An Input-Delay Method for Event-Triggered Control of Nonlinear Systems

# An Input-Delay Method for Event-Triggered Control of Nonlinear Systems

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###### Abstract

The paper proposes a novel event-triggered control scheme for nonlinear systems based on the input-delay method. Specifically, the closed-loop system is associated with a pair of auxiliary input and output. The auxiliary output is defined as the derivative of the continuous-time input function, while the auxiliary input is defined as the input disturbance caused by the sampling or equivalently the integral of the auxiliary output over the sampling period. As a result, it forms a cyclic mapping from the input to the output via the system dynamics and back from the output to the input via the integral. The event-triggering law is constructed to make the mapping contractive such that the stabilization is achieved and an easy-to-check Zeno-free condition is provided. Within this framework, we develop a theorem for the event-triggered control of interconnected nonlinear systems which is employed to solve the event-triggered control for lower-triangular systems with dynamic uncertainties.

A]Lijun Zhu , B] Zhiyong Chen, A] David J. Hill , C] Shengli Du

Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong

School of Electrical Engineering and Computing, The University of Newcastle, Callaghan, NSW 2308, Australia

College of Automation, Faculty of Information Technology, Beijing University of Technology, Beijing, 100124, China

Key words:  Event-triggered control, Zeno behavior, lower-triangular systems, nonlinear systems, input-delay method.

11footnotetext: This research was supported under The University of Hong Kong Research Committee Post-doctoral Fellow Scheme

## 1 Introduction

The majority of modern control systems reside in microprocessors and need more efficient implementation in order to reduce computation cost, save communication bandwidth and decrease energy consumption. Sampled-data control has been developed to fulfill these tasks where the execution of a digital controller is scheduled among sampling instances periodically or aperiodically. As a type of aperiodic sampling, event-triggered control suggests scheduling based on the state and/or sampling error of the plant and achieves more efficient sampling pattern than periodic sampling. Event-triggered control has been developed for stabilization and tracking of individual systems [18, 12, 17, 11] and cooperative control of networked systems [6, 16].

The two-step digital emulation is a common technique for analysis and design of sampled-data control systems especially for nonlinear systems. A continuous-time controller is first designed for a continuous-time system such that the control objective is fulfilled, which is then discretized into a sampled-data controller for digital implementation. For periodic sampled-data control, it becomes an efficient tool to explicitly compute maximum allowable sampling period that guarantees asymptotic stability of sampled-data systems with the emulated version of the given continuous-time controller [2, 9, 15]. Emulation is also commonly adopted for the design of event-triggered laws where the continuous-time controller is usually assumed to render the closed-loop system have the input-to-state stability (ISS) property [1, 5, 11, 13, 10, 17] with the sampling error as the external input. Specifically, the max-form ISS condition is assumed in [11] for the closed-loop system and the event-triggered controller is designed using small gain conditions. In [1, 10, 13, 17], the continuous-time controller is assumed to render the closed-loop system admit an ISS-Lyapunov function and the event-triggering law is designed to ensure the derivative of Lyapunov function to be negative. Cyclic small gain theorem has been proved effective for event-triggered control of large-scale systems [5, 10, 11]. Despite these progresses, explicit event-triggered controller design and exclusion of Zeno behavior for complex nonlinear systems still remains a challenging problem. For instance, event-triggered control of nonlinear lower-triangular systems with dynamic uncertainties and a higher relative degree has yet to be fully addressed, except for [10] where the dynamic uncertainties only appear at first relative degree level.

Recently, the input-delay approach was proposed for the controller design and performance analysis of linear systems [7, 14] and nonlinear sampled-data systems [3] using the emulation technique. This paper aims to extend the input-delay approach to the event-triggered control of nonlinear systems. The contribution of this paper is two-fold. First, a novel event-triggered control design is proposed to achieve stabilization of individual and interconnected nonlinear systems. Specifically, a pair of auxiliary input and output is associated with the closed-loop system. The auxiliary output is defined as the derivative of the continuous-time feedback input function, while the auxiliary input is defined as the feedback input error caused by the sampling or equivalently the integral of the auxiliary output over the sampling period. Consequently, a closed-loop mapping is formed from the input to the output via the system dynamics and back from the output to the input via the integral function. The event-triggering law is constructed to make the mapping contractive. As opposed to [10, 11, 17], the local (global) stabilization without Zeno behavior does not rely on a local (global) Lipschitz condition on the system dynamics. Moreover, the new event-triggered control ensures that the sampling interval converges to a constant as time approaches infinity. Secondly, we solve event-triggered control for lower-triangular nonlinear systems with dynamic uncertainties. Compared to [10], more complicated systems are dealt with where dynamic uncertainties appear at each relative degree level.

