Event-triggered and self-triggered stabilization of distributed networked control systems

Event-t rigger ed and self-trigg ered stabiliza tion of distrib uted networke d contr ol systems Romain Po stoyan, Paulo T abuada, Dragan Ne ˇ si ´ c, Adolfo Ant a Abstract — Event-trigger ed and self-triggere d control ha ve recently been p roposed as impl ementation strategies that con- siderably reduce the reso urces r equired for control. A lthough most of the work so far has f ocused on closing a single control loop, some resea rchers ha ve started to in ves tigate how these new implementation strategies can be applied when closing multipl e- feedback loops in the presence of physically distributed sens ors and actuators. In th is paper , we consid er a scenario where the distributed sens ors, actuators, and controllers communicate via a shared wired channel. W e use our recent prescriptiv e framew ork f or the ev ent-trigger ed contr ol of nonlinear systems to d ev elop nov el policies su itable for the consid ered di stributed scenario. Afterwards, we explai n how se lf-triggering rules can be deduced from the d ev eloped eve nt-triggered strategies. I . I N T RO D U C T I O N T oda y’ s control systems ar e freq uently implem ented over networks as these typ es of structures present many ad - vantages in terms of flexibility and c ost. I n this setup, controllers commun icate with sensors and actuat ors through the n etwork, n ot in a continuo us fashion but rather at discrete time instants when the channel is available for the control s ystem. T r aditionally , t he t ime in terval between two successi ve transmissions is c onstrained to be less than a fixed constant , which is called the maxim um allowable transmissi on interval (MA TI) (see e.g. [6], [14] , [2 1]). In order to ach iev e a desired performa nce, is genera lly chosen as s mall as technology and netw ork load p ermit. Th is strategy , although easy t o implement and analyze, represents a conservati ve solution th at m ay unnecessarily overload the commun ication channel. Indee d, one would expect that the transmission instants should not satisfy a prefixed bo und but rathe r be bas ed on the cur rent state of the system, the channel o ccupan cy and the des ired p erform ance. Drawing intuition from t his idea, event-triggere d contr ol has been dev eloped to reduce t he need for f eedback while guaran - teeing satisfa ctory lev els of perfo rmance. It in v olves closing the loop wh enever a pred efined state-depe ndent tr iggering condition is satisfied, e.g . [3], [4], [9], [20]. This technique reduces resourc e usage s uch as communication ban dwidth or compu tational time and provi des a high degree of ro bus tness This technical repor t i s an extende d versio n of [18 ]. This work was supporte d by the Austra lian Research Council unde r the Future Fello wship and D isco very Grants Sch emes, the NSF awards 08 20061 and 0834771 and the Alexand er von Humboldt Foundat ion. R. Postoyan is with the Centre de Reche rche en Automatique de Nanc y , UMR 7039, Nanc y-Uni versit ´ e, CNRS, France romain.postoyan@ cran.uhp-nancy .fr P . T ab uada is with the Department of Electrical Engineering, Univ er- sity of Californi a at Los Angel es, Los Ang ele s, CA 90095-1594, USA tabuada@ee.ucla. edu D. Ne ˇ si ´ c is with t he D epartmen t of Elect rical and Electron ic En- gineer ing, the Univ er sity of Mel bourne, Parkvi lle, VIC 3010, Austral ia d.nesic@ee.unime lb.edu.au A. Anta is wit h the T echnische Uni v ersit ¨ at Ber lin & Max Planck Institu te f ¨ ur Dynamik komple xe r technische r Systeme, Germany anta@control.tu- berlin.de since the st ate is continuo usly monitored. Most w ork in this direction has focused so far o n the so-called one packet transmission pr oblem, which corres ponds to the case when all the states are sent together in a sing le packet. This generally implies the collocation of all sensors and, for multiple-inpu t control system s, collocati on of all actuators as well. Such an assumption do es n ot ho ld in many cases. Thus, significant w ork on distributed e vent-triggered control has r ecently appeared i n [13], [23]. The se papers focu s on a setup where t he sensors decide locally when they need to transmit their me asurements. Proposed solutions may howe ver be conservati ve as they allow several sensors to transmit simultan eously . In t his paper , we consider a n etwork setup where a central coordin ator i s av ailable and on ly one g roup of sensors o r actuators ( that f orms a node ) can tr ansmit at each trans- mission instant. This central coord inator gr ants access to the no de selected accor ding to a g i ven schedulin g protocol whenever a predefined triggerin g r ule is satisfied. W e follow the same appr oach as in [ 17], which is a part icular case of this study since we considere d netw orks that ha ve one node only . Mo deling the prob lem using the hybr id formalism of [8], we apply the prescrip ti ve framew ork in [17] to synthesize ev ent-triggerin g rules f or