# Input-to-State Stabilization of a Class of Uncertain Nonlinear Systems via Observer-Based Event-Triggered Impulsive Control

Authors: Xiangru Xing; Jin-E Zhang
Journal: Complexity
 (2020)

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## Abstract

This article concerns the problem of input-to-state stabilization for a group of uncertain nonlinear systems equipped with nonabsolutely available states and exogenous disturbances. To appropriately cope with these partially measurable state variables as well as dramatically minimize controller updating burden and communication costs, an event-triggered mechanism is skillfully devised and an observer-based impulsive controller with the combination of sample control is correspondingly presented. By resorting to the iterative method and Lyapunov technology, some sufficient criteria are established to guarantee the input-to-state stability of the newly uncertain controlled system under the employed controller, in which an innovative approximation condition as to the uncertain term is proposed and the linear matrix inequality technique is utilized for restraining sophisticated parameter uncertainties. Furthermore, the Zeno behavior in the proposed event-triggered strategy is excluded. The control gains and event-triggered mechanism parameters are conjointly designed by resolving some inequalities of linear matrix. Eventually, the availability and feasibility of the achieved theoretical works are elucidated by two simulation examples.


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## Body

## 1. Introduction

Since it is originally put forward by [1, 2], input-to-state stabilization has caught widespread attention [3–5], attributing to its performance in characterizing dynamical systems reaction to exogenous disturbances with bounded magnitude. The property of input-to-state stabilization, crudely speaking, symbolizes that the system state will ultimately approach the origin neighborhood whose dimension is in direct proportion to the size of the system input regardless of the magnitude of the initial state. With this characteristic, a system is asymptotically stabilizable under disturbance-free condition and has the evolution of bounded state in the bounded perturbation circumstance. Indeed, input-to-state stability behavior can characterize robustness and stability on dynamic systems possessed disturbances, in which the corresponding stabilization problem has a great signality for the control issue of [5–7]. Input-to-state stability is incipiently introduced for continuous systems to evaluate dynamical behaviors, which is especially a fundamental conception for investigating robust dynamics on nonlinear systems influenced by noise, inputs, or interferences [8]. Afterwards, it is diffusely capitalized for stabilizing controller synthesis and stabilization analysis of diverse discontinuous systems, to name a few, switched systems [9, 10], stochastic systems [5, 11], and fuzzy systems [12, 13].Accompanied by the prompt development of some technologies such as digital control for resource-limited models and sensors incorporated embedded microprocessors, event-triggered impulsive control strategy, more recently, has been highly valued. On the one hand, the impulsive system, composed of discrete dynamics and continuous dynamics, is an important hybrid system in which the uncontinuous behavior is a momentary state jump occurring at given moments, while the consecutive behavior is usually expressed as differential equation. Correspondingly, impulsive control is a control approach that the control signals are transmitted to a system only at certain moments. In comparison with continuous control [14, 15], it has the advantages of only discrete control which is required for deriving the desired performance, discontinuity, and stronger robustness. Consequently, the control approach is extensively applied in practice, such as ecosystem management [16, 17], satellite orbit transfer [18], secure communication [19], pharmacokinetics [20], and complex switched network [21]. Furthermore, controlling the operation of systems all the while is unnecessary or even impossible in practice. In population model [16], for example, it is merited to release predators at appropriate discrete circumstances, rather than the continuous instances for controlling the amount of a category insect. Moreover, as [22] amply demonstrated, impulsive control allows utilizing small control impulses as much as possible to stabilize a type of chaotic system. Not merely does it reduce redundant information transmission, but it increases the robustness of disturbances rejection. On the other hand, event-based control, as the name implies, is the strategy that event is triggered by some elaborate state-based or output-based event conditions to update the control input, which compared to the conventional time-triggered control is capable of avoiding unnecessary communication since a system adjusts the sampling rate adaptively according to the current situation [23, 24]. Specifically, the issue of self-learning optimal supervision on discrete systems via event-driven formulation is investigated in [24]. And the critics learning standard is improved for the design of nonlinear H∞ state-feedback control based on events [25]. Distinct from the extant achievements involving sectionally continuous or consecutive control inputs, event-triggered impulsive control is able to dramatically minimize communication load and communication cost as well as enhance robustness, which, for these reasons, is deserving increasing attention. Simultaneously, the integration of two control strategies also creates tremendous challenges in designing appropriate controller.As yet, some (but few) significant accomplishments about the event-triggered impulsive control such as [26–29] have been reported. Taking [26] as an instance, the synchronization issue on multiple neural networks with disconnected switching topology and delay under this control strategy is studied. Nevertheless, systems are generally affected by some uncertain factors such as human error, random disturbance, information loss, inherent deviation, or environmental noise. The uncertainty caused by these factors is referred to as the parametric uncertainty that is perhaps foremost provenance of model uncertainty [30]. Without taking model uncertainty into account, it seems to be far-fetched and preposterous in reality for analyzing performances of various systems like estimating the property indexes on steady state. In this condition, none of the before-mentioned results are valid. Besides, in the control engineering application, when it comes to the fact that the system states may not be fully available because of implementation costs or physical restrictions, it becomes crucial and inevitable to formulate the event-triggered impulsive control strategy according to practical observer measurements. At this juncture, once the incomplete testability of states and the uncertainty of parameters are incorporated into the characterization of nonlinear systems, then these uncertainties may give rise to a totally new rule with more uncertain antecedents and results. What is exhilarating is that there is no work on the observer-based event-triggered impulsive control strategy to achieve the input-to-state property of uncertain nonlinear systems. After all, it is of more difficulty to find a feasible analytical framework compared with the nominal nonlinear systems. Moreover, in comparison with the previous methods, the robust handling for uncertain parameters during the course of system performance implementation becomes increasingly tricky as the number of uncertain parameters surges. Therefore, the theoretical challenges and technical deficiencies urge us to explore the actual performance evaluation for nonlinear systems with parametric uncertainties under observer-based control.The abovementioned analysis motivates us to focus on issues of both input-to-state stability and event-triggered impulsive control scheme design on a type of uncertain nonlinear systems with incomplete measurable state variables and exogenous disturbances in this paper. Firstly, we establish a category of newly uncertain nonlinear systems, where the uncertainty terms are legitimately estimated by capitalizing on a creationary approximation condition of uncertainties, matrix synthesis method, and some inequalities of linear matrix. Secondly, a novel observer is constructed on the uncertain nonlinear system, in which the information between plant and observer is transmitted as impulses. In particular, the impulsive controllers are dependent upon the partial measurement output of observer and plant, which can eliminate the adverse effects of output data loss attributed to the external environment. Thirdly, an applicable observer-based event-triggered mechanism is designed and an event-triggered impulsive control strategy is correspondingly constructed, which could lessen burden of sampling and information transmission. At last, several sufficient criteria on excluding the Zeno behavior and analyzing the input-to-state stability property are developed, meanwhile, which suggest that a more extensible framework in complex dynamics can be explored through taking full advantage of a range of the employed ideas and methods.The content of the remaining sections is summarized as follows. Section2 puts forward the model and preparatory works for a kind of uncertain nonlinear systems. Section 3 furnishes primary research results. In addition, Section 4 corroborates the validity of the derived results by two numerical simulations. Finally, conclusion is exhibited in Section 5.


## 2. Preliminaries and Model Description

### 2.1. Notations

Throughout this article,ℝq×p, ℝq, and ℕ+ are separately the set of all q×p real matrices and q−dimensional Euclidean space and the set of positive integers. I stands for an identity matrix with matched dimensionality in matrices or matrix inequalities. 0 in matrices is a zero matrix of appropriate dimensions. Let D and DS denote the 2-norm of matrix D and the supremum of D on the interval S, respectively. For a matrix D, D−1, DT, λmaxD, and λminD represent severally its inverse, transposition, maximum eigenvalue, and minimum eigenvalue. The symbol ∗ is defined as the symmetric term in a matrix. s∨d and s∧d represent the maximum and minimum of s and d, respectively. D>0 and D<0 mean that D are symmetric positive definite and symmetric negative definite separately. Let HeD=D+DT, K∞=ϕ∈Cℝ+,ℝ+ϕ0=0,lims⟶∞ϕs=∞, and ϕs is strictly increasing in s, and Kℒ=ψ∈Cℝ+×ℝ+,ℝ+ψs,j, for each fixed j≥0, belongs to the function of class K as regards s, but ψs,j, for each fixed s≥0, is strictly decreasing to 0 as j⟶∞.


