Event-triggered Control of Linear Systems With

Asian Journal of Control, Vol. 17, No. 4, pp. 1–13, July 2015Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.928

EVENT-TRIGGERED CONTROL OF LINEAR SYSTEMS WITHSATURATED INPUTS

Wei Ni, Ping Zhao, Xiaoli Wang, and Jinhuan Wang

ABSTRACT

This paper investigates the event-triggered control of linear systems with saturated state feedback and saturatedobserver-based feedback, respectively. The problem of simultaneously deriving stabilizing event-triggered controllersand tackling saturation nonlinearity is cast into a standard linear matrix inequalities problem. Key topics are studied,such as event-triggered observer design and event-triggered saturated observer-based feedback synthesis. Importantissues are touched on, including the existence of the positive lower bound for inter-event times, and self-triggeredalgorithms.

Key Words: Event-triggered control, event-triggered observer, linear systems, saturated inputs.

I. INTRODUCTION

Modern control systems are often implementedin digital platforms by using the microprocessors. Tra-ditional digital control techniques often assume thatcontrollers execute periodically. This is usually calledtime-triggered control [2,10,20], where the stability ofthe resulting sampled-data system under time triggeredcontrol can be achieved if the worst situation, that thesampling period is included in a certain interval, is con-sidered. This paradigm may result in unnecessarily highworkloads since control does not utilize the resources inan optimum way, and furthermore the control task is exe-cuted after the elapse of a certain amount of time regard-less of whether anything significant has occurred in thesystem. Realizing this, one hopes that the control is exe-cuted only when necessary rather than periodically; morespecifically, one may adapt the control sampling sequenceto some events on demand; this is usually achieved bysampling and computing the controller only when a cer-tain threshold condition on the state is violated. This isso-called event-trigged control which has appeared as ahot field of control theory in recent years; the readers

Manuscript received March 29, 2013; revised September 13, 2013; acceptedNovember 17, 2013.

W. Ni is with School of Sciences, Nanchang University, Nanchang, China.P. Zhao (corresponding author, e-mail: [email protected]) is with School

of Electrical Engineering, University of Jinan, Jinan, China.X. Wang is with School of Information Science and Engineering, Harbin

Institute of Technology at Weihai, China.J. Wang is with School of Sciences, Hebei University of Technology, Tianjin,

China.This work is supported by the NNSF of China (61304161, 61104096, 61374074,

61203142, 11361043), the JXNSF (20132BAB211037, 20114BAB201002), theYouth Foundation of Jiangxi Provincial Education Department of China(GJJ12132), and the Project-sponsored by SRF for ROCS, SEM.

are referred to [1,6,12,18,21] for more details. As analternative to the more traditional periodic execution ofcontrol tasks, the event-triggered scheme has the meritsof reducing the number of executions while guarantee-ing desired levels of performance, which makes them veryappealing in the context of sensor/actuator networks.Although the event-triggered scheme is less simple andpredictable than the time-triggered one, it is the engi-neer’s first choice for implementation when consideringthe usage of the system’s resources; refer to [9,15] fordetailed justifications.

Another very important issue which is inherentto control systems is the presence of control satura-tions. From an engineering point of view, most physicalactuators, sensors and interfacing devices are subject tosaturation because of the existence of hard limitation.Considering these, one usually applies a saturation func-tion to the control, resulting a control system with sat-urated inputs. As for saturated constraints on system’sinput, suitable tools should be pursued to handle the sat-uration nonlinearity. There are many approaches to dealwith this, such as the polytopic representation approach,sector nonlinearity model approach, saturation avoid-ance approach, parameterized algebraic Riccati equationapproach, and the anti-windup approach; interestedreaders are referred to the books [7,19]. For the sta-bilization of linear systems with saturated inputs byusing event-triggered control, there are very few resultsreported in the literature; see for example [8,11]. Thework of [8] investigates the stabilization of linear systemsvia saturated and event-trigged state feedback, wherecontrollers are synthesized and stability properties arederived using linear matrix inequalities. Event-triggeredPI control for scalar linear systems subject to input

© 2014 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

Asian Journal of Control, Vol. 17, No. 4, pp. 1–13, July 2015

Fig. 1. Event-triggered sensor-observer and observer-controller communications for saturated control systems.

saturation was considered in [11]. This paper is furtherdevoted to the event-triggered control of linear systemswith saturated inputs, but toward a more general frame-work, where the control is implemented from the view-point of networked control systems, shown in Fig. 1.

Note that the networked control system consid-ered here is composed of a set of sensors, observers,and control processing units, with them being embed-ded in a sensor–observer–controller network throughwhich the output measurements by sensors are sent to theobservers only at discrete instants when a certain condi-tion, call a triggering condition, is violated; the states esti-mated by observers are synthesized into control signalsand transmitted, also at discrete instants when anotherevent-triggered condition is violated, through a satura-tion unit to the control controller. In this event-triggeredway of sensor–observer–controller transition of signal,the number of times that a feedback law and an observerstrategy are executed is reduced, leading to a reductionin transmissions and thus a reduction in energy expendi-tures. Obviously, event-triggered saturated state feedbackis a special case of this scenario, and our paper extendsthe work of [8,11].