networked contr ol systems (NCS ). It h as to be noted that existing strategies are not d irectly applicable since only a subset of sensors and actuato rs g et access to the network a t a tran smission instant. Hence, we adapt a policy developed in [17] an d dev elop new event- triggering rules fo r classes of NCS governed b y unifor mly globally asymp totically stable ( UGAS) proto cols (see [15] ). Examples of UGAS pro tocols are the ro und-r obin pr otocol or the try-on ce-discard pr otocol [21] as shown in [14]. T o the best of th e autho rs’ knowledge, we belie ve tha t this is the first t ime t hat ev ent-trigger ed co ntrol is addr essed for such NCS. The scheme we introd uce i n this p aper can be seen as a centralized event-triggeri ng policy for p hysically distributed control systems, which in p ractice would re quire con stant commun ication between the sensors/actuators and the co - ordinator (since the deci sion depends on the current v alue of the system states). T o overco me this pr oblem, w e show how self-trigg ered i mplementati ons (see [1], [2], [22]) can be derived from a kn own e vent-triggering strategy by app lying the technique s in [2]. Under th is paradi gm, t he coordin ator decides the ne xt time instant at which commu nication should occur based on the la st r ecei ved information. The remaind er of the p aper is organi zed a s follows. The problem we solve in this paper is described in Section II. In Section II I we recall the hybrid L yapu nov-based framew ork propos ed in [17] to analyse the stability o f NCS. Th is framework is used as a b aseline to derive event-triggering conditions in Section IV . Section V descri bes how such ev ent-triggere d conditions can be emu lated by means of the self-triggere d for mulation. Finally , an illustrative example of a distrib uted sy stem is covered in Section VI to show the benefits of this strategy . Th e proofs are giv en in the Append ix. I I . P RO B L E M S T A T E M E N T W e omit the d escription o f the notation an d defin itions used througho ut thi s paper and refer the reader to Section II in [17]. Consider the pl ant: (1) where is the p lant s tate a nd the c on- trol inpu t. The following stabilizing dy namic state-feedback controller is designed: (2) where is the controller state. W e cons ider a scenario where the controller (2) co mmunic ates with the plant (1 ) via a shared central ized network. Since state measurements and contro l inputs are n o lo nger continuo usly av ailable but only at given transmission instants , , systems ( 1) and (2) become: (3) where and denote the variables respecti vely generated from the most rec ent transmitted plant state and c ontrol input throug h the network. Between two tran smission i nstants, they are generated by the i n-network pro cessing algorith m modeled by functions and : (4) Zero-o rder-hold devices are o ften used so that and are kept co nstant on i.e. and . Ne vertheless, we allow for o ther typ es of i mplementation s. Sensors and actuators are grouped into nodes depen ding on th eir spatial location. At e ach transmission instant, a single node get s acc ess to the chan nel according to th e scheduling protocol and transmit its data. W e model this process as f ollows: (5) where , with and , denoting t he net worked-induced error . Thus, vector is partitioned as . At each tran smission instant , functions are t ypically such that if the n ode gets access to the netw ork, the corr espondin g error experiences a jump while th e other compo nents of remain unchang ed; usually but this assumption is not n eeded in general. In that w ay , and can be u sed to model common sched uling protocol s such as ro und-ro bin ( RR) or try-once-d iscard (TOD) [21], see [ 14] for mo re de tails. The sequence o f transmi ssion instants , , is traditionally defined such that (as i n [21], [14], [6]), w here denotes the MA TI an d is an arbitra ry small con stant which models the fact that there exists a minim um amount of time between two tra nsmissions. In this stud y , we resort to a dif ferent paradigm : transmissions are trig gered accord ing to a criterion t hat d epends on the variables of the overall system. The un derly ing idea be hind this parad igm i s to r educe the usage of the commu nication bandwidth by transmitting data on ly when needed to ensure the de sired stability p ropertie s. W e consider the fo llo wing impl ementation architectur e. Sensors, actua tors an d con trollers exchange info rmation throug h a network, where the schedu le is dyn amically de- cided by a central c oordina tor . The o rder at which the network is assigned t o each node is defined by means of the protoco l. The time instants at which co mmunic ation needs to be established are decided by the central coordin ator according to the current state of the dyn amical system, in the spir it of event-triggered control. T his central no de receiv es information fr om the sensors and the controller and evaluates a so-called triggeri ng rule in ord er to decide whether