### 2.2. Some Preliminaries and Problem Formulation

A class of uncertain nonlinear systems incorporated exogenous disturbances is of the following form:(1)x˙t=B+ΔBxt+Bd+ΔBdfxt+A+ΔAut+C+ΔCνt,yt=D+ΔDxt,in which t≥t0, xt∈ℝn, yt∈ℝp, and νt∈ℝn are the system state, the measurement output, and measurable locally bounded exogenous disturbances, respectively; ut=u1t+u2t means the control input in which u1t is the sample control input and u2t is the Dirac delta control input; a nonlinear vector-valued function f:ℝn⟶ℝn satisfies some conditions that will be provided in the sequel, and x˙t represents the right-hand derivative of xt. B, Bd, A, C, and D are constant matrices, and ΔB, ΔBd, ΔA, ΔC, and ΔD are the norm-bounded uncertain parameters.Given that the incompletely procurable system states can generate the ineffectiveness of state-feedback controllers, an observer-based controller is considered in this paper, and the state observer for uncertain system (1) is constructed by(2)x¯˙t=Bx¯t+Bdfx¯t+Au3t+Cνt,y¯t=Dx¯t,where t≥t0; x¯t∈ℝn; and y¯t∈ℝp are separately the estimated state and the estimated output. The control input u3t of observer is described as(3)u3t=∑a=1∞K¯yt+Ky¯t−K¯y¯tδt−ta,where t∈ta,ta+1; K and K¯ are control gains; δ is the Dirac delta function, which is also called the impulsive control function. And the impulsive time sequence taa=1∞ satisfies t1<t2<⋯<ta<⋯ and lima⟶∞ta=+∞. It is well-known that the Dirac delta function has two properties: for any constants c and Δ1>0 and function gt, (1) δt−c=0 only when t≠c; (2) ∫c−Δ1c+Δ1gtδt−cdt=gc. Then, by virtue of (2) and (3) and the properties of function δ, what we can see is that u3t=0 at t≠ta and a∈ℕ+, and for any constant Δ>0 that is small enough,(4)x¯ta+Δ−x¯ta−Δ=∫ta−Δta+ΔBx¯s+Bdfx¯s+Cνsds+∫ta−Δta+Δ∑a=1∞AK¯ys+Ky¯s−K¯y¯sδs−tads=∫ta−Δta+ΔBx¯s+Bdfx¯s+Cνsds+AK¯yta+Ky¯ta−K¯y¯ta,where ∫ta−Δta+Δ∑a=1∞AK¯ys+Ky¯s−K¯y¯sδs−tads, which can be regarded as the convolution in the interval ta−Δ,ta+Δ based on the properties of function δ, represents the sum of the effects of all unit impulses on the observer state over ta−Δ,ta+Δ.LetΔ⟶0 and Δx¯t=x¯ta+−x¯ta−; then we can infer that(5)Δx¯t=AK¯yta+Ky¯ta−K¯y¯ta,t=ta.By means of the above calculation, the controlleru3t with function δ can make the observer state change instantaneously in the discrete time sequence taa=1∞ so as to achieve the impulsive effect. Thus, observer (2) is converted into an impulsive control system as follows:(6)x¯˙t=Bx¯t+Bdfx¯t,t≠ta,Δx¯t=AK¯yt−+Ky¯t−−K¯y¯t−,t=ta,y¯t=Dx¯t,where the left-continuous case of the estimate of xt is always considered; that is, x¯t=x¯t−≜limΔ⟶0x¯t−Δ.According to the aforementioned observer, the control inputsu1t and u2t are devised as(7)u1t=Lx¯ta,t∈ta,ta+1,u2t=∑a=1∞Ky¯tδt−ta,t∈ta,ta+1,where L is the gain of u1t. In a similar way, uncertain system (1) can be transformed into(8)x˙t=B+ΔBxt+Bd+ΔBdfxt+A+ΔALx¯ta+C+ΔCνt,t≠ta,Δxt=A+ΔAKy¯t−,t=ta,yt=D+ΔDxt.Without loss of generality, we always assume that the state of system (8) is left-continuous. For forward complete impulsive systems, there is no fundamental difference between utilizing left-continuous model and employing right-continuous model. Then by defining the tracking error et=xt−x¯t, the error system can be expressed as(9)e˙t=Bet+Bdfxt−fx¯t+A+ΔALx¯ta+C+ΔCνt+ΔBxt+ΔBdfxt,t≠ta,Δet=−ΔAKD−AK¯Det−+ΔAKD−AK¯ΔDxt−,t=ta.Remark 1.
Since uncertain system (1) is a category of impulsive systems, coupled with the incomplete measurability of the system state, we need to construct an appropriate observer and subsequently establish an applicable error system related to plant and observer. Based on the prerequisite of ensuring real-time monitoring, fault-tolerant control, easy realization, and so on, as a result, it is necessary and natural to construct observer (2) which is only influenced by impulsive. One more point needs noting that the controllers u2t and u3t designed by us can exert positive effects on the unstable systems and meanwhile control them only at the impulsive instant. In this way, u2t and u3t can stabilize systems (1) and (2), respectively, for ages, while reducing unnecessary computing costs.
Let(10)℘t=xTt,eTtT,ζt=νTt,νTtT,ℱt=fTxt,fTxt−fTx¯tT,and then the argument system can be deduced as(11)℘˙t=B˜℘t+B˜dℱt+A˜L˜℘ta+C˜ζt,t≠ta,℘t=K˜℘t−,t=ta,where(12)B=B+ΔB0ΔBB,B˜d=Bd+ΔBd0ΔBdBd,A˜=A+ΔA00A+ΔA,L˜=L−LL−L,C˜=​C+ΔC00C+ΔC​,K˜=I+A+ΔAKD−A+ΔAKDΔAKD−AK¯ΔDI−ΔAKD−AK¯D.
Furthermore,∀t∈ta,ta+1,a∈ℕ+, argument system (11) is rewritten as(13)℘˙t=A1℘t+A2ℱt+A3ϱt+A4ζt,t≠ta,℘t=K˜℘t−,t=ta,where ϱt=℘ta−℘t;(14)A1=B+ΔB+A+ΔAL−A+ΔALΔB+A+ΔALB−A+ΔAL,A2=B˜d=Bd+ΔBd0ΔBdBd,A3=A˜L˜=A+ΔAL−A+ΔALA+ΔAL−A+ΔAL,A4=C˜=​C+ΔC00C+ΔC​.
An adaptive event-triggered mechanism, determining the continuously updated controller works at the instantstaa∈ℕ+ known as the triggered time sequence, is introduced to decrease the burden of updating and communication in control. It is notable that the system states are imperfectly accessible, so the event-triggered mechanism included exogenous disturbances as well as the system and observer output is designed. By defining(15)ℑt=yTt,yt−y¯tTT,D˜=D+ΔD0ΔDD,ϱ¯t=D˜ϱt,the event-triggered mechanism is formulated as(16)ta+1=ta+1⋆∧ta+ℓ,ta+1⋆=inft≥taℋt≥0,where the event generator function ℋt=ϱ¯t2−ηℑta2−ρζtt0,t2,t∈ta,ta+1; parameters η>0, ρ>0, and ℓ>0 in which ℓ is a forced triggered constant. Denote by ta⋆a=1∞ the event-triggered time sequence that it is determined by function ℋt. For t≥taa∈ℕ+, the next event ta+1⋆ will be triggered only when the correlative measurement reaches or surpasses the stated threshold, and then, the next triggered instant (impulsive instant) will be generated by comparing the obtained event-triggered time with the forced triggered time. In addition, it is worth mentioning that the aforesaid two sequences may differ depending on the selected parameters ℓ, η, and ρ.
In what follows, two assumptions are proposed around the uncertain terms and the nonlinear function.Assumption 1.
(17)ΔBΔBdΔAΔCΔD=MEtF1F2F3F4F5,in which Et is the unknown time-varying matrix with αI≤ETtEt≤I, the adjustment coefficient of the uncertain term α∈0,1, and F1, F2, F3, F4, and F5 are constant matrices with compatible dimensionality.Remark 2.
Different from existing achievements, such as [21, 30, 31], this paper has more uncertain parameters. Note that, in the practical application, each program will inevitably be subjected to the actual limitation of imprecise modeling for controlled plant and affected by external factors like environmental noise. Therefore, it is favorable and urgent for increasing the number of uncertainties to describe a larger range nonlinear system.Remark 3.
Only the norm-bounded uncertainties are taken into account in this article to efficaciously avoid needlessly intricate notations and restrain parameter uncertainty. In accordance with Assumption1 and several linear matrix inequalities, the uncertainties ΔB, ΔBd, ΔA, ΔC, and ΔD can be reasonably eliminated. Moreover, compared with the conventional constraint conditions of uncertain terms, the adjustment coefficient α is added in this paper, which not merely does not change the norm value range of the uncertain terms but also can ingeniously resolve the input-to-state stability problem of fairly sophisticated system. Even though the uncertainty parameters are also present at other singular structures, the subsequent results could be popularized to this circumstance in parallel.Assumption 2.
Suppose that there exists a scalarβ>0 such that the nonlinearity f satisfies fy1−fy2≤βy1−y2, ∀y1,y2∈ℝn. Particularly, f0=0.
Hereafter, a definition and several lemmas are introduced for latter use.Definition 1.
For every initial conditiont0,℘0 and each measurable locally bounded exogenous disturbance ζt (see [1]), system (13) is said to be input-to-state stabilizable under the given event-triggered mechanism (16) if there exist functions Y∈Kℒ and Z∈K∞ such that the solution ℘t satisfies(18)℘t≤Y℘0,t−t0+Zζtt0,t,t≥t0.Lemma 1.
Given constant matricesU, P, and V with suitable dimensionality and a matrix function ℳt (see [23, 32]),(1)
∀ε1>0 and ℳTtℳt≤I, then(19)PℳtV+VTℳTtPT≤1ε1PPT+ε1VTV.(2)
∀ε2>0 such that ε2VTV<I and ℳTtℳt≤I, then(20)U+PℳtVU+PℳtVT≤UI−ε2VTV−1UT+1ε2PPT.Particularly, whenU≡0, we obtain(21)PℳtVPℳtVT≤1ε2PPT.Lemma 2.
∀s1,s2∈ℝq (see [33]), the inequality(22)s1Ts2+s2Ts1≤s1TRs1+s2TR−1s2,holds, where R∈ℝq×q is a positive definite matrix.Lemma 3.
Given constant matricesℬ1,ℬ2,andℬ3 (see [34]), where ℬ1=ℬ1T and ℬ2>0, then(23)ℬ1+ℬ3Tℬ2−1ℬ3<0,if and only if(24)ℬ1ℬ3Tℬ3−ℬ2<0,or−ℬ2ℬ3ℬ3Tℬ1<0.