For linear systems with saturated inputs, the syn-thesis in our paper is carried out in a two-step way: anevent-triggered observer is designed to estimate the state,and then another event-triggered observer-based feed-back subject to saturation is synthesized to stabilize thesystem. To make our basic idea clear, we first considerthe case of event-triggered state feedback, where the statefeedback signal in the saturation function is implementedin an event-triggered way in the sense that the control is

piece-wise constant and it chooses instants to update bychecking the violation of a certain triggering condition.Results are extended to a more general case where the fullstate is not available. More specifically, the sensed outputsare discretilized by an event-triggered mechanism, andtransmitted to an observer, which is properly designed togive an estimate of the plant state. Then the observer’sstates are transmitted also in an event-triggered way tothe saturation unit, giving rise to saturated discrete-timesignals, which are synthesized into a controller to sta-bilize the systems. Here, the problem of simultaneouslyderiving stabilizing event-triggered controllers and tack-ling saturation nonlinearity is cast into standard linearmatrix inequality (LMI) problem. Different from [8,11],to reduce the dimension of the LMIs, we do not apply thetechnique of enlarging the design space to deal with sat-uration nonlinearity, but instead we adopt the method ofplacing saturation nonlinearity into the convex hull of agroup of linear feedbacks [7].

It is well known, on the one hand, that Zeno phe-nomena should be avoided in the event-triggered scheme.To this end, event-triggered algorithms in observer designand controller design have been shown to bear positivelower bounds for inter-execution times. It is also wellknown, on the other hand, that event-trigger algorithmsrequire continuously checking of the violation of certaintriggering conditions. To reduce the computation com-plexity, we propose a self-triggered scheme in the sensethat the next event time, at which the control law updates,can be calculated by using only the system’s state eval-uated at previous event times, without requiring statevalues between the control updates.


W. Ni et al.: Event-Triggered Control of Linear Systems with Saturated Inputs

The rest of this paper is organized as follows.Section II contains some some preliminaries andSection III, the problem formulation. The main results ofthe paper are included in Section IV and Section V, whereSection IV concerns event-triggered control via saturatedstate feedback and Section V deals with event-triggeredcontrol via saturated observer-based feedback. Illustra-tive examples are presented in Section VI. Section VIIpresents a brief conclusion.

II. PRELIMINARIES

Notation. Let N denote the set of nature numbers, i.e.,N = {0, 1, 2, · · ·}. Let R denote the set of real numbers,and Rn,Rm×n denote the sets of n-dimensional real vec-tors and m×n real matrices, respectively. Let I denote theidentity matrix whose dimension is clear from the con-text. By P > 0 (P < 0) we mean that the matrix P issymmetric and positive (negative) definite. For a vectorx = (x1, · · · , xn)T ∈ Rn, let ‖x‖ denote its Euclidean

norm, i.e., ‖x‖ =√

x21 + · · · + x2

n, and let ‖x‖∞ denote itsinfinity norm, i.e., ‖x‖∞ = maxi=1,···,n |xi|. For a squarematrix M, let ‖M‖ denote the matrix norm induced bythe Euclidean vector norm. The scaler saturation func-tion with upper bound c, satc(⋅), is defined as satc(s) =sign(s)min{c, |s|}, and the vector saturation functionis defined as satc(u) = [satc(u1), satc(u2), · · · , satc(um)]Twith u = [u1, u2, · · · , um]T ∈ Rm. For a group of points,p1, p2, · · · , pl, the convex hull of these points is defined as

co{pk|k=1, 2, · · · , l}={∑l

k=1 𝛼kpk|∑lk=1 𝛼k =1, 𝛼k ≥ 0

}.

We also present some lemmas for later use. Let be the set of m × m diagonal matrices whose diagonalelements are either 1 or 0. There are 2m elements in .Suppose that all elements of are labeled as D+

j , j ∈ Δ=

{1, 2, · · · , 2m}. Denote D−j = I −D+

j , j ∈ . With these, aproperty with regard to the saturation function is shownas follows.

Lemma 1. [7] Let u, v ∈ Rm be given with u =(u1, u2, · · · , um)T and v = (v1, v2, · · · , vm)T . Suppose‖v‖∞ ≤ c, where c is the upper bound of saturationfunction defined before. Then

satc(u) ∈ co{

D+j u + D−

j v|j ∈

}.

The following lemma is a local version of the corre-sponding global result of “converging input convergingstate”. The readers are referred to [16,17] for details.

Lemma 2. [16,17] Let x = 0 be an equilibrium point forthe system x(t) = f (x(t)) and suppose the existence and

uniqueness conditions of solutions on [t0,+∞) are satis-fied. If x = 0 is locally asymptotically stable in ⊂ Rn,then , for any given signal x(t) satisfying limt→∞ x(t) = 0,one can conclude that the trajectory of the differentialsystem x = f (x) + x(t) with initial state in satisfieslimt→∞ x(t) = 0.

III. PROBLEM FORMULATION

The problems of event-triggered control of linearsystems with both saturated state feedback and saturatedobserver-based feedback are formulated.

3.1 Event-triggered control via saturated state feedback

Consider the following linear system with saturatedinputs,

x(t) = Ax(t) + Bsat(u(t)), (1)

where x ∶ [0,+∞) → Rn is the state, u ∶ [0,+∞) → Rm

is the control input, and A ∈ Rn×n,B ∈ Rn×m are systemmatrices. For ease of presentation, we assume in what fol-lows that the upper bound for the saturation function isc = 1 and the subscript c is dropped. The signals of statefeedback controller Kx(t) with a user-designed matrixK ∈ Rm×n are transmitted in a discrete manner to theactuator at times tk, k ∈ N, producing an event-triggeredcontroller

u(t) = Kx(tk), t ∈ [tk, tk+1), k ∈ N. (2)

When k = 0, t0 denotes the initial instant and x(t0)denotes the initial condition. One sees that this schemegenerates a sporadic sequence of controller updating.The times tk, k ∈ N, called event times, are specified bythose instants when a certain triggering condition is vio-lated. The triggering condition adopted in this paper isbased on the state error and takes the form of

‖ee(t)‖ <√𝜌‖xx(t)‖, (3)

where e(t) = x(tk) − x(t), t ∈ [tk, tk+1) is the gap associ-ated with the kth event time and e,x are user-designedmatrices and 𝜌 is a user-designed positive parameter.Then, the event times are produced as follows:

t0 = 0,

tk+1 = inf{

t|t > tk, ‖ee(t)‖ ≥√𝜌‖xx(t)‖} , k ∈ N.