communicatio n is needed to gu arantee st ability for the contro l system . Since the triggering condition usually depends on the state of th e plant and the controller, such setup requir es in general continuou s comm unication bet ween the sensors, the controller and th e c entral c oor dinator . T o overcome this hand icap, we propose in this p aper a self- triggered implemen tation that emu lates the d esigned event- triggered po licy , where the next transmission i s decid ed based on the last d ata received b y th e coordi nator (see Section V ). Since transmission times ar e kn own in advance under this polic y , the self-triggered policy also facilitates the schedulab ility analysis for the network. W e model the problem using the hybrid formalism of [8], similar to [6], [7]. W e group together the states of the plant and the control ler in th e variable and we den ote by the co unter variable that m ay b e required to m odel protocols su ch as round-rob in for instance (see [ 14]). It h as to be n oted t hat ad ditional variables may also b e introdu ced for d esigning the triggering rule. F or instance, we will see in Sec tion IV -A that th e event-triggering strategy in [20] is n ot applicab le to the co nsidered distributed NCS unless we in troduce an app ropriate auxiliary variable. W e will a lso sho w i n Sectio n IV -B that the time-triggered policy in [6] can be m odified t o obtain e ve nt-trigg ering rules thanks to the u se of a clock- like variable. Thu s, we deno te by all au xiliary variables. The model can b e w ritten as: (6) where . W e u se and to denote (6). The sets and are closed, includ ed in ( ) an d respe ctiv ely denote the flow an d the jump set. T ypically , the system flo ws on and experiences a jump o n , where the tri ggering condition is s atisfied. When , the system can e ither jump or flow , the latter only if flowing keep s in . W e call the prot ocol where as in [ 14], [6]. Func tions , , (7) where , are assumed to be continuous. The main prob lem a ddressed in this paper is to de fine appropri ate event-triggering rule s, that is , to define ap propri - ate flow and jump sets and for system (6) i n order to ensure asymptotic stability properti es of (6) while re ducing the num ber of transmi ssions as much as possible. Afterwards, we exp lain how self-triggerin g conditions may be derived from a kno wn event-triggering criterion. I I I . A P R E S C R I P T I V E F R A M E W O R K F O R T H E E V E N T - T RI G G E R E D C O N T RO L O F N C S W e recall in this s ection the framew ork of [1 7], originally dev eloped fo r sampled-data systems. I t is based o n the following theo rem that p rovide s sufficient con ditions t hat ensure asymptotic stability properti es for system (6). It can be regarded as a v ariation of the genera l results in [5]. Theorem 1. Consider system (6) and sup pose and that ther e exist a locally Lipschitz function and a co ntinuou s function with such that the following co nditions ho ld: (i) Ther e exist such tha t fo r a ny : . (ii) Ther e exists such that for all 1 : . (iii) F or all , . (i v) Solution s to (6) have a s emiglobal dwell time 2 on , wher e . Then the set is S-GAS. Theorem 1 c an be used as a frame work for the synthesis of ev ent-triggerin g r ules for (6). The main idea is to design the triggering cri terion so that there exists a L yapunov function for the overall system (6) that decreases on flows, does not i ncrease at jumps an d g uarantees th e existence of a minimal interval of times between two j umps o utside the stable set. This approach has be en used to in vest igate the stability of oth er types of h ybrid systems ( e.g. , see [16], [6]). G eneral guidelines on how to a pply Theorem 1 to synthesize triggering conditions for system (6) can be found in Section IV in [1 7]. The ma in d ifference with [17] is t hat the - depende ncy of the L yapunov function will depen d on the considered scheduling protocol as we sho w it in Section IV. The follow ing result is used to verify the exi stence of semiglobal dwell times in Section IV. It is a corolla ry of Lemma 1 in [17]. Lemma 1. Consider system (6) and assume that and items (i) -(iii) of Theorem 1 a re s atisfied. I f 1 W e co nsider the , the Clarke d eri v ati ve of (see [17 ]), by abu se of nota tion, alt hough is not nec essarily loca lly Lipschi tz in . This is justified s ince the component of corresp onding to i s . 