Robust Stabilization of Nonlinear Systems Using Periodic Event-triggered Control

# Robust Stabilization of Nonlinear Systems Using Periodic Event-triggered Control

Xiangru Xu, Adam M. Tahir, Behçet Açıkmeşe111William E. Boeing Department of Aeronautics & Astronautics, University of Washington, Seattle, WA, USA. Emails: {xiangru,tahiram,behcet}@uw.edu.
###### Abstract

Periodic event-triggered control (PETC) has the advantages of both sampled-data control and event-triggered control, and is well-suited for implementation in digital platforms. However, existing results on PETC design mainly focus on linear systems, and their extension to nonlinear systems are still sparse. This paper investigates PETC design for general nonlinear systems subject to external disturbances, and provides sufficient conditions to ensure that the closed-loop system implemented by PETC input-to-state stable, using state feedback and observer-based output feedback controllers, respectively. For incrementally quadratic nonlinear systems, sufficient conditions for PETC design are provided in the form of linear matrix inequalities. The sampling period and triggering functions for all the cases considered are provided explicitly. Two examples are given to illustrate the effectiveness of the proposed method.

## 1 Introduction

Digital control systems are normally executed in a time-triggered fashion, under which the sensors and actuators are accessed periodically. In contrast, event-triggered control (ETC) executes the sensing and actuation only when certain triggering rules are satisfied; this can be seen as adding feedback to the sensing and actuation processes (see a recent survey paper [18] and references therein). The ETC paradigm is designed to avoid unnecessary waste of computation/communication resources by reducing the number of sensing/actuation executions, while still guaranteeing a desirable closed-loop performance [5, 31, 29, 33, 24, 15, 7]; this shows potential in applications for systems with limited energy resources, such as networked control systems and embedded control systems. In reality, the full state information is hard to obtain, so observer-based output feedback ETC design has also been considered; however, output feedback ETC can not be extended straightforwardly from state feedback ETC, especially for nonlinear systems, for which the separation principle does not hold in general [12, 2, 1, 34].

Since the triggering condition of ETC has to be monitored continuously, it is difficult to implement ETC in digital platforms directly. To overcome this problem, periodic event-triggered control (PETC) was proposed [20, 17, 19]. By evaluating the triggering conditions periodically and deciding whether to update the sensing/actuation at each sampling time, PETC inherits both the benefits of ETC and sampled-data control, and can be implemented on a standard digital platform. Furthermore, Zeno phenomenon is avoided since the sampling period is a lower bound for the minimum inter-execution time. Note that although ETC for discrete-time models can be considered as PETC (e.g., see [17, 13]), the inter-sample behavior of the original continuous-time systems are not captured in the discrete-time analysis.

PETC design for continuous-time linear systems was investigated in [20] where three approaches were presented by considering the closed-loop system as an impulsive system, a piecewise linear system, and a perturbed linear system, respectively. PETC design for continuous-time nonlinear systems is more difficult because of an intrinsic difficulty: the discrete-time dynamics of a nonlinear system can not be exactly known from its continuous-time dynamics in general [28, 14, 6, 32, 4, 36]. State feedback PETC design for nonlinear systems was given in [32] to ensure the globally asymptotically stability of the closed-loop system, with the sampling period bound provided explicitly; state feedback PETC design for nonlinear systems was investigated in [6] by redesigning the event function of an existing continuous ETC system using overapproximation techniques, such that the control performance guarantees for the continuous ETC system can be preserved; output feedback PETC for nonlinear Lipschitz systems was considered using impulsive observers in [14], to guarantee practical stabilization of the closed-loop system. In spite of these interesting results, many PETC design problems for nonlinear systems are largely open and deserve to be further explored; for instance, output feedback PETC design for general nonlinear dynamics subject to external disturbances has not been investigated, and systematic methods to determine the sampling period are still rare.

