An InputDelay Method for EventTriggered Control of Nonlinear Systems
Abstract
The paper proposes a novel eventtriggered control scheme for nonlinear systems based on the inputdelay method. Specifically, the closedloop system is associated with a pair of auxiliary input and output. The auxiliary output is defined as the derivative of the continuoustime 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 eventtriggering law is constructed to make the mapping contractive such that the stabilization is achieved and an easytocheck Zenofree condition is provided. Within this framework, we develop a theorem for the eventtriggered control of interconnected nonlinear systems which is employed to solve the eventtriggered control for lowertriangular 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: Eventtriggered control, Zeno behavior, lowertriangular systems, nonlinear systems, inputdelay method.
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. Sampleddata 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, eventtriggered control suggests scheduling based on the state and/or sampling error of the plant and achieves more efficient sampling pattern than periodic sampling. Eventtriggered control has been developed for stabilization and tracking of individual systems [18, 12, 17, 11] and cooperative control of networked systems [6, 16].
The twostep digital emulation is a common technique for analysis and design of sampleddata control systems especially for nonlinear systems. A continuoustime controller is first designed for a continuoustime system such that the control objective is fulfilled, which is then discretized into a sampleddata controller for digital implementation. For periodic sampleddata control, it becomes an efficient tool to explicitly compute maximum allowable sampling period that guarantees asymptotic stability of sampleddata systems with the emulated version of the given continuoustime controller [2, 9, 15]. Emulation is also commonly adopted for the design of eventtriggered laws where the continuoustime controller is usually assumed to render the closedloop system have the inputtostate stability (ISS) property [1, 5, 11, 13, 10, 17] with the sampling error as the external input. Specifically, the maxform ISS condition is assumed in [11] for the closedloop system and the eventtriggered controller is designed using small gain conditions. In [1, 10, 13, 17], the continuoustime controller is assumed to render the closedloop system admit an ISSLyapunov function and the eventtriggering law is designed to ensure the derivative of Lyapunov function to be negative. Cyclic small gain theorem has been proved effective for eventtriggered control of largescale systems [5, 10, 11]. Despite these progresses, explicit eventtriggered controller design and exclusion of Zeno behavior for complex nonlinear systems still remains a challenging problem. For instance, eventtriggered control of nonlinear lowertriangular 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 inputdelay approach was proposed for the controller design and performance analysis of linear systems [7, 14] and nonlinear sampleddata systems [3] using the emulation technique. This paper aims to extend the inputdelay approach to the eventtriggered control of nonlinear systems. The contribution of this paper is twofold. First, a novel eventtriggered 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 closedloop system. The auxiliary output is defined as the derivative of the continuoustime 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 closedloop 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 eventtriggering 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 eventtriggered control ensures that the sampling interval converges to a constant as time approaches infinity. Secondly, we solve eventtriggered control for lowertriangular 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 eventtriggered control design for individual and interconnected systems is introduced in Section 2. The method is applied to solved eventtriggered control of lowertriangular systems in Section 3. Numerical simulation is presented in Section 4 and the paper is concluded in Section 5.
2 A New Eventtriggered Control Scheme
2.1 EventTriggered control
Consider a nonlinear system
(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 continuoustime state feedback controller
(2) 
for a continuously differentiable function . In this paper, we will study the eventtriggered version of (2.1) as follows
(3) 
where is a sequence of sampling time instances with and triggered by the condition
(4) 
with the triggering law to be designed. The objective of eventtriggered control is to design the triggering law (2.1) such that the closedloop system composed of (2.1) and (2.1) achieves

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

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 eventtriggered stabilization problem can be solved by a small gain theorem [11] if the controlled system has a certain inputtostate 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 eventtriggered control that requires the controlled system have a certain ISS property from the input disturbance to be defined to the state . By the inputdelay method, the closedloop system composed of (2.1) and (2.1) can be written as follows
(5) 
with the auxiliary input and output defined as follows
(6) 
Different from most eventtriggered control design, we assume the continuoustime state feedback controller in (2.1) renders the closedloop system has the following ISS and IOS conditions.
Assumption 2.1
The closedloop system (2.1) with as the piecewise continuous bounded external input and as the output has following inputtostate stability (ISS) and inputtooutput stability (IOS) properties
(7)  
(8) 
for and .
Then, a new eventtriggered control scheme is proposed as follows.
Theorem 2.1
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
(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
(13) 
The closedloop 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
(14) 
Since and hence are bounded for , using (8) and (10) shows that
(15) 
Due to
(16) 
For , since Zeno behavior is avoided, one can find a such that . Using inequalities (16) and (13) leads to
which is a contradiction against . So, is bounded, i.e., . Substituting (16) into (13) obtains
(17) 
Thus, and hence are bounded for due to (7), i.e.,
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
(18) 
for some constant , which implies that and thus
(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
(20) 
Note that there exists integers such that where , and . Then, it follows from (20) and (10) that
(21)  
where the second inequality uses (13), the third one uses (16) and the last one uses . Denote , (21) can be rewritten as
(22) 
Next, we will show that . Otherwise, there exists a positive such that, for any , there exists such that . Pick a positive integer satisfying
(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 eventtriggered control are achieved.
Remark 2.1
Remark 2.2
The following proposition shows that the sampling interval converges to a constant and the eventtriggered control tends to be periodic sampling control as .
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
(24) 
Due to by Theorem 2.1, for any , there exists a such that . In what follows, we consider the behavior of for . Let
(25) 
We choose such that is satisfied for and such that . Consequently, (24) implies
which together with the eventtriggering 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 eventtriggered control of a nonlinear interconnected system described as follows
(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 eventtriggered control of (26) is to construct an eventtriggered controller (2.1) such that state stabilization with and Zenofree 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 eventtriggering law design even when only is used for the feedback.
Here, we will adopt the new eventtriggered control scheme proposed in Section 2.1 to solve the problem and explicitly show how the dynamics affect the eventtriggered control law. The closedloop system composed of (26) and (2.1) can be written as follows
(27) 
with the auxiliary input and output defined as follows
(28) 
The following ISS and bounded state and input to bounded output (BSIBO) conditions are assumed for the closedloop 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 lowertriangular systems.
Assumption 2.2
The closedloop 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.,
(29) (30) for some functions and .

It is BSIBO viewing and as states, as the input and as the output, i.e.,
(31) for some functions .
A useful lemma is presented as follows.
Lemma 2.1
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 eventtriggered control design. On the other hand, under the small gain condition , we can derive the ISS property for the dynamics
(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 eventtriggering law. As will be explained in Remark 2.5, 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
(35) 
Let and be a function satisfying
(36) 
The objectives of the eventtriggered control are achieved if the eventtriggering law (2.1) is
(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 closedloop system (27) are bounded. Since all signals are bounded, we can substitute (32) and (33) into (31) and obtain
(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 Zenofree behavior for both aforementioned cases. Specifically, in both cases, it is not necessary to redesign the eventtriggering 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 eventtriggering law rather than (32) and (33), we can derive the following inequality similar to (38),
(39) 
with
where and . It follows from the proof of Theorem 2.2 that should be
(40) 
The fact makes the choice of more conservative.
3 LowerTriangular Systems
In this section, we consider eventtriggered control for a class of lowertriangular systems
(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 continuoustime 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
(42) 
where the functions satisfying are virtual controllers to be designed at each recursive step. Under the coordinate (42), the system (41) becomes
(43) 
where , and are sufficiently smooth functions satisfying and . For continuoustime stabilization ([4, 8]), the controller is by setting . For eventtriggered control,