2 See Defini tion 2 in [17] for any there e xists a loca lly Lipschit z fun ction in , wher e such that: (i) Ther e exists such that for any with : . (ii) Th er e exists such tha t for any with dom : . (iii) Ther e exist s a continuous no n-decreasing function such t hat for all : . Then so lutions to (6) have a semiglo bal dwe ll time on wher e . I V . E V E N T - T RI G G E R E D S T R AT E G I E S W e apply the framew ork o f Section III to synthesize event- triggering ru les for NCS. T wo stra tegies are propo sed but others can be developed by using T heorem 1. A. Using a thres hold-like variable First, we sho w that the ev ent-trigge ring s trategy proposed in [20] f or samp led-data systems is n ot dire ctly applica ble to distributed NCS. Hence, we redesign th is technique as in [17] by introducing an aux iliary v ariable. It has to be noted that the meth od in Section V .B in [1 7] c annot be applied ‘off-the-shelf ’ here as we need to adapt the strategy to the protocol. W e suppose that the controller (2) has been designed to m ake the closed- loop system (1) in put-to-stab le w .r .t. to networked-ind uced error , which is equivalent to the following assumption (see Theorem 1 in [19]). Assumption 1. There exists a smooth Lyapunov function , such t hat for all : (8) and fo r all : (9) W e suppos e that the pr otocol is unifo rmly globally asymp - totically stable (UGAS) [15], i.e ., that t he fo llowing holds. Assumption 2. There exist , and such t hat for all the fol lowing is sat isfied: (10) (11) The ro und-ro bin an d try- once-discard protocol s hav e been shown to satisfy this prop erty in [14], as well as other protoco ls (see [15] f or instance). W e note that when we rec over the situation i n [ 17] where all no des transmit at each transmission instant (i.e. in (6 )) (no tice that ( 10) implies ( 11) in that case). In view of (9), for any with for , we have that implies: (12) Instead o f co mparin g and to de ri ve the triggering rule as in [20], [17], we use the L yap unov function which is characteristic of the p rotocol. According to (10), we hav e that . Theref ore we ca n conclud e that (where , f or ) i mplies , tha t in return ensures (12). F ollowing the mai n idea of [2 0 ], a first attempt to define the tri ggering rule is: (13) In this way , the fl ow and the jump sets of the corresponding system (6 ) are: (14) The pro blem with this policy is that we have n o guaran tee that enters into after a jump. Indeed, while in [20] after each ju mp is reset to , h ere ty pically only a su bvector is res et to z ero after each tran smission ( see Section II).This may not b e enou gh for to be come less than . As a consequence, the tr iggerin g rule (13 ) may g enerate se veral tran smissions in a row befo re enterin g into that is un realistic and co ntradicts item (i v) of Theorem 1. T o overcome this dr awback, we intr oduce a variable with the f ollowing dynamics: (15) where is any locally Lipschitz class- function. The system is n ow modeled as: (16) where and the se ts and are defined as and and (17) The v ariable can be regarded as a decreasing threshold on in view of (15 ), and thu s we en ter into after a jump. Indeed, we have that according to (1 1), (15) and since and is strictly in creasing, therefore . W e are now able to apply Theorem 1 to guarantee st ability propert ies f or sy stem (16) and th e existence o f dwell times. Theorem 2. Consider system (16) and sup pose the fol - lowing cond itions h old. (i) A ssumptions 1 -2 are satisfied. (ii) Function is loca lly Lipschitz in . (iii) F or a ny compa ct set , th er e e xist such that f or any , , (i v) Ther e exists such that . Then is S -GAS a nd solutions to (16) have a semiglob al dw ell time on . Items (i ii) and (iv) nee d to be added to guara ntee the existence of dwell times c ompared to T heorem 3 in [17]. It is easy to che ck that condition (iv) o f T heorem 2 holds when is polyn omial or ho mogeneou s of degree . B. Using a clock-like va riable In [6], NCS with time-triggered ex ecution are modeled as a hybrid system similar to (6) by in troducing a clock variable that would co rrespon d to in ( 6). The flow and the jump sets are defined as bein g bigger or not than a given fixed boun d known as MA TI. T his consta nt co rrespond s to the time it takes f or the solution o f the ordin ary differential equation to decrease from to , where and are some co nstants (see (5) in [6]) an d is gi ven by Assump tion 2 , which is assumed to h old. In this subsec tion, we reduce the con servati veness of the strategy in [6] by making the ordinary differential equation th at define s state-d ependent. This allows us to consider a larger class of systems an d to potenti ally enlarge the int er-e xecution i ntervals comp ared to [6]. W e supp ose that Ass umptio n 2 is satisfied with locally Lipschitz in and that the following holds. Assumption 3. There e xist a locally Lipschitz function and continu ous functions , , , con- tinuous, p ositive defini te such tha t the following conditions holds. (i) F o r al l , (18) wher e comes fr om Assumption 2. (ii) F or all : (19) (iii) F or all : (20) In [6], and are supposed to be co nstant that implies that syste m is -ga in stable from to . Making and state-dependent allows us to enlarge the studi ed class o f systems and to e ventually obtain less conservativ e upper bounds in (18) and (20) that will help to enlarge the inter-ev ent intervals. Model (6) be comes her e: (21) where , is any constant in and plays the role of mentioned above and is call ed a clock-like variable (see [6]). Th e sets and are: (22) Note that, in stead of setting to at jumps as in [ 