In this paper, event-triggering state-norm estimators are studied for nonlinear delayed sys- temsalsonormed-observabilityconceptisextendedforthesesystems. Moreover, weestablished a Lyapunov–Krasovskii functional to obtain normed observability of the delayed nonlinear sys- tems. Furthermore, an adaptive observer is developed such that convergence of the error system is guaranteed. The proposed event-triggering algorithm yields an event-based observer that ensures uniform ultimate boundedness of the tracking error. 1 Introduction Sampling continuous-time signals plays an important role in modern engineering and various disciplines such as digital communications and signal processing beneﬁt from it. Practically, there is an ever-increasing demand for removing expensive analog circuits and replacing them with cheap, multi-purpose and integrated digital frameworks which in turn aid us to prevent problems of frequency aliasing and so on. Besides many other numerous applications, make it forming a vital research direction in the modern technological world, [1-3]. asadbeigi.ayyob@gmail.com 1 In control theory event-triggering strategy is based on a well-known sampling method called Lebesgue sampling, a none constant sampling law that occurs when a deﬁnite event takes place. This irregular control technique in the digital approach yields closed-loop system stability. Moreover, in this strategy, less mathematical operations are accomplished because the signal input is implemented when an event is triggering, which adds to the merits of it, [4]. Another outstanding and practical advantage of this technique is the designing of sensors. Sensors are the indispensable part of the control systems which valuate the system information and transfer it to the control, [5]. While transmitting data over a lossy channel, considering a controlapproachwhichrequiresminimalstatetransmission, isinteresting. Theevent-triggering method addresses this issue and enables sensors to decide the transmission instant, [5-7]. Diﬀerentcontroltechniqueshavebeenappliedwithevent-triggeringcontrolstrategy,namely in [8-11]. In [8], the authors have considered an event-triggered controller together with delay conditions in a networked system. The subject of [9] is power systems, which are studied again under delays by introducing an event-triggering controller. The event-triggering technique also has studied through observer design, in [10] the main focus is studying linear multi- agentsystemsbyapplyinganobserver-basedevent-triggeringstrategytoachievetheconsensus. Again in the case of linear systems in [11], an observer-based approach for event triggering control of the system is studied. On many occasions, in the absence of the full system state, state estimation is implemented while only speciﬁc output information is available, but the access to the full state is required. Theusualapproachtohandlethissituationisintroducinganobserverthatestimatesthesystem state without observing outputs and inputs. Designing such an observer is a tough challenge for complicated nonlinear systems such as time-delayed systems. The manifestation of this concept was in [12] where authors provided the foundation of input-output-to-state stability (IOSS), for nonlinear systems. Also, the existence of these types of observers is proved in [12, 13]. Input-to-state stability (ISS) concepts have applied to an event-triggering feedback control method in [14, 15]. In [15], a discrete-time version ofthe ISS technique forthe consensus problem of a multi-agent stochastic system is discussed. Feedback event-triggering technique can be applied to overcome the over-provisioning