## 2.1. Notations

Throughout this article,ℝq×p, ℝq, and ℕ+ are separately the set of all q×p real matrices and q−dimensional Euclidean space and the set of positive integers. I stands for an identity matrix with matched dimensionality in matrices or matrix inequalities. 0 in matrices is a zero matrix of appropriate dimensions. Let D and DS denote the 2-norm of matrix D and the supremum of D on the interval S, respectively. For a matrix D, D−1, DT, λmaxD, and λminD represent severally its inverse, transposition, maximum eigenvalue, and minimum eigenvalue. The symbol ∗ is defined as the symmetric term in a matrix. s∨d and s∧d represent the maximum and minimum of s and d, respectively. D>0 and D<0 mean that D are symmetric positive definite and symmetric negative definite separately. Let HeD=D+DT, K∞=ϕ∈Cℝ+,ℝ+ϕ0=0,lims⟶∞ϕs=∞, and ϕs is strictly increasing in s, and Kℒ=ψ∈Cℝ+×ℝ+,ℝ+ψs,j, for each fixed j≥0, belongs to the function of class K as regards s, but ψs,j, for each fixed s≥0, is strictly decreasing to 0 as j⟶∞.


## 2.2. Some Preliminaries and Problem Formulation

A class of uncertain nonlinear systems incorporated exogenous disturbances is of the following form:(1)x˙t=B+ΔBxt+Bd+ΔBdfxt+A+ΔAut+C+ΔCνt,yt=D+ΔDxt,in which t≥t0, xt∈ℝn, yt∈ℝp, and νt∈ℝn are the system state, the measurement output, and measurable locally bounded exogenous disturbances, respectively; ut=u1t+u2t means the control input in which u1t is the sample control input and u2t is the Dirac delta control input; a nonlinear vector-valued function f:ℝn⟶ℝn satisfies some conditions that will be provided in the sequel, and x˙t represents the right-hand derivative of xt. B, Bd, A, C, and D are constant matrices, and ΔB, ΔBd, ΔA, ΔC, and ΔD are the norm-bounded uncertain parameters.Given that the incompletely procurable system states can generate the ineffectiveness of state-feedback controllers, an observer-based controller is considered in this paper, and the state observer for uncertain system (1) is constructed by(2)x¯˙t=Bx¯t+Bdfx¯t+Au3t+Cνt,y¯t=Dx¯t,where t≥t0; x¯t∈ℝn; and y¯t∈ℝp are separately the estimated state and the estimated output. The control input u3t of observer is described as(3)u3t=∑a=1∞K¯yt+Ky¯t−K¯y¯tδt−ta,where t∈ta,ta+1; K and K¯ are control gains; δ is the Dirac delta function, which is also called the impulsive control function. And the impulsive time sequence taa=1∞ satisfies t1<t2<⋯<ta<⋯ and lima⟶∞ta=+∞. It is well-known that the Dirac delta function has two properties: for any constants c and Δ1>0 and function gt, (1) δt−c=0 only when t≠c; (2) ∫c−Δ1c+Δ1gtδt−cdt=gc. Then, by virtue of (2) and (3) and the properties of function δ, what we can see is that u3t=0 at t≠ta and a∈ℕ+, and for any constant Δ>0 that is small enough,(4)x¯ta+Δ−x¯ta−Δ=∫ta−Δta+ΔBx¯s+Bdfx¯s+Cνsds+∫ta−Δta+Δ∑a=1∞AK¯ys+Ky¯s−K¯y¯sδs−tads=∫ta−Δta+ΔBx¯s+Bdfx¯s+Cνsds+AK¯yta+Ky¯ta−K¯y¯ta,where ∫ta−Δta+Δ∑a=1∞AK¯ys+Ky¯s−K¯y¯sδs−tads, which can be regarded as the convolution in the interval ta−Δ,ta+Δ based on the properties of function δ, represents the sum of the effects of all unit impulses on the observer state over ta−Δ,ta+Δ.LetΔ⟶0 and Δx¯t=x¯ta+−x¯ta−; then we can infer that(5)Δx¯t=AK¯yta+Ky¯ta−K¯y¯ta,t=ta.By means of the above calculation, the controlleru3t with function δ can make the observer state change instantaneously in the discrete time sequence taa=1∞ so as to achieve the impulsive effect. Thus, observer (2) is converted into an impulsive control system as follows:(6)x¯˙t=Bx¯t+Bdfx¯t,t≠ta,Δx¯t=AK¯yt−+Ky¯t−−K¯y¯t−,t=ta,y¯t=Dx¯t,where the left-continuous case of the estimate of xt is always considered; that is, x¯t=x¯t−≜limΔ⟶0x¯t−Δ.According to the aforementioned observer, the control inputsu1t and u2t are devised as(7)u1t=Lx¯ta,t∈ta,ta+1,u2t=∑a=1∞Ky¯tδt−ta,t∈ta,ta+1,where L is the gain of u1t. In a similar way, uncertain system (1) can be transformed into(8)x˙t=B+ΔBxt+Bd+ΔBdfxt+A+ΔALx¯ta+C+ΔCνt,t≠ta,Δxt=A+ΔAKy¯t−,t=ta,yt=D+ΔDxt.Without loss of generality, we always assume that the state of system (8) is left-continuous. For forward complete impulsive systems, there is no fundamental difference between utilizing left-continuous model and employing right-continuous model. Then by defining the tracking error et=xt−x¯t, the error system can be expressed as(9)e˙t=Bet+Bdfxt−fx¯t+A+ΔALx¯ta+C+ΔCνt+ΔBxt+ΔBdfxt,t≠ta,Δet=−ΔAKD−AK¯Det−+ΔAKD−AK¯ΔDxt−,t=ta.Remark 1.
Since uncertain system (1) is a category of impulsive systems, coupled with the incomplete measurability of the system state, we need to construct an appropriate observer and subsequently establish an applicable error system related to plant and observer. Based on the prerequisite of ensuring real-time monitoring, fault-tolerant control, easy realization, and so on, as a result, it is necessary and natural to construct observer (2) which is only influenced by impulsive. One more point needs noting that the controllers u2t and u3t designed by us can exert positive effects on the unstable systems and meanwhile control them only at the impulsive instant. In this way, u2t and u3t can stabilize systems (1) and (2), respectively, for ages, while reducing unnecessary computing costs.