6], we consider any that may help generating larger inter-ex ecution intervals. Remark 1. W e focus in this pape r o n NCS for which only one node a mong the nodes commu nicates at ea ch transmission in stant so that . Wh en , as in [17], we can redefine the sets in ( 22) as follows: wher e some consta nts. The following theorem ensu res the stability of system (2 1) and the existenc e of dwel l times. Theorem 3. Consider system (21) a nd suppose Assump- tion 2-3 hold with locally Lipschitz i n . Then the set is S-GAS and solutions to (21) have a semig lobal dwell tim e on . V . S E L F - T RI G G E R E D C O N T RO L As mentio ned before, th e pro posed e vent-trigg ering schemes in Section IV require the coord inator to contin- uously ev aluate the triggering co ndition . This may induce a significant cost in terms of co mmunication , co mputatio n time a nd p ower . T o overcome these drawbacks, a poss ible solution lies in the self-trig gered strategy . Self- triggered con- trol considers the mathematical model of the contro l system and the last measuremen t of the plant states and/or the last control input in order to d eriv e the next transmission inst ant. It represents a mo del-based emulation of e vent-triggered control in the sense that it iden tifies the time ins tants at which the jum p cond ition is satisfied. In this section, we suppo se that a n e ven t-triggering s trategy has been designed such that the fo llowi ng condition s hold. Assumption 4. The conditions of Theorem 1 hold an d item (iii) is satisfied on fo r system (6) with flow and jump sets of the form and wher e . Formally speak ing, the event-triggered control strategy in Assumption 4 requires data tra nsmission at the following time instant , for : dom where is a solution to (6 ). I n order to guarant ee stability , data transmission needs to occu r no later than . When only the previous measuremen t of the state is available, the compu tation o f in an exact way is in mo st cases not possible. T he self-trig gered strategy com putes a lower bo und for at w hich the next jump will occur . In [2], this lower bound i s taken to b e , w here is strictly positiv e and satisfies for each dom : (23) where we ha ve d enoted the th Lie deri v ativ e o f along as , and . The parameters and are coefficients c omputed from and . W e assume that and are smooth functions i n . B y abuse of notation we con sider the th Lie deriv ati ve e ven thoug h and ma y no t be differentiable in ( this is justified since on flo ws). Equation (23) correspo nds to the bound p rovided in Theorem V .4 in [2]. Guidelines for the design parameter are provided in Section VIII-A in [2], and the set of parameters have to b e cho sen to satisfy inequality (V .1 2) in [2]. W e refer t o [ 2] fo r a more de tailed description of the rol es played by th ese coefficients and how to co mpute them from the expressions of and . The parameter in (23) repre sents a d esign choice that trades the accuracy of t he bound for the c omputat ional complexity . In other words, h igh values of imply times closer to (and there fore less transmissions), but a t th e co st of solving a mo re comp lex algebraic e quation. Since we may now transmit befo re , we su ppose item (iii) o f Theorem 1 h olds on (and not on ly ) i n order to guaran tee that the considered L yapunov fun ction still does not in crease at jumps. This ad ditional co ndition typically comes fo r free as it is the case i n Section IV for instance. The pr oblem is mo deled as follows: (24) where is a clock v ariable and is used to define t he n ext tran smission instant, , an d the sets and are: (25) It is shown in [2] that the form ula in (23) can b e used to design th at is very close to . Ne vertheless, to prevent fro m the situation where the self-triggerin g techn ique of [2] generates con servati ve times because of an i nadeq uate parameter s cho ice in (23), we introduce an arbitrary small constant in (24) to guarantee the existence of a minimal interval of time between tw o transmissions. This is justified since Assumption 4 implies the existence o f such a constant time (sem iglobally), in view of item (iv) of T heorem 1. T he following theorem shows that the p roperties ensured by the conside red e vent-triggering strategy are maintained under the proposed self-trigg ering rules. Theorem 4. Let an d be smo oth fun ctions in . Under Assumpt ion 4, the s et is S- GAS f or system (24) and solutions to (24) ha ve a semigloba l dwell time on . Self-triggering r ules c an be derived for the event-triggered strategies developed in Section IV as follows. For Section IV -B, we t ake . The function is smooth and we assume that are smoo th fun ctions in . Embed ding system (21) into (24) allow s us to obtain a self-triggered cont rol techniqu e. The conclu sions of Theo rem 4 app ly as all the required conditio ns hold. For the event-triggered control of Section IV -A, we can not define as will not be smooth. Therefor e, we d efine a nd . By assuming that and are smooth in and resp ecti