Zhiqian LIU, Xuyang LOU, Jiajia JIA

Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education),Jiangnan University, Wuxi 214122, China

Abstract: This paper investigates the problem of dynamic output-feedback control for a class of Lipschitz nonlinear systems. First, a continuous-time controller is constructed and sufficient conditions for stability of the nonlinear systems are presented. Then, a novel event-triggered mechanism is proposed for the Lipschitz nonlinear systems in which new event-triggered conditions are introduced. Consequently, a closed-loop hybrid system is obtained using the event-triggered control strategy. Sufficient conditions for stability of the closed-loop system are established in the framework of hybrid systems. In addition, an upper bound of a minimum inter-event interval is provided to avoid the Zeno phenomenon. Finally, numerical examples of a neural network system and a genetic regulatory network system are provided to verify the theoretical results and to show the superiority of the proposed method.

Key words: Lipschitz nonlinear system; Dynamic output-feedback control; Event-triggered control; Global asymptotic stability

1 Introduction

Nonlinear systems have received much attention for decades due to their broad requirements in scientific and engineering fields such as electrical circuits, chemical processes, and biomedical engineering (Casey et al., 2006; Angulo et al., 2019; Chen J et al., 2019). However, due to the complexity of nonlinear systems, their analysis is still a challenging topic. Specifically, different from linear systems,nonlinear systems must find a suitable real-time solution, even though there may be many such solutions or none (Collins et al., 2006). Note that nonlinear systems with Lipschitz characteristics can avoid this problem, because the local Lipschitz property of the system dynamics ensures local existence and uniqueness of the solution, and even guarantees global existence and uniqueness with some extra conditions (Khalil, 2014). Indeed, nonlinear systems with Lipschitz characteristics are in wide demand in the field of system modeling. For instance, Rehan et al.(2018)presented a class of one-sided Lipschitz nonlinear multi-agent systems that combine linear and Lipschitz nonlinear features and have many applications in synchronization,formation, flocking,and so on. In addition, multi-agent systems with Lipschitz conditions have been studied (Zhang Z et al., 2020) owing to their potential applications in many areas, including autonomous underwater vehicles and distributed sensor networks. Pham et al.(2019)concerned a class of nonlinear Lipschitz systems applied to a real electro-rheological (ER)automotive suspension. Therefore, the analysis of nonlinear systems with Lipschitz characteristics is meaningful and general.

Among the analyses of nonlinear systems with Lipschitz conditions,the stability issue has attracted significant attention. For the stabilization issue,feedback control as a basic control law has been presented in many applications(e.g.,Zuo et al.(2016)).On one hand,state-feedback control,one of the most widely used control methods, has been adopted to solve stability issues by combining some other control strategies(Qian and Lin, 2001;Tabuada,2007).One of the limitations of this control strategy is that it requires full state measurements,which is often not possible in many scenarios. Therefore,some outputfeedback control methods have been established to avoid the full state measurement requirement. Currently,many kinds of output-feedback control strategies exist, such as observer-based output-feedback control,static output-feedback control,and dynamic output-feedback control; see Andrieu and Praly(2009), Zhou et al. (2012),Kammogne et al. (2020),and the references therein. Explicitly, observerbased output-feedback control constructs so-called observers to estimate full states from measured outputs, and uses such estimation to control the system. In Pertew et al. (2006), a Lipschitz observer was introduced and some sufficient conditions were presented for the asymptotic convergence of the closed-loop system. Hamid et al. (2019) considered the design of a regional observer-based controller for the locally Lipschitz nonlinear systems.Note that observer-based output-feedback control still requires that the controlled system is observable. On the other hand,static output-feedback control(Ekramian,2020)and dynamic output-feedback control do not have such limitations. Moreover,compared with static output-feedback control, the dynamic output-feedback control strategy contains the memory of the output trajectory by introducing the integral of an auxiliary variable and thus leads to better control effects(such as shorter convergence time).Many researchers have devoted effort to the analysis of dynamic output-feedback control for nonlinear systems. For instance,Chen PN et al.(2006)focused on the problem of local stabilization of nonlinear systems by dynamic output-feedback control. Liu W et al. (2016) proposed a dynamic output-feedback control method for fast sampling of discrete-time singularly perturbed systems. Dong and Yang (2008)proposed a dynamic output-feedback controller for continuous-time Takagi–Sugeno(T-S)fuzzy systems.However, to our knowledge, analysis of dynamic output-feedback control for Lipschitz nonlinear systems is still absent.