Let(10)℘t=xTt,eTtT,ζt=νTt,νTtT,ℱt=fTxt,fTxt−fTx¯tT,and then the argument system can be deduced as(11)℘˙t=B˜℘t+B˜dℱt+A˜L˜℘ta+C˜ζt,t≠ta,℘t=K˜℘t−,t=ta,where(12)B=B+ΔB0ΔBB,B˜d=Bd+ΔBd0ΔBdBd,A˜=A+ΔA00A+ΔA,L˜=L−LL−L,C˜=​C+ΔC00C+ΔC​,K˜=I+A+ΔAKD−A+ΔAKDΔAKD−AK¯ΔDI−ΔAKD−AK¯D.
Furthermore,∀t∈ta,ta+1,a∈ℕ+, argument system (11) is rewritten as(13)℘˙t=A1℘t+A2ℱt+A3ϱt+A4ζt,t≠ta,℘t=K˜℘t−,t=ta,where ϱt=℘ta−℘t;(14)A1=B+ΔB+A+ΔAL−A+ΔALΔB+A+ΔALB−A+ΔAL,A2=B˜d=Bd+ΔBd0ΔBdBd,A3=A˜L˜=A+ΔAL−A+ΔALA+ΔAL−A+ΔAL,A4=C˜=​C+ΔC00C+ΔC​.
An adaptive event-triggered mechanism, determining the continuously updated controller works at the instantstaa∈ℕ+ known as the triggered time sequence, is introduced to decrease the burden of updating and communication in control. It is notable that the system states are imperfectly accessible, so the event-triggered mechanism included exogenous disturbances as well as the system and observer output is designed. By defining(15)ℑt=yTt,yt−y¯tTT,D˜=D+ΔD0ΔDD,ϱ¯t=D˜ϱt,the event-triggered mechanism is formulated as(16)ta+1=ta+1⋆∧ta+ℓ,ta+1⋆=inft≥taℋt≥0,where the event generator function ℋt=ϱ¯t2−ηℑta2−ρζtt0,t2,t∈ta,ta+1; parameters η>0, ρ>0, and ℓ>0 in which ℓ is a forced triggered constant. Denote by ta⋆a=1∞ the event-triggered time sequence that it is determined by function ℋt. For t≥taa∈ℕ+, the next event ta+1⋆ will be triggered only when the correlative measurement reaches or surpasses the stated threshold, and then, the next triggered instant (impulsive instant) will be generated by comparing the obtained event-triggered time with the forced triggered time. In addition, it is worth mentioning that the aforesaid two sequences may differ depending on the selected parameters ℓ, η, and ρ.
In what follows, two assumptions are proposed around the uncertain terms and the nonlinear function.Assumption 1.
(17)ΔBΔBdΔAΔCΔD=MEtF1F2F3F4F5,in which Et is the unknown time-varying matrix with αI≤ETtEt≤I, the adjustment coefficient of the uncertain term α∈0,1, and F1, F2, F3, F4, and F5 are constant matrices with compatible dimensionality.Remark 2.
Different from existing achievements, such as [21, 30, 31], this paper has more uncertain parameters. Note that, in the practical application, each program will inevitably be subjected to the actual limitation of imprecise modeling for controlled plant and affected by external factors like environmental noise. Therefore, it is favorable and urgent for increasing the number of uncertainties to describe a larger range nonlinear system.Remark 3.
Only the norm-bounded uncertainties are taken into account in this article to efficaciously avoid needlessly intricate notations and restrain parameter uncertainty. In accordance with Assumption1 and several linear matrix inequalities, the uncertainties ΔB, ΔBd, ΔA, ΔC, and ΔD can be reasonably eliminated. Moreover, compared with the conventional constraint conditions of uncertain terms, the adjustment coefficient α is added in this paper, which not merely does not change the norm value range of the uncertain terms but also can ingeniously resolve the input-to-state stability problem of fairly sophisticated system. Even though the uncertainty parameters are also present at other singular structures, the subsequent results could be popularized to this circumstance in parallel.Assumption 2.
Suppose that there exists a scalarβ>0 such that the nonlinearity f satisfies fy1−fy2≤βy1−y2, ∀y1,y2∈ℝn. Particularly, f0=0.
Hereafter, a definition and several lemmas are introduced for latter use.Definition 1.
For every initial conditiont0,℘0 and each measurable locally bounded exogenous disturbance ζt (see [1]), system (13) is said to be input-to-state stabilizable under the given event-triggered mechanism (16) if there exist functions Y∈Kℒ and Z∈K∞ such that the solution ℘t satisfies(18)℘t≤Y℘0,t−t0+Zζtt0,t,t≥t0.Lemma 1.
Given constant matricesU, P, and V with suitable dimensionality and a matrix function ℳt (see [23, 32]),(1)
∀ε1>0 and ℳTtℳt≤I, then(19)PℳtV+VTℳTtPT≤1ε1PPT+ε1VTV.(2)
∀ε2>0 such that ε2VTV<I and ℳTtℳt≤I, then(20)U+PℳtVU+PℳtVT≤UI−ε2VTV−1UT+1ε2PPT.Particularly, whenU≡0, we obtain(21)PℳtVPℳtVT≤1ε2PPT.Lemma 2.
∀s1,s2∈ℝq (see [33]), the inequality(22)s1Ts2+s2Ts1≤s1TRs1+s2TR−1s2,holds, where R∈ℝq×q is a positive definite matrix.Lemma 3.
Given constant matricesℬ1,ℬ2,andℬ3 (see [34]), where ℬ1=ℬ1T and ℬ2>0, then(23)ℬ1+ℬ3Tℬ2−1ℬ3<0,if and only if(24)ℬ1ℬ3Tℬ3−ℬ2<0,or−ℬ2ℬ3ℬ3Tℬ1<0.