The time sequence {tk}k∈N represents the instants atwhich the controller (2) is updated.

The above event-triggered control scheme showsthat the actuation is performed when needed rather than



continuously or periodically. Specifically, the control isexecuted when the gap between the current and the mostrecently measured state, which equals zero at each eventtime and grows soon after, exceeds a specified threshold.The constraint in the triggering condition (3) can beviewed as a state-dependent threshold condition, whichis chosen in a way that preserves stability. There aremany types of triggering conditions such as those basedon state error [18], input error [4], or Lyapunov functions[13]. Now, we formulate the problem as follows:

Problem A. Design a feedback gain matrix K and anevent-triggering condition (3) such that the system (1)under event-triggered controller (2) is asymptoticallystable.

3.2 Event-triggered control via saturatedobserver-based feedback

This section considers the more general case wherethe full state is not available. Refer to Fig. 1 and considernow the following saturated control system:

x(t) = Ax(t) + Bsat(u(t)),y(t) = Cx(t),

(4)

where y ∈ Rp is the output which is transmitted only atsome discrete times sk, k ∈ N, to an observer

x(t) = Ax(t) + L[y(sk) − Cx(t)

]+ Bsat(u(t)), (5)

where L ∈ Rn×p is the observer gain matrix to be designedlater. The times, sk, k ∈ N, are specified when the follow-ing triggering condition

‖‖‖eyey(t)

‖‖‖ <√𝜌y

‖‖‖y (y(t) − Cx(t))‖‖‖ (6)

is violated, i.e.,

s0 = 0,

sk+1 = inf{

t|t > sk,‖‖‖ey

ey(t)‖‖‖

≥√𝜌y

‖‖‖y (y(t) − Cx(t))‖‖‖} , k ∈ N,

where ey(t) = y(sk) − y(t), t ∈ [sk, sk+1),ey,y are

matrices of appropriate dimensions, and 𝜌y is a positivenumber to be designed later. Note the information for thetriggering condition design includes only the output andthe observer state.

The observer state is further transmitted in a dis-crete manner by another event-triggered mechanism tothe saturated controller, taking the form of

u(t) = Kox(𝜏k), t ∈ [𝜏k, 𝜏k+1), k ∈ N, (7)

where Ko ∈ Rm×n is a matrix to be designed later, and theevent times 𝜏k, k ∈ N, are specified when the followingtriggering condition

‖‖‖ee(t)‖‖‖ <√𝜌o

‖‖‖xy(t)‖‖‖ (8)

is violated, i.e.,

𝜏0 = 0,

𝜏k+1 = inf{

t|t > 𝜏k,‖‖‖ee(t)‖‖‖ ≥

√𝜌o

‖‖‖xy(t)‖‖‖} , k ∈ N,

where e(t) = x(𝜏k)− x(t), t ∈ [𝜏k, 𝜏k+1), e, x are matricesof appropriate dimensions, and 𝜌o is a positive number.Now the problem in this subsection can be formulatedas follows:

Problem B. Design the triggering conditions (6), (8) andgain matrices Ko and L such that the closed-loop system(4) under (7) and (5) is asymptotically stable.

We now divide Problem B into two sub-problems.Defining the error between observer’s state and theplant’s state as x(t) = x(t) − x(t), the dynamics of x canbe obtained as

x(t) = (A − LC)x(t) + Ley(t), (9)

and the dynamics of x can be obtained as

x(t) = Ax(t) + Bsat(Kox(t) + Koe(t) + Kox(t)). (10)

Putting (9) and (10) together and adopting the ideaincluded in Lemma 2, it can be seen that solving ProblemB is transformed into the stabilization of (9) and (11)below:

x(t) = Ax(t) + Bsat(Kox(t) + Koe(t)). (11)

Note that the stabilization of (11), similar to ProblemA, is roughly an event-triggered state feedback problem,where the triggering condition is imposed on e(t) andit is output based. Based on the above consideration,solving Problem B is divided into the following two sub-problems: an event-triggered state estimation problemand an event-triggered state feedback problem.

Problem B-I. Design an observer gain matrix L and anevent-triggering condition (6) such that the system (9) isasymptotically stable.



Problem B-II. Design a feedback matrix Ko andan event-triggering condition (8) such that (11) isasymptotically stable.

Remark 1. Note that the equation (1) in Problem A canbe rewritten as x(t) = Ax(t) + Bsat(Kx(t) + Ke(t)), whichtakes a similar form as the equation (11) in ProblemB-II. The difference is that Problem A imposes trigger-ing condition on e(t) and it is state based, while ProblemB-II imposes triggering condition on e(t) which is outputbased. The method of transforming Problem B into thepair of Problem B-I and Problem B-II is similar to theseparation principle in optimal control.

IV. EVENT-TRIGGERED CONTROL VIASATURATED STATE FEEDBACK

This section is devoted to Problem A. Furthermore,a positive lower bound for inter-event times will be pro-vided, showing the proposed event-triggered mechanismcan be implemented without causing Zeno phenomena.Self-triggered control is also considered.