The rest of this paper is organized as follows. The event-triggered control design for individual and interconnected systems is introduced in Section 2. The method is applied to solved event-triggered control of lower-triangular systems in Section 3. Numerical simulation is presented in Section 4 and the paper is concluded in Section 5.

## 2 A New Event-triggered Control Scheme

### 2.1 Event-Triggered control

Consider a nonlinear system

 ˙x(t)=f(x(t),u(t)), (1)

where is the state and the input. The function satisfies such that is the equilibrium point of the uncontrolled system. Suppose stabilization of the equilibrium point can be fulfilled by the continuous-time state feedback controller

 u(t)=g(x(t)) (2)

for a continuously differentiable function . In this paper, we will study the event-triggered version of (2.1) as follows

 u(t)=g(x(tk)),t∈[tk,tk+1),k∈I (3)

where is a sequence of sampling time instances with and triggered by the condition

 tk+1=Ξ(tk,x(tk),x(t)),t∈[tk,tk+1),k∈I−0, (4)

with the triggering law to be designed. The objective of event-triggered control is to design the triggering law (2.1) such that the closed-loop system composed of (2.1) and (2.1) achieves

1. Stabilization: The state is globally asymptotically stable at the origin.

2. Zeno Behavior free: The intervals between sampling time instants are lower bounded by a positive constant, i.e., .

Except for the main Objective 1, Objective 2 guarantees that infinitely fast sampling is avoided. The event-triggered stabilization problem can be solved by a small gain theorem [11] if the controlled system has a certain input-to-state stability (ISS) property from the measurement disturbance to the state . However, this ISS property is not always achievable for nonlinear systems, and we will propose a new event-triggered control that requires the controlled system have a certain ISS property from the input disturbance to be defined to the state . By the input-delay method, the closed-loop system composed of (2.1) and (2.1) can be written as follows

 ˙x(t) = f(x(t),g(x(t))−r(t)), (5)

with the auxiliary input and output defined as follows

 r(t) = t∫tkξ(s)ds,t∈[tk,tk+1), ξ(t) = dg(x(t))dt. (6)

Different from most event-triggered control design, we assume the continuous-time state feedback controller in (2.1) renders the closed-loop system has the following ISS and IOS conditions.

###### Assumption 2.1

The closed-loop system (2.1) with as the piecewise continuous bounded external input and as the output has following input-to-state stability (ISS) and input-to-output stability (IOS) properties

 ∥x(t)∥ ≤ max{~β(∥x(t0)∥,t−t0),~γ(∥r[t0,t]∥)}, (7) ∥ξ(t)∥ ≤ max{β(∥x(t0)∥,t−t0),γ(∥r[t0,t]∥)},∀t>t0 (8)

for and .

Then, a new event-triggered control scheme is proposed as follows.

###### Theorem 2.1

Consider the system (2.1) with the controller (2.1). Suppose the closed-loop system satisfies Assumption 2.1 and the gain function satisfies

 lims→0+γ(s)s<∞. (9)

Let and be a function satisfying

 ¯γ(s)≥ϵγ(s),∀s>0andlims→0+¯γ(s)s>0 (10)

The objectives of the event-triggered control are achieved if the event-triggering law (2.1) is

 tk+1=inft>tk{(t−tk)¯γ(∥r[tk,t]∥)=∥r[tk,t]∥ and ∥r[tk,t]∥≠0}. (11)

Proof: The proof will be divided into two steps. First, we prove Zeno behavior is avoided. Due to (11), it suffices to prove that , For each , there exists a such that the signal is

If , (11) implies . If , we will see there exists a such that and hence also holds. In fact, according to (9), we can find a function such that . As a result, for any , there exists an such that

 ∥r[tk,t]∥<ϵ⟹∣∣∣¯γ(∥r[tk,t]∥)∥r[tk,t]∥−C∣∣∣<δ (12)

