggering conditions in Propositions 2-4 can be accu-rately checked only when the variable w = v−h(y) is exactlycomputed. For many systems, h(y) can only be found uptoan approximation by solving (10). This brings into questionthe utility of the change of coordinates from (y; v) to (y;w)and the following questions arise naturally - 1) Can triggeringconditions be designed without resorting to the change ofvariables, directly in the (y; v) coordinates? 2) Can we use theavailable partial knowledge of the center manifold to designtriggering conditions? In the next two sections, we take arelook at the problem formulation, the assumptions and theproofs presented so far and address the questions posed above.

V. NEED FOR THE CHANGE OF VARIABLES FROM v TO w

The brief discussion following Lemma 2 in Section IVon the nature of the trajectories of nonlinear systems withcenter manifolds suggested that (y;w) coordinates is theappropriate coordinate system to understand the behaviourof nonlinear systems with center manifolds. The change ofvariables from v to w = v − h(y) results in the functions Nisatisfying conditions (15), which are crucial for the proofsof Propositions 1-4. On proceeding without the change ofvariables from (8), we see that the functions

N1 = g1(y, v + Ey,K(y + ey; v + ev + E(y + ey))

− g1(y, h(y) + Ey,K(y;h(y) + Ey))

11

N2 = g2(y, v + Ey,K(y + ey; v + ev + E(y + ev)))

satisfy

Ni(0, 0, 0) = 0,∂Ni∂y

(0, 0, 0) = 0, (46)

∂Ni∂v

(0, 0, 0) = 0 and∂Ni∂e

(0, 0, 0) = 0. (47)

In a small neighborhood of the origin

S3 = {(y; v) | ||(y; v)|| < δyv} (48)

we have for i = 1, 2,

||Ni|| ≤ ki||(y; v; e)|| ≤ ki(||y||+ ||v||+ ||e||) (49)

where ki are positive constants which can be made arbitrarilysmall by decreasing δyv . The difference between (49) andinequalities (17) is the presence of the term ||y||. The absenceof this term in (17) was crucial for all the proofs presentedso far. Next, we analyse the local input-to-state stability ofsystem (12) without the change of coordinates, by choosingthe LISS Lyapunov function V4 = V1 +

√v>Pv (which is

(18) with w replaced by v). Taking the time derivative of V4

along the trajectories of system (12), we have

V ≤− (1− sy)α4(||y||)− (1− sf )λmin(Q)

2√λmax(P )

||v||

+

(kvk1 + k2

λmax(P )√λmin(P )

+||PB2K1||√λmin(P )

)||e||

+

((kvk1 + k2

λmax(P )√λmin(P )

)||y|| − syα3(||y||)

)

+

(kvk1 + k2

λmax(P )√λmin(P )

− sfλmin(Q)

2√λmax(P )

)||v||

(50)

where sy, sf ∈ (0, 1). When δyv defining the set S3 is chosensuch that k1 and k2 in (49) lead to((

kvk1 + k2λmax(P )√λmin(P )

)||y|| − syα4(||y||)

)≤ 0 (51)

and(kvk1 + k2

λmax(P )√λmin(P )

− sfλmin(Q)

2√λmax(P )

)≤ 0, (52)

we obtain

V ≤− (1− sy)α3(||y||)− (1− sf )λmin(Q)

2√λmax(P )

||v||

+

(kvk1 + k2

λmax(P )√λmin(P )

+||PB2K1||√λmin(P )

)||e||.

(53)

By making S3 smaller, the constants k1 and k2 can be chosensuch that (52) is satisfied. Such a choice for the constants alsoleads to (51) only if α4(||y||) ∈ O(||y||p) with p ≤ 1. In thiscase, the origin of system (8) is locally input-to-state stableby Definition 1. If p > 1, (51) is not satisfied close to the

origin for any choice of k1 and k2 and (53) holds only in theset S3 \ S4 where

S4 = {y ∈ Rk | ((kvk1+k2λmax(P )√λmin(P )

)||y||

− syα4(||y||)) ≥ 0}.(54)

Therefore, LISS of only the set S4 can be concluded in thiscase. With the chosen LISS Lyapunov function in the (y, v)coordinates, LISS of the origin of system (8) cannot be shownfor all comparison function α4 given by the converse Lya-punov theorem. The function α4 guaranteed by the converseLyapunov theorem (under the assumption that the reducedsystem is locally stable) can be any class-K function and mayor may not belong to O(||y||p), p ≤ 1. The change of variablesfrom v to w helps in establishing LISS of the origin for allcomparison functions α4.

