A robust adaptive event-triggered control scheme for dynamic output-feedback systems

Accepted Manuscript

A Robust Adaptive Event-Triggered Control Scheme for Dynamic Output-Feedback Systems Hassan Adloo , Mohammad Hosein Shafiei PII: DOI: Reference:

S0020-0255(18)30814-4 https://doi.org/10.1016/j.ins.2018.10.017 INS 13998

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Information Sciences

Received date: Revised date: Accepted date:

22 April 2018 7 October 2018 13 October 2018

Please cite this article as: Hassan Adloo , Mohammad Hosein Shafiei , A Robust Adaptive EventTriggered Control Scheme for Dynamic Output-Feedback Systems, Information Sciences (2018), doi: https://doi.org/10.1016/j.ins.2018.10.017

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A Robust Adaptive Event-Triggered Control Scheme for Dynamic OutputFeedback Systems Hassan Adloo and Mohammad Hosein Shafiei Department of Electrical and Electronics Engineering, Shiraz University of Technology, Modares Blvd., Shiraz, Iran ([email protected], [email protected])

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Abstract

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In this paper, a new output-based adaptive event-triggered control (AETC) method is introduced for a general class of nonlinear systems. A crucial aspect of the AETC method is to design an event-triggering mechanism (ETM) determining when the sampling operation should be done. The proposed ETM, which is derived from the finite-gain L2 stability condition, reduces the number of updating operations while the performance of the closed-loop system is maintained. Thanks to the Lipschitz property, the avoidance of Zeno behavior is also guaranteed by introducing a positive lower bound on the inter-event times. This lower bound is obtained by solving a set of proposed ordinary differential equations. In addition, the proposed method is developed into robust control of uncertain linear time invariant (LTI) systems through some matrix inequalities (such as linear matrix inequalities (LMIs)). Finally, several numerical simulations are presented to demonstrate the ability of the proposed method to achieve greater average inter-event times and to preserve the system performance rather than other references.

1. Introduction

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Keywords: Adaptive Event-Triggered Control; Dynamic Output Feedback; Zeno-Free; Minimum Inter-Event Time; Robust Control Design

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The event-triggered control (ETC) method has been introduced for digital control systems to overcome communication challenges arising from some limitations in processing resources [17]. In ETC methods, when a special event occurs, the control signal is updated through a specific mechanism, called event-triggering mechanism (ETM). The ETM is generally pre-designed in terms of stability conditions and actually determines sporadic updating times at which the controller is executed [17]. This leads to save much of computational and network resources.

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Although ETC methods have been introduced to diminish the burden of computations and also to improve the efficiency of the communication network, they might decrease the performance of the control system, considerably [19]. In addition, in some cases, two consecutive events might occur at the same time, which deteriorates the system performance more. This phenomenon is called Zeno behavior. There are some earlier works in the literature, which have tried to remove these deficiencies. For instance, the authors of [1] have presented an event-based output feedback controller with Zenofree analysis. Moreover, in [10], an output-based ETM has been designed to guarantee the existence of a positive minimum inter-event time and at the same time, to achieve a better

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performance using some manually adjusted parameters in numerical simulations. In continuing, the reference [5] has proposed an observer-based event-triggered scheme in which the existence of a lower bound on inter-sample times has been investigated. However, parameters of the mechanisms of these references are assumed to be fixed and thus they could not be adapted with measured signals.

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Recently, other interesting ideas have been suggested to improve the performance criteria based on adaptive techniques. These approaches are commonly named adaptive event-triggered control (AETC) methods. Broadly speaking, in AETC methods, one or more parameters of the ETM might be adaptively changed in order to get a higher level of system performance. This technique can also lead to a considerable reduction in the number of sampling operations.

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Along with the existing literature, in [6] an ETM has been proposed for an event-based networked control system, in which the threshold parameter changes adaptively with regard to the squared error norm. Moreover, an adaptive event-based feedback controller has been suggested in [27] to balance between communication resources and the control performance for uncertain nonlinear interconnected systems. On the other hand, the consensus problem and the analysis of distributed interconnected systems have been investigated by the authors of [24] and [9], respectively. In both works, some parameters of ETM are adapted in each sampling operation depending on the measured signals. However, the aforementioned works have not presented the Zeno-free analysis for their approaches. In this regard, the authors of [13] have presented a lower bound on the minimum inter-event time for a special case of strict form nonlinear systems. Moreover, in [18], as a sense of min-max optimal control policy, an eventbased adaptive approximator has been presented with a guaranteed Zeno-free behavior. A robust analysis for a networked control system has been studied in [26] whereby the threshold of the proposed ETM has been adaptively changed to manage communication network resources. Moreover, to achieve a faster consensus agreement for linear multi-agent systems, the authors of [8] have presented an AETC method by introducing a time-varying exponential bound on the measured error signal. After that, in [7], this method has been developed for a class of uncertain interconnected multi-agent systems. In both of these papers, it is guaranteed that no Zeno behavior can occur while the closed-loop system is asymptotically stable. Lastly, in [2], the ETC method of [1] has been developed to reduce the number of samplings more and to guarantee the avoidance of Zeno behavior. However, in all the aforementioned works the whole system states should be employed in the ETM, while the state variables are rarely accessible in practical situations. An alternative approach to remove this problem is to use output-based controllers or observer-based system designs. In the case of output-based AETC methods, a few researches have been presented. For instance, [14] has proposed an adaptive fuzzy observer based on AETC techniques in a backstepping design scheme. However, the considered system is a special class of single input-single output systems in the strict form. Moreover, authors of [22] have presented an optimal AETC method for stochastic linear time-varying multi-agent systems. An event-based fault detection algorithm is suggested in [21] for time-delayed networked control systems. Furthermore, in [23], an H∞ tracking controller in an adaptive event-driven manner has been designed for networked control