Let such that and there exists a satisfying and (which always exists due to ). Consequently, the inequality (12) leads to . Thus, Zeno behavior is avoided and the following holds

 (t−tk)¯γ(∥r[tk,t]∥)≤∥r[tk,t]∥,∀t∈[tk,tk+1) (13)

The closed-loop system (2.1) can be regarded as the interconnection of the -subsystem and the -subsystem. The second step is to show that the state approaches zero asymptotically by the spirit of small gain theorem where we first need to show signals , and are bounded. We start with proving that is bounded. If this is not true, for every number , there exist a finite time such that . Let us choose

 R:=β(∥x(t0)∥,0) (14)

Since and hence are bounded for , using (8) and (10) shows that

 ∥ξ[t0,T]∥≤max{β(∥x(t0)∥,0),¯γ(∥r[t0,T]∥)}. (15)

Due to

 ∥r[tk,t]∥≤(t−tk)∥ξ[tk,t]∥,∀t∈[tk,tk+1) (16)

For , since Zeno behavior is avoided, one can find a such that . Using inequalities (16) and (13) leads to

 ∥ξ[t0,T]∥ ≤ max{R,¯γ(∥r[t0,t1]∥),¯γ(∥r[t1,t2]∥),⋯,¯γ(∥r[ti,T]∥)} ≤ max{R,1t1−t0∥r[t0,t1]∥,1t2−t1∥r[t1,t2]∥,⋯,1T−ti∥r[ti,T]∥} ≤ R

which is a contradiction against . So, is bounded, i.e., . Substituting (16) into (13) obtains

 ¯γ(∥r[tk,t]∥)≤∥ξ[tk,t]∥,∀t∈[tk,tk+1). (17)

Thus, and hence are bounded for due to (7), i.e.,

 ∥r[t0,∞)∥ ≤ ¯γ−1(β(∥x(t0)∥,0)) ∥x[t0,∞)∥ ≤ max{~β(∥x(t0)∥,0),~γ(¯γ−1(β(∥x(t0)∥,0)))}.

For any , one can always find an such that if then . It proves that the equilibrium point is stable.

Finally, we will show the state approaches zero asymptotically

Due to (10), one has

 infs∈(0,r∞)¯γ(s)s≥1/Tmax (18)

for some constant , which implies that and thus

 tk+1−tk≤Tmax,∀t∈I. (19)

i.e., is the upper bound of the sampling interval. Consider the system behaviors of among interval for any . First, the inequality (8) with implies the signal satisfies

 ∥ξ[t∗/2,t∗]∥≤max{β(∥x(t∗4)∥,t∗4),γ(∥r[t∗/4,t∗]∥)}. (20)

Note that there exists integers such that where , and . Then, it follows from (20) and (10) that

 ∥ξ[t∗/2,t∗]∥ ≤ max{β(∥x(t∗4)∥,t∗4),1/ϵ¯γ(∥r[ti,t∗]∥)} (21) ≤ max{β(∥x(t∗4)∥,t∗4),1ϵ(ti+1−ti)∥r[ti,ti+1]∥,⋯,1ϵ(t∗−tj)∥r[tj,t∗]∥} ≤ max{β(∥x(t∗4)∥,t∗4),1/ϵ∥ξ[t∗/8,t∗]∥} ≤ max{β(∥x∞∥,t∗4),1/ϵ∥ξ[t∗/8,t∗/2]∥},

where the second inequality uses (13), the third one uses (16) and the last one uses . Denote , (21) can be rewritten as

 ζ(t∗)≤max{β(∥x∞∥,t∗4),1ϵζ(t∗2),1ϵζ(t∗4)},∀t∗≥8Tmax. (22)

Next, we will show that . Otherwise, there exists a positive such that, for any , there exists such that . Pick a positive integer satisfying

 δϵN>R (23)

and a such that

So, there exists such that . As a result,

which together with (22) implies where or . By repeating this manipulation times, one has

As a result, (23) further leads to

which is a contradiction against proved in the second step. Consequently, the fact that holds which in turn implies , , and hence . Thus, Objective 1 and 2 of the event-triggered control are achieved.

###### Remark 2.1

It is not difficult to select a function that satisfies the condition (10) in Theorem (2.1). For instance, for and .