The triggering conditions in Propositions 2-4 can be usedby replacing w by v, but Zeno-free local asymptotic stabilityof the origin is guaranteed when α4 ∈ O(||y||p), p ≤ 1 andonly Zeno-free local asymptotic stability to the set (54) canbe guaranteed when α4 ∈ O(||y||p), p > 1.

The insights gained in the above discussion and in thediscussion presented in the next section are formalized inPropositions 5-7 presented in section VI. Next, we considerthe case of systems for which only approximate knowledge ofthe center manifold is available.

VI. EVENT-TRIGGERED CONTROL WITH APPROXIMATEKNOWLEDGE OF THE CENTER MANIFOLD

In this section, we analyse the applicability of Propositions3-4 and the satisfaction of Assumptions 1, to see if triggeringrules can be designed that require only partial knowledge ofthe center manifold. We begin with an illustrative example thatbrings out the differences in local stability analysis of systemsfor which only an approximation of the center manifold can befound and those for which the center manifold can be exactlyfound (Example 1).

A. Example 2

Consider the system

y = −y(z1 − 4z2)[z1

z2

]=

[0 1−2 3

] [z1

z2

]+

[01

]u+

[y2

0

].

(55)

A linear state-feedback controller u = [1 − 4]z is usedto place the poles of the linear part of the z-subsystem at−0.5± j0.0866. The closed-loop system is

y = −y(z1 − 4z2)[z1

z2

]=

[0 1−1 −1

] [z1

z2

]+

[y2

0

]. (56)

The one-dimensional center manifold satisfying (10), foundaccurately up to order two, is given by[

z1

z2

]=

[h1(y)h2(y)

]=

[y2 +O(|y|4)−y2 +O(|y|4)

].

12

The dynamics on the center manifold is governed by

y = g1(y, h(y)) = −5y3 +O(|y|5). (57)

Consider the nonsmooth Lyapunov function V1(y) = |y|. Onthe set R \ {0},

V1 =yy

|y|= −5|y|3 +O(|y|5).

In a small neighbourhood of the origin

V1 ≤ −5(1− θ)|y|3 − 5θ|y|3 + kp|y|5

≤ −5(1− θ)|y|3 < 0 ∀ |y|2 ≤ 5θ

kp

(58)

for some kp > 0 and local asymptotic stability of (57) is con-cluded. The partial knowledge of the center manifold sufficesfor the local stability analysis of this system. Through (57) and(58), we see that Assumption 1 is satisfied. However, sinceg1(y, h(y)) in (57) cannot be obtained in closed form, onlycrude estimates of the neighbourhoods where the conditions inAssumption 1 hold can be found. Similarly, good estimates ofthe neighborhoods where (15) are satisfied cannot be obtained.These estimates however, improve as the center manifold iscomputed to higher powers.

Remark 4. If establishing the local stability of the equilibriumpoint is of sole interest, approximate knowledge of the centermanifold suffices. In practice, as in [13], [14], only the localanalysis is performed and it is argued that the region ofattraction is much larger than the estimates derived as above.

Next, we analyse the consequences of using the triggeringconditions of the form ||e|| ≥ σ(||wa|| + ||y||) and ||ev|| ≥σ(||wa|| + ||y||(p+1)), which possess the same structure asthe triggering conditions in Propositions 2 and 3 respectively,but with w = v − h(y) replaced by the approximation wa =v−ha(y). Here, ha(y) is a polynomial approximation of h(y)of degree r, found by solving (10) and they are related byh(y) = ha(y) +O(||y||s) with s > r.