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systems with application to satellite systems. Moreover, a developed adaptive strategy based on the method of [1] has been proposed in [4] wherein to prevent Zeno incident, the proposed eventbased mechanism will be allowed to perform only when a specific time is elapsed. Furthermore, [11] has introduced an AETC approach for linear and nonlinear discrete-time systems, in which the stability of the closed-loop system and the existence of a lower bound on inter-event times have been investigated.

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This paper offers a new output-based AETC method for a general nonlinear continuous-time system. In the proposed method, the ETM has enriched with a non-negative function of an adaptive variable to decrease the number of control updates and at the same time, to preserve system performance as much as possible. Moreover, a Zeno-avoidance analysis is considered by investigating the existence of a positive lower bound on the inter-sample times. The robustness analysis against uncertainties of the system has not yet been considered for this field, thus in this paper, the proposed method is applied to uncertain LTI systems with guaranteed robustness. Furthermore, asymptotical stability of the closed-loop system has been proved through solving some matrix inequalities (especially LMIs). Finally, some numerical examples are presented which show the effectiveness of the proposed method in comparison with a recent work.

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The rest of the paper is organized as follows: Section 2 reviews some required notations and lemmas. In Section 3, system statements with some necessary assumptions are introduced. To exclude Zeno behavior, a minimum inter-event time analysis will be presented in Section 4. Moreover, a robust design method for uncertain linear systems is presented in Section 5. Then in Section 6, some numerical simulations will be considered to verify and compare the proposed method. Finally, the paper is concluded in Section 7.

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2. Preliminaries notations and lemmas

Notations and Lemmas

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In this paper, |·| indicates the Euclidean norm of a vector or matrix. Rm×n and R≥0 stand for all m×n real matrices and real non-negative numbers, respectively. The expression f : Rm×Rn → Rp means a real vector function, f (.,.), belonging to Rp where input arguments are vectors defined in Rm and Rn. A function α : [0, a) → R≥0, a > 0 is said to be a class K function, if it is continuous, strictly increasing and α(0) = 0. Besides, if a = +∞ and α(r) → +∞ as r → +∞, α is called a class K∞ function. A function f : Rm×Rn → Rp is said to be Lipschitz continuous on compact set S  Rm×Rn, if there exists a constant L > 0 such that | f(x1, y1) – f(x2, y2) | ≤ L | (x1, y1) - (x2, y2) | for every x1, x2Rm and y1, y2Rn. Matrix I(n) is defined to show a unity matrix belongs to Rn×n. Lemma 1 (L1) [16]: For any vector r = [r1, …, rn]T and positive constants αi for i = 1, ..., n, the following inequalities are held ri  r : for i  1,..., n

)1(

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n

r   ri

)2(

i 1

n

  i 1

i

 n  ri      i  r  i 1 

)3(

Lemma 2 (L2) [25] (Schur complement): Consider the matrix M as shown below

 A B M  T  B C 

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)4(

where A, B and C are matrices with suitable dimensions. Thus, M < 0 if and only if one of following expressions are fulfilled; C  0 & A  BC 1B T  0

)5(

1

A0 & CB A B 0

)6(

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Lemma 3 (L3) [3] (Elimination lemma): Let W, R, S and are some matrices or vectors with appropriate dimensions and suppose that WT = W. As a consequence, the inequality W  R S  ST

is held for any

T

RT  0 T

 I , if and only if there exists a real positive parameter β, such that

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W   RRT   1S T S  0

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3. Systems representation

Consider a general form of an output-feedback ETC system that is, y(t )  g p ( x p (t ))

)7(

u(t )  gc ( xc (t ), y(ti ))

)8(

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x p (t )  f p ( x p (t ), u(ti )),

xc (t )  f c ( xc (t ), y(ti )),

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where xpRn and xcRm are the states of plant and controller respectively. Also, uRp is the control input and yRq stands for the system output (Fig 1). In addition, t represents time variable and ti indicates the i’th index of updating time thus u(ti) and y(ti) stand for the measured signals of u and y, at the time ti. Also, fp : Rn × Rp → Rn, fc : Rm × Rq → Rm, gp : Rn → Rq, gc : Rm × Rq → Rp are smooth vector-valued functions. In this paper, it is assumed that the event-triggering mechanism (it will be proposed later) is based on an extra parameter, ηR≥0, which is governed by the following dynamic:

 (t )  f ( (t ), e(t ), y(t ))

)9(

where e = [eyT , euT ]T denotes the measurement error vector such that ey = y(ti) – y(t) and eu = u(ti) – u(t).