This paper investigates PETC design for continuous-time nonlinear control systems affected by disturbances using the state feedback and observer-based output feedback controllers, respectively, to ensure input-to-state stability of the closed-loop system implemented by the corresponding event-triggering mechanism (ETM). Specifically, assuming that the continuous-time state feedback controller (respectively, the continuous-time observer and output feedback controller) for the nonlinear system is given, sufficient conditions to determine the sampling period and triggering functions are provided explicitly. Based on that, PETC design for incrementally quadratic nonlinear systems, which is a broad class of nonlinear systems whose nonlinearity is characterized by incremental multiplier matrices and includes Lipschitz nonlinear systems and sector-bounded nonlinear systems as special cases, is considered. A systematic and constructive way to design the sampling period and triggering functions are given as LMI-based sufficient conditions that guarantee input-to-state stability of the closed-loop system. Discussion on applying the results to continuous-time linear control systems is also given. Compared with existing results on PETC design (e.g., [14, 6, 32, 4, 36]), this paper considers more general plant dynamics (i.e., general nonlinear model subject to disturbances) and more general PETC setups (i.e., ETMs exist in both the sensing and actuation channels for the output feedback case), provides verifiable LMI-based sufficient conditions for a broad class of nonlinear systems (i.e., incrementally quadratic systems), and gives explicitly the interval that the sampling period can be chosen from. In addition, although the analysis techniques in this paper also utilize some results from the emulation approach, they do not make the same type of assumptions that was proposed in [8, 26] and used in [2, 32] (c.f., Remark 4).

The remainder of the paper is organized as follows: in Section 2, the problem setup and statement are given; in Section 3, state feedback and output feedback PETC design for continuous-time general nonlinear models are presented individually; in Section 4, state feedback and output feedback PETC design for incrementally quadratic nonlinear systems are provided, with corresponding sufficient conditions given in the form of LMIs, respectively, and followed by the discussion on applying the results to continuous-time linear control systems; in Section 5, two simulation examples are used to illustrate the effectiveness of the proposed method.

Nomenclature. Denote the set of real numbers, non-negative real numbers and non-negative integers by , and , respectively. Denote the 2-norm by . Given a non-empty and closed set , the point-to-set distance from to is denoted by . Denote the identity matrix of size by . Denote the zero matrix of size by and the zero vector of size by ; the subscripts will be omitted when clear from context. Given a matrix , , , and means that is positive definite positive semi-definite, negative definite, and negative semi-definite, respectively; given two matrices and , means . Denote the block diagonal matrix by where are square matrices in the diagonal block. For symmetric matrices, will be used to stand for entries whose values follow from symmetry. A signal is called left-continuous if for all . “” means for every except for a set of zero Lebesgue-measure in .

A continuous function belongs to class (denoted as ) if it is strictly increasing and ; belongs to class (denoted as ) if and as . A continuous function belongs to class (denoted as ) if for each fixed , function with respect to (w.r.t.) and for each each fixed , function is decreasing w.r.t. and as . is the gradient of a function . Given a system where is the state, is a measurable essentially bounded input, and is a locally Lipschitz function, it is called input-to-state stable (ISS), if there exist functions , such that for every initial state , the solution of the system, , satisfies for all , where .

## 2 Problem Statement

At first, consider the setups shown in Fig. 1 where the full-state information is available and the state feedback control is used.

Fig. 1 (a) shows the closed-loop system in continuous time. The plant is a nonlinear system given as

 ˙x(t) =f(x(t),u(t),w(t)) (1)

where is the state, is the control input, is the disturbance, is a locally Lipschitz continuous function. The state feedback controller is given as where is a continuous function. Assume that is known and designed such that the closed-loop system in continuous time is ISS.

Fig. 1 (b) shows the closed-loop system implemented with ETM. In the following, denote the sampling period to be , and define the sampling times as for any . The state of the plant, , is sampled at each sampling time . The input to the controller, , is updated only when the event-triggering condition for the state is satisfied. Specifically, is a left-continuous, piecewise constant signal that is defined for as

 ~xc(t)={x(tk),ifΓx(x(tk),e(tk))≥0,~xc(tk),ifΓx(x(tk),e(tk))<0, (2)

where

 e(t)=x(t)−~xc(t) (3)

and is the triggering function that will be determined later. The triggering times are given by and

 txk+1=mini∈Z≥0{ih∣ih>txk,Γx(x(ih),e(ih))≥0}.