4.1 Event-triggered stabilization

The system (1) under event-triggered state feedbackbecomes

x(t) = Ax(t) + Bsat(Kx(tk)). (12)

To deal with the saturation nonlinearity, Lemma 1 is used

to give sat(Kx(tk)) ∈{

D+j Kx(tk) + D−

j Hx(t)|j ∈

}if‖Hx‖∞ ≤ 1 for a matrix H ∈ Rm×n. Let P be a posi-

tive definite matrix to be designed such that the ellipsoid{x|xT Px ≤ 1} is an invariant set of the closed-loopsystem (12) and such that{

x|xT Px ≤ 1}⊂{

x| ‖Hx‖∞ ≤ 1}. (13)

If one chooses initial state x0 ∈ {x|xT Px ≤ 1},then anytrajectory x(t) with initial state x0 stays in {x|xT Px ≤ 1}and consequently ‖Hx(t)‖∞ ≤ 1 is satisfied for all t >

0. Therefore, the system (12) can be put in the followingform:

x ∈ co{[

A + B(

D+j K + D−

j H)]

x + BD+j Ke|j ∈

}.

We now give a sufficient condition for the stability ofthis differential inclusion. Let V (x) = xT Px, whose timederivative along the trajectories of above system satisfy:

V ≤ maxj∈

{xT

{P[A + B

(D+

j K + D−j H

)]+[A + B

(D+

j K + D−j H

)]TP}

x

+ 2xT PBD+j Ke

}.

Using the fact that

2xT PBD+j Ke ≤ xT PBD+

j BT Px + eT KT Ke,

and imposing the triggering condition

‖Ke(t)‖ <√𝜌‖Px(t)‖, (14)

one has

V ≤ maxj∈

xT{

P[A + B

(D+

j K + D−j H

)]+[A + B

(D+

j K + D−j H

)]TP

+ PBD+j BT P + 𝜌P2

}x.

A sufficient condition for negative definiteness of V is

P[A + B

(D+

j K + D−j H

)]+[A + B

(D+

j K + D−j H

)]TP

+ PBD+j BT P + 𝜌P2 < 0, j ∈ ,

which, by left and right multiplying P−1 on both sides ofthe inequality and by denoting X = P−1,Y = KX , andZ = HX , can be transformed into the following LMIs:

AX + XAT + BD+j Y + Y T D+

j BT + BD−j Z

+ ZT D−j BT + BD+

j BT + 𝜌I < 0, j ∈ . (15)

Therefore, the saturated control of the system (12)via event-triggered state feedback can be tackled by find-ing solutions to (13) and (15). Before presenting the resultin this section, we also reformulate the condition (13) interms of LMIs. Denote by hi the ith row of the matrix H,where 1 ≤ i ≤ m. Note that the condition (13) holds ifand only if all the hyperplanes hix = 1, 1 ≤ i ≤ m, liecompletely outside of the ellipsoid {x|xT Px ≤ 1}, i.e., ateach point x on the hyperplane hix = 1, 1 ≤ i ≤ m, wehave xT Px ≥ 1. This means that the constraint (13) isequivalent to

minx∈Rn

{xT Px|hix = 1

}≥ 1, 1 ≤ i ≤ m.

By using the Lagrangian multiplier method the minimumcan be calculated as

minx∈Rn

{xT Px|hix = 1

}=[hiP

−1hTi

]−1, 1 ≤ i ≤ m.



Consequently, constraint (13) is equivalent to

hiP−1hT

i ≤ 1, 1 ≤ i ≤ m.

By Schur complement [22], the above inequalities areequivalent to(

1 hiP−1

P−1hTi P−1

)≥ 0, 1 ≤ i ≤ m.

Noting that P = X−1,K = YX−1,H = ZX−1, the aboveconstraint is exact(

1 zi

zTi X

)≥ 0, 1 ≤ i ≤ m, (16)

where zi is the i-th row of the matrix Z.The above analysis is summarized in the following

theorem.

Theorem 1. For the system (12), if there exists a setof solutions {X > 0,Y , 𝜌 > 0} to the LMIs(15) and (16), then under the event-triggered feed-back u(t) = Kx(tk), t ∈ [tk, tk+1), k ∈ N withthe gain matrix K = YX−1 and with the eventtimes tk, k ∈ N given by violating the triggeringcondition (14), the closed-loop system (12) is locallyasymptotically stable.

Remark 2. As for the event-triggered control of linearsystems with saturated inputs proposed in [8,11], thefeedback gain matrix K is pre-given based on the crite-rion that A + BK is Hurwitz, and then the rest of desireddesign variables are obtained by solving some LMIs inwhich K is viewed as a constant matrix. This is simi-lar to the emulation-based method [14]. However, in ourpaper, the matrix K, together with other design variables,is computed simultaneously by solving LMIs. This treat-ment allows us to have more design freedom to obtain anoptimal design (for example, achieving minimum num-ber of controller executions) of the combined feedbackcontroller and triggering mechanism. This optimizationproblem deserves intensive study (see for example [5]) andit is not the focus of this paper.

4.2 Lower bound for inter-event times

In order that the event-triggered control policy inthe last subsection is applicable, it is necessary to showthat the inter-event times tk+1 − tk, k ∈ N, where

t0 = 0,

tk+1 = inf{

t|t > tk, ‖Ke(t)‖ ≥√𝜌‖Px(t)‖} , k ∈ N,

are lower bounded by a positive number. This is provedin the following theorem.