prob- lemasduringtheconsideredperiodmayhappenbecauseofthesystem’sperformance. Thepro- posed method involves a sensor transmitting the over-ﬂowing information of the measurements and the information is transformed when the diﬀerence between measurements is observed so that the stability is veriﬁed. Consider the following systems 2 _x(t) =f(t;x;x ;u);a.e.tt0 (1) y=h(x(t)) (2) here Lipschitz function f(t;x;u )onxis almost everywhere continuous on (x;u)2RnRm, f(0) = 0andhis a a real diﬀerentiable function. _x=f(t;x;x ;u) (3) Inwhichhere x =x(t )for 2[t0;tf]andt >0. Ifthecomputationof k(x)issuﬃciently fast, also if the sensors and actuators are suﬃciently accurate, then the implementation will converge to the speciﬁcation. Moreover, for the deﬁnite values of xwe suppose@-Assumption as follows Assumption 1. We assume @=@0[@0 (4) where@0=Inffrjjxjrjx jg;@0=Inffrjjx jrjxjg: (5) 2 Problem Description and Required Prelimaniries We investigate control problem (1), for u2Rmandx2Rmfor which a feedback controller: u=K(x) (6) Herej:j,j:j1,j:j0are respectively the Euclidian norm, Supremum norm, and the absolute value. We need to remind some categories of functions as what follows Deﬁnition 1. A strictly increasing nonnegative function :R+Sf0g!R+Sf0gis said to be aK- function if it is continuous, and (0) = 0. Deﬁnition 2. A strictly increasing K-function :R+Sf0g!R+Sf0g, is said to beK1 function if lims!1 (s) =1. Deﬁnition 3. A strictly decreasing nonnegative function :R+Sf0g!R+Sf0gis said to be anL- function if it is continuous, and lims!1(s) =1. 3 Deﬁnition 4. A strictly decreasing nonnegative function (s;t) :R+Sf0gR+Sf0g ! R+Sf0gis said to be an KL- function if if, for each ﬁxed t0, the function (:;t)is a K-function, and for each ﬁxed s0, the function (s;:)isL- function. Deﬁnition 5. The system (1) is Input-to State Stable (ISS) if there exist KL- function and K- functions so that jx(t)jmaxf (jx j;tt0); (jyj))g (7) Deﬁnition 6. The system (1) is Input/Output-to State Stable (IOSS) if there exist KL- func- tion andKL- functions 1; 2so that jx(t)jmaxf (jx j;tt0); 1(kuk); 2(ky(t;x;u )k))g (8) the control uis considered on t2[t0;tf), whichtf=tf(x ;u). Now we deﬁne the diﬀerence functionalDa real continuous function Lipschitz as Deﬁnition 7. _(t;x;x ) = lim sup h!0+h(x)(x) h(9) where h(x) = (t+h;x+h;x +h) (10) Fort2[t0;tf). Deﬁnition 8. Assumption 2. An absolutely continuous function Vwhich holds the following condition is said to be an Lyapunov candidate for system (1)-(2): 1.maxf 1(jx (0)j); 1(jx(t)jg<V(t;)< 2(jx j). 2. There areK1- functions 1;2so that _V(t;)(t)V(t;) +1(juj) +2(jyj) 3(jx j) (11) Wherey=y(t;x;u ),u=u(t;x),(t0) 2(jx j)< 3(jx j). Now we introduce the IOSS Lyapunov-Krasovskii functional as W(t;) = exp(Zt t0(s)ds)V(s;) (12) 4 In the above deﬁnition we consider restriction of positive none-increasing function (t)on[t0;t] so that 0<m(t)Mandmt<Rt t0(s)ds<Bin whichB=Ml,mt0<and lis the length of [t0;tf]. Without loss of generality, gently we will consider the following Assumption Assumption 3. Consider all of the previous assumption conditions, are veriﬁed. Moreover, for positive integers q0,q1,qwe have _V(t;)q2(t)V(t;) +1(juj) +2(jyj) 3(jx j) (13) and W(t;) =q1exp(Zt t0(s)ds)V(s;) (14) Such thatq=q1+q1q2,q0=q1q3,q<q 0,q(t)q0(t0)<0,q(t0) 2(jx j)< 3(jx j). Lemma 1. Suppose that n is non-negative and x veriﬁes the following inequality x(t)m(t) +Zt t0n(s)x(s)ds; (15) fort2[t0;t]. Then x(t)m(t) +Zt t0m(s)n(s)expZt sn(r)dr ds: (16) Deﬁnition 9. A normed observer for system (1)-(2) is _z=g(t;z;z ;u;y) (17) here inputs and outputs of the systems coincide with each others, moreover the following conditions should be veriﬁed 1. System (1)-(2) should be Input to State Stable (ISS) . 