The control law is , and the actuation input to the plant, , is equal to , i.e.,

 u(t)=k(~xc(t)). (4)

Next, consider the setups shown in Fig. 2 where only the output information is available and output feedback control is used.

Fig. 2 (a) shows the closed-loop system in continuous time where the plant is given in (1) and the output is given as

 y(t) =g(x(t)) (5)

with and a continuous function. The observer is given as

 ˙^x=φ(^x,u,y) (6)

where , is a continuously differentiable function, and the observer-based controller is given as where is a continuous function. Assume that and are designed for (1) and (5) such that asymptotically converges to when , and the system (1) implementing the observer-based controller is ISS.

Fig. 2 (b) shows the closed-loop system implemented with ETMs in both the sensing and actuation channels. The output of the plant, , is sampled at each sampling time . The input to the observer, , is updated only when the event-triggering condition for the output is satisfied. Specifically, is a left-continuous, piecewise constant signal that is defined for as

 yc(t)={y(tk),ifΓy(y(tk),ye(tk))≥0,yc(tk),ifΓy(y(tk),ye(tk))<0, (7)

where

 ye(t)=yc(t)−y(t) (8)

and is the triggering function of the output that will be determined later. The triggering times are given by and

 tyk+1=mini∈Z≥0{ih∣ih>tyk,Γy(y(ih),ye(ih))≥0}.

Under ETMs, the observer (6) becomes

 ˙^x=φ(^x,u,yc) (9)

and the observer-based controller generates a continuous control law which is sampled at each sampling time . The input to the plant, , is updated only when the event-triggering condition for the input is satisfied. Specifically, define a left-continuous, piecewise constant signal for as

 (10)

where

 xe(t)=^xc(t)−^x(t) (11)

and is the triggering function of the input that will be determined later. The triggering times are given by and

 tuk+1=mini∈Z≥0{ih∣ih>tuk,Γu(^x(ih),xe(ih))≥0}.

The input to the plant, , is given as

 u(t)=k(^xc(t)). (12)

Systems that are implemented with ETMs are impulsive systems, which evolve continuously based on ODEs most of the time and exhibit impulses at some instances [16, 37]. Clearly, for systems implemented with ETMs, the impulses happen when the triggering conditions are met.

Inspired by [21] and [23], the input-to-state stability of impulsive systems w.r.t. a given set is defined below.

###### Definition 1.

Consider the following impulsive system

 {˙x(t)=f(x(t),u(t)),t∈(Ti,Ti+1],x+(t)=g(x(t),u(t)),t=Ti, (13)

where , is a sequence of impulsive times with , the state is absolutely continuous between impulses, is a locally bounded Lebesgue-measurable input, and . Given a time sequence , the impulsive system (13) is ISS w.r.t. a given non-empty and closed set if there exist functions and , such that for every initial condition and every admissible input , the solution to (13) exists globally and satisfies

 ∥x(t)∥A≤β(∥x(T0)∥A,t−T0)+γ(∥u∥[T0,T]) (14)

where denotes the supremum norm on an interval . The impulsive system (13) is uniformly ISS w.r.t. over a given class of admissible sequences of impulse times if there exist functions and that are independent of the choice of the time sequence, such that (14) holds for every time sequence in .

In the following, the closed-loop system implemented with ETMs is called uniformly ISS, or just ISS for short, w.r.t. a given (non-empty and closed) set , if it is uniformly ISS over all impulsive times generated by the periodic event-triggering mechanisms. It should be noted that the impulsive times generated by the periodic event-triggering mechanisms have no accumulation point (i.e. Zeno phenomenon is avoided) since the inter-execution times are lower bounded by the sampling period.

The PETC design problems that will be investigated in this paper are the following:
1. Given the setup in Fig. 1 (b) where the full-state information is available, design the sampling period and the triggering function such that the closed-loop system is ISS;
2. Given the setup in Fig. 2 (b) where the output information is available, design the sampling period and the triggering functions such that the closed-loop system is ISS.