Theorem 2. Suppose the conditions of Theorem 1 hold.Then the inter-event times, {ti+1 − ti}, k ∈ N, definedabove are lower bounded by a positive number 𝜏𝜙.

Proof. Denote e=Ke, x=Px, and y(t)=‖e(t)‖2 ∕ ‖x(t)‖2.Then

dydt

= 2eT exT x − 2eT exT x‖x‖4

≤ 2‖e‖ ‖‖ e‖‖‖x‖2

+ 2‖e‖2 ‖‖ x‖‖‖x‖3

= 2‖e‖ ‖‖ x‖‖‖x‖2

+ 2‖e‖2 ‖‖ x‖‖‖x‖3

= 2(√

y + y) ‖‖ x‖‖‖x‖ .

It follows from K = YP,H = ZP and the definition ofx, e that

x =∑j∈

{kjP

[A + B

(D+

j K + D−j H

)]x + PBD+

j Ke}

=∑j∈

{kjP

[AP−1 + B

(D+

j Y + D−j Z

)]x + PBD+

j e},

where∑

j∈ kj = 1, kj ≥ 0, j ∈ . Therefore,

dydt

≤ 2(√

y + y)∑

j∈

{kj

(‖‖‖‖P[AP−1 + B

(D+

j Y + D−j Z

)]‖‖‖‖+ ‖‖‖PBD+

j‖‖‖√y

)}≤ 2

(√y + y

) (c + d

√y)

= 2a1

√y + 2a2y + 2a3y3∕2,

where a1 = c, a2 = c + d, a3 = d, c = maxj∈‖‖‖‖P

[AP−1+

B(

D+j Y + D−

j Z)]‖‖‖‖, d = maxj∈

‖‖‖PBD+j‖‖‖. Thus the

solution y(t) of this differential inequality with initialcondition y(0) = 0 is, according to comparison prin-ciple, dominated by the solution 𝜙(t) of the differentialequation �� = 2a1

√𝜙 + 2a2𝜙 + 2a3𝜙

3∕2 with the sameinitial condition 𝜙(0) = 0, i.e., y(t) ≤ 𝜙(t). Noting thatthe minimal time 𝜏 between events is given by the time ittakes for y(t) to evolve from the value 0 to 𝜌, this time 𝜏

is obviously no smaller than that time 𝜏𝜙 for 𝜙(t) evolv-ing from the value 0 to 𝜌. Since the solution 𝜙(t) can beobtained as



t =⎧⎪⎨⎪⎩||||| ln|||c√𝜙+c|||−ln|||d√𝜙+c|||

c−d

||||| , c ≠ d;

2c+d

− 2

c+d+2d√𝜙, c = d,

one has

𝜏𝜙 =⎧⎪⎨⎪⎩|||| ln|c√𝜌+c|−ln|d√𝜌+c|

c−d

|||| , c ≠ d;

2c+d

− 2c+d+2d

√𝜌, c = d.

Consequently, 𝜏 ≥ 𝜏𝜙 > 0, which shows that inter-eventtimes have a strict positive lower bound.

4.3 Self-triggered control

Obviously, Theorem 1 requires continuous checkingof the violation of the triggering condition (14). To reducethe computation complexity, we propose a self-triggeredscheme: the next event time, at which the control lawupdates, can be calculated by using only the system’s stateevaluated at the previous event time, without requiringstate values between the control updates. Denote Πj =D+

j K +D−j H, j ∈ . Under this scheme, the sequence of

event times {tk}k∈N satisfy

‖‖Ke(tk+1−)‖‖ = 𝜌‖Px(tk)‖, k ∈ N, (17)

where e(

tik+1

−) Δ= limt→tk+1− ei(t).

The first inequality uses the fact that d‖𝜉(t)‖dt

≤‖‖‖ d𝜉(t)dt

‖‖‖; indeed, one the one hand, d𝜉T (t)𝜉(t)dt

= d‖𝜉(t)‖2

dt=

2‖𝜉(t)‖ d‖𝜉(t)‖dt

; one the other hand, d𝜉T (t)𝜉(t)dt

= 2𝜉(t) d𝜉(t)dt

≤

2‖𝜉(t)‖ ⋅ ‖‖‖ d𝜉(t)dt

‖‖‖. Therefore, d‖𝜉(t)‖dt

≤‖‖‖ d𝜉(t)

dt‖‖‖.

Note that

ddt‖e(t)‖ ≤ ‖e(t)‖ = ‖x(t)‖

=‖‖‖‖‖‖∑j∈

kj

[(A + BΠj

)x(t) + BD+

j Ke(t)]‖‖‖‖‖‖

=‖‖‖‖‖‖∑j∈

kj

[(A+BΠj

)(x(tk)−e(t)

)+BD+

j Ke(t)]‖‖‖‖‖‖

≤ maxj∈

‖A + BΠj‖ ⋅ ‖x(tk)‖+(

maxj∈

‖A + BΠj‖ + maxj∈

‖‖‖BD+j K‖‖‖

)‖e(t)‖.Also noting that the initial condition is ‖e(tk)‖ = 0,

solving the above differential inequality for ‖e(t)‖ yields

‖e(t)‖ ≤maxj∈ ‖A + BΠj‖ ⋅ ‖x(tk)‖

maxj∈ ‖A + BΠj‖ + maxj∈‖‖‖BD+

j K‖‖‖×(

exp((

maxj∈

‖A + BΠj‖+ max

j∈‖‖‖BD+

j K‖‖‖)(t − tk)

)− 1

).