Consider (27) in the proof of Proposition 3. With the pro-posed event-triggering condition, ||ev|| ≤ σ(||wa||+||y||(p+1))is ensured throughout the implementation. Using the inequality||wa|| = ||w+ (h(y)−ha(y))|| ≤ ||w||+ ||(h(y)−ha(y))|| ≤||w||+O(||y||s), we arrive at

V ≤− k6(1− sy)||y||p − (1− sf )λmin(Q)

2√λmax(P )

||w||

+O(||y||s).(59)

The difference between (59) and (31) in the proof of Proposi-tion 3 is the presence of O(||y||s). Close to the origin, the sumof the first and third term, which can be any polynomial inO(||y||s), is less than zero. Therefore V < 0 close to the originand local asymptotic stability of the origin is guaranteed. Thenon-existence of Zeno behaviour can be shown along the samelines as in the proof of Proposition 3. Similar argument holdswhen using the triggering conditions presented in Propositions2 and 4 with w replaced with wa.

The insights gained above and in Section V are formalisedin the propositions to follow. The proofs of the Propositionsare omitted, as they follow along the same lines as the proofs

of Propositions 2-4 and differ as discussed in the present andthe previous section.

Proposition 5. Consider the system (8). Under the assump-tions of Theorem 1, if the origin y = 0 of the reducedsystem (11) is locally asymptotically stable and there existsa LISS Lyapunov function such that in (20), α4 ∈ O(||y||p),p ≤ 1, then the origin of the overall system (14) is locallyasymptotically stable under event-triggered implementationswith relative triggering rules ||e|| ≥ σ(||y|| + ||v||) and||e|| ≥ σ(||y|| + ||wa||), with σ chosen as in (25). Moreover,the inter-execution times (ti+1 − ti) are lower bounded by apositive constant for all i ≥ 0.

The next proposition caters to the class of systems for whichAssumption 1 is satisfied.

Proposition 6. Consider the system (8). Under the assump-tions of Theorem 1, if the origin y = 0 of the reduced system(11) is locally asymptotically stable and the conditions inAssumption 1 are satisfied, then

1) If sets S3 in (48) and S4 in (54) are such that S4 ⊂ S3,then the set S4 containing the origin of the overallsystem (14) is locally asymptotically stable under theevent-triggered implementation with relative threshold-ing ||ev|| ≥ σ(||v|| + ||y||(p+1)), with σ chosen as in(25).

2) The origin of the overall system (14) is locally asymp-totically stable under the event-triggered implementationwith relative thresholding ||ev|| ≥ σ(||wa|| + ||y||p+1),with σ chosen as in (25).

Moreover, in both the cases above, the inter-execution times(ti+1 − ti) are lower bounded by a positive constant for alli ≥ 0.

The event-triggered implementation in case 1) of Proposi-tion 6 must be as in (38) with no triggering in the interior of theset S4 to ensure Zeno-free implementation. Unlike Proposition4, the set for which asymptotic stability can be guaranteedcannot be made arbitrarily small.

Proposition 7. Consider the system (8). Under the assump-tions of Theorem 1, if the origin y = 0 of the reducedsystem (11) is locally asymptotically stable and the condi-tions in Assumption 2 are satisfied, then the trajectories ofthe system (14) are locally ultimately bounded by Brs ⊂Sv (a sub-level set of (26) where (59) holds), under theevent-triggered implementation (38) with relative thresholding||e|| ≥ σ(||wa|| + ||y||(p+1)) and σ chosen according to(25). Moreover, the inter-execution times (ti+1− ti) are lowerbounded by a positive constant for all i ≥ 0.

Propositions 5-7 are analogues of the Propositions 2-4 fromSection IV. Next, we implement the controller designed forExample 2 using the relative thresholding rule from Proposi-tion 6.

B. Example 2 revisited

The center manifold h(y) was found up to order two andha1(y) = [y2 −y2]>. The Lyapunov function from Proposition

13

0 50 100

0

2

4

10-2

81.32 81.36

25.66

25.6810

-3

(a) Evolution of the states of theclosed-loop system

0 0.01 0.02 0.03 0.04

0

5

10

1510

-3

0.032 0.034 0.036

1

1.5

10-3

(b) Evolution of the system in they − z1 plane

0 0.01 0.02 0.03 0.04

-10

-5

0

5

1010

-3

0.03 0.035

-1.5

-1

-0.510

-3

(c) Evolution of the system in they − z2 plane

0 50 100

-1

0

1

2

0 5

0.02

0.04

0.06

(d) Inter-event times

Fig. 2: Simulation results for the event-triggered implemen-tation of a controller designed for system (55) with thethresholding rule ||ez|| ≥ 0.03(y4 + ||z − ha1(y)||), σ = 0.03.