###### Remark 2.2

The Zeno-free condition (9) can be easily checked and ensured while the event-triggered law is constructed (see example in Section 4). It does not require locally (globally) Lipschitz condition on the system dynamics as normally needed in the event-triggered control [10, 11, 17].

The following proposition shows that the sampling interval converges to a constant and the event-triggered control tends to be periodic sampling control as .

###### Proposition 2.1

Consider the event-triggering law (11) in Theorem 2.1. Suppose and . Then, .

Proof: It suffices to prove for any there exists a such that , . Without loss of generality, we only consider the case of . Due to , for any , there exists an such that

 ∥r[tk,t]∥<ϵ⟹∣∣∣¯γ(∥r[tk,t]∥)∥r[tk,t]∥−1/T∣∣∣<δ. (24)

Due to by Theorem 2.1, for any , there exists a such that . In what follows, we consider the behavior of for . Let

 δ>max{δ∗T2+Tδ∗,δ∗T2−Tδ∗}>0. (25)

We choose such that is satisfied for and such that . Consequently, (24) implies

which together with the event-triggering law (11) shows that

By selection of in (25), one has

which implies that , . Thus, the proof is complete.

### 2.2 Interconnected Systems

In this section, we consider event-triggered control of a nonlinear interconnected system described as follows

 ˙z(t) = q(z(t),x(t),u(t)), ˙x(t) = f(z(t),x(t),u(t)), (26)

where and are the states of the two subsystems, and is the input. The functions and satisfy and such that is the equilibrium point of the overall system with . The state is assumed not available for the feedback. The event-triggered control of (26) is to construct an event-triggered controller (2.1) such that state stabilization with and Zeno-free behavior are achieved. This problem was solved in [10] using the cyclic small gain theorem, provided that the controlled system has an ISS property from the measurement disturbance to the state . It also shows that the -dynamics must be taken into account for the event-triggering law design even when only is used for the feedback.

Here, we will adopt the new event-triggered control scheme proposed in Section 2.1 to solve the problem and explicitly show how the -dynamics affect the event-triggered control law. The closed-loop system composed of (26) and (2.1) can be written as follows

 ˙z(t) = q(z(t),x(t),g(x(t))−r(t)), ˙x(t) = f(z(t),x(t),g(x(t))−r(t)), (27)

with the auxiliary input and output defined as follows

 r(t) = t∫tkξ(s)ds,t∈[tk,tk+1), ξ(t) = dg(x(t))dt. (28)

The following ISS and bounded state and input to bounded output (BSIBO) conditions are assumed for the closed-loop system (27). The assumption will be matched through proper controller design in real applications, for example, the specific design approach is discussed in Section 3 for lower-triangular systems.

###### Assumption 2.2

The closed-loop system (27) with as the piecewise continuous bounded external input and as the output has following ISS properties

• The -dynamics and -dynamics are ISS, i.e.,

 ∥z(t)∥ ≤ max{βz(∥z(t0)∥,t−t0),γxz(∥x[t0,t]∥),γrz(∥r[t0,t]∥)},∀t>t0 (29) ∥x(t)∥ ≤ max{βx(∥x(t0)∥,t−t0),γzx(∥z[t0,t]∥),γrx(∥r[t0,t]∥)},∀t>t0 (30)

for some functions and .

• It is BSIBO viewing and as states, as the input and as the output, i.e.,

 ∥ξ(t)∥≤max{γzξ(∥z[t0,t]∥),γxξ(∥x[t0,t]∥),γrξ(∥r[t0,t]∥)},∀t>t0 (31)

for some functions .

A useful lemma is presented as follows.

###### Lemma 2.1

([19]) Consider the system (27) with . Suppose is the piecewise continuous bounded input. If the first condition of Assumption 2.2 and the small gain condition are satisfied, the system (27) is ISS in the sense of

 ∥z(t)∥ ≤ max{¯βz(∥Λ(t0)∥,t−t0),¯γrz(∥r[t0,t]∥)},t≥t0 (32) ∥x(t)∥ ≤ max{¯βx(∥Λ(t0)∥,t−t0),¯γrx(∥r[t0,t]∥)},t≥t0 (33)

for some functions and class functions

###### Remark 2.3

For the similar reason justified in [10], we deliberately consider the ISS property for the and -dynamics separately rather than consider that for as a whole. On one hand, it facilitates examination of and -dynamics’ individual effect on the event-triggered control design. On the other hand, under the small gain condition , we can derive the ISS property for the -dynamics

 ∥Λ(t)∥≤max{βΛ(∥Λ(t0)∥,t−t0),γΛ(∥r[t0,t]∥)},t≥t0 (34)

for functions and . By Lemma 2.1, we can use the ISS properties (32) and (33) with less conservative gain functions and instead of (34) with to design the event-triggering law. As will be explained in Remark, it may lead to a better sampling pattern.