Letting t approach from left to tik+1

, one obtains

‖‖e(tk+1−)‖‖ ≤maxj∈ ‖A + BΠj‖ ⋅ ‖x(tk)‖


j K‖‖‖×(

exp((

maxj∈

‖A + BΠj‖+ max

j∈‖‖‖BD+

j K‖‖‖)k

)− 1

),

where k = tk+1 − tk. This, together with (17), implies

𝜌‖Px(tk)‖ ≤ ‖K‖‖e(tk+1−)‖≤

‖K‖maxj∈ ‖A + BΠj‖ ⋅ ‖x(tk)‖maxj∈ ‖A + BΠj‖ + maxj∈

‖‖‖BD+j K‖‖‖

×(

exp((

maxj∈

‖A + BΠj‖+ max

j∈‖‖‖BD+

j K‖‖‖)k

)− 1

).

Solving the above algebraic inequality for k gives

k ≥

ln

(𝜌‖Px(tk)‖(maxj∈ ‖A+BΠj‖+maxj∈

‖‖‖BD+j K‖‖‖)∑

j∈ ‖K‖maxj∈ ‖A+BΠj‖⋅‖x(tk)‖ + 1

)(


j K‖‖‖) .

Obviously, the sampling periods k, k ∈ N can never bezero. Each k can be viewed as the lower bound for eachinter-event time tk+1 − tk, k ∈ N. Let the event timesgiven by

t0 = 0, tk+1 = tk + k, k ∈ N. (18)

Based on these analyses, we have the following theorem.

Theorem 3. For the system (12), if there exists a set ofsolutions {X > 0,Y , 𝜌 > 0} to the LMIs (15) and (16),then under the self-triggered feedback u(t) = Kx(tk),t ∈ [tk, tk+1), k ∈ N with K = YX−1 and tk, k ∈N given by (18), the closed-loop system (12) is locallyasymptotically stable.



V. EVENT-TRIGGERED CONTROL VIASATURATED OBSERVER-BASED FEEDBACK

This section is devoted to Problem B; that is, theevent triggered control of linear systems with saturatedobserver-based feedback. As pointed out before, thisproblem can be divided into Problem B-I and ProblemB-II.

Problem B-I concerns the design of a gain matrixL and a triggering condition with the aim to achieve thestability of the system (9). It can be approached as fol-lows. By solving the following LMIs with variables X1,Y1and 𝜌y

X1A + AT X1 − Y1C − CT Y T1 + I + 𝜌yCT C < 0,

X1 > 0, 𝜌y > 0, (19)

the matrix L is obtained as L = X−11 Y1, and by defining

P1 = X1, the triggering condition is proposed as

‖P1Ley(t)‖ ≤√𝜌y ‖Cx(t) − y(t)‖ . (20)

We show that these designs render the stability of thesystem (9), shown in the following theorem.

Theorem 4. If there is a set of solution X1,Y1 and 𝜌y tothe LMIs (19), then the event-triggered observer (5) withthe event times sk, k ∈ N given by violating the triggeringcondition (20) gives an asymptotical estimate of the stateof the system (4).

Proof. In order to show that the event-triggered observer(5) under the triggering condition (20) gives an asymptot-ical estimate of the state of the system (4), we only need toprove the asymptotical stability of their error system (9)under the triggering condition (20). To this end, considerthe Lyapunov function candidate V1 (x) = xT P1x for thesystem (9), and its time derivative can be calculated as

V1 = xT [P1(A − LC) + (A − LC)T P1]x + 2xT P1Ley

≤ xT [P1(A − LC) + (A − LC)T P1]x + xT x

+ (P1Ley)T (P1Ley)≤ xT [P1(A − LC) + (A − LC)T P1 + I + 𝜌yCT C]x= xT [

X1A + AT1 X1 − Y1C − CT Y T

1 + I + 𝜌yCT C]

x

< 0,∀x ≠ 0.

Therefore, the system (9) is asymptotically stable.

Remark 3. Unlike the result concerning theevent-triggered observer design in [3], where the matrixL is pre-given such that the matrix A − LC is Hurwitzand the rest of design variables are obtained by solving

some LMIs, our joint design method is more plausibleto achieve an optimal design than the above sequentialdesign procedure would.

We proceed to Problem B-II. Note that by Lemma 1one has

sat(Kox(t) + Koe(t)

)∈ co

{D+

j

(Kox(t) + Koe(t)

)+ D−

j Hox(t)|j ∈

}if ‖Hox(t)‖∞ ≤ 1, where Ho ∈ Rm×n is a matrix to bedesign later. Therefore,

x(t) ∈ co{[

A + B(

D+j Ko + D−

j Ho

)]x(t)

+ BD+j Koe(t)|j ∈

}. (21)

A sufficient condition for ‖Hox(t)‖∞ ≤ 1, similar to theanalysis in Section 4.1, is that there exists a positive def-inite matrix P2 such that the ellipsoid {x|xT Px ≤ 1} isan invariant set of the closed-loop system (4) and at thesame time{

x|xT P2x ≤ 1}⊂ {x| ‖Hox‖∞ ≤ 1}. (22)

To obtain the matrix P2 > 0 satisfying (22) and thegain matrices Ko,Ho, together with a triggering condi-tion, such that the system (21) is asymptotically stable,we first concern the stabilization for the dual system ofsystem (21), i.e.,

x(t) ∈ co{[

A + B(

D+j Ko + D−

j Ho

)]Tx(t)

+ BD+j Koe(t)|j ∈

}. (23)

These matrices and the triggering condition can be con-structed as follows. By solving the following LMIs withvariables X2,Y2,Z2 and 𝜌o,