## 3. Main Results

This section is devoted to the following tripartite through theoretical analysis and demonstrates the following:(T1) The presence of the lower bound of adjacent impulse instants is testified, whereafter, the Zeno behavior can be excluded.(T2) The resultant augmented system which is equipped with parameter uncertainties and exogenous disturbances is input-to-state stabilizable where the uncertainties are tactfully subdued.(T3) The control gains and event-triggered scheme parameters are devised without strong constrained condition under system (13) corresponding stability.Before verifying the above statements, it is necessary to introduce some symbols:(25)ℬ=B00B,Δℬ=ΔB0ΔB0,ℬd=Bd00Bd,Δℬd=ΔBd0ΔBd0,A=A00A,ΔA=ΔA00ΔA,C=C00C,ΔC=ΔC00ΔC,D=D00D,ΔD=ΔD0ΔD0,ℳ=M00M,ℰt=Et00Et,K1=I+AKD−AKD0I−AK¯D,K2=KD−KDKD−KD,K3=000−AK¯.According to Assumption1, we have(26)ΔℬΔℬdΔAΔCΔD=ℳℰtℱ1ℱ2ℱ3ℱ4ℱ5,where(27)ℱ1=F10F10,ℱ2=F20F20,ℱ3=F300F3,ℱ4=F400F4,ℱ5=F50F50.Theorem 1.
Under event-triggered scheme (16), then (T1) holds, where the positive lower bound J of adjoining impulsive moments conforms to J=lnφ1η∧ρ/φ2∨φ3+1/φ1∧ℓ, constants η, ρ, and ℓ are specified in (16), and(28)φ1=2ℬ+AL˜+2ℳ⋅ℱ1+2ℳ⋅ℱ3L˜+2βℬd+2βℳ⋅ℱ2+2AL˜+2ℳ⋅ℱ3L˜φ3,φ2=ℬ+AL˜+ℳ⋅ℱ1+ℳ⋅ℱ3L˜+βℬd+βℳ⋅ℱ2,φ3=‖C‖⋅‖D‖+‖C‖⋅‖ℳ‖⋅‖ℱ5‖+‖D‖⋅‖ℳ‖⋅‖ℱ4‖+‖ℳ‖2⋅‖ℱ4‖⋅‖ℱ5‖.Proof.
For the sake of checking on (T1), we consider the following three cases:Case (i). The generation of the triggered time sequence taa=1∞ depends entirely on the event-triggered time sequence ta∗a=1∞. Based on ϱ¯t=D˜ϱt=ℑta−ℑt, in this case, the upper right Dini derivative of ϱ¯t in the interval ta,ta+1 is calculated as(29)D+ϱ¯t2=2ϱ¯tD+ϱ¯t=2ϱ¯t⋅D˜℘˙t≤2ϱ¯t⋅D˜A1℘t+D˜A2ℱt+D˜A3ϱt+D˜A4ζt≤2ϱ¯t⋅A1⋅ℑt+2βϱ¯t⋅A2⋅ℑt+2A3⋅ϱ¯t2+2ϱ¯t⋅D˜⋅A4⋅ζt≤A1ϱ¯t2+ℑt2+βA2ϱ¯t2+ℑt2+2A3⋅ϱ¯t2+D˜⋅A4ϱ¯t2+ℑt2≤A1ϱ¯t2+ℑta−ϱ¯t2+βA2ϱ¯t2+ℑta−ϱ¯t2+2A3⋅ϱ¯t2+D˜⋅A4ϱ¯t2+ζt2≤2A1+2βA2+2A3+D˜⋅A4ϱ¯t2+A1+βA2ℑta2+D˜⋅A4⋅ζt2≤2ℬ+AL˜+2ℳ⋅ℱ1+2ℳ⋅ℱ3L˜+2βℬd+2βℳ⋅ℱ2+2AL˜+2ℳ⋅ℱ3L˜+D+ℳ⋅ℱ5⋅C+ℳ⋅ℱ4ϱ¯t2+ℬ+AL˜+ℳ⋅ℱ1+ℳ⋅ℱ3L˜+βℬd+βℳ⋅ℱ2ℑta2+D+ℳ⋅ℱ5⋅C+ℳ⋅ℱ4⋅ζt2≤φ1ϱ¯t2+φ2ℑta2+φ3ζtt0,ta+12.Letφa=φ2ℑta2+φ3ζtt0,ta+12; then,(30)D+ϱ¯t2≤φ1ϱ¯t2+φa,∀t∈ta,ta+1.Usinge−φ1t−ta to premultiplication and postmultiplication in (30) results in(31)D+e−φ1t−taϱ¯t2≤φae−φ1t−ta.By consideringϱ¯t0=0 and integrating both sides of (31) from ta to t, we have(32)ϱ¯t2≤φaφ1eφ1t−ta−1,∀t∈ta,ta+1.When mechanism (16) is triggered, we can derive ηℑta2+ρζtt0,ta+12=ϱ¯ta+12. Together with φa=φ2ℑta2+φ3ζtt0,ta+12, it follows that(33)η∧ρℑta2+ζtt0,ta+12≤eφ1ta+1−ta−1φ1×φ2∨φ3×ℑta2+ζtt0,ta+12,which indicates that(34)ta+1−ta≥lnφ1η∧ρ/φ2∨φ3+1φ1>0,a∈ℕ+.Case (ii). Only the forced triggered time sequence t0+nℓa=1∞ exists in the sequence taa=1∞. Obviously, in this case, ta+1−ta=ℓ>0,a∈ℕ+.Case (iii). The sequence taa=1∞ is composed of the event-triggered instants ts∗s∈ℕ+ and the forced triggered instants tjj∈ℕ+. If the Zeno behavior lives in argument system (13), it must be that a finite time interval owns infinite impulse jumps in this case. To exhibit this phenomenon, suppose that T presents the accumulation time (or Zeno time) on the finite time interval t0,T. By defining ς=T−ℓ/2∨t0, it is apparent that countless of impulsive instants appears in the interval ς,T. Denote by tN0+ss=1∞∈ς,T the subsequence of taa=1∞ satisfying tN0+s⟶T as s⟶∞, where integer N0≥0. If there is no forced triggered moment in ς,T, similar to the discussion of Case (i), we conclude that tN0+s⟶∞ as s⟶∞, which contradicts the definition of the accumulation time T. If there exists tj∈tN0+ss=1∞ for some j∈ℤ+ over the interval ς,T, recalling the definition of ς, it can be deduced that only one tj∈tN0+ss=1∞, which implies that the triggered moments totally consist of the event-triggered moments in tj,T. Then it follows from Case (i) that tN0+s⟶∞ as s⟶∞. Hence, the Zeno behavior is precluded in Case (iii).
From the foregoing discussion, the lower bound of neighboring impulsive instants is ultimately acquired, which symbolizes that the Zeno behavior can be eliminated.Remark 4.
When designing event-triggered mechanism (16), ζtt0,t2 cannot be superseded by ζt2. It is on account of a discovery that the correlation ζt2 over ta,ta+1 and ζta+12 is not easily obtained in the abovementioned reasoning process of excluding the Zeno phenomenon. Therefore, it is essential for the device of (16) to take the supremum of ζt2 in t0,t. In addition, the study in [29] assists us in detecting the indispensability of the forced trigger condition in (16).Remark 5.