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y(t )

u (ti )

Fig. 1: Output-based AETC system

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y (ti )

u (t )

x p  f p ( x p , u  eu ),

y  g p ( xp )

xc  fc ( xc , y  ey ),

u  gc ( xc , y  ey )

  f ( , e, y)

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Therefore, the integrated closed-loop system can be written as follows,

)11(

)12(

)11(

where for simplicity, the variable time t, has been ignored in (10)-(12).

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Assumption 1 (A1): Consider that for any vectors xpi  Xp  Rn, xci  Xc  Rm , ui  U  Rp, yi  Y  Rq , ei  E  Rq+p and ηi  H  R≥0 for i = 1 and 2, functions fp, fη, gc and fη are Lipchitz such that,

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f p ( x p1 , u1 )  f p ( x p 2 , u2 )  Fp x p1  x p 2  Fp u1  u2 f c (x c 1 , y 1 )  f c (x c 2 , y 2 )  Fc x c 1  x c 2  Fc y 1  y 2 g c (x c 1 , y 1 )  g c (x c 2 , y 2 )  G c x c 1  x c 2 G c y 1  y 2

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f (1 , e1 , y1 )  f (2 , e2 , y2 )  F 1 2  F e1  e2  F y1  y2

)13( )14( )15( )16(

g p

 Gp

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x p

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where Fp, Fc, Gc and Fη are Lipschitz constants. In addition, assume that gp is a continuously differentiable bounded function, such that,

)17(

where Gp is a positive constant. Finally, in this paper it is assumed that the dynamical equation of adaptation parameter η (equation (9)) and its initial condition are chosen such that η to be positive for all t ≥ 0. Indeed, this assumption will be used later to design a procedure for determining a minimum inter-event time. In addition, from (17), it can be concluded that [12],

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g p ( x p1 )  g p ( x p 2 )  Gp x p1  x p 2

)18(

Assumption 2 (A2): Given q = [xpT , xcT , η]T ∈ Rn+m+1 , f = [fpT , fcT , fηT]T. Suppose that there exists a locally Lipschitz Lyapunov function V : R n+m+1 → R≥0, such that for all q ∈ R n+m+1, 1 ( q )  V ( q )   2 ( q )

)19(

V , f   ( q )   ( y )   ( e )

)21(

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where α1, α2, α, δ, γ are class K∞ functions. Additionally, assume that there are positive constants, Δ and Γ such that γ(|e|) ≤ Γ|e| and Δ|y| ≤ δ(|y|).

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Note that A2 provides finite-gain L2 stability condition devoted to the closed-loop system (1012). In this paper, according to emulation method [15], it is supposed that the controller is predesigned in the continuous time case (without any sampling), such that the whole closed-loop system (10-12) is finite-gain L2 stable from e to y. However, ETC methods have benefited from the event-based sampling operations, for which according to (20), the next event time, ti+1 might be generated from the following traditional mechanism: ti 1  inf  t  R : t  ti   1 ( y )   ( e )  0 

)21(

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where ζ1∈(0, 1]. Generally, (21) is called static event-triggering mechanism. Nevertheless, in this paper, the goal is to introduce a new adaptive event-triggering mechanism to reduce the number of updating operations as well as to improve the system performance as possible. Remark 1 (R1): Regarding (20), another finite-gain L2 stability condition can be expressed as

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V , f   ( q )   ( y )  l ( )   ( e )

)22(

where l(.) is a class K∞ function and Lη ≤ l(η) ≤ L'η for L ≥ 0 and L' ≥ 0.

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Actually, with respect to the fact that l(.) is a nonnegative function, it is evident that )23(

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 ( q )   ( y )  l ()   ( e )   ( q )   ( y )   ( e )

Thus, condition (22) implies condition (20) and consequently the closed-loop system (10-12) is finite-gain L2 stable.

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According to the condition (22), in order to guarantee asymptotic stability of the system, sampling operations should be performed when the following triggering criteria is violated:

  e   1  y    2l ( )

)24(

where ζi ∈ (0, 1] for i = 1,2. Therefore, a new AETM can be proposed as below, ti 1  inf  t  R : t  ti  1 ( y )   2l ( )   ( e )  0 

)25(

Clearly, if l ( )  0 , equation (25) leads to the static mechanism (21). 6

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Remark 2 (R2): Suppose that last update has been occurred at the time ti. Now, the goal is to show that if the next update time is determined using AETM, the corresponding time will be greater than the case of using the static ETM. Define tis1 and tia1 as next event times, respectively produced by static and adaptive event-triggering mechanisms (21) and (25). Thus, it is not hard to show that tis1  tia1 (See Fig 2). This is because of the fact that for same initial values of the output signal in a time duration between any two successive events, violation of static eventtriggering condition,   e (t )   1  y (t )  takes less time rather than using adaptive event-triggering