In Section 3, problem 1 and problem 2 will be studied for the plant being a general nonlinear system. In Section 4, problem 1 and problem 2 will be studied for the plant being an incrementally quadratic nonlinear system, for which constructive ways based on convex programs to determine the sampling period and the triggering functions will be provided.

## 3 Robust Stabilization of Nonlinear Systems Using PETC

In this section, PETC design for the nonlinear system (1) with output (5) will be investigated.

One key technique that will be used is from the emulation approach, which has been widely used to analyze the stability property of a system under sampling [22, 25]. Computation of the maximum allowable sampling period in the emulation approach was investigated in [2, 8, 26], which will be also used later to determine the sampling period in PETC design.

The following lemma from [8] will be used in analysis later. This lemma gives the explicit time for the solution of a special ODE to decrease monotonically from to where .

###### Lemma 1.

[8] Let be the solution of the following ODE:

 ˙ϕ=−2μϕ−γ(ϕ2+1) (15)

with , , and

 ~T(μ,γ,λ) =⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩1μrarctan(r(1−λ)2λ1+λ(γμ−1)+1+λ),γ>μ,1μ1−λ1+λ,γ=μ,1μrarctanh(r(1−λ)2λ1+λ(γμ−1)+1+λ),γ<μ, (16) r = ⎷∣∣∣(γμ)2−1∣∣∣. (17)

Then, for all , and .

With given in (17), define as

 T(μ,γ)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩1μrarctan(r),γ>μ,1μ,γ=μ,1μrarctanh(r),γ<μ. (18)
###### Remark 1.

Clearly, the functions and are both positive, and . Furthermore, for fixed , is a strictly decreasing function, and as .

### 3.1 State Feedback PETC Design For Nonlinear Systems

In this subsection, PETC design is considered for the setup in Fig. 1 (b) where the plant is given by (1) and the state feedback controller is given by (4). The dynamics of the closed-loop system can be expressed as an impulsive model as follows:

 ˙xs =Fs(x,e,w):=⎛⎜ ⎜⎝~fs(x,e,w)~fs(x,e,w)1⎞⎟ ⎟⎠,t∈(tk,tk+1], (19) x+s =Gs(x,e):=⎛⎜⎝xgs(x,e)0⎞⎟⎠,t=tk, (20)

where is a clock variable, is defined in (3), and

 xs(t) =⎛⎜⎝x(t)e(t)τ(t)⎞⎟⎠,x+s=⎛⎜⎝x(t+)e(t+)τ(t+)⎞⎟⎠, (21) ~fs(x,e,w) =f(x,k(x−e),w), gs(x,e) ={0,ifΓx(x,e)≥0,e,ifΓx(x,e)<0.

Recall that above is the sampling times defined as for any ; the subscript in notations above stands for state.

Under these notations, the following theorem is given for the state feedback PETC design.

###### Theorem 1.

Consider the setup shown in Fig. 1 (b), in which the plant and the controller are given by (1) and (4), respectively. Suppose that there exist positive numbers , and a differentiable, positive definite, radially unbounded function such that , ,

 ∇V(xs)Fs(x,e,w)≤−αV(xs)+d∥w∥2 (22)

where , , is the solution of ODE (15). Choose positive numbers satisfying , and

 log(1+s)α0

where and are defined in (16) and (18), respectively. Let the initial condition of be . If the triggering function is chosen as

 Γx(x,e)=(λ−1−(1+s)λ)∥e∥2−sV1(x), (26)

then the closed-loop system (19)-(20) is ISS w.r.t. the set .

###### Proof.

By Lemma 1, for any , and . Because and are both positive definite, the function is positive definite w.r.t. and (i.e., for any , and when , otherwise). Furthermore, is differentiable and radially unbounded for any .

During the continuous dynamics when , inequality (22) implies

 V(xs(t))≥dα−α0∥w(t)∥2⇒ ˙V(xs(t))≤−α0V(xs(t)),∀t∈(tk,tk+1]a.e. (27)

where is the derivative of along the trajectory of (19).

At the impulse time when , there are two cases. Note that implies .