vely , we see that and are smooth in . W e mod ify the jump equation for in (24) as fo llo ws: where satisfies (23) with , . Under this setup, the conditions of Theorem 4 are satisfi ed provide d that are smoot h. n : 4 : 3 - T ABLE I P A R A M E T E R S F O R T H E S E L F - T R I G G E R E D T E C H N I QU E . V I . I L L U S T R A T I V E E X A M P L E T o illustrate the pr oposed st rategy , we consider the c ontro l of a jet e ngine compressor . W e borrow the model from [ 10]: (26) where represent s the mass flo w , is the p ressure r ise and is the throttle mass flo w . In th is model t he ori gin has been translated to the desire d equilibrium p oint, h ence the objective is to steer to zero . The control law is designed to stabilize the system. Th is con troller is connec ted to th e two sensors measuring a nd thro ugh a network under the T OD protoco l. For simplicity , we cons ider that the controll er is connec ted to the actua tor . Nonethel ess, as po inted out previously in this paper, the de veloped framework accou nts for the mo re realistic case of the con troller and the a ctuator not bei ng collocated. W e implement the event-triggerin g strategy proposed in Secti on IV -A. A ssumption 1 i s satisfied with , , for . A ssumption 2 is satisfied f or for the TOD protocol, where and represent the network-induced error s for and , an d with as the nu mber of nod es is (see Proposition 5 in [1 4]). W e select in (1 2) and thus we ob tain . The Y almip sof tware [1 2] was used to co mpute , an d . W e now constru ct a linear differential equation for the auxiliary variable : , with initial condition . It is expected th at large r values of would enla rge the transmission times, at t he cost of a d egradation in perf orman ce. It can b e v erified th at items (ii)-(iv) of T heorem 2 h old (we use the fact that is a pol ynomial fun ction to show item ( i v) of The orem 2) . W e derive a s we ll a self-t riggered emu lation o f t he ev ent- triggered technique of Section IV - A. As detailed at the end of Sec tion V, we con sider and . The design parameters for the self-triggerin g cond ition are reported in T able I. In both cases we con sidered . W e comp are the event-triggered strategy herein p ropo sed with a period ic implementation . In order to compute a period, we a pply the technique in [ 6]. For an operating ball of radius 1, the obtained p eriod is . F or the comparison, we consider 20 0 different initial conditions ran domly distributed in a ball of r adius 1. T abl e II shows the a verage inter- transmission time und er the three d ifferent strategies. Both the ev ent-trigger ed and the s elf-trigger ed strategy outperf orm the periodic ap proach. Th e g ap b etween the event-triggered and the self-trigg ered in ter-transmission times is due to the conservati veness of the technique in [2]. T o further illu strate the p roposed appr oach, we d epict as well the ev olution of the tr ansmission times and the network-induced error un der Periodic [6] Event-tr iggered Self-triggere d 0.010 0.061 0. 046 T ABLE II A VE R AG E I N T E R - T R A N S M I S S I O N T I M E F O R 2 0 0 I N I T I A L C O N D I T I O NS . the self-tr iggered strategy fo r a particular initial condition ( , ). The p lot shows how the net work grants access to the nod e with the largest error . Likewise, the transmission times vary acco rding to the cu rrent state o f the plant. This fact suggests th at the rigid period ic paradigm overloads u nnecessarily the network, and the flexibility and adaptability of e vent-triggered con trol (and consequ ently self-tr iggered control) is able to relax this requirem ent and reduce network usage. 0 0 .2 0. 4 0.6 0 .8 1 1.2 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 Transmission times[s] 0 0 .2 0. 4 0.6 0 .8 1 1.2 −0.1 −0.05 0 0.05 0.1 Network−induced error time[s] e 1 e 2 Fig. 1. E voluti on of the tra nsmission times and the networ ked- induce d error under the self-tri ggered strate gy . V I I . C O N C L U S I O N W e have used the p rescriptive framework in [ 17] or ig- inally dev eloped fo r sampled-data systems to synthesize novel e vent-triggering rules fo r distributed NCS. W e hav e then shown how self-triggeri ng cond itions can then been derived (under some con ditions) by applying the tech niques in [2]. This work represents the first step towards a mo re fundam ental question in distrib uted networked co ntrol: given a set of phy sically distributed sensors and actuators, how should commun ication bet ween the different nodes be sched- uled? Ad-hoc solutions to this problem includ e the TOD protoco l, where th e node with the largest network- induced error info rmation is gran ted access to the commu nication channel. Th e presen ted fr amework can b e furth er explored to design comm unication proto cols t hat decide the order at which each n ode n eeds to sen d info rmation . V I I I . A P P E N D I X Proof o f Theorem 1. Let where and be a sol ution to (6 ) with . De fine the set where is such that (take for i nstance