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condition   e (t )   1  y (t )    2l ( (t )) . According to the principle of contradiction, assume that t ia1  t is1 , thus, at the time t ia1 , triggered time tis1 is not evoked yet using (21). Therefore, it can be

concluded that







 e (t ia1 )  1 y (t ia1 )



)26(









 e (t ia1 )  1 y (t ia1 )   2l  (t ia1 ) 

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On the other hand, noting to adaptive event-triggered rule (25), we know that

Since the function  2l  (t ia1 )  is a nonnegative function, thus







 e (t ia1 )  1 y (t ia1 )



)27(

Here, we find that (26) contradicts to (27), thus t i 1  t i 1 . This fact has been depicted in Fig. 2 graphically. a

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s

e

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 1  y   1  y 

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1  y    2l ( ) 1  y    2l  

t

tis tia

tis1 tia1

Fig. 2: Time history of the function γ(|e|); in static ETM (dashed line) and in adaptive ETM (Solid line)

4. Inter-event time analysis As mentioned earlier, the deleterious effects of Zeno phenomenon disrupts fundamentally the system operation and causes misleading information for the plant. Here, to cancel out the Zeno behavior, regarding some properties of the system, it is shown that there exists a positive

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minimum time interval between any two sequential events. Indeed, first suppose that the following condition is established

 e  1 y   2 L

)28(

where Γ, Δ and L are previously defined in A2 and R1. Then, it can be concluded that

  e    e  1 y   2 L  1  y    2l ( )

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)29(

Obviously, from (29), it is realized that condition (28) will be satisfied earlier than (24). Therefore, as a sense of prevention of Zeno behavior, interval times between any two consecutive events generated from violation of (28) is less or equal than interval times produced by (25). Because of this and to have simpler derivations, let us focus on the adaptive eventtriggering condition (28) in order to investigate minimum time elapsed between two consecutive updating instants. Therefore, according to (26), any sampling time is occurred when

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 e  1 y   2 L

)31(

Proposition 1: Consider the closed-loop event-triggering system (10-12) satisfying A1 and R1. Also, assume that the adaptive event-triggering condition (28) is used to evaluate the updating times. Thus, a lower bound on inter-event time is obtained by intersecting the response of following differential equations with line  (t , 0)   (t ,0 )  1 :

)31(

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  ( M   N  )(1   )     F  F  F   M   N       H p  Fc (1  G p )   ( I p  Fc )   M   N  

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N  Gp I p .

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where θ, θ and ψ are real positive variables with θ0 = |e0| / |y0|, θ0 = η0 / |y0| and ψ0 = |x0| / |y0|. In addition, μ = Γ / ζ1Δ, ν = ζ2 L / ζ1Δ, H p  Fp 1  Gc (1  Gp )  , I p  Fp (1  Gc ) , M  Gp H p and

Proof: First, rewrite (30) as below e(t )  (t ) (t )   1, y(t ) y(t )

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t  [ti , ti 1 ) )32(

Without loss of generality, suppose that ti = 0 and ti+1 = η, thus, the goal is to study the equation (32) over the region, t  [0, η). Moreover, it is clear that determining the lower bound of intersample times is concerning with the behavior of |e| / |y| and η / |y|. In this regard, for the former we have, e  y d e  eT y yT y e   2  1       dt  y  e y y y y y 

)33(

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Also, from (10) and (17), we can conclude that y 

g p d g p (xp )  x p  Gp x p  dt x p

)34(

On the other hand, according to A1, it is convenient to show that

 Fp x  Fp e  Fp gc ( xc , y  ey )

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x p  f p ( x p , u  eu )  Fp x p  Fp u  eu  Fp x  Fp e  Fp u

 Fp x  Fp e  FpGc xc  FpGc y  FpGc ey  Fp 1  Gc (1  G p )  x  Fp (1  Gc ) e

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Thus, xp  H p x  I p e

)35(

)36(

where H p  Fp 1  Gc (1  Gp )  and I p  Fp (1  Gc ) . Hence, from (34) and (36) we obtain )37(

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y M x N e

where M  Gp H p and N  Gp I p . Accordingly, it follows that

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e  x e  d  e   M  N     1    dt  y   y  y y

)38(

d   dt  y

 yT y    y       2 y y y  y y y

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

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On the other hand, dynamic behavior of η / |y| should be also considered. Regarding to (32), the worst case of minimum inter-event time is occurred when the lower bound of dynamical behavior η / |y| or equivalently the upper bound of (– η / |y|) is happened. From this point of view, one can write

)39(

According to (36) we have 

d   dt  y

  x e    N      M y y y y  

)41(

In addition, from A1, it follows that

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

d   dt  y

  e x e    F  F   M N   F  y y y y  y  

)41(

It is clear that solutions of (38) and (41) depend also on the behavior of |x|/|y|. Therefore, we have x y x d  x  xT x y T y x  2      dt  y  x y y y y y y