In this study, we aim to establish dynamic output-feedback control methods for Lipschitz nonlinear systems. Generally speaking, the continuoustime control strategy(Park,2005;Molaei,2008)has been widely used to stabilize nonlinear systems and it always guarantees a better control effect than discontinuous strategies. Therefore, we first introduce a continuous-time dynamic output-feedback control law that can stabilize Lipschitz nonlinear systems under certain conditions. However, in many scenarios, Lipschitz nonlinear systems may be networked and large-scale,and may suffer from communication and control resource limitations. In such cases, the proposed continuous-time dynamic output-feedback control is not applicable;hence,an alternative strategy,namely the event-triggered control,is often considered to get rid of communication and control resource constraints. More explicitly, event-triggered control allows considerable reduction of resource usage while guaranteeing the stability and maintaining a certain level of control performance by determining event-triggered conditions accordingly (e.g.,Goebel et al. (2012), Yu and Antsaklis (2013), Peng and Yang(2013),Zhang JH and Feng(2014),Zhang XM and Han (2016), Gu et al. (2018), and Shu and Zhai (2020)). Note that event-triggered control involves the control updating problem; that is, the event-triggered control system is a hybrid system with both continuous-time and discrete-time dynamics. Research on stability and stabilization of nonlinear systems using event-triggered control is thus challenging and has received much attention in recent years(Donkers and Heemels,2012;Abdelrahim et al., 2016; Liu S et al., 2017; Zhang XM and Han,2017; Theodosis and Dimarogonas, 2019). Specifically, in Theodosis and Dimarogonas (2019), the problems of event-triggered and self-triggered control of nonlinear systems were addressed. The authors exploited certain stability assumptions for the continuous-time system as well as the Lipschitz properties of system dynamics, and presented a strategy to guarantee the stability of the system. Liu S et al.(2017)addressed an event-triggered dynamic outputfeedback robust model predictive control strategy,but for a class of discrete polytonic systems rather than nonlinear systems with Lipschitz continuoustime conditions. Donkers and Heemels (2012) studied an event-triggered mechanism which includes a dynamic output-feedback controller. While the system considered in Donkers and Heemels (2012)is a linear system, the controller adopts a simplified form of dynamic output feedback, where the controller reflects only the effect of the integral of the output rather than the usual dynamic outputfeedback controller in the form of proportionalintegral control. Moreover, the event-triggered condition in Donkers and Heemels (2012) was designed based on the networked output and input signals.Hence, to reduce the number of event-triggered times further, our goal is to establish a new eventtriggered condition based on the output and a timer variable.

Motivated by the above results, in this study,we establish an event-triggered dynamic outputfeedback controller, under which the hybrid system obtained from a Lipschitz nonlinear system is globally asymptotically stable. In summary, the main contributions are as follows:

1. A continuous-time dynamic output-feedback controller is designed for the considered Lipschitz nonlinear system, to ensure the global asymptotic stability of the closed-loop system.

2. We establish an event-triggered dynamic output-feedback controller with some eventtriggered conditions and an upper bound of the minimum inter-event interval. Moreover, by means of hybrid system theory, the obtained closed-loop system is proven to be globally asymptotically stable.