in vie w of item (i) of Theorem 1). The s et is f orward in v ariant for system (6) since . Hence, in view o f items ( i)-(iii) of Theo rem 1, for any dom . F rom item (i i) of Theo rem 1 and by using standard co mparison pr inciples, there exists that satisfies, for all : (27) and such that, for all dom , (28) where means that . From item (iii) of Theorem 1, it follows that: (29) for all such that do m fo r some . Combining (2 7)-(29 ), w e ob tain: dom (30) Now let dom , if then according to item (i) o f Theorem 1. If then that means that for all dom (i.e. and ) since i s forward inv ariant for system (6 ) ( according to items (i)-(iii) of Th eorem 1 and since ). As a co nsequenc e, we have that for all dom , wh ere is a mi nimal inter val of t imes between two jump s on whose e xistence is ensured by item (iv) of Theorem 1. It fo llow s that: (31) By using item (i) of Theo rem 1, we dedu ce that, for all dom : (32) denoting (since , ), for all dom : (33) Hence, the set is S-GAS according to Definition 1 in [17]. Proof of Lemma 1. L et and define and where is such that (take for i nstance in vie w of item (i) of Theorem 1). According to items (i)-(iii) of T heorem 1, for any dom . W e notice that . Denote the fir st jump instan t (if no ju mp ever occurs i.e. , (2) in [17] is obviously satisfied). W e have that and . When then for any dom (i.e. and ) as is forward in variant fo r system (6) an d (2 ) in [17] is ensured. When , then according to item (i) of Lemma 1, and therefor e a jump can not occu r immediately in view of item (iii-b) of Lemma 1. Let denote with d om the n ext jum p instant and su ppose , oth erwise th e desired result holds. By the continuity o f and t he solution to system (6) on flows, ther e exists s uch that for any with dom as . Accord ing to item (ii) of L emma 1, we deduce that for any . In view of item (iii) of Lemma 1, in v oking st andard comparison principles, we deduce tha t for a ny where is the solution of satisfying . Th e next j ump can not o ccur before the t ime it takes f or to e volv e fro m to (which is indep endent of ) has elapsed. By i nduction , we d educe that the inter-jump inte rval on is lower bound ed by . Hence solutions to (6) ha ve a semiglobal dwell time on according to D efinition 2 in [17]. Proof of Theorem 2 . W e verify the co nditions of T heorem 1 and a pply it to ob tain the desired results. W e note th at and . W e consider the fo llowing candidate L y apunov function: (34) It can be shown that item ( i) of Theorem 1 h olds with by using (8 ), (10) an d Rem ark 2. 3 in [1 1] and n oting t hat on . On , we have that , as a consequ ence, when , we get in vie w of (12): , and wh en , . T hus, item (ii) of Theorem 1 is en sured with for . Let , (35) according to A ssumption 2, (36) since and is increasing: item (iii) o f Theorem 1 is ensured. W e now prove that item (iv) of T heorem 1 is satisfied using Lemma 1. Let that is defined on with ( comes fr om Lemma 1). Let such t hat , using Assump tion 2 we have: (37) since , (38) Define . Function is defined and continuous on since is continuou s and onl y cancels at the or igin. Moreover, for any since is strictly incr easing as a class- function and . On the other han d, in view of item (iv) of Theorem 2 , , therefore we can write that . In that way , from (38 ), the fol low ing holds: (39) and item (i) of Lemm a 1 is satisfied w ith . Sinc e after eac h ju mp, for , we deduce f rom the d efinition of that item (ii) of L emma 1 holds with , since as long as , . Let , we have that: (40) in vie w of item (iii) of Th eorem 2 and denotin g the Lipschitz co nstant of on and , we ob tain: (41) applying the d efinition of in (34 ), (42) hence (43) On the o ther h and, fro m item (iii) of Theo rem 2, (44) Consequently , from (40 ), (43) an d (44 ): (45) where is continuo us and non-d ecreasing. W e have proved that item (iii) of Lemm a 1 applies: so lutions to (16) hav e a semiglobal dwell time o n . As a c onsequ ence, the set is S-GAS accordi ng to Theorem 1. Proof of Theorem 3. W e show that the cond itions o f Th e- orem 1 holds. W e note that an d and . W e consider the can didate L ya punov function: (46) that sat isfies i tem (i) of Theorem 1 with and for by iden tifying and and using ( 10), (19) , the fact that o n and Remark 2.3 in [1 1]. For any , we h av e that: from items (i) and (iii) of Assumption 3, using the fact that , we get : (47) consequ ently , i tem (i i) of The orem 1 is satisfied with for . L et , (48) W e obtain using (11): (49) since on , we hav e: (50) and item (iii) of Theor em 1 is ensured. Finally , we apply Lemma 1 to prove the existence o f dwell times. L et that is defined on wit h (see L emma 1). For , we have that so it em (i ) of Lemma 1 h olds w ith . Moreover , it can b e seen that item (i i) of Lem ma 1 is satisfied with . L et , denoting (that i s well d e- fined since and are continu ous), (51) W e see that is non-d ecreasing and co ntinuou s on . As a conseque nce, it em (iii) of Lemma 1 holds that implies that item (i v) o f The orem 