Note that, from (36) and (14), it is concluded that

x  ( x p , xc )  x p  xc  H p x  I p e  Fc x  Fc e  Fc y

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  H p  Fc (1  Gp )  x  ( I p  Fc ) e

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)42(

Consequently,

)43(

x e  x e  x d x  N      H p  Fc (1  G p )   ( I p  Fc )   M dt  y  y y  y y y

)44(

p  1  p  ( M r  Np)

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q  F q  F p  F   M r  Np  q

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Finally, if the following three inequalities are held

r   H p  Fc (1  Gp )  r  ( I p  Fc ) p   M r  Np  r

)45( )46( )47(

CE

PT

where, p = |e|/|y|, q = η/|y| and r = |x|/|y|, then using the Comparison Lemma [12], we obtain that q(t )   (t ,0 ) and r (t )   (t , 0 ) , where  (t , 0 ) ,  (t ,0 ) and  (t , 0 ) are solutions of equations (31) where the initial conditions are 0   (0, 0 )  0 , 0   (0,0 ) and  0   (0, 0 ) .

AC

In this regard, finding a lower bound on inter-event times i.e. η is concerned with the intersection of solutions of differential equations (31) with following static equation which is derived from (32):  (t , 0)   (t ,0 )  1,

t [0, )

This is the end of proof.

)48(

■

5. Robust control design for linear systems In this section, the proposed method is applied to an event-based LTI system in which the dynamic of the plant is; 10

ACCEPTED MANUSCRIPT

x p (t )  Ap x p (t )  B p u(ti )

)49(

y(t )  C p x p (t )

)51(

Moreover, consider that the controller is governed by xc (t )  Ac xc (t )  Bc y(ti )

)51(

u(t )  Cc xc (t )  Dc y(ti )

)52(

CR IP T

where xpRn, xcRm, uRp and yRq are respectively the state of the plant, state of the controller, input control and output of the plant. Moreover, Ap, Ac, Bp, Bc, Cc, Cp and Dc are matrices with proper dimensions. Assumption 3 (A3): Consider that Ap = Apn + ΔAp and Cp = Cpn + ΔCp, where Apn and Cpn are known parts of Ap and Cp, also ΔAp and ΔCp are considered as uncertain parts. Moreover, suppose that there exist known matrices H1, H2 and E with proper dimensions satisfying,

AN US

 Ap   H1   C     F (t ) E  p  H2  T

where F (t) F(t) < I.

)53(

According to definitions ey = y(ti) – y(t) and eu = u(ti) – u(t), the closed-loop system (49-52) can be rewritten as follows: x p  Ap x p  Bp  eu  u  ,

M

y (t )  C p x p

u  Cc xc  Dc  ey  y 

ED

xc  Ac xc  Bc  ey  y  ,

)54( )55(

Moreover, after some simplifications, we have

x  Ax  Be

)56(

PT

where x = [xpT , xcT ] T, e = [eyT , euT ] T and Bp Cc  B D , B p c  Ac   Bc

CE

 Ap  Bp DcC p A BcC p 

Bp  0 

)57(

On the other side, let the dynamic of the adaptation variable, η is chosen as

AC

   ( )  1 ( y )   ( e ),

0  0

)58(

where  is a class K∞ function. Proposition 2: Consider the AETM (25) for which the η-dynamic is considered as (58) is used. It is shown that  (t )  0 for any 0  0 . Proof: According to AETM (25), for all t  R≥0 –{ti}, we have

 ( e )  1 ( y )   2l ( )

)59(

11

ACCEPTED MANUSCRIPT

Therefore,

 2l ( )  1 ( y )   ( e )

)61(

On the other hand, from Lipschitz property of the function l(.), it can be concluded that  2l ( )   2 L

)61(

 2 L   1 ( y )   ( e )

CR IP T

Thus, )62(

Now, subtracting (η), from both sides, leads to      2 L       1 ( y )   ( e )

)63(

Then, regarding (56) and (61), it can be concluded that

AN US

     2 L  

)64(

Now, define ˆ   ˆ    2 Lˆ with ˆ0  ˆ  0 . It is easily shown that ˆ (t )  0 for all t  R≥0. Therefore, according to Comparison lemma [12] and because of ˆ   , one can conclude that ≥0 ■ 0  ˆ (t )   (t ) for 0  ˆ0  0 and t  R –{ti}.

y(t )  C pn x p

where

BpCc  Ac 

PT

 Apn  Bp DcC pn An   BcC pn 

)65(

ED

x  An x  Be,

M

In what follows, two cases are considered, the closed-loop system with uncertainty and without uncertainty. First, consider that the nominal system (without uncertainty) is defined as

)66(

CE

In this case, the following theorem gives conditions of Input-Output finite-gain L2 stability of system (65).