X2AT + AX2 + BD+j Y2 + Y T

2 D+j BT + BD−

j Z2

+ ZT2 D−

j BT + I + 𝜌oCT C < 0, j ∈ ,(1 zi

2(zi

2

)TX2

)≥ 0, 1 ≤ i ≤ m,

X2 > 0, 𝜌o > 0, (24)

where zi2 is the i-th row of the matrix Z2, the matrix

Ko,Ho are obtained as Ko = Y2X−12 ,Ho = Z2X−1

2 , and bydefining P2 = X2, the triggering condition is proposed as

maxj∈

‖‖‖P2BD+j Koe(t)‖‖‖ ≤

√𝜌o‖y(t)‖. (25)



We show that these designs render the stability of thedifferential inclusion (23). To this end, consider the Lya-punov candidate V2(x) = xT P2x, and calculate its timederivative along the trajectory of system (23)

V2 ≤ maxj∈

{xT

[P2

[A + B

(D+

j Ko + D−j Ho

)]T+[A + B

(D+

j Ko + D−j Ho

)]P2

]x + 2xT P2BD+

j Koe}

≤ maxj∈

{xT

[P2

[A + B

(D+

j Ko + D−j Ho

)]T+[A + B

(D+

j Ko + D−j Ho

)]P2

]x

+ xT xT +(

P2BD+j Koe

)T (P2BD+

j Koe)}

≤ maxj∈

{xT

{[P2

[A + B

(D+

j Ko + D−j Ho

)]T+[A + B

(D+

j Ko + D−j Ho

)]P2

]+ I + 𝜌oCT C

}x}

.

By referring to the LMIs on the first line of (24), whichis equivalent to

P2

[A+B

(D+

j Ko + D−j Ho

)]T+[A+B

(D+

j Ko+D−j Ho

)]P2

+ I + 𝜌oCT C < 0, j ∈ , (26)

one sees that V2 < 0. Thus, system (23) is asymptoticallystable.

0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

Time(s)

The

sta

te

(a) Time evolution of the state under the event−triggered state feedback

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

Time(s)

||Ke(

t)||

and

ρ1/2 ||P

x(t)

||

(b) Illustration of the triggering condition under the event−triggered state feedback

0 0.5 1 1.5 2 2.5 3 3.5 4

−1

−0.5

0

0.5

1

Time(s)

The

sat

urat

ed in

put

(c) Time evolution of saturated input under the event−triggered state feedback

Fig. 2. Event-triggered control under state feedback.



We then come back to the design for the originalsystem (21) in Problem B-II. Let N ∈ Rn×m be a matrixsuch that rank

(P2CT

)= rank

(P2CT ,N

), which ensures

the existence of a matrix solution M ∈ Rp×m to the matrixequation P2CT M = N. We show that the stability of thesystem (21) is achieved under the above designed matricesKo,Ho and the triggering condition

maxj∈

‖‖‖BD+j Koe(t)‖‖‖ ≤

√𝜌o‖MT y(t)‖. (27)

Problem B-II can be solved in the followingtheorem.

Theorem 5. If there exists a set of solutions X2,Y2, 𝜌o tothe LMIs (24), then the gain matrix Ko = Y2X−1

2 ,Ho =Z2X−1

2 and the triggering condition (27) render the sta-bility of the system (21).

Proof. For the system (21), consider the Lyapunov can-didate V = xT P−1

2 x, whose time derivative satisfies

V = maxj∈

{xT

[P−1

2

[A + B

(D+

j Ko + D−j Ho

)]+[A + B

(D+

j Ko + D−j Ho

)]TP−1

2

]x + 2xT P−1

2 BD+j Koe

}≤ max

j∈

{xT

[P−1

2

[A + B

(D+

j Ko + D−j Ho

)]+[A + B

(D+

j Ko + D−j Ho

)]TP−1

2

]x

+ xT P−22 x +

(BD+

j Koe)T (

BD+j Koe

)}≤ max

j∈xT

{P−1

2

[A + B

(D+

j Ko + D−j Ho

)]+[A + B

(D+

j Ko + D−j Ho

)]TP−1

2 + P−22 + 𝜌oP−1

2 CT CP−12

}x

= maxj∈

xT P−12

{[A + B

(D+

j Ko + D−j Ho

)]P2 + P2

[A + B

(D+

j Ko + D−j Ho

)]T+ I + 𝜌oCT C

}P−1

2 x.

0 0.5 1 1.5 2 2.5 3 3.5−8

−6

−4

−2

0

2

4

Time(s)

Th

e s

tate

(a) Time evolution of the state under self−triggered state feedback

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

Time(s)

Th

e n

orm

of

Ke

(t)

(b) Illustration of the triggering condition under self−triggered state feedback

0 0.5 1 1.5 2 2.5 3 3.5

−1

−0.5

0

0.5

1

Th

e s

atu

ra

ted

in

pu

t

(c) Time evolution of saturated input under the self−triggered state feedback

Fig. 3. Self-triggered control under state feedback.



0 2 4 6 8 10 12−10

−8

−6

−4

−2

0

2

4

Time(s)

The

sta

te(a) Time evolution of the state under the event−triggered observer−based feedback

The first component of the stateThe second component of the state

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

Time(s)

Fun

ctio

ns in

trig

gerin

g co

nditi

on

(b) Illustration of the triggering condition for sensor−observer communication under the event−triggered observer−based feedback

Function on the left hand side of triggering condition Function on the right hand side of triggering condition

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

Time(s)

Fun

ctio

ns in

the

trig

gerin

g co

nditi

on

(c) Illustration of the triggering condition for the observer−controller communication under event−triggered observer−based feedback

Function on the left hand side of the triggering conditionFunction on the right hand side of the triggering condition

0 0.5 1 1.5 2 2.5 3 3.5 4

−1

−0.5

0

0.5

1

Time(s)

The

sat

urat

ed in

put

(d) Time evolution of saturated input under the event−triggered observer−based feedback

Fig. 4. Event-triggered control under observer-based feedback.