3 is of the form V = |y| +√w>Pw, where P � 0 is

found by solving the Lyapunov equation for Q = I2. Withthe thresholding rule from Proposition 6, with sf = 0.5, wearrive at

V ≤ −|y|3 − λmin(Q)

4√λmax(P )

||w||+O(|y|)4.

Local asymptotic stability of the origin and the non-existenceof Zeno behaviour is guaranteed through Proposition 6. Simu-lation results using the triggering condition ||ez|| ≥ 0.03(y4 +||z − ha1(y)||) with σ = 0.03 chosen to satisfy (25) arepresented in Figure 2. As the transformation v = z−Ey is notnecessary for system (56), the variable z appears in the trigger-ing condition. The system is initialized at (0.04, 0.01, 0.01).Exponential convergence of the states z1 and z2 to the centermanifold can be seen in Figures 2b and 2c. As in Example 1,minimum inter-event time (MIETs) from ten initial conditionswas found to be 11.9 ms. The mean time between two eventswas found to be 541.3 ms. In Figure 2d, we see that the inter-event times are lower bounded by MIETs. With MIETs as thesampling time, time-triggered implementation was performedand in Figure 2a we see that the performance of the event-triggered system is a close match to that of the time-triggeredsystem. On an average, the control is updated every 541.3 msin event-triggered control and every 11.9 ms in time-triggeredcontrol, which makes a case in favour of event-triggeredcontrol. From Figures 2b and 2c, we see that the trajectoriesof the system show oscillations about the center manifold, asthey tend to the origin along the center manifold. When theexact knowledge was available and was employed in checkingthe triggering condition in Example 1, the trajectories showed

no oscillations along the center manifold.Next, we analyse the effects of using better approximations

of the center manifold in the triggering condition. Solving thePDE (10) for h(y) upto orders four and six we get

ha2(y) =

[y2

−y2 − 10y4

]ha3(y) =

[y2 − 200y6

−y2 − 10y4 − 80y6

].

Using the two approximations, simulations were performedfor two sets of initial conditions, P1 and P2. The initialconditions in P1 are close to the origin and those in P2 arerelatively farther. The initial conditions (0.04, 0.01, 0.01) and(0.4, 0.1, 0.1) are sample initial conditions from sets P1 andP2 respectively. As the approximation for h(y) improves, itis found that, for initial conditions in both P1 and P2, thereis no discernible difference in the system performance andthe qualitative properties of trajectories. For initial conditionsin P1, no change in the triggering behaviour is observed butfor initial conditions in P2, the mean inter-event times usingha1(y), ha2(y) and ha3(y) are found to be 0.2021 s, 0.2018 sand 0.2014 s respectively. This indicates that use of a betterapproximation leads to more frequent triggers, as the triggeringcondition is checked in a more refined manner.

Next, we use the triggering condition from Proposition 6for the event-triggered implementation of a position stabilizingcontroller for the Mobile Inverted Pendulum (MIP) robot.

C. Example 3 : Position stabilization of MIP robotThe MIP robot is a four degree-of-freedom robot with

two independently driven wheels and a pendulum-like centralbody that has unstable pitching motion under the influence ofgravity. A controller that asymptotically drives the robot tothe origin of the (x, y) plane, while maintaining an uprightposition, is considered here for event-triggered implementa-tion. There is no specification on the final orientation θ ofthe robot at the origin. This is called position and reducedattitude stabilization [13]. For simplicity, we refer to this taskas position stabilization.

1) Mathematical model: The model of the robotfrom [13] and [31] is considered, with the statesx = (x1, x2, x3, x4, x5, x6) = (tanh(x sin θ −y cos θ), tanh(x cos θ + y sin θ), α, α, v, θ)> ∈ Qc ,(−1, 1)2 × S1 × R3, where S1 denotes the unit circle, (x, y)represents the position of the robot in the plane, θ is theorientation of the robot at this position, α and α are the pitchand pitch velocity, v and θ = ω are the linear and angularvelocities of the robot. The state-space model of the MIProbot presented in [13], [31] is

x = f1(x) + h1(x)u1 + h2(x)u2

f1(x) =

(1− x21)(tanh−1 x2)x6

(1− x22)(−(tanh−1 x1)x6 + x5)

x4(sin (2x3)(b3b1x

26 − b25x2

4) + 2b5b3ag sinx3

)m1(x)(

sin (2x3)(−b1b5x25 cosx3 − b25ag) + 2b2b5x

24 sinx3

)m1(x)