2. There must be found respectively KandKLfunctions as ; such that jx(t)j (jx j+jz j;tt0) +(t;z;u;y (t;x;u )) (18) for allt2[t0;tf). In this text arbitrary real function zis restricted to the interval [t0;t]. Deﬁnition 10. (Normed Observer Stabilization ) 5 The system (1)-(2) is said to be Normed Observer Stabilized if there exist a Kfunction (s) , aKLfunction (s;t), and a real function ]such that](z)Im(z), then je(t)j (je(t)j+jz j;tt0) +(t;z](z);u;y(t;x;u )) (19) heree(t)is the error of the proposed system. Assumption 4. In system (1)-(2) the function f satisﬁes the following inequality f(t;x;x ;u)f0(t;x;x ) +g0(t;x;x )u (20) whereg0;f0are none-zero continously diﬀerentiable monotone increasig functions. 3 Existence of the Normed Observer Theorem 2. The following statements for the system (1)-(2) are equivalent: 1. The system is IOSS. 2. There is a normed observer for the system. Proof.Suppose that the system is IOSS we claim that _z=AzBz +1(juj) +2(jyj)is a state normed observer. Input-State-Stability immediadely is achieved, as it can be considered as a linear stable system with the input 1(juj) +2(jyj). We need to prove that the condition (16) is valid. Taking the ﬁrst derivative of the W(t)we have _W(t) = exp(Zt t0(s)ds)_V(t;xt) +(t) exp(Zt t0(s)ds)V(t;xt) (21) By applying inequalities in Assumption (2), we will have _W(t)2(t)W(t) + exp(Zt t0(s)ds)( _z+z+z ) (22) Also by considering Lemma 1 as well as the upper bound on exp(Rt t0(s)ds)on we will lead to W(t)W (t0)+Zt t0eB z+z + _z +2Zt t0(s) W(t0)+Zt t0eB z+z + _z exp(Zt s2(v)dv)ds (23) For@-property in Assumption 1, the integral involving z+z + _z, converts to z +rz +r_z . Besides, restriction of the functions on [t0;t]gives an approporiate positive constant upper boundMfor the terms including z . 6 W(t) W(t0) +eB(Ml(Ar+B) +reBjz j + 2Zt t0(s) exp(Zt s2(v)dv) W(t0) +Zt t0eB Az+Bz + _z ds(24) Again according to the Assumption 2, W(t) 1+reBjz j + 2Be2Bh 0(jz j+ 2(jx j)) +2i (25) Here1=W(t0) +Ml(Ar+B)eB,2= 2lBe2BW(t0) + 2lBe3BMl(Ar+B),0 maxfleB;lreBg. Obviously there are strict increasing functions 0(t), 1(t)such that1+ reBjz j 0(jz j)and0(jz j+ 2(jx j)) +2 1(jz j+ 2(jx j)), so the relation (24) can be stated as W(t) 0(jz j) + 2Be2Bh 1(jjz j+ 2(jx j)i (26) Finally applying the deﬁnition of W(t), and the Assumpsion 1 on the bounds of V(t;x ) jx(t)j  1 1 2 0(jz j) + 1 1(4l0emth 1(jjz j+ 2(jx j)i ) (27) Bytaking:= 1 1(2(:))and := 1 1(4l0e!t 1(:)), inwhichl0= 2Be2B+theexistence of the norm observer is proved. Now, Suppose that there is a normed observer (16), and we need to show that the system is IOSS. By setting z = 0and inserting the following relation in (16) jz(t)jmaxf (0;tt0); 1(kuk); 2(ky(t;z;u )k))g (28) We will obtain the following inequality jx(t)j (jx j;tt0) +(t;z;u;y (t;x;u )) (29) The equations (22),(23) complete the proof. 4 Event Triggering Policy In this section, event-triggering algorithm in the presence of delay is discussed. Most of the times the delay role in the control input is neglected so its presence it seems unrealistic in some 7 positions Despite mainly because one can. Meanwhile if he delay is large enough in comparison with the sampling periods then certainly it must be regarded in the computations. By applying of the feedback law (3) on an embedded processor and sampling the state at time instants we get u(ti) =Kx(ti)for the samples tiwhichi2Z+Sf0g. Letibe the delay associated with the triggering instant ti. Besides we take into account updating the actuator values at introduced time nodes as fti+ighereiis the delay time to load the state information from the sensors, which is a sequence of instants that updates control input associated with the sequence ti. Consequently, this