(i) If , the triggering condition is not met. Since implies , the following holds:

 V(x+s) =V1(x)+λ−1∥e∥2<(1+s)V(xs).

(ii) If , the triggering condition is met. Then from (20) and since , it holds that

 V(x+s) =V1(x)≤V(xs).

In summary, at the impulse time when ,

 V(x+s)≤(1+s)V(xs)=elog(1+s)V(xs). (28)

Then a bound for can be shown using (27) and (28) as follows. Clearly, there exists a sequence of times such that

 V(xs(t))≥dα−α0∥w∥2[t0,t], ∀t∈(^ti,ˇti+1],i=0,1,2,... (29) V(xs(t))≤dα−α0∥w∥2[t0,t], ∀t∈(ˇti,^ti],i=1,2,... (30)

Now consider the case when the first interval is non-empty, i.e., . If , then between any two consecutive impulses , from (27) and (29), it follows that , a.e., which implies that

 V(xs(tk))≤e−α0hV(xs(tk−1)).

From (28), it follows that

 V(xs(t+k))≤elog(1+s)V(xs(tk)).

Therefore, for any , it holds that

 V(xs(t)) ≤elog(1+s)t−t0he−α0(t−t0)V(xs(t0)) =elog(1+s)−α0hh(t−t0)V(xs(t0)). (31)

If , then it is easy to see that (31) holds for any . Note that by the choice of in (23).

Next, consider the case when . For any subinterval where , inequality (30) holds. If is not an impulse time, then (30) holds for . If is an impulse time, then (28) implies that

 V(xs(^t+i))≤elog(1+s)dα−α0∥w∥2[t0,^ti]. (32)

In either case, inequality (32) holds. For any subinterval , , where , it is easy to see that inequality (32) also holds. In summary, (32) holds for any subinterval .

For any subinterval , using the same argument that derives (31), the following inequality holds for any :

 V(xs(t)) ≤elog(1+s)−α0hh(t−^ti)V(xs(^ti)) ≤elog(1+s)dα−α0∥w∥2[t0,^ti]. (33)

Therefore, combing (31), (32) and (33), the following bound can be shown for :

 V(xs(t)) ≤max{elog(1+s)−α0hh(t−t0)V(xs(t0)),elog(1+s)dα−α0∥w∥2[t0,t]},∀t≥t0.

Since , the function is strictly decreasing for . Since is positive definite and radially unbounded for any , by the standard argument for ISS (e.g., see [30, 23, 21]), it can be concluded that (14) holds. This completes the proof.∎

Theorem 1 provides a systematic way to determine the sampling period and triggering function for the state feedback PETC design. The key is to find a function , which is in fact an exponential ISS-Lyapunov function of the impulsive system (19)-(20), such that (22) holds. The given structure of makes it possible to construct in a systematic way. If, for example, the dynamics (1) is polynomial, the sum-of-squares optimization can be used to find (see Example 1 in Section 5); the case when (1) is an incrementally quadratic nonlinear system will be discussed in Section 4.

###### Remark 2.

A set of Lyapunov-based sufficient conditions for the input-to-state stability of impulsive systems were given in [21, 11]. Particularly, when the continuous dynamics are exponentially ISS but the impulses are destabilizing, it was shown in [21] that the impulsive system is uniformly ISS if some average dwell-time condition is satisfied, which was relaxed to be a generalized average dwell-time condition in [11]. These important results rely on the existence of (exponential) ISS-Lyapunov functions. Part of the proof of Theorem 1 is inspired by Theorem 1 in [21], and the sampling period in PETC design is a lower bound for the dwell-time.

###### Remark 3.

In Theorem 1, there always exist that satisfy (23), (24) and (25). Specifically, since as , there always exist satisfying (23). Because and have the properties stated in Remark 1, there always exists satisfying (24). If (25) does not hold with such and , then it is always possible to find a smaller such that (25) holds, while still guaranteeing that (23) holds. Therefore, the values in Theorem 1 always exist.