Notions and notations used throughout the paper are as follows: Denote Rnas then-dimensional Euclidean space. Denote R≥0as the set of nonnegative real numbers, i.e., R≥0:= [0,+∞). Denote N as the set of natural numbers, i.e., N :={0,1,...}. Given a vectorx ∈Rn,‖x‖denotes the Euclidean norm. In addition, the distance between a vectorxand a subsetA ?Rnis denoted by‖x‖A:= inf{‖x-y‖:y ∈A}. Given a continuously differentiable functionh: Rn →R and a functionf: Rn →Rn, the Lie derivative ofhatxin the direction offis denoted by〈?h(x),f(x)〉.DenoteA-1,AT, andλ(A) as the inverse, transpose, and eigenvalues of any square matrixA, respectively. LetA ＞0 (A ＜0) represent that matrixAis positive definite (negative definite).

2 Preliminary results and problem formulation

Consider a class of nonlinear control systems as follows:

wherex ∈Rnxdenotes the state vector,u ∈Rnuthe input, andy ∈Rnythe output. MatricesA,B,C, andEare real matrices of appropriate dimensions. In addition, the functionf(x) =[f1(x),f2(x),...,fnx(x)]Tis the nonlinear part such that for eachi(i= 1,2,...,nx), the following socalled Lipschitz conditions (Zhang Z et al., 2020)hold:

1.fi(0)=0;

2. There exist constant numbersli ＞0 satisfying‖fi(m(t))-fi(n(t))‖≤li‖m(t)-n(t)‖for anym(t),n(t)∈Rnx.

It follows immediately from the above conditions that system (1) has at least one equilibriumx*=0. However,in general,the equilibrium of system (1) may not be globally asymptotically stable.Therefore, we aim to find some suitableuto guarantee the global asymptotic stability of system (1)atx*. In addition, in many scenarios, some entries ofxcannot be controlled; i.e., some rows ofEhave all “0” elements. Without loss of generality,we assume thatE= [0I]T, where 0 represents a zero matrix of appropriate dimensions andIrepresents an identity matrix of appropriate dimensions.Correspondingly, we partition matricesA,B, and

Next, we will provide conditions under which the above closed-loop system (4) is globally asymptotically stable.whereqis a positive constant. Then, we have that closed-loop system (4) is globally asymptotically stable.

The proof of the above theorem can be found in Appendix.

Remark 1 Theorem 1 is given to establish sufficient conditions for the closed-loop system under continuous-time dynamic output-feedback control.In fact, it is also a new result compared to the literature related to the control of Lipschitz nonlinear systems.

Up to now, we have established conditions for the asymptotic stability of system (4) under continuous-time dynamic output-feedback control.Although continuous-time control performs well, in many scenarios, nonlinear systems may have limited communication and control resources because of their large-scale and underlying interconnection networks. To get rid of this obstacle,one needs to adopt other control strategies. Specifically,in the following,we will consider the so-called event-triggered control strategy to improve the communication efficiency.Moreover, event-triggered control allows considerable reduction of resource usage while guaranteeing stability and maintaining a certain level of control performance by determining event-triggered conditions accordingly (Peng and Yang, 2013). Hence,compared with the continuous-time control method,event-triggered control is more feasible and practical in many system applications. According to this kind of control strategy,one first provides event-triggered conditions and then uses the event detector to continuously monitor the event-triggered conditions to determine whether an “event” occurs or not. Once an event occurs,the event detector will transmit the newest output measurement to the event-triggered controller. Specifically, denote the time instants when an event occurs by a sequence

In summary, the schematic of the above event-triggered dynamic output-feedback control is depicted in Fig.1b with the continuous-time control method depicted in Fig. 1a.

Fig. 1 Continuous-time (a) and event-triggered (b)dynamic output-feedback control schematics (ZOH:zero-order holder)

whereτ ≥0 is a timer variable introduced to reduce the event frequency. For the sake of notation simplicity, we define

In the sequel, a hybrid system defined as above will be represented by the notationH=(F,F,G,J), or briefly,H.

In event-triggered control, transmissions occur whenever the event-triggered conditions are satisfied.Hybrid model (8) is well-equipped to describe dynamic systems with event-triggered control. Indeed,a transmission can be modeled as a jump of system (8) that will be generated whenever the eventtriggered condition is satisfied. This indicates that the state of system (8) enters inJat the transmission instant. When events are not generated, it means that the state of system (8) is inF, and the system evolves along with flows.