1 is guar anteed: solutions to (21) hav e a semiglob al dwell time on . W e obtain the desired re sult by appl ying Theor em 1. Proof of Theo rem 4. Consider system ( 24) and the candidate L yapunov functio n for , where co mes from Theorem 1. W e hav e that and , therefore for any in view item (i) o f Theorem 1 . According t o [2], in (24) is such that fo r any , . Hence, implies . Consequently , we have that in view of ite m (ii) of Theore m 1. Item (iii) of Th eorem 1 is assumed to hold on , ther efore we ha ve that . Soluti ons to (2 4) obviou sly have a semiglobal dwell ti me on since a jump cannot occur on after seconds has elapsed which i s a semig lobal dwell-time exhibited by the event-triggered strategy . By following the similar lines as in the proof of Theorem 1 and noting that , (since in view of Assumptio n 4) , the set is S-GAS. R E FE R E N C E S [1] A. Anta and P . T ab uad a. T o sample or not to sample: self-trigger ed control f or nonline ar systems. IEEE T r ansaction s on A utomatic Contr ol , 55(9):20 30–2042, 2010. [2] A. Anta and P . T ab uad a. Exploiting isochrony in self-trig gered c ontrol. Pr ov isional ly accepted for publicati on. arXiv 1009.5208 , 2011. [3] K.E . Arz ´ en. A s imple ev ent-ba sed PI D controll er. In 14 IF AC W orld Congr ess, Beijing , China , 1999. [4] K.J. Astrom and B. M. Bernhardsson. Comparison of Riema nn and Lebesgue sampling for first order stochastic systems. In CDC (IEE E Confe r ence on Decision and Contro l), Las V e gas, U.S.A. , 2002. [5] C. Cai, A.R. T eel, and R. Goeb el. Smooth Ly apuno v functions for hybrid systems pa rt II: (pre)asy mptotica lly stable compac t sets. IEEE T ransactions on Automati c Contr ol , 53(3):734–748, 2008. [6] D. Carne v al e, A. R. T e el, and D. Neˇ si ´ c. A Lyapunov proof of an impro ved maximum allo wable transfer interv a l for networke d contro l systems. IEEE T ransac tions on Automatic Contr ol , 52(5), 2007. [7] M.C.F . Donkers and W .P .M.H. Heemels. Output-based e vent-trig gered cont rol with guarante ed -gain and improve d e ve nt-trigge ring. In IEEE Confere nce on Decision and Contr ol, Atlan ta , 2010. [8] R. Goebel and A.R. T eel. Solution to hybrid inclusions via s et and graph ical con v er gence with stability theory applicat ions. Automati ca , 42:57 3–587, 2006. [9] W .P .M.H . Heemels, J .H. Sande e, and P .P .J. van d en Bosch. Analy sis of eve nt-dri v en controllers for linear systems. Internat ional Journal of C ontr ol , 81(4):571–590, 2009. [10] M. Krsti ´ c an d P . Kok oto vi ´ c. L ean backstepping desi gn for a jet engine compressor model. 4th IEEE Conf. on Contr ol Applicati ons , 1995. [11] D.S. Laila and D. Ne ˇ si ´ c. Lyap unov based small-ga in theorem for para meterize d discrete-ti me interconn ecte d IS S systems. In CDC (IEEE Confer ence on Decision and Contr ol), Las V e gas, U.S.A. , pages 2292–22 97, 2002. [12] J. L ¨ ofber g. Pre- and post-processing sum-of-squares programs in prac tice. IEEE T ransac tions on Automatic Contr ol , 54(5), 2009. [13] M. Mazo and P . T abua da. Decentraliz ed ev en t-trigge red control ov er wirele ss sensor/actuat or networks. Accepted for publicati on in IEE E T ransactions on Automati c Contr ol. arXiv:1004.0477 . [14] D. Neˇ si ´ c and A. R. T eel. Input-outp ut stabi lity p ropert ies of netw orke d cont rol systems. IEEE T ransa ction s on Aut omatic Contro l , 2004 . [15] D. Neˇ si ´ c and A.R. T ee l. I nput-to-s tate stability of n etw ork ed control systems. Automat ica , 40:2121–2128, 2004. [16] D. N eˇ si ´ c, L . Zaccarian, and A. R. T eel . Stabilit y propert ies of r eset systems. Automat ica , 44:2019–2026, 2008. [17] R. P ostoy an, A. Anta, D. N eˇ si ´ c, and P . T ab uada. A unifying Lya punov- based f rame work for the e v ent-tr iggered control of nonl inear systems. In I EEE Conf er enc e on Decision a nd Contr ol and Eur opean C ontr ol Confe r ence , Orlando, U .S.A. , 2011. [18] R. Pos to yan, P . T abuada, D. Ne ˇ si ´ c, and A. A nta. Event-trigg ered and self-tr iggered stabil izati on of distrib ute d netw orked control systems. In I EEE Conf er enc e on Decision a nd Contr ol and Eur opean C ontr ol Confe r ence , Orlando, U .S.A. , 2011. [19] E.D . S onta g and Y . W an g. On charac teriz ation s of the input- to-state stabili ty proper ty . Systems & Contr ol Le tters , 24(5):3 51–359, 1995. [20] P . T ab uada. Event-t riggered real -time sc heduling of sta bilizi ng control tasks. IEEE T ransa ction s on A utomatic Contr ol , 52(9), 2007. [21] G.C. W alsh, O. Bel diman, a nd L .G. Bushnell. Asymptotic behavior of nonlin ear networked control systems. I EEE T ransa ctions on Automatic Contr ol , 46:1093–1097 , 2001. [22] X. W ang a nd M.D. L emmon. Self-trigg ered feedback control syste ms with finite-gai n stabil ity. IEEE T ransacti ons on Automatic Contr ol , 45:45 2–467, 2009. [23] X. W a ng and M.D. Lemmon. Asymptoti c stability in distrib uted eve nt- trigge red networ ked control systems with delays. In AC C (American Contr ol Confer ence ), Baltimor e, U.S.A. , 2010.