−b1x4x6 sin (2x3)m2(x)

14

h1(x) = (0, 0, 0,−2(b3r + b5 cosx3)

m1(x),

2(b2 + b5r cosx3)

m1(x), 0)

h2(x) =

(0, 0, 0, 0, 0,

b

m2(x)r

),

m1(x) = 2(b2b3 − b25 cos2 x3) and m2(x) = (b4 + b1 sin2 x3)

where m1(x),m2(x) > 0,∀ x ∈ Qc, and b, r, bi, i = 1, ..., 5are constants dependent on the robot parameters. The controlinputs are u = (u1, u2) = (τr + τl, τr − τl), where (τr, τl) arethe torques applied to the right and left wheels respectivelyand ag is the acceleration due to gravity. Linearizing about(x;u) = (0; 0) yields

x = Ax+

[0B1

]u1 +

[0B2

]u2 + G(x, u) (60)

where

A =

[0 01×5

05×1 A1

], A1 =

0 0 0 1 00 0 1 0 00 a1 0 0 00 a2 0 0 00 0 0 0 0

,B1 =

[0 0 a3 a4 0

]>, B2 =

[0 0 0 0 a5

]>and G are the nonlinearities satisfying conditions in (3). Thecontrollability matrix of the linearized system (60) has rankfive, with x1 being the uncontrollable state.

2) Choice of a controller: A linear state-feedback controllaw with the following structure

u1 = −K1[x2 x3 x4 x5]>, u2 = −K2[x6 x1]> (61)

where, K1 = [k1i]i=1,...,4 ∈ R4,K2 = [k2j ]j=1,2 ∈ R2 wasproposed in [13] to achieve the control objective of reducedattitude stabilization. For ease of notation, the controller isrepresented as u = K(x). The controller parameters are cho-sen so as to assign negative real eigenvalues to the fifth-ordercontrollable subsystem of (60). The eigenvalue correspondingto the x1 dynamics is zero and cannot be influenced by thecontroller (61). The state x1 is used in the control u2 toindirectly influence and stabilize the dynamics on the centermanifold.

3) Center manifold analysis of the closed-loop system:Using the partitioning of the states p = (p1, p2) ∈ Qc, p1 ,x1, p2 , (x2, x3, x4, x5, x6), we arrive at

p1 = Acp1 +G1(p1, p2)

p2 = Asp2 +B2k21p1 +G2(p1, p2,K(p1, p2))(62)

where, As ∈ R5×5 is Hurwitz, Ac = 0, G1 ∈ R5 and G2 ∈ R.As G1 is independent of u, the control directly influences onlythe p1-subsystem.

As discussed in the problem formulation in Section III, thecross-coupling linear term, B2k21p1 between the p1 and p2

subsystems must be eliminated for the results from centermanifold theory to hold. This term is removed using thechange of variables p2 = p2 − Ep1, E ∈ R5×1, where thematrix E is obtained by solving (7) and the matrix E isE = [0 0 0 0 − k22

k21]>. Using the notations

g1(p1, p2 + Ep1)) = G1(p1, p2 + Ep1))

g2(p1, p2 + Ep1,K(p1; p2 + Ep1))

=G2(p1, p2 + Ep1,K(p1; p2 + Ep1))

− EG1(p1, p2 + Ep1,K(p1; p2 + Ep1))

we arrive at

p1 = Acp1 + g1(p1, p2 + Ep1)

˙p2 = Asp2 + g2(p1, p2 + Ep1,K(p1; p2 + Ep1))(63)

where p2 , (x2, x3, x4, x5, x6 + (k22/k21)x1). By The-orem 1, there exists locally, a smooth map h :R → R5 such that p2 = h(p1) is a center man-ifold for the system (63). The choice of h(p1) =(c1p