AC

Theorem 1: Consider nominal closed-loop event-triggered system (65) with adaptive ETM (25) and dynamic (58). If there exist positive parameters ζ1, ε1, ε2, ε3, εy, εe, εη, ρ where ζ1 ∈ (0, 1] and also a positive definite matrix PRn+m satisfying  AnT P  PAn   y 1 (   1)C TpnC pn  1I nx  0   BT P 

where C pn  C pn stable.

0

 2   3   0

 PB  0 0  e (   1) I ny 

)67(

0 , then the nominal closed-loop event-triggering system is finite-gain L2

12

ACCEPTED MANUSCRIPT

Proof: Consider a candidate Lyapunov function V as, V (q)  xT Px  

)68(

where PRn+m and ρ is a positive real number. Now, the derivation of the Lyapunov function V along with the time is V  xT Px  xT Px  

Now, according to (58) and (65), (69) can be written as V  xT  AnT P  PAn  x  2 xT PBe   ( )  1 ( y )   ( e )

CR IP T

)69(

)71(

In this case, let us define  ( )   ,  ( e )   e e and  ( y)   y y , where εη > 0 , εe > 0 and 2

2

AN US

εy > 0, thus

V  xT  AnT P  PAn  x  2 xT PBe    1 y y   e e 2

2

Referring to this fact that y  yT y  xT C TpnC pn x where C pn  C pn 2

)71(

0 , we have )72(

M

V  xT  AnT P  PAn   y1C TpnC pn  x  2 xT PBe     e eT e

Now, in order to establish the finite-gain L2 stability condition presented in L1, let us define

 (q)  1 x   2 and l ( )   3 , thus, from (22), it is necessary to have

ED

2

V  xT  AnT P  PAn   y1C TpnC pn  x  2 xT PBe     e eT e

PT

 1 xT x   2  xT  yC TpnC pn x   e eT e   3

)73(

Since 0 < ζ1 < 1, therefore,

CE

xT  AnT P  PAn   y1 (   1)C TpnC pn  1I nx  x  2 xT PBe  ( 2   3   )   e (   1)eT e  0

AC

Then it is evident that (74) leads to the (67).

)74(

■

Now let us consider the uncertain system (56-57) satisfying A3. The main result in this case has been presented in the following theorem. Theorem 2: Consider the uncertain closed-loop system represented in (56-57) with Assumption 3 where the sampling strategy is governed by (25) and the adaptive variable η is updated using (58). If there exist positive parameters β1, β2, ζ1, εy, εe, εη and ρ where ζ1 ∈ (0, 1] and also a positive definite matrix PRn+m satisfying

13

ACCEPTED MANUSCRIPT

PX 1 0 0 0  11I (r) *

CR IP T

 T C Tpn  1 T 0 PB  A n P  PA n  1 Y 1 Y 1  1I (n)    0    *  2   3   0 0  * *  e (   1)I (p q) 0   1 * * * ( y (  1  1)) I (q)   2 H 2 H 2T   * * * *  * * * * 

 ET   0   0 0 0   0     2 I (s) 

)75(

where C p  C p

0 then the closed-loop system is finite-gain L2 stable.

Proof:

   I    0 

 0   0 

)76(

ED

 A p  B p Dc     C p   A p  Bc     C p 

0 0

M

 A p  B p Dc C p A   B c C p 

AN US

First, note that A  An  A and C p  C pn  C p , thus

Moreover, regarding to (53) we have

PT

H  B p Dc   1  F (t )E H 2  H  B c   1  F (t )E H 2 

CE

   I A     0 

AC

 H 1    I B p Dc     H 2    F (t )  E  H1    0 Bc       H 2    H  B p Dc H 2   1  F (t )  E 0 Bc H 2  

 0   0 

0

)77(

Now, for simplicity, assume that

14

ACCEPTED MANUSCRIPT

A  X1F (t )Y1

)78(

 H 1  B p Dc H 2   , Y 1  E Bc H 2  

where X 1  

0 . Hence, from A1, the finite-gain L2 stability is

established if xT  AT P  PA   y ( 1  1)C Tp C p  1I nx  x  2 xT PBe  ( 2   3   )   e (   1)eT e  0

CR IP T

To do this, note that AT P  PA  AnT P  PAn  (A)T P  P(A)  AnT P  PAn  Y1T F T X1T P  PX1FY1

)79(

Therefore, according to Elimination Lemma (L3), there exists a parameter β1>0, such that

)81(

AN US

AT P  PA  AnT P  PAn  11Y1T Y1  1PX 1 X1T P

Thus,

V  xT  AnT P  PAn  11Y1T Y1  1PX1 X1T P   y ( 1  1)C Tp C p  1I nx  x 2 xT PBe  ( 2   3   )   e (   1)eT e