As described earlier, mechanism (16) is presented to select the optimal triggered moment according to the steady state of resulting system (13). Based on the results of Theorem 1, it can be proved that the designed event-triggered mechanism is effective. By comparison with the existing results of the uncertain models, such as [21, 35, 36], although they reduce the transmission of information, impulsive controller with fixed impulsive moments, in design, is still conservative. Now, in this paper, the controller based on event-triggered mechanism only is updated at the triggered moment. By this means, the burden of controller update can be decreased without affecting accurate control. Moreover, the sampling control adopted in this paper is only dependent on the state information of observer at the triggered moment, which can reduce the communication between the equipment under test and the observer.Assumption 3.
K˜ is a nonsingular matrix, and there exists a constant γ>0 such that λmaxK˜≤e−1/2γ.Theorem 2.
Uncertain system (13) is input-to-state stable via the event-triggered scheme (16); suppose that for given parameters η∈0,1/2, ρ>0, and ℓ>0 and the control gains K, K¯∈ℝq×p, and L∈ℝq×n, there exist matrices P>0 and Hι>0ι=1,2,3 and constants ϑ^>0, ϑ>0, γ>0, and εb>0b=1,2,3,…,58, such that(35)Ψ100∗Ψ20∗∗Ψ3>0,(36)Ω1DTPDℬdDTPDAL˜DTPDC∗−H100∗∗−H20∗∗∗−H3<0,(37)Ω2ℳK2Tℱ3TK3ℳℱ5T∗−ε57I000∗∗−1ε57I00∗∗∗−ε58I0∗∗∗∗−1ε58I≤0,(38)γ−ϑ˜ℓ>0,where(39)Ψ1=Iℱ5ℳ∗1ϖ1I,Ψ2=IΛ2ℳ∗1ϖ2I,Ψ3=IℳTPℳ∗1ϖ3I,Ω1=−ϑI+ℛ1,Ω2=HeK1−2e−1/2γI,ϑ˜=ϑ^+1−2ηD+αℳ⋅ℱ52ϑ+2ηλmaxℛ21−2ηD+αℳ⋅ℱ52λminP,ℛ1=HeDTPDΛ1+μ1Λ3TΛ3+μ2Λ2TΛ2+μ3ℱ5Tℱ5+μ4ℱ1Tℱ1+μ5L˜Tℱ3Tℱ3L˜+μ6Λ1Tℱ5Tℱ5Λ1+1ε8Λ1TΛ2TΛ2Λ1+μ7βℱ2Tℱ2β+μ8βℬdTℱ5Tℱ5ℬdβ+1ε25βℬdTΛ2TΛ2ℬdβ+βH1β,ℛ2=μ9L˜Tℱ3Tℱ3L˜+μ10L˜TATℱ5Tℱ5AL˜+1ε37L˜TATΛ2TΛ2AL˜+H2,Λ1=ℬ+AL˜,Λ2=ℳTPD,Λ3=ℳTDTPD,ϖ1=maxε5,ε7,ε17,ε20,ε24,ε32,ε36,ε44,ε48,ε56,ϖ2=maxε10,ε12,ε27,ε39,ε51,ϖ3=maxε14,ε16,ε19,ε29,ε31,ε41,ε43,ε53,ε55,μ1=ε1+ε2+ε21+ε33+ε45,μ2=ε3+ε4+ε6+ε22+ε23+ε34+ε35+ε46+ε47,μ3=ε8+ε9+ε11+ε13+ε15+ε18+ε25+ε26+ε28+ε30+ε37+ε38+ε40+ε42+ε49+ε50+ε52+ε54,μ4=1ε1+1ε4ε5+1ε9ε10+1ε15ε16ε17,μ5=1ε2+1ε6ε7+1ε11ε12+1ε18ε19ε20,μ6=1ε3+1ε13ε14,μ7=1ε21+1ε23ε24+1ε26ε27+1ε30ε31ε32,μ8=1ε22+1ε28ε29,μ9=1ε33+1ε35ε36+1ε38ε39+1ε42ε43ε44,μ10=1ε34+1ε40ε41.Proof.
Suppose that℘t=℘t,t0,℘0 is the solution of system (13) with the initial value t0,℘0. Due to the imperfect measurability of the system states, the Lyapunov functional related to the relevant outcomings is considered as Vt=ℑTtPℑt. ∀t∈ta,ta+1, a∈ℤ+, the derivative of Vt can be calculated that(40)V˙=2ℑTtPℑ˙t=2℘TtD˜TPD˜℘˙t≤2℘TtD˜TPD˜A1℘t+ϑ^℘TtD˜TPD˜℘t+2℘TtD˜TPD˜A2ℱt+2℘TtD˜TPD˜A3ϱt+2℘TtD˜TPD˜A4ζt.
It follows from Lemma1 and conditions (26) and (35) that(41)HeDTPDΔℬΔAL˜≤1ε1ℱ1Tℱ11ε2L˜Tℱ3Tℱ3L˜+Λ3TΛ3ε1Iε2I,HeDTPΔDΛ1ΔℬΔAL˜≤1ε3Λ1Tℱ5Tℱ5Λ11ε4ε5ℱ1Tℱ11ε6ε7L˜Tℱ3Tℱ3L˜+Λ2TΛ2ε3Iε4Iε6I,HeΔDTPDΛ1≤ε8ℱ5Tℱ5+1ε8Λ1TΛ2TΛ2Λ1,HeΔDTPDΔℬΔAL˜≤1ε9ε10ℱ1Tℱ11ε11ε12L˜Tℱ3Tℱ3L˜+ℱ5Tℱ5ε9Iε11I,HeΔDTPΔDΛ1ΔℬΔAL˜≤1ε13ε14Λ1Tℱ5Tℱ5Λ11ε15ε16ε17ℱ1Tℱ11ε18ε19ε20L˜Tℱ3Tℱ3L˜+ℱ5Tℱ5ε13Iε15Iε18I,2℘TtDTPDΔℬdℱt≤1ε21ℱTtℱ2Tℱ2ℱt+ε21℘TtΛ3TΛ3℘t,2℘TtDTPΔDℬdℱtΔℬdℱt≤1ε22ℱTtℬdTℱ5Tℱ5ℬdℱt1ε23ε24ℱTtℱ2Tℱ2ℱt+℘TtΛ2TΛ2℘tε22Iε23I,2℘TtΔDTPDℬdℱtΔℬdℱt≤1ε25ℱTtℬdTΛ2TΛ2ℬdℱt1ε26ε27ℱTtℱ2Tℱ2ℱt+℘Ttℱ5Tℱ5℘tε25Iε26I,2℘TtΔDTPΔDℬdℱtΔℬdℱt≤1ε28ε29ℱTtℬdTℱ5Tℱ5ℬdℱt1ε30ε31ε32ℱTtℱ2Tℱ2ℱt+℘Ttℱ5Tℱ5℘tε28Iε30I,2℘TtDTPDΔAL˜ϱt≤1ε33ϱTtL˜Tℱ3Tℱ3L˜ϱt+ε33℘TtΛ3TΛ3℘t,2℘TtDTPΔDAL˜ϱtΔAL˜ϱt≤1ε34ϱTtL˜TATℱ5Tℱ5AL˜ϱt1ε35ε36ϱTtL˜Tℱ3Tℱ3L˜ϱt+℘TtΛ2TΛ2℘tε34Iε35I,2℘TtΔDTPDAL˜ϱtΔAL˜ϱt≤1ε37ϱTtL˜TATΛ2TΛ2AL˜ϱt1ε38ε39ϱTtL˜Tℱ3Tℱ3L˜ϱt+℘Ttℱ5Tℱ5℘tε37Iε38I,2℘TtΔDTPΔDAL˜ϱtΔAL˜ϱt≤1ε40ε41ϱTtL˜TATℱ5Tℱ5AL˜ϱt1ε42ε43ε44ϱTtL˜Tℱ3Tℱ3L˜ϱt+℘Ttℱ5Tℱ5℘tε40Iε42I,2℘TtDTPDΔCζt≤ε45℘TtΛ3TΛ3℘t+1ε45ζTtℱ4Tℱ4ζt,2℘TtDTPΔDCζtΔCζt≤1ε46ζTtCTℱ5Tℱ5Cζt1ε47ε48ζTtℱ4Tℱ4ζt+℘TtΛ2TΛ2℘tε46Iε47I,2℘TtΔDTPDCζtΔCζt≤1ε49ζTtCTΛ2TΛ2Cζt1ε50ε51ζTtℱ4Tℱ4ζt+℘Ttℱ5Tℱ5℘tε49Iε50I,2℘TtΔDTPΔDCζtΔCζt≤1ε52ε53ζTtCTℱ5Tℱ5Cζt1ε54ε55ε56ζTtℱ4Tℱ4ζt+℘Ttℱ5Tℱ5℘tε52Iε54I,and from Assumption 2 and Lemma 2, we derive(42)ℱt≤β℘t2℘TtDTPDℬdℱt≤℘TtDTPDℬdH1−1ℬdTDTPD℘t+℘TtβH1β℘t,2℘TtDTPDAL˜ϱt≤℘TtDTPDAL˜H2−1L˜TATDTPD℘t+ϱTtH2ϱt,2℘TtDTPDCζt≤℘TtDTPDCH3−1CTDTPD℘t+ζTtH3ζt.And then we infer that(43)2℘TtD˜TPD˜A1℘t+2℘TtD˜TPD˜A2ℱt=2℘TtD+ΔDTPD+ΔDΛ1+Δℬ+ΔAL˜℘t+2℘TtD+ΔDTPD+ΔDℬd+Δℬdℱt≤℘Ttℛ1℘t+℘TtDTPDℬdH1−1ℬdTDTPD℘t,(44)2℘TtD˜TPD˜A3ϱt=2℘TtD+ΔDTPD+ΔDAL˜+ΔAL˜ϱt≤ϱTtℛ2ϱt+℘TtDTPDAL˜H2−1L˜TATDTPD℘t,(45)2℘TtD˜TPD˜A4ζt=2℘TtD+ΔDTPD+ΔDC+ΔCζt≤ζTtℛ3ζt+℘TtDTPDCH3−1CTDTPD℘t,where(46)ℛ3=1ε45+1ε47ε48+1ε50ε51+1ε54ε55ε56ℱ4Tℱ4+1ε46+1ε52ε53CTℱ5Tℱ5C+1ε49CTΛ2TΛ2C+H3.
Consequently, under Lemma3 and (35), substituting (43)–(45) into (40) gives that, ∀t∈ta,ta+1,a∈ℕ+,(47)V˙≤ϑ℘Tt℘t+ϑ^℘TtD˜TPD˜℘t+ϱTtℛ2ϱt+ζTtℛ3ζt≤ϑ^Vt+ϑ℘t2+λmaxℛ2ϱt2+λmaxℛ3ζt2.
∀t∈ta,ta+1, it holds from Assumption 1, (26), and the fact ϱ¯t2≤ηℑta2+ρζtt0,t2 that(48)ϱt2≤2η1−2ηD+αℳ⋅ℱ52ℑt2+ρ1−2ηD+αℳ⋅ℱ52ζtt0,t2,which supplied into (47) yields that(49)V˙t≤ϑ˜Vt+ψζtt0,t,∀t∈ta,ta+1,where(50)ψc=λmaxℛ3+ρλmaxℛ21−2ηD+αℳ⋅ℱ52c2.
By virtue of condition (38), a positive constant ϑ¯>ϑ˜ could be spotted with ease such that γ−ϑ¯ℓ>0, and it could be derived that(51)V˙t≤ϑ¯Vt,whenever Vt≥σψζtt0,t, σ=1/ϑ¯−ϑ˜.
Whent=ta+1, it is deduced from (13) that(52)Vt=ℑTtPℑt=℘TtD˜TPD˜℘t=℘Tt−K˜TD˜TPD˜K˜℘t−.
By Assumption3, there exists an invertible matrix ℜ such that(53)ℜ−1K˜ℜ≤e−1/2γI.
In accordance with Lemma1, we discover that(54)HeΔAK2≤1ε57ℳℳT+ε57K2Tℱ3Tℱ3K2,HeK3ΔD≤1ε58K3ℳℳTK3T+ε58ℱ5Tℱ5.
Afterwards, combined with Lemma3, (37), (52), and (53), we obtain(55)Vt≤e−γVt−.