21+O(|p1|4),O(|p1|4),O(|p1|4),−c2p2

1+O(|p1|4), c3p31+

O(|p1|4)) (derived in [13]), where

c1 =k14k22

k11k21, c2 =

k22

k21, c3 = −b4k14k

322r

k11k421b

(64)

qualifies as a local center manifold, since it satisfies (10) in asmall neighbourhood of the origin (p1; p2) = 0. The dynamicson the one-dimensional center manifold is governed by

p1 = (c3p31 − c2p1)(1− p2

1) tanh−1(c21p21) +O(|p1|6). (65)

The local asymptotic stability of p1 = 0 of (65) can be inferredusing the Lyapunov function V2 = 1

2p21, which yields

V2 = −c2c21p41 + k|p6

1| < 0 ∀ |p1| < c1

√c2k

(66)

where the constant k > 0. By Theorem 2 and Proposition1, local asymptotic stability and local input-to-state stabilityof the overall closed-loop system (60)-(61) respectively isconcluded.

4) Event-triggered implementation: The controller (61) thatplaces the closed-loop poles of the fifth order controllablesubsystem of (60) at {−2± j0.5,−0.9,−0.5,−1} is

u1 = 0.1091x2 + 7.0089x3 + 1.0014x4 + 0.4302x5

u2 = −0.1929x6 − 0.09645x1.(67)

Th constants c1, c2 and c3 in (64) are functions of the robotparameters and the controller gains kij . For the controllerchosen, it can be checked that the dynamics on the reducedsystem (65) is locally asymptotically stable. The conditions inAssumption 1 are satisfied and we use the relative thresholding||ep2 || ≥ σ(||p2 − ha(p1)||+ ||p1||4) from Proposition 3 withha(p1) , [c1p

21, 0, 0, −c2p2

1, c3p31]> for event-triggered im-

plementation. Under this implementation, asymptotic stabilityand the non-existence of Zeno behaviour is guaranteed throughProposition 6.

The pair of matrices (P,Q) required to compute the boundon the thresholding parameter σ from (25) are found bysolving the Lyapunov equation As>P + PAs + Q = 0 bychoosing Q = I5. For the controller (67) and the (P,Q) pairchosen as above, the bound for the relative threshold is foundto be σ ≤ 10−4. Choosing σ = 10−4, the simulation resultsof event-triggered position stabilization of the MIP robot arepresented in Figures 3 and 4.

The MIP robot is initialized at (x, y, θ) = (2, 2, π2 ) withthe pitch upright, that is, α = α = 0 and v = θ = 0. The

15

0 1 2

-2

-1.5

-1

-0.5

0

0.88 0.9

-0.62

-0.6

(a) Position (x, y) of the robot.

0 10 20 30 40 500

1

2

3

4

510

-3

Threshold

Error

10 10.1 10.2

0

0.5

1

10-3

(b) Evolution of the error and thethreshold

0 10 20 30 40 50

-6

-4

-2

0

2

410

-2

(c) Evolution of α and α

0 10 20 30 40 50

0

0.5

1

0 10 20

0

0.05

(d) Inter-event times

Fig. 3: Simulation results for the event-triggered positionstabilization of an MIP robot.

evolution of the position of the robot is shown in Figure 3a.The robot asymptotically reaches the origin of the (x, y) plane.In Figure 3b, the evolution of the norm of the error ||ep2 || andthe threshold 10−4(||p2 − ha(p1)|| + ||p1||4) are shown. Anevent occurs when the norm of the error rises from zero tomeet the threshold. In Figure 3c, we see that the pitch angle αand pitch velocity α tend asymptotically to zero as guaranteedby Proposition 6. The evolution of the inter-event times isshown in Figure 3d. The inter-event times are lower boundedby 2.4 ms. Considering ten initial conditions inside a circle ofradius 3 m in the (x, y) plane, it is found that the minimumtime between two events (MIETs) is 2.4 ms. With MIETs asthe sampling time for time-triggered control, simulations wereperformed and as can be seen in Figure 3a, the performanceof event-triggered control is a close match to the performanceof time-triggered control. In achieving this close similarity inperformance, event-triggered control requires fewer closings ofthe control-loop than time-triggered control. From Figures 4a,4b and 4c, we see that the trajectories of the event-triggeredclosed-loop system converge rapidly to the center manifold,while evolving slowly along the center manifold.