)81(

M

On the other side, the inequality matrix form of (81) is,

ED

 AnT P  PAn  11Y1T Y1  1PX 1 X 1T P   y (  1  1)C Tp C p  1I nx  qT  0  BT P 

0

 2   3   0

 PB  0 q  0  e (   1) I ne 

PT

)82(

where q  [x T ,  , e T ]T . The inequality (82) can be rewritten as

AC

CE

 AnT P  PAn  11Y1T Y1  1 PX 1 X 1T P  1 I nx  0   BT P 

0

 2   3   0

T

 C Tp  C Tp  PB     0    y (  1  1)  0   0   0  0  0   e (   1) I ne    

Here, thanks to Schur Complement Lemma (L2), we have  0   0 W  0   0  C  p

0 0 0 0 0

0 0 0 0 0

0 C Tp   0 0  0 0 0  0 0  0 0 

)83(

15

ACCEPTED MANUSCRIPT

where 0

PB

 2   3   0

0  e (   1) I ne

0

0

Now, according to A3, we can conclude that T

 0       0   0 0 0 0    0  F  E 0 0 0 0   0     0    H    2 

AN US

 0   0    W   0  F E    0   H 2 

       1 ( y (  1  1)) I ny  C Tpn     0  0 0

CR IP T

 T 1 T T  An P  PAn  1 Y1 Y1  1PX 1 X 1 P  1 I nx  0 W   BT P   C pn 0  

)84(

Consequently, refereeing to Elimination Lemma (L3), there is a parameter β2 > 0, such that T

ED

M

 0  0   0  0     T W   2  0   0    21  E 0 0 0 0  E 0 0 0 0   0     0  0   H 2   H 2 

And finally

PB 0  e (   1) I ne 0

   0   1 T ( y (  1  1)) I ny   2 H 2 H 2  C Tpn     0  0 0

AC

CE

PT

 T 1 T T 1 T 0  An P  PAn  1 Y1 Y1  1PX 1 X 1 P  1I nx   2 E E   0  2   3    T B P 0   C pn 0 0 

)85(

In addition, here, Schur Complement Lemma (L2) can be used to show that

16

ACCEPTED MANUSCRIPT

0  e (   1) I ne

C Tpn     0  0 0

0

( y (  1  1)) 1 I ny   2 H 2 H 2T

0

0

PB

Again using Schur complement lemma (L2), we have (75).

CR IP T

 T 1 T 1 T 0  An P  PAn  1 Y1 Y1  1I nx   2 E E   0  2   3    T B P 0   C pn 0 0   X 1T P 0 

 PX 1   0  0 0   0   11I r 

■

Remark 3: Note that nonlinear matrix inequalities (67) and (75) in Theorems 1 and 2 can be converted to a LMI system if parameters ζ1 and ρ are supposed to be fixed values.

AN US

6. Numerical simulations

Two numerical examples are considered in this section. In the first example, a nonlinear system is presented and in the second example includes an uncertain LTI system. Note that both examples are taken from [1]. Moreover, to indicate capability of the proposed method, simulation results will be compared with results of the method represented in [1].

ED

 x1  10( x1  x2 )   x2  28 x1  x2  x1 x3  u  x  x x  8x / 3 1 2 3  3 y  x1

M

Example 1: Consider a nonlinear third-order system as shown in (86);

)86(

PT

Assume that the static output feedback controller u  28.67 y is used that was proposed by [1] such that P = diag(2, 30, p3),   e    e e and   y    y y 2 where e(t) = y(ti) – y(t),  e  1572 and 2

CE

 y  10 and p3 > 0. Here regarding to L4, we can consider l     with   25 . Moreover, consider that the sampling operations are performed based on triggering condition (24) where ζ1=0.9 and the adaptive dynamic (58) is applied where  3  0.1 and η0 = 0. The time duration of

AC

the simulation is 0 to 1 seconds with initial condition x0 = [-2, -2, 3]T. Fig. 3.a. illustrates control inputs and output responses resulted from the proposed method and the method of [1]. Besides, inter-event times with respect to time, t is depicted in Fig. 3.b.

17

CR IP T

ACCEPTED MANUSCRIPT

AN US

a) b) Fig. 3: a) Output response, control input and b) Inter-event times in example 1 for the proposed method (solid lines) and the method in [1] (dashed lines).

M

As observed in Fig. 3. a, the quality of output responses as well as control inputs generated by both methods are very similar while sampling time intervals obtained by the proposed method are remarkably greater than sampling time intervals in the method of [1]. Additionally, Table 1 can verify this claim where it shows a comparison between the average inter-event time and the minimum inter-event time of the proposed method and the method of [1]. Table 1: Average and minimum inter-event times as well the number of sampling operations in example 1 Average inter-event time 13.997 4.5866

ED

Proposed method Method in [1]

Minimum inter-event time 8.2500 3.5700

Number of samplings 141 460

0  Bp    , 1 

CE

 2 3 Apn   , 1 3

PT

Example 2: Let us consider the uncertain LTI system (49-50) with following matrices

AC

 0.01  H1   ,  0.01

H 2  0.01,

C pn  1 0 E  0.1 0.2 ,

F (t )  0.01sin(t ) )87(

In addition, assume that the following observer-based state feedback controller is applied which is suggested by [20], xˆ (t )  ( Apn  FC pn ) xˆ(t )  Bp K xˆ(ti )  Fy(ti )

u(t )  K xˆ (t )