###### Theorem 2.2

Consider the system (26) with the controller (2.1). Suppose Assumption 2.2 is satisfied with the small gain condition . Let where and are given in Lemma 2.1. Suppose satisfies

 lims→0+γ(s)s<∞. (35)

Let and be a function satisfying

 ¯γ(s)≥ϵγ(s),∀s>0andlims→0+¯γ(s)s>0. (36)

The objectives of the event-triggered control are achieved if the event-triggering law (2.1) is

 tk+1=inft>tk{(t−tk)¯γ(∥r[tk,t]∥)=∥r[tk,t]∥ and ∥r[tk,t]∥≠0}. (37)

Proof: By Lemma 2.1, one has (32) and (33). Following the similar argument in Theorem 2.1, we can prove signals , , and of the closed-loop system (27) are bounded. Since all signals are bounded, we can substitute (32) and (33) into (31) and obtain

 ∥ξ(t)∥≤max{β(∥Λ(t0)∥,t−t0),γ(∥r[t0,t]∥)},∀t≥t0 (38)

for some . Note that (32), (33) and (38) are similar to conditions of Theorem 2.1. The rest of the proof can easily follow from that of Theorem 2.1.

###### Remark 2.4

Let us consider two special cases: (1) and do not appear in the -dynamics, i.e., (); (2) does not appear in the -dynamics but does, and does not appear in -dynamics, i.e, (). It follows from Theorem 2.2 that in (36) should be

As opposed to the method in [10], we explicitly show that the variation of -dynamics does not affect Zeno-free behavior for both aforementioned cases. Specifically, in both cases, it is not necessary to re-design the event-triggering law (37) when the -dynamics vary.

###### Remark 2.5

For a given signal , it is observed from (37) that less conservative selection of increases the sampling interval , which could lead to a desirable sampling pattern that less number of control executions are taken within a given period. If the ISS property (34) is used to derive an event-triggering law rather than (32) and (33), we can derive the following inequality similar to (38),

 ∥ξ(t)∥≤max{β(∥Λ(t0)∥,t−t0),~γ(∥r[t0,t]∥)},∀t≥t0 (39)

with

where and . It follows from the proof of Theorem 2.2 that should be

 ¯γ(s)≥ϵ~γ(s). (40)

The fact makes the choice of more conservative.

## 3 Lower-Triangular Systems

In this section, we consider event-triggered control for a class of lower-triangular systems

 ˙zj(t) = qj(→zj(t),→xj(t),w), ˙xj(t) = fj(→zj(t),→xj(t),w)+bjxj+1,j=1,⋯,ℓ (41)

where with and with are the states, is the control input, ’s are constants and is the relative degree. represents system uncertainties (such as unknown parameters) in a known compact set , while -dynamics are called dynamic uncertainties where the function ’s are not precisely known and is not available for feedback. The functions and are assumed to be sufficiently smooth and satisfy and such that the equilibrium point of the system with is . The continuous-time stabilization of such system has been solved using Lyapunov function method [4] and small gain theorem [8] based on backstepping technique. In spirit of backstepping, we introduce the coordinate transformation

 ¯x1 = x1 ¯xj = xj−ϑj−1(¯xj−1),j=2,⋯,ℓ+1 (42)

where the functions satisfying are virtual controllers to be designed at each recursive step. Under the coordinate (42), the system (41) becomes

 ˙zj(t) = ¯qj(→zj(t),→¯xj(t),w) ˙¯xj(t) = ¯fj(→zj(t),→¯xj(t),w)+ϑj(¯xj(t))+¯xj+1(t),j=1,⋯,ℓ (43)

where , and are sufficiently smooth functions satisfying and . For continuous-time stabilization ([4, 8]), the controller is by setting . For event-triggered control,