VII. CONCLUSIONS

In this work, we presented event-triggered implementationof control laws designed for nonlinear systems with centermanifolds. The proposed methods ensured Zeno-free localultimate boudedness of the trajectories, and under some as-sumptions on the controller structure, Zeno-free asymptoticstability of the origin. Systems for which the center manifoldis exactly computable were considered first and triggeringconditions were presented, the checking of which requiresthe exact knowledge of the center manifold. Then, we con-

-1 -0.5 0 0.5 1

-1

0

1

2

9.34 9.36

10-3

0

0.5

110

-3

(a) Evolution in (x1, x2) plane

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

9.34 9.35

10-3

-5.5

-5

-4.5

-410

-5

(b) Evolution in (x1, x5) plane

-1 -0.5 0 0.5 1

-0.5

0

0.5

1

9.35 9.4

10-3

-4

-2

0

210

-5

(c) Evolution in (x1, p2(5))plane

Fig. 4: The trajectories of the states x2, x5 and p2(5) = x6 +(k22/k21)x1 of the event-triggered closed-loop system.

sidered systems for which the center-manifold can only beapproximately computed and showed that the same triggeringconditions could be used with the available approximateknowledge of the center manifold.

When the triggering conditions employed the approximateknowledge of the center manifold, the trajectories of theclosed-loop system showed oscillations about the center man-ifold, as they tend to the origin along the center manifold.When the exact knowledge was employed, the trajectoriesconverged to the origin without any oscillations about thecenter manifold. For initial conditions close to the origin, asimulation study showed that use of a better approximationof the center manifold leads to more frequent triggers (as thetriggering condition is checked in a more refined manner) andyields no significant improvement in the system performance.The minimum inter-event time from multiple simulations(MIETs) was used as sampling time for time-triggered controland it was found that event-triggered control yields similarperformance as time-triggered control but with significantlylesser control updates.

As part of our future work, we look to relax the assumptionson the controller structure, to guarantee Zeno-free asymptoticstability of the origin for a larger class of nonlinear systemswith center-manifolds.

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Akshit Saradagi is currently a Postdoctoral re-searcher in the Department of Electrical Engineer-ing, Indian Institute of Technology Madras, Chennai,India. He received the B.E degree in Electronics andCommunication Engineering from P.E.S Institute ofTechnology, Bengaluru in 2013 and the M.S andPh.D Dual degree from Indian Institute of Technol-ogy Madras, Chennai in 2020. His current researchinterests are in the areas of networked dynamicalsystems, control under resource constraints and bio-inspired control.

Vijay Muralidharan obtained his PhD in roboticsand control from IIT Madras, India. His experienceincludes postdoc and product development with LTUSweden and SigTuple Technologies India, respec-tively. He is currently an Assistant Professor inthe Department of Electrical Engineering at IndianInstitute of Technology Palakkad, India. His researchinterests include dynamics modeling, nonlinear con-trol, state estimation for robotic systems, medicaldevices and aerospace systems.

Arun D Mahindrakar (M’93-SM’19) received theBachelor of Engineering in Electrical and Electron-ics from Karnatak University, Dharwad, India in1994, the Master of Engineering in Control Sys-tems from VJTI, Mumbai in 1997 and the Ph.D inSystems & Control from IIT Bombay in 2004. Heworked as a Postdoctoral Fellow at the Laboratory ofSignals and Systems at Supelec, Paris from 2004 to2005. He is currently a Professor in the Departmentof Electrical Engineering, Indian Institute of Tech-nology Madras, India. His research interests include

nonlinear stability, geometric control and nonsmooth optimization.

Pavankumar Tallapragada (S’12-M’14) receivedthe B.E. degree in Instrumentation Engineeringfrom SGGS Institute of Engineering & Technology,Nanded, India in 2005, M.Sc. (Engg.) degree inInstrumentation from the Indian Institute of Sciencein 2007 and the Ph.D. degree in Mechanical Engi-neering from the University of Maryland, CollegePark in 2013. He was a Postdoctoral Scholar in theDepartment of Mechanical and Aerospace Engineer-ing at the University of California, San Diego from2014 to 2017. He is currently an Assistant Professor

in the Department of Electrical Engineering and the Robert Bosch Centrefor Cyber Physical Systems at the Indian Institute of Science. His researchinterests include networked control systems, distributed control, multi-agentsystems and networked transportation systems.