Different choices of will result in different triggering frequencies. For instance, if is chosen, which implies is fixed, then a smaller will render the triggering condition (26) easier to be met, which will tend to increase the triggering frequency, while a larger will tend to decrease the triggering frequency; if is fixed, then a smaller will results in a larger , which will tend to decrease the triggering frequency, while a larger will tend to increase the triggering frequency. The effect of choosing different parameters will be demonstrated by Example 1 in the simulation section.

### 3.2 Output Feedback PETC Design For Nonlinear Systems

In this subsection, PETC design is considered for the setup in Fig. 2 (b) where the plant is given by (1), the output is given by (5), the observer is given by (9) and the observer-based output feedback controller is given by (12).

Define the estimation error of the observer as

 ^e(t) =x(t)−^x(t) (34)

and

 ξ(t) =(x(t)^e(t)). (35)

Define the sampling induced error as

 η(t) =(ye(t)xe(t)) (36)

where are defined in (8) and (11), respectively.

Then dynamics of the closed-loop system can be expressed as an impulsive model as follows:

 ˙xo =Fo(ξ,η,w):=⎛⎜ ⎜⎝~f1o(ξ,η,w)~f2o(ξ,η,w)1⎞⎟ ⎟⎠,t∈(tk,tk+1], (37) x+o =Go(ξ,η):=⎛⎜⎝ξgo(ξ,η)0⎞⎟⎠,t=tk, (38)

where is a clock variable, and

 xo(t) =⎛⎜⎝ξ(t)η(t)τ(t)⎞⎟⎠,x+o=⎛⎜⎝ξ(h+)η(h+)τ(h+)⎞⎟⎠, (39) ~f1o(ξ,η,w) =(f(x,k(^xc),w)f(x,k(^xc),w)−φ(^x,k(^xc),w)), ~f2o(ξ,η,w) =(∇g(x)⋅f(x,k(^xc),w)−f(x,k(^xc),w)+φ(^x,k(^xc),w)), go(ξ,η) =(g1o(ξ,η)g2o(ξ,η)), g1o(ξ,η) ={0,ifΓy(y,ye)≥0,ye,ifΓy(y,ye)<0, g2o(ξ,η) ={0,ifΓu(^x,xe)≥0,xe,ifΓu(^x,xe)<0.

The subscript in notations above stands for output.

Under these notations, the following theorem is given for observer-based output feedback PETC design.

###### Theorem 2.

Consider the setup shown in Fig. 2 (b) where the plant, output, observer and controller are given by (1), (5), (9) and (12), respectively. Suppose that there exist positive numbers , and differentiable, positive definite, radially unbounded functions , and such that , ,

 ∇V(xo)Fo(ξ,η,w)≤−αV(xo)+d∥w∥2, (40) c1V3(g(x))+c2V4(^x)≤V1(ξ), (41)

where

 V(xo) =V1(ξ)+V2(η,τ), V2(η,τ) =c1ϕ1y⊤eye+c2ϕ2x⊤exe,

and are the solutions of the following ODEs:

 ˙ϕi=−2μiϕi−γi(ϕ2i+1). (42)

Choose positive numbers satisfying , , , and

 log(1+s)α0

where and are defined in (16) and (18), respectively. Let the initial condition of be for . If the triggering functions are chosen as

 Γy(y,ye) =(λ−11−(1+s)λ1)∥ye∥2−sV3(y), (46) Γu(^x,xe) =(λ−12−(1+s)λ2)∥xe∥2−sV4(^x), (47)

then the closed-loop system (37)-(38) is ISS w.r.t. the set .

###### Proof.

By Lemma 1, for any , and , . Because are both positive definite, the function is positive definite w.r.t. and . (i.e., for any , and when , otherwise). Furthermore, is differentiable and radially unbounded for any .

During the continuous dynamics when , the inequality (40) holds. Hence,

 V(xo(t))≥dα−α0∥w(t)∥2⇒ ˙V(xo(t))≤−α0V(xo(t)),∀t∈(tk,tk+1]a.e. (48)

where is the derivative of along the trajectory of (37).

At the impulse time when , there are four cases regarding satisfaction of the input and output triggering conditions. Note that implies , for .

(i) If and , the output and input triggering conditions are not met. Since , ; since ,