Pay attention thatVt0>σψζt0. Accordingly, we can define t^1=inft≥t0Vt≤σψζtt0,t, which symbolizes that Vt≥σψζtt0,t,∀t∈t0,t^1. Consider this case that t^1<∞ initially. If there is no impulse over t0,t^1, it gives from (51) that(56)Vt≤eϑ¯t−t0Vt0,∀t∈t0,t^1.
If there exist a few impulsive momentst11<t12<⋯<t1m<⋯ on t0,t^1, we can deduce that(57)Vt≤eϑ¯t−t0Vt0,t∈t0,t^1.
Based on (55), we get(58)Vt11≤e−γVt11−≤e−γ+ϑ¯t11−t0Vt0,which illustrates that(59)Vt≤eϑ¯t−t11Vt11≤e−γ+ϑ¯t11−t0Vt0,t∈t11,t12.
Equally, the inequalityVt12≤e−2γ+ϑ¯t12−t0Vt0 holds. Paralleling to the above deduction, we could get an inference that, ∀t∈t0,t^1,(60)Vt≤e−nt,t0γ+ϑ¯t−t0Vt0≤e−nt,t0γ¯+ϑ¯ℓVt0,where nt,t0 denote the number of impulsive moments in the corresponding interval and γ¯=γ−ϑ¯ℓ>0.
As a result, whent^1<∞, whether impulse exists in t0,t^1 or not, the following inequality is always attainable:(61)Vt≤e−nt,t0γ¯+ϑ¯ℓVt0,∀t∈t0,t^1.
Whent^1=∞, executing the similar argument as the case of t^1<∞, we could reason out(62)Vt≤e−nt,t0γ¯+ϑ¯ℓVt0,∀t∈t0,∞.
Associating with the situation oft^1<∞, let tˇ1=inft≥t^1Vt≥σψζtt0,t; the priority concern is the case of tˇ1<∞. Evidently, whether impulse appears on t^1,tˇ1 or not, it can be invariably obtained that(63)Vt≤σψζtt0,t,∀t∈t^1,tˇ1.
Similarly, whentˇ1=∞; Vt≤σψζtt0,t, ∀t∈t^1,∞.
Combined with the circumstance thattˇ1<∞, it follows that(64)Vt≤e−nt,t0γ¯+ϑ¯ℓVt0+σψζtt0,t,∀t∈t0,tˇ1,and then let t^2=inft≥tˇ1Vt≤σψζtt0,t. When t^2<∞, we give priority to the case of no impulse over tˇ1,t^2; we could derive from Vtˇ1=σψζtt0,tˇ1 that(65)Vt≤eϑ¯t−tˇ1Vtˇ1≤e−ϑ¯ℓσψζtt0,t,t∈tˇ1,t^2.
If there exist impulsive momentst21<t22<⋯<t2m<⋯, it holds that(66)Vt≤eϑ¯t−tˇ1σψζtt0,t,∀t∈tˇ1,t21.
According to (55), we have(67)Vt21≤e−γVt21−≤e−γ+ϑ¯t21−tˇ1σψζtt0,t,which suggests that(68)Vt≤e−γ+ϑ¯t−tˇ1σψζtt0,t,∀t∈t21,t22.
On account ofγ−ϑ¯ℓ>0, performing semblable operation, we get(69)Vt≤eϑ¯ℓσψζtt0,t,∀t∈tˇ1,t^2.
Whent^2=∞, it can be inferred in repeating similar iterations that(70)Vt≤eϑ¯ℓσψζtt0,t,∀t∈tˇ1,∞.
Together witht^2<∞, we derive(71)Vt≤e−nt,t0γ¯+ϑ¯ℓVt0+eϑ¯ℓσψζtt0,t,∀t∈t0,t^2.
Arguing in the identical manner, we conclude that(72)Vt≤e−nt,t0γ¯+ϑ¯ℓVt0+eϑ¯ℓσψζtt0,t,∀t∈t0,∞,which fulfills the proof of (T2) based on Definition 1.Remark 6.
Obviously, Assumption3 and condition (37) play a complementary role in the proof, which simultaneously explains the rationality and significance of Assumption 3.
Next, for the purpose of achieving (T3), the following hypothesis needs to be proposed.Assumption 4.
MatricesD and A with appropriate dimensionality are of full-row rank and full-column rank separately.Theorem 3.
Assume that, for given constantsειι=1,2,3, there exist matrices P>0, Si>0, Ξi, and Θi,i=1,2, and constants ϑ^>0, ϑ>0, γ>0, and εb>0b=1,2,3,…,58 such that (35) and(73)HeΣ1ℳΣ2Tℱ3TΣ3ℳℱ5T∗−ε57I000∗∗−1ε57I00∗∗∗−ε58I0∗∗∗∗−1ε58I<0,(74)ΩSDTPDℬdSSDTPDAΘSDTPDCS∗−ε1S00∗∗−ε2S0∗∗∗−ε3S<0,where(75)Σ1=1−e−12γI+AΞ1−AΞ101−e−12γI−Ξ2D,Σ2=Ξ1−Ξ1Ξ1−Ξ1,Σ3=000−Ξ2,S=S100S2,Θ=Θ1−Θ2Θ1−Θ2,Ω=−ϑS2+ℛ¯1,ℛ¯1=μ1SDTPDℳℳTDTPDS+μ2SDTPℳℳTPDS+μ3Sℱ5Tℱ5S+μ4Sℱ1Tℱ1S+μ5ΘTℱ3Tℱ3Θ+μ6SℬTℱ5Tℱ5ℬS+μ6ΘTATℱ5Tℱ5AΘ+1ε8SℬTDTPℳℳTPDℬS+1ε8ΘTATDTPℳℳTPDAΘ+μ7βSℱ2Tℱ2Sβ+μ8βSℬdTℱ5Tℱ5ℬdSβ+HeSDTPDℬS+1ε25βSℬdTDTPℳℳTPDℬdSβ+HeSDTPDAΘ+ε1βSβ,μ1=ε1+ε2+ε21+ε33+ε45,μ2=ε3+ε4+ε6+ε22+ε23+ε34+ε35+ε46+ε47,μ3=ε8+ε9+ε11+ε13+ε15+ε18+ε25+ε26+ε28+ε30+ε37+ε38+ε40+ε42+ε49+ε50+ε52+ε54,μ4=1ε1+1ε4ε5+1ε9ε10+1ε15ε16ε17,μ5=1ε2+1ε6ε7+1ε11ε12+1ε18ε19ε20,μ6=1ε3+1ε13ε14,μ7=1ε21+1ε23ε24+1ε26ε27+1ε30ε31ε32,μ8=1ε22+1ε28ε29,and then system (13) is input-to-state stabilizable under event-triggered strategy (16). In addition, the control gains K, K¯, and L and parameters η, ρ, and ℓ are jointly devised by(76)K=Ξ1DTDDT−1,K¯=ATA−1ATΞ2,L=Θ1S1−1=Θ2S2−1,0<η<κ2+2κ,0<ℓ<λminPγλminPϑ^+ϑ,ρ>0,where(77)κ=λminPλmin2Sγ/ℓ−ϑ^−ϑ/λminPD+αℳ⋅ℱ52λmaxℛ¯2,ℛ¯2=μ9ΘTℱ3Tℱ3Θ+μ10ΘTATℱ5Tℱ5AΘ+1ε37ΘTATDTPℳℳTPDAΘ+ϵ2λmin2SS−1,μ9=1ε33+1ε35ε36+1ε38ε39+1ε42ε43ε44,μ10=1ε34+1ε40ε41.Proof.
The controller gainsK and K¯, under Theorem 2, could be determined by (73), which leads to (37). Meanwhile, let Hι=ειS−1ι=1,2,3; then we use diagS−1,S−1,S−1,S−1 to postmultiplication and premultiplication in inequality (74); condition (36) can be derived. Furthermore, condition (38) is linearized to (76). This accomplishes the proof.Remark 7.
More recently, the design problems of adaptive controller are studied for uncertain nonlinear systems with diverse structures, such as lower triangular structure [37], nonlower-triangular structure [38, 39], and nonstrict-feedback structure [40, 41]. Inspired by these interesting and pioneering works, the event-triggered impulsive control scheme designed and the approach adopted in this paper may be extended in these areas. In addition, through the enlightenment of some excellent works like [42–44], whether our model can be extended to various memristive neural network models would be a probable and rewarding research topic.