)88(

where xˆ is state estimation vector and according to A1, we can assign F = – [20, 28] and K = – [30, 20] T. By comparing (88) with (49-50), it is clear that

18

ACCEPTED MANUSCRIPT

Ac  Apn  FC pn  Bp K ,

Bc   F ,

Cc  K ,

Dc  0

Note that, this simulation is regarded in two cases, without uncertainty and with uncertainty. In both cases, adaptive ETM (24) with η-dynamic (58) are considered where   e    e e 2 ,   y    y y , l     ,      3 and e = (ey, eu). Also, we run simulations in [0, 5] seconds 2

CR IP T

and select initial conditions as xp0 = [2, 3]T, xc0 = [1, -1]T and η0 = 1. It is noticeable that to reduce further the number of updating operations we can alternatively use eˆx  xˆ (t i )  xˆ (t ) instead of the measurement error eu, [1]. Thus, in this example, both definitions are considered. Case 1 (Nominal system): Consider the nominal closed-loop ETC system (49-50) with matrices represented in (87). According to Theorem 1, if we select 1  0.95 ,  2  0.15 and   2 thus (67) will be a LMI expression thus its solutions are e  2.886 ,  y  0.059 ,   3.717 and  3  3.720 .

PT

ED

M

AN US

Fig. 4 shows system Outputs generated from two methods for eu and eˆx .

AC

CE

a) b) Fig 4: Output responses, control inputs in example 2, case 1 for the proposed method (solid lines) and the method in [1] (dashed lines), a) when eu is used, b) when eˆx is used.

19

CR IP T

ACCEPTED MANUSCRIPT

a)

b)

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Fig 5: Inter-event times in example 2, case 1 for the proposed method (solid lines) and the method in [1] (dashed lines), a) when eu is used, b) when eˆx is used.

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In addition, in Fig 5, inter-event times versus time have been depicted where eu and eˆx are used respectively in (a) and (b). As it is observed, input error measurement, eu can help to increase inter-event times rather than when eˆx is used. Tables 2 and 3 compare average of inert-event times, minimum inert-event times and the number of samplings in both methods. Clearly, the average inter-event time and MIET of the proposed approach are greater than results of [1] while the number of sampling operations is less than results obtained in [1]. On the other hand, from Fig. 4, it is observed that the system performance is almost kept. Table 2: Average and minimum inter-event times as well the number of sampling operations in example 2, case 1, when eu is used

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Proposed method Method in [1]

Average inter-event time 18.195×10-3 12.601×10-3

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eu

Minimum inter-event time 2.500×10-3 1.00×10-3

Number of samplings 276 397

Table 3: Average and minimum inter-event times as well the number of sampling operations in example 2, case 1, when eˆx is used.

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eˆx

Proposed method Method in [1]

Average inter-event time

Minimum inter-event time

Number of samplings

46.031×10-3 35.561×10-3

21.200×10-3 14.500×10-3

110 142

Case 2 (Uncertain system): In the robust controller design, assume that 1  0.7 ,  2  0.5 and

  5 . Accordingly, matrix inequality (75) becomes to a LMI that is resulted in  e  99.286 ,

 y  10.152 ,   68.303 and

 3  45.803 . Fig. 6 shows input and output of the system from two

methods. 20

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a) b) Fig 6: Output responses, control inputs and inter-event times in example 2, case 2 for the proposed method (solid lines) and the method in [1] (dashed lines), a) when eu is used, b) when eˆx is used.

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a) b) Fig 7: Inter-event times in example 2, case 2 for the proposed method (solid lines) and the method in [1] (dashed lines), a) when eu is used, b) when eˆx is used.

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The numerical results are also summarized in Table 4 and 5 for different measurement errors. Table 4: Average and minimum inter-event times as well the number of sampling operations in example 2, case 2, when eu is used

eu

Proposed method Method in [1]

Average inter-event time 17.550×10-3 10.773×10-3

Minimum inter-event time 0.800×10-3 0.400×10-3

Number of samplings 286 466

Table 5: Average and minimum inter-event times as well the number of sampling operations in example 2, case 2, when eˆx is used

eˆx

Average inter-event time

Minimum inter-event time

Number of samplings

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55.427×10-3 36.463×10-3

Proposed method Method in [1]

24.500×10-3 15.500×10-3

91 138

As shown in Figs 6 as well as Table 4 and 5, the proposed method in comparison with the method suggested by [1] has almost the same performance, while the average time and lower bound of inert-events obtained by the method presented here, is significantly greater. These facts have been also shown by Fig 7, in a) and b).

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7. Conclusions

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In this paper, to expand the average time of inter-events for a nonlinear output feedback control system, a novel adaptive ETM was introduced. The Zeno behavior exclusion was also investigated using Lipschitz properties of functions of the system. Moreover, the proposed method was applied to uncertain LTI systems and conditions of finite gain L2 stability were obtained in terms of some matrix inequalities. Finally, numerical examples showed a decrease in the number of sampling operations and keeping the system performance compared with a recent work.

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