delay will increase the error of the system till applying the next control. Here at ﬁrst step we will determine the tolerance of the proposed delay and its eﬀect on the error of the system. It must mentioned that each time interval the previous index is acting for instance on [ti;ti+i)so the input u(ti1)is considered. To sum up the control input is deﬁned as u(t) =K(x(ti1));fort2[ti;ti+i) (30) u(t) =K(x(ti));fort2[ti+i;ti+1+i+1): (31) The strategy considered here for the sampling instants is as ti+1=minftti+Tjjej>1 r 1 2q1 1(juj) +2(jyj)  (32) In sequel, the boundedness of the triggering method for the delayed case is studied. Here1;2are as in Deﬁnition 8, (t) :=q(t)q0(t0). Also, we apply this notation + max:=q0(t0)qm,+ min:= (q0q)M,is a constant tuning positive parameter and ﬁnally e(t) =x(t)x(ti): (33) Theorem 3. Let us consider the system (1)–(2) and the control law (24)-(25). Then, the error in the plant remains bounded by provided (3.38) holds. 1. For positive constant parameters ,a,b, and eacht2[t0;tf]suppose that 2(+ min;+ max)(a;b)((0;1). Deﬁne i=1 L(1 +r)LnA(e ?1) (34) 0:=L(r+ 1) r(35) 8 1:=L(r+ 1) 1 2r 1 2q1 1(juj) +2(jyj)  (36) 0:= 0lpL(r+ 1) r(37) whereA> 0is a positive constant and the last relation is deﬁned for a ﬁxed t2[t0;tf], andmaxfa + min;a + maxg<  < maxfb + min;b + maxg,06=lp ,1<  0<b . We also choose , ?, respectively in the following ranges maxf0;0g<< minf1g (38) Ln(1 +1 A)< ?<Ln (1 + plLA (r+ 1)) (39) Also consider i=1 L(r+ 1)Lnh1 r 1 21(juj)+2(jyj)  p?i (40) 2(;b ) (41) here p?= p+  (42) in which  L(r+ 1)q1 1juj+2jyj 2r(43) lq1 1juj+2jyj 2rjpj0(44) Then,ti+1tii>0ifii. 2. The error in the plant (ep)(t) =x(ti)x(t)jt2[ti+1;ti+1+i+1)remains bounded by the following inequality under the previous assumptions and stability of the closed loop system. ep(t) p+ L(r+ 1) L(r+ 1) (45) 9 Proof. 1. The derivative of the error system (28) is j_e(t)j=j_xjLjxj+Ljx j+Ljyj+Ljuj Lje+x(ti)j+Ljr(e+x(ti)))j+Ljuj1+Ljyj1 jej(L(1 +r)) + (Ljx(ti)j+Lrjx(ti)j+Ljuj1+Ljyj1)(46) By using Lemma 1 for t2[ti;ti+i)we have je(t)jpl:exp ((L(1 +r))(tti)) (47) Wherep=Ljx(ti)j+Lrjx(ti)j+Ljuj1+Ljyj1+ 1. and easily we lead to je(t)jpl:exp ((L(1 +r))(i)) (48) Considering the tolerance ias in (35) je(t)j L(r+ 1)(49) Again by using the same approach for t2[ti+i;ti+1+i+1)we will obtain je(t)jp:exp[L(r+ 1)(ttii)] (50) Here p= L(r+1)+e(ti+i) +pl. By considering t=ti+1and applying condition (33) in (50) we get to 1 r 1 2q1 1(juj) +2(jyj)  p:exp[L(r+ 1)(ti+1tii)] (51) By computing ti+1tiifrom the previous relation we get to ti+1tii1 L(r+ 1)Lnhr 1 2 1(juj)+2(jyj)  p?i(52) In whichp?is deﬁned as (43), andare repectively given by (44),(45).Without loss of generality we suppose that p?= p+so thatpp?, and the coeﬃcients are determined so that the p?and alsos 2(s). and this will lead us to ti+1tii>0. 10 2. Here the changes of the error of the system while t2[ti+i;ti+1+i+1)is studied, for nonenegative integer i. Since in [ti+i;ti+1+i+1), the control signal is not updated, the error in the plant grows and becomes maximum during the [ti+i;ti+1+i+1), the plant error increases and gets the maximum value during this interval.The error in the plant during [ti+i;ti+1+i+1)isep(t)and denoted by ep(t) =e(t)e(ti). Fort2[ti;ti+i), the errorjejrises from zero to L(r+1)and overﬂow this quantity before the input signal is enetered. For about the interval [ti+1;ti+1+i+1]wheniithe error satisﬁes L(r+1) By applying Lemma 1 by the initial value L(r+1)for the proposed interval we get to the following inequality jepj(p+ L(r+ 1))exp[L(r+ 1)(tti+1)] (p+ L(r+ 1))exp(L(r+ 1)i+1)  p+ L(r+ 1) L(r+ 1)(53) Wherepis as the previous section. 