To conclude this section, we formally state the research problem of this study as follows:

Problem statement Consider the hybrid systemH= (F,F,G,J) given by Eq. (8). Our goal is to design the flow mapF,jump mapG,flow setF,and jump setJto guarantee that the hybrid systemHis uniformly globally asymptotically stable.

3 Main results

In this section, recalling the hybrid systemH= (F,F,G,J) in Eq. (8), we will first design the controller gainsAc,Bc,Cc, andDcand the event-triggered conditions. Second,based on the design of the controller and event-triggered conditions,we separately determine the flow mapF, jump mapG, flow setF, and jump setJfor the hybrid systemH. Moreover,we establish the stability criterion under which the obtained closed-loop hybrid system is globally asymptotically stable by the hybrid system theory without the Zeno phenomenon.

Before presenting the main results, some preliminaries and concepts related to the framework for stability analysis of hybrid systems will be reviewed first, following the conventions (Nesic et al., 2009;Goebel et al.,2012;Meslem and Prieur,2015).

1. For allj ∈N and for almost allt ∈Ij, whereIjis such thatIj×{j}:=domz ∩(R≥0×{j}), we havez(t,j)∈Fand ˙z(t,j)=F(z(t,j)).

2. For all (t,j)∈domzsuch that (t,j+1)∈domz, we havez(t,j)∈ Jandz(t,j+ 1) =G(z(t,j)).

Then, we have the following definition for analyzing the stability of hybrid system(8):

Definition 1 For hybrid system (8), the setA={(ζ,e,τ) :ζ= 0, e= 0, τ ∈R≥0}is uniformly globally pre-asymptotically stable, if the following properties hold:

1. Uniform global stability

There exists a functionα: R≥0→R≥0belonging to classK∞such that for any solutionz(t,j) to system(8),‖z(t,j)‖A ＜α(‖z(0,0)‖A)for all(t,j)∈domz.

2. Uniform global pre-attractivity

For eachε,r ＞0, there exists a positive real numberTsuch that for any solutionz(t,j) to system (8) with‖z(0,0)‖A ＜r, (t,j)∈domzandt+j ≥Timply‖z(t,j)‖A ＜ε, andAis said to be uniformly globally asymptotically stable(UGAS)when, in addition, the maximal solutions to system (8)are complete.

Furthermore, a functionV: domV →R is said to be a Lyapunov function candidate for the hybrid systemH= (F,F,G,J) if the following conditions hold:

1. ˉF ∪J ∪G(J)?domV;

2.Vis continuously differentiable on an open set containing ˉF, where ˉFdenotes the closure ofF.

Based on the aforementioned notions,we can introduce the following condition,under which a given closed setAis UGAS for systemH:

Lemma 1 (Goebel et al. (2012), Proposition 3.27) Consider a hybrid systemH=(F,F,G,J) and a closed setA ?Rn. Suppose thatVis a Lyapunov function candidate forHand there existα1,α2∈K∞andρ ∈PDsuch that the following inequalities hold:

If for eachr ＞0, there existγr ∈K∞andNr ≥0 such that for every solutionztoH,‖z(0,0)‖A ∈(0,r], (t,j)∈domz, andt+j ≥Timply thatt ≥γr(T)-Nr, thenAis uniformly globally preasymptotically stable forH.

Consequently,setAis UGAS for the hybrid systemH.

Proof According to Definition 1, we can obtain that setAis UGAS ifAis uniformly globally pre-asymptotically stable and the maximal solutions toHare complete. Clearly, because domzis unbounded, the maximal solutions toHare complete.Hence, it remains to show thatAis uniformly globally pre-asymptotically stable by adopting Lemma 1.