By referring to (26), one sees that V is negative definiteand therefore, the system (21) is asymptotically stable.

Remark 4. Event-triggered control via saturatedobserver-based feedback is realized by combiningTheorem 4 with Theorem 5, where the observer is alsoevent-triggered type. Available results only considerevent-triggered control via state feedback.

Remark 5. By arguing in a similar manner as inSection IV, the event-triggered sensing and event-triggered controller proposed in this section can beshown to have positive lower bounds, and further-more, self-triggered schemes for these two triggeringapproaches can also be obtained. To avoid the repeatabil-ity of the method used before and to save space, all theseissues are not treated again in this section.

VI. SIMULATION RESULTS

Consider the saturated linear control system (4)with

A =(−1.7741 0.4815−7.6837 2.0741

),B =

(88

),C =

(6 9

).

The upper bound for the saturation function is cho-sen to be c = 1. For the state feedback case, the gainmatrices K ,H and the parameter 𝜌 can be, by solv-ing the LMIs (15)-(16) with D+

1 = diag(1, 1),D+2 =

diag(0, 1),D+3 = diag(1, 0),D+

4 = diag(0, 0),D−1 =

diag(0, 0),D−2 = diag(1, 0),D−

3 = diag(0, 1),D−4 =

diag(1, 1) in (15) and i = 1 in (16), obtained as fol-lows K = (−46.8096, 23.6373),H = (−4.3415, 2.6915),and 𝜌 = 0.3004. And for the observer-based feed-back case, the gain matrices L,Ko,Ho and the param-eters 𝜌y, 𝜌o can be, by solving the LMIs (19) and (24)with j = 1, 2, 3, 4, i = 1 in (24), obtained as followsL = (0.4233, 7.6559)T ,Ko = (−1.8476, 1.2671),Ho =(−1.2527, 0.6547), and 𝜌y = 0.95, 𝜌o = 0.82. Note thata saturated inputs system is only locally stabilizable. Forsimulation, we choose the initial condition as x0 = (4, 4)Tand x0 = (3, 5)T .

The simulation result for the case of event-triggeredstate feedback is shown in Fig. 2, from which one seesthat the trajectory is convergent, shown in 2a, and thecontrol input is actually saturated, shown in 2c. The trig-gering condition (14) is illustrated in 2b which depicts thetime evolution of functions on both sides of (14), with themeaning that the controller (2) is invoked when the bluecurve hits the red one.

The simulation for the case of self-triggered statefeedback is shown in Fig. 3. Although the self-triggered

control, compared with the event-triggered case, has theadvantage that it does not continuously check the trig-gering condition (14) but determines the next executiontime by calculating a function of the last measurementof the state, it samples more frequently than with anevent-triggered control since the inter-event times areestimated in a conservative way, shown in Fig. 3b. Themore frequently the controller samples, the higher per-formance (for example, the higher convergent rate) thesystem achieves; this fact has been pointed out in [5] andit is illustrated in Fig. 3a. The control in self-triggeredcase is also saturated, shown in 3c.

For the case of event-triggered observer-based feed-back, Fig. 4 depicts the time evolution the state of theplant and the observer (Fig. 4a), time evolution of thefunctions on both sides of the triggering conditions (20)(Fig. 4b), time evolution of the functions on both sidesof the triggering conditions (27) (Fig. 4c), and time evo-lution of the saturated input (Fig. 4d).

VII. CONCLUSION

For linear systems with saturated inputs, event-triggered state feedback is firstly considered. We alsostudy the event-triggered control via observer-basedfeedback, where the observer is constructed also in anevent-triggered form and it is used to estimate the sys-tem’s state. Positive lower bounds for inter-event times areobtained and self-triggered schemes are presented. Themethod of placing saturation nonlinearity into the con-vex hull of a group of linear feedbacks is adopted to dealwith saturated control which is challenge to control the-ory. The convergence analysis is carried out both withproofs and simulations.

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Wei Ni received his Ph.D. in systems sci-ence from Academy of Mathematics andSystems Science, Chinese Academy of Sci-ences in 2010. He is currently Lecturerwith School of Science, Nanchang Uni-versity, Nanchang, China. His researchinterests include control of switched and

impulsive systems, complex systems, etc.

Ping Zhao received his Ph.D. from theAcademy of Mathematics and SystemsScience, Chinese Academy of Sciences, in2008. He is currently a teacher at the Uni-versity of Jinan. His research interests arein stability theory and control of stochas-tic and nonlinear systems.

Xiaoli Wang received her Ph.D. fromthe Academy of Mathematics and Sys-tems Science, Chinese Academy of Sci-ences, in 2010. She is currently Lecturerwith School of Information and Electri-cal Engineering, Harbin Institute of Tech-nology at Weihai, Weihai, China. Her

research interests include system modeling and multia-gent systems.

Jinhuan Wang received her Ph.D. degreefrom the Academy of Mathematics andSystems Science, Chinese Academy ofSciences in 2008. She is currently Asso-ciate Professor of the School of Sciences,Hebei University of Technology, China.Her research interests include complex

systems control, switched systems.


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Event-triggered Control of Linear Systems With