5 Normed Observer Stablization Now we are going to represent our main theorem in this paper which is studying normed observer sablizability of the system (1)-(2). Theorem 4. Consider the system (1)-(2) under the control laws (55) and the update laws (56) , then the system is normed observer stablized. u=!0h Kx(ti) +e++P1eTQ jej2+ [xTKTPe+eTPKx ] +Gh (fTPe+eTPf+ (Az?)TQi +Gh TQ(A+rBjzj?+rjzjTBTQ(ex))i P1eW 2jej2i (54) 11 _^G(t) = exp(t) fTPe+eTPf+ (Az?)TQ)TPe+eTP fTPe+eTPf+ (Az?)TQ) + exp(t) TQ(A+rBjzj?+rjzjTBTQ(ex))TPe+eTP TQ(A+rBjzj? +rjzjTBTQ(ex) exp((t ))(t )TS1(t ) + exp(t)TS1 exp((t ))(t )TS1(t )exp((t ))(t )TS2(t ) (55) , and !0=1 jg0j+ 1: (56) Where ^G1(0) =G10;^G2(0) =G20are the initial value of the parameters ^G1;^G2. Proof.The general format of the observer is considered as _z=AzBz +1(juj) +2(jyj) (57) Consider the Lyapunov-Krasovskii functional as V(e) = exp(t) eTPe+TQ +Zt t (eTS0e + exp(s)[TS1+ TS2])ds+1 2~GT~G(58) Heree(t)is as (34), is the delay of the system (1)-(2), ,,is the error of the observation which is (t) =x(ti)z(t); (t) =z(ti)z(t);(59) and ^Gis the parameter estimation, also ^G= G1;G2;G3 is the estimation of the unknown parameter G and ~G=G^G. Moreovere P,Q,S0,S1are symmetric matrices of approporiate dimensions so that eTS0e+ [TPe+eTP] +TQ=W(t). Taking time derivative of (58) along the error trajectories (34) we get to _V(e) = exp(t)h _eTPe+eTP_e+ _eTPe+ _TQ+TQ_i exp(t)h eTPe+TQi +eT(t)S0e(t)eT(t )S0e(t ) + exp(t)TS1exp((t ))(t )TS1(t ) + exp(t)TS2exp((t ))(t )TS2(t ) + ( ^G(t)G)T_^G (60) 12 After that considering the error dynamics (34), (59) and applying @r(x )-property for the system (1)-(2) together with Assumption (4) we have _V(e)exp(t)h (f0+g0u)TPe+eTP(f0+g0u) +TQ(Az+Bz 1(juj)2(jyj)) + (Az+Bz 1(juj)2(jyj) + (Az+Bz 1(juj)2(jyj)TQ)i + (1r2)eTS0e+ exp(t)TS1 exp((t ))(t )TS1(t ) + exp(t)TS2exp((t ))(t )TS2(t ) + (^G(t)G)T_^G (61) By considering1(juj0)2(jyj0) =?wherey0= minjyj,u0= minjuj, then by employing the following relation G_^G=h (fTPe+eTPf+TQ(A+rBjzj?)i +h (Az?)TQ+rjzjTBTQ(ex)i exp((t ))(t )TS1(t ) + exp(t)TS1exp((t ))(t )TS1(t ) exp((t ))(t )TS2(t ) (62) We obtain the next inequality _V(e)exp(t)h uTPe+eTPui +rjz jTBTQjz j+jjTS2jj+ (1r2)eTS0e+^G(t)T_^G(t) (63) At this stage by utilizing the control input (55) we get to _V(e)exp(t)h !0 Kx(ti) +e++P1eTQ jej2+ [xTKTPe+eTPKx ] +G1h (fTPe+eTPf+ (Az?)T Qi +Gh TQ(A+rBjzj?+rjzjTBTQ(ex))i P1eW 2jej2T Pe+eTP!0 Kx(ti) +e++1 jej2 +P1eTQ+ [xTKTPe+eTPKx ] +Gh (fTPe+eTPf+ (Az?)TQi +G2h TQ(A+rB jzj?+rjzjTBTQ(ex))i P1eW 2jej2i +rjz jTBTQjz j+ (1r2)eTS0e +jzjTS1jzj+^G(t)T_^G(t) (64) Using 13 ^G(t)T_^G(t) =G exp(t) fTPe+eTPf+ (Az?)TQ)TPe+eTP fTPe+eTPf + (Az?)TQ) +G exp(t) TQ(A+rBjzj?+rjzjTBTQ(ex))TPe +eTP TQ(A+rBjzj? +rjzjTBTQ(ex) +G(exp((t ))(t )TS1(t ) + exp(t)TS1exp((t ))(t )TS1(t )) +G(exp((t ))(t )TS2(t )) (65) From Lemma 2, and if satisﬁes the following inequality KTP+PK+P+ <0 (66) then we lead to the desired result _V(e)exp(t)[min( )eTe+rmin((BTQ)jz jTjz j) +min(S2)jjTjj(67) Consequently, it yields je(t)j 1 1(2[exp(t)[min( )eTe+min(BTQ)jz jTjz j) + 1 1(2[min(S2)jjTjj])(68) By taking := 1 1(2[:]),:= 1 1(2[:])normed observer stablizability of the system (1)-(2) is proved. Corollary 4.1. If!0ast!1then the system (1)-(2) with the contol input (55) and adaptive laws (56) is asymptotically normed observer. Proof.The proof is easily achieved by considering the assumptions on (68). Example 1 6 Conclusion The Lyapunov–Krasovskii functional to investigate the problem of the event-triggered normed estimator of nonlinear delay systems is generalized. In this paper, an event-based observer method for nonlinear time-delayed systems by applying an adaptive controller is introduced which ensures the stability of the event triggering technique. 14 7