## 4. Illustrative Examples

For the purpose of corroborating the merit and effectiveness of the developed results, two numerical examples are considered in this section.Example 1.
Consider a dynamic model of a vehicle radar servo system. Following [45, 46], the nonlinear dynamic model with parametrized uncertainties can be modeled as(78)x˙1=x2,x˙2=−b+Δbx2+c+Δcu−d+ΔdFc+fVf,y=x1,where x1 and x2 are the angular and the angular speed of the motor, respectively; b=Br/J+mr2/G2, Δb=ΔBr/ΔJ+mr2/G2, c=Kf/J+mr2/G2, Δc=ΔKf/ΔJ+mr2/G2, d=r/GJ+mr2/G2, and Δd=r/GΔJ+mr2/G2; J, m, r, G, Br, Kf, iq, Fc, and f are the rotor inertia, the mass of radar antenna, the radius of the gearing wheel, the gearing ratio, the rotor shaft friction, the torque constant, the current in q axis, the dry friction force, and the wind resistance coefficient, respectively; Vf is a nonlinear function in regard to radar antenna velocity. Besides, the parameters ΔBr, ΔJ, and ΔKf are not precisely known, which are dependent on several factors in the nonlinear dynamics; for instance, the direction of the radar antenna changes with vehicle vibration and gusty winds. Suppose that Fc is an interfering variable that varies over time. To stabilize this system in the input-to-state sense, now, some parameters are selected as(79)Br=1.7×10−4N⋅m⋅s,J=48×10−3kg⋅m2,m=1000kg,r=50mm,G=1,Kf=1.2N⋅mA,f=1.4,Fc=tanht,Vf=0.2tanhx2,ΔBr=0.2×10−4sintN⋅m⋅s,ΔJ=3×10−3costkg⋅m2,ΔKf=0.2sintN⋅mA.
Based on Theorem2 proposed in this paper, let εi=20,i=1,2,…,29, εj=25,j=30,31,…,49, εl=30,andl=50,51,…,58, and choose ϑ^=0.45, ϑ=3×10−6, and γ=0.03. Then, the control gains and the event-triggered mechanism parameters can be designed by utilizing MATLAB toolbox as follows:(80)K=−0.2164−0.1922,K¯=0.21490.1974,L=−1.6450−1.1250,0<η<0.2253×10−3,0<ℓ<0.56.
In order to achieve simulation, we chooseη=0.18×10−3, ρ=0.1, and ℓ=0.4. When the designed controller does not act on system (78), as shown in Figure 1, then the system is unstable. However, under control, the change trend of error between the observer state and the system state, the dynamics of the controller updates (the points on the t-axis mean that the controller does not update), and the sampling dynamics of the event triggering mechanism are illustrated in Figures 2–4, respectively. Thereinto, in Figure 4, the point with a value of 1 corresponds to the event trigger sampling time, and the point with a value of 0.5 corresponds to the forced trigger sampling time. Assume that the total elapsed time of system is 20s, if the output signal is transmitted by a fixed impulsive time sequence or time-triggered scheme, and the sampling period is 0.05s; then the number of data traffic will be 400, whereas, as shown in Table 1, the amount of data communication can be dramatically decreased by adopting the designed event-triggered scheme. In conclusion, the aforementioned simulation outcomes reflect that when the system state is not completely measurable and uncertainties and external interferences exist in the system, the vehicle radar servo system can accurately estimate the corresponding input in real time and save unnecessary communication resources.Figure 1
Responses ofx1t and x2t.Figure 2
Responses ofe1t and e2t.Figure 3
The dynamics of control inputs.Figure 4
The dynamics of event-triggered mechanism.Table 1
Amount of date communication under varyingℓ, η, and ρ.
ℓ,η0.1,0.000430.2,0.000240.4,0.00018​ρ=0.01856855​ρ=0.05746151​ρ=0.1715948Example 2.
To ensure the participation of more uncertainties and the establishment of Assumption4, uncertain nonlinear system (1) incorporating two subsystems is taken into consideration, whose parameters are set as follows:(81)B=1001,Bd=−1.700−1.7,A=0.20.2,C=420.381900420.3819,D=−3526.200−3526.2,fx=0.3tanhx10.2tanhx2,νt=sinttanht,M=−149.575900−149.5759,F1=0.11000.11,F2=1001,F3=−0.000545−0.000545,F4=F5=0.001000.001.
If system (1) has no the control input, as portrayed in Figure 5, its dynamic behavior is unstable under exogenous disturbance and uncertainties. Next, without capturing the information of system states, an observer-based event-triggered impulsive controller will be designed to implement the input-to-state stabilization attribute on system (1). In accordance with Theorem 3, let εi=15,i=1,2,…,29; εj=18,j=30,31,…,49; εl=20,l=50,51,…,58 and choose ϑ^=0.1, ϑ=10−10, γ=0.001, ε1=160, ε2=36520, ε3=36971, and β=18.9505; then we devise the control gains as(82)K=2.64462.6446,K¯=−3.8464−3.8464,L=0.010,and the parameter of event-triggered mechanism as 0<η<0.0267, ρ>0, and 0<ℓ<0.643, so as to achieve the control performance of the input-to-state stabilization. To facilitate the simulation, we select η=0.02, ρ=0.4, and ℓ=0.4; then event-triggered mechanism (16) can be given by(83)ta+1=ta+1⋆∧ta+0.4,ta+1⋆=inft≥taℋt≥0,where(84)ℋt=ϱ¯t2−0.2ℑta2−0.4ζtt0,t2.
By exploiting such mechanism and controller gains, the trajectories of states and errors are separately illustrated in Figures6 and 7, where the evolution of control inputs is exhibited in Figure 8. Moreover, viewed from the triggered dynamics of Figure 9, the event-triggered scheme lessens unnecessary sampling and avoids repeated sampling.
The advantage of the presented strategy in reducing data traffic, in the following, will be further verified. Assume that the total elapsed time of the closed-loop system (13) is 20s, analyzing in the same way as Example 1, then we can clearly observe from Table 2 that the event-triggered scheme proposed in this paper is superior to the periodic sampling scheme put forward in [21, 35, 36]. This table also exhibits that the sampling amount varies with the parameters ℓ, η, and ρ. It can be seen that the greater the value of ℓ or ρ is, the less the data communication would be generated and thus the more resources would be saved.Figure 5
Responses ofx1t and x2t.Figure 6
Responses ofx1t and x2t.Figure 7
Responses ofe1t and e2t.Figure 8
The dynamics of control inputs.Figure 9
The dynamics of event-triggered mechanism.Table 2
Amount of date communication under varyingℓ, η, and ρ
ℓ,η0.1,0.390.2,0.130.4,0.02ρ=0.2826652ρ=0.3725949ρ=0.4695646


## 5. Concluding Remarks

This article investigates innovatively the input-to-state stabilization and controller design for a type of uncertain nonlinear systems included partially measurable states and exogenous disturbances. A suitable event-triggered mechanism is constructed which decides when the controllers are updated. With the combination of sample control, subsequently, an observer-based impulsive controller has been devised to warrant the performance of input-to-state stabilization on the uncertain controlled system. Particularly, the Zeno behavior in presented control scheme has been eliminated. With the aid of several analysis strategies and the linear matrix inequality technology, some sufficient criteria are deduced to guarantee the input-to-state stabilization. In addition, substantial uncertain parameters are reasonably estimated by exploiting some constant matrix inequalities and an innovative hypothesis. Although the reasoning process is slightly more complicated, fortunately, the result of theoretical analysis and simulations can demonstrate that the observer-based event-triggered impulsive scheme can work well. In some circumstances, the controller perhaps oversteps its physical restrictions that possibly lead to the controlled system subjected to rigorous performance degradation. Consequently, it is a probable research work for observer-based event-triggered saturated impulsive control on a category of nonlinear controlled systems in the future.



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