To begin with, assume that the system matrices in Eqs. (12) and (14) satisfy LMI (15). Let us consider the following Lyapunov function candidate:

To do so,we first separately deal with the Lie derivatives〈?V1(ζ),?fa(ζ,e)〉and〈?U(e),?fb(ζ,e)〉. Substituting ?fa(ζ,e) into〈?V1(ζ),?fa(ζ,e)〉yields

where Combining inequalities (17), (20), and(23)leads to it follows that inequality (25) is equivalent to inequality(15). Therefore,inequality(15)implies thatρ1belongs toPD.

Due to the nature ofF,G,F, andJ, scaling an initial condition by a positive constant causes solutions that are scaled by the same constant. Therefore, it follows that, except for solutions starting at the origin, each jump is followed by flowing for at least ˉτunits of time. In particular, (t,j)∈domzimplies thatj ≤1+t/ˉτ. In turn, (t,j)∈domzandt+j ≥ Timplyt ≥(T- 1)ˉτ/(ˉτ+ 1) =Tˉτ/(ˉτ+1)- ˉτ/(ˉτ+1). Letγr(T) =Tˉτ/(ˉτ+1)andNr= ˉτ/(ˉτ+1). Clearly,γr ∈K∞andNr ≥0.This satisfies the additional condition in Lemma 1.In summary,we have proven that setAis UGAS for the hybrid systemH.

Remark 4 It is worth pointing out that in Theorem 2, the controller gains are determined by the positive definite matricesN1,N2,N3,M1,M2,Π1,Π2,Π3, ?X4and LMI (15). In practice, to simplify the computation, one can chooseM1=q1I,M2=q2I,N1=q3I,N2=q4I,N3=q5I,Π1=q6I,Π2=q7I,Π3=q8I, and ˉX3=q9I,whereqi ＞0 (i= 1,2,...,9). Consequently, we can achieve the desired control effect by adjusting the selection ofqi(i=1,2,...,9).

Remark 5 Note that the LMI condition in Theorem 1 is less conservative than that in Theorem 2.According to the LMI condition in Theorem 2, the feasible solutions ofAc,Bc,Cc,andDcare included in the feasible solutions of Theorem 1. If no feasible solutions of LMI in Theorem 2 exist, we can use Theorem 1, but bear a higher control burden.

4 Numerical simulations

In this section,we will present two numerical examples to illustrate the effectiveness and applicability of our proposed methodologies. More explicitly,we first introduce an unstable neural network system and show that both the proposed continuoustime and event-triggered dynamic output-feedback controllers are able to stabilize this network system.Later,we adopt a genetic regulatory network system,which is also unstable,to illustrate the difference between event-triggered dynamic output-feedback controllers and static controllers.

Example 1 Consider a neural network of the following form(Sanchez and Perez,2003):

whereQis any positive definite matrix. This implies that system(26)has an equilibriumx=[0,0]Twhen the input is absent. One can verify that this equilibrium is unstable for system (26). Take the initial states as

The evolution trajectory of statesx1andx2are depicted in Fig. 2a.

Recall the proposed event-triggered dynamic output-feedback controller design method and construct the following event-triggered output-feedback controller:

whereˉy=y(tk)withtkbeing thekthevent-triggered time,andAc,Bc,Cc,andDcare scalers here,which will be determined later. Combining Eqs. (26) and(27), we have the following hybrid system:

Take the initial states as [x1(0),x2(0)]T=[3,4]T, [ξ(0),e(0),τ(0)]T= [0,0,0]T. It can be seen from Fig. 2b that the statesx1andx2converge to zero gradually under event-triggered dynamic output-feedback control, which means that the closed-loop hybrid system(28)is globally asymptotically stable. Define the event intervals asδk=tk+1-tkwithk= 0,1,...(Fig. 2c). The figure reflects the relationships between the triggering instants and the triggering intervals. It turns out that the sequence of{δk}is a combination of two kinds of intervals: one is larger than ˉτand the other is equal to ˉτ. Intuitively, the former is caused by the condition 240.3072eTe ≥yTy,and the latter one is a result of the minimum event-triggered interval constraint,which also prevents the Zeno phenomenon of system(28).

Fig. 2 Simulation results of Example 1: (a) evolutions of x(t) of the original system; (b) evolutions of x(t) of the closed-loop system under event-triggered dynamic output-feedback control; (c) evolutions of the intervals of event-triggered instants (the height of each vertical line shows the triggering interval between the current triggering instant and the last triggering instant); (d) evolutions of x(t) of the closed-loop system under continuous-time dynamic output-feedback control

In addition, to compare the effectiveness of continuous-time and event-triggered control, consider the following continuous-time dynamic outputfeedback controller:

Because the designed controller gains in the continuous-time dynamic output-feedback controller also satisfy condition(5)in Theorem 1,we have that the obtained continuous-time closed-loop system is globally asymptotically stable (Fig. 2d). Note that the controller gains in Eq.(29)are the same as those in the event-triggered dynamic output-feedback controller. This situation reflects the fact that these two closed-loop systems have similar convergence time.However,the event-triggered control method reduces the number of transmission times and the control burden.

Example 2 Consider a genetic regulatory network system of the following form(Li and Sun, 2010):

whereζ:= col(x,ξ),τ ≥0 is the timer variable,

Take the initial states as [x1(0),x2(0)]T=[30,80]Tand[ξ(0),e(0),τ(0)]T=[0,0,0]T. It can be seen that the statesx1andx2converge to zero gradually (Fig. 3b). This implies that the hybrid system is asymptotically stable and reflects the effectiveness of the proposed control strategy.

Again, to compare the effectiveness of our proposed dynamic output-feedback controller with that of a static output-feedback controller, consider a static output-feedback controller designed asu=Kˉy. To compare these two control strategies more fairly, we takeK=Dc=-21.6673 and the same event-triggered condition in the static outputfeedback method as those in the dynamic case. The simulations results in Figs. 3c and 3d show that the evolutions of states in the dynamic output-feedback control system have shorter convergence time. One intuitive reason for this phenomenon is that different from the static output-feedback controller, the dynamic output-feedback controller introduces an auxiliary variableξwhich contains the memory information ofyand hence guarantees a more effective control result.

Fig. 3 Simulation results of Example 2: (a) evolutions of p(t) of the genetic regulatory network system; (b)evolutions of x(t) of the closed-loop error system under event-triggered dynamic output-feedback control;(c) evolutions of x1(t) under dynamic output-feedback (red line) and static output-feedback (blue-circle line)control; (d) evolutions of x2(t) under dynamic output-feedback (red line) and static output-feedback (bluecircle line) control. References to color refer to the online version of this figure

5 Conclusions

In this paper, we addressed the problem of dynamic output-feedback control for a class of Lipschitz nonlinear systems. Both continuous-time dynamic output-feedback control and event-triggered dynamic output-feedback control have been investigated for such Lipschitz nonlinear systems. For the continuous-time dynamic output-feedback control strategy,sufficient conditions for stability of the obtained closed-loop system have been established.In addition,for the event-triggered dynamic outputfeedback control strategy, based on the design of the dynamic output-feedback controller gains and the event-triggered conditions, a hybrid system has been presented. Sufficient conditions have been proposed to ensure the stability of the hybrid system and an upper bound of the minimum inter-event interval has been provided to avoid the Zeno phenomenon. Finally, simulation results of two control strategy classes for nonlinear systems have been provided to confirm the efficiency of the proposed results.

One of our future research topics will be the study of robust event-triggered dynamic outputfeedback control for nonlinear systems with external disturbance. In addition, the extension of the theoretical results to generalized Lipschitz systems is another future topic.

Contributors

Zhiqian LIU and Xuyang LOU designed the research.Zhiqian LIU processed the data. Zhiqian LIU and Xuyang LOU drafted the paper. Jiajia JIA helped organize and polish the paper. Zhiqian LIU and Xuyang LOU revised and finalized the paper.

Compliance with ethics guidelines

Zhiqian LIU, Xuyang LOU, and Jiajia JIA declare that they have no conflict of interest.

Appendix: Proof of Theorem 1