I. INTRODUCTION WIRELESS sensing and control systems have received

Transcription

1 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER Mutpe-Loop Sef-Trggered Mode Predctve Contro for Network Schedung and Contro Erk Henrksson, Dane E. Quevedo, Senor Member, IEEE, Edwn G. W. Peters, Henrk Sandberg, and Kar Henrk Johansson, Feow, IEEE Abstract We present an agorthm for controng and schedung mutpe near tme-nvarant processes on a shared bandwdth-mted communcaton network usng adaptve sampng ntervas. The controer s centrazed and not ony computes at every sampng nstant the new contro command for a process but aso decdes the tme nterva to wat unt takng the next sampe. The approach rees on mode predctve contro deas, where the cost functon penazes the state and contro effort as we as the tme nterva unt the next sampe s taken. The atter s ntroduced to generate an adaptve sampng scheme for the overa system such that the sampng tme ncreases as the norm of the system state goes to zero. Ths paper presents a method for syntheszng such a predctve controer and gves expct suffcent condtons for when t s stabzng. Further expct condtons are gven that guarantee confct free transmssons on the network. It s shown that the optmzaton probem may be soved offne and that the controer can be mpemented as a ookup tabe of state feedback gans. The smuaton studes whch compare the proposed agorthm to perodc sampng ustrate potenta performance gans. Index Terms Networked contro systems, predctve contro, process contro, schedung, sef-trggered contro, stabty. I. INTRODUCTION WIRELESS sensng and contro systems have receved ncreased attenton n the process ndustry n recent years. Emergng technooges n ow-power wake-up rado enabe engneerng of a new type of ndustra automaton systems where sensors, controers, and actuators communcate over a wreess channe. The ntroducton of a wreess medum n the contro oop gves rse to new chaenges, whch need to be handed [1]. The am of ths paper Manuscrpt receved December 4, 2013; revsed October 31, 2014; accepted January 10, Date of pubcaton Apr 30, 2015; date of current verson October 12, Manuscrpt receved n fna form February 9, Ths work was supported n part by the Austraan Research Counc Dscovery Project Fundng Scheme under Project DP , n part by the Knut and Ace Waenberg Foundaton, n part by the Swedsh Governmenta Agency for Innovaton Systems through the WComP Project, and n part by the Swedsh Research Counc under Grant and Grant Recommended by Assocate Edtor O. Stursberg. E. Henrksson s wth Öhns Racng AB, Uppands Väsby, Sweden e-ma: erke02@ee.kth.se. H. Sandberg and K. H. Johansson are wth the ACCESS Lnnaeus Centre, Schoo of Eectrca Engneerng, Roya Insttute of Technoogy, Stockhom , Sweden e-ma: hsan@ee.kth.se; kaej@ee.kth.se. D. E. Quevedo s wth the Department of Eectrca Engneerng EIM-E, Unversty of Paderborn, Paderborn, Germany e-ma: dquevedo@eee.org. E. G. W. Peters s wth The Unversty of Newcaste, Caaghan, NSW 2308, Austraa e-ma: edwn.g.w.peters@gma.com. Coor versons of one or more of the fgures n ths paper are avaabe onne at Dgta Object Identfer /TCST Fg. 1. Actuators A and processes P are wred to the controer C whe the sensors S communcate over a wreess network, whch n turn s coordnated by the network manager. s to address the probem of how the medum access to the wreess channe coud be dvded between the oops, takng the process dynamcs nto consderaton. We nvestgate the possbtes to desgn a sef-trggered controer that adaptvey chooses the sampng perod for mutpe contro oops. The am s to reduce the amount of generated network traffc, whe mantanng a guaranteed eve of performance n respect of drvng the nta system states to zero and the contro effort needed. Consder the networked contro system n Fg. 1, whch shows how the sensors and the controer are connected through a wreess network. The wreess network s controed by a network manager that aocates medum access to the sensors and trggers ther transmssons. Ths setup s motvated by current ndustry standards based on the IEEE standard [2], [3], whch utzes ths structure for wreess contro n process ndustry. Here, the trggerng s n turn generated by the controer that, n addton to computng the approprate contro acton, dynamcay determnes the tme of the next sampe by a sef-trggerng approach [4]. In dong so, the controer gves varyng attenton to the oops dependng on ther state, whe tryng to communcate ony few sampes. To acheve ths, the controer must, for every oop, trade contro performance aganst ntersampng tme and gve a quanttatve measure of the resutng performance. The man contrbuton of ths paper s to show that a seftrggerng controer can be derved usng a recedng horzon contro formuaton, where the predcted cost s used to jonty determne what contro sgna to be apped as we as the tme of the next sampng nstant. Usng ths formuaton, we can guarantee a mnmum and a maxmum tme between sampes. We w ntay consder a snge-oop system. We w then extend the approach to the mutpe-oop case, whch can be anayzed wth addtona constrants on the IEEE. Persona use s permtted, but repubcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 2168 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER 2015 communcaton pattern. The resuts presented heren are extensons to [5]. These resuts have been extended to hande mutpe contro oops on the same network whe mantanng contro performance and smutaneousy guaranteeng confct-free transmssons on the network. The deveopment of contro strateges for wreess automaton has become a arge area of research n whch, up unt recenty, most efforts have been made under the assumpton of perodc communcaton [6]. However, the dea of adaptve sampng s recevng ncreased attenton. The efforts wthn ths area may coarsey be dvded nto the two paradgms of event- and sef-trggered contro. In event-trggered contro [7] [15], the sensor contnuousy montors the process state and generates a sampe when the state voates some predefned condton. Sef-trggered contro [4], [16] [18] utzes a mode of the system to predct when a new sampe needs to be taken to fuf some predefned condton. A possbe advantage of event-over sef-trggered contro s that the contnuous montorng of the state guarantees that a sampe w be drawn as soon as the desgn condton s voated, thus resutng n an approprate contro acton. The sef-trggered controer w nstead operate n open oop between sampes. Ths coud potentay be a probem as dsturbances to the process between sampes cannot be attenuated. Ths probem may, however, be avoded by good choces of the ntersampng tmes. The possbe advantage of sef-over event-trggered contro s that the transmsson tme of sensor packets s known aprorand hence we may schedue them, enabng sensors and transmtters to be put to seep n between sampes and thereby savng energy. The research area of jont desgn of contro and communcaton s currenty very actve, especay n the context of event-trggered contro. In [19], a jont optmzaton of contro and communcaton s soved usng dynamc programmng by pacng a communcaton scheduer n the sensor. In [20] [22], the contro aw and event-condton are codesgned to match performance of perodc contro usng a ower communcaton rate. In [23], ths dea s extended to decentrazed systems. The use of predctve contro s aso ganng popuarty wthn the networked contro communty [24] [27]. In [28], predctve methods and vector quantzaton are used to reduce the controer-to-actuator communcaton n mutpe nput systems. In [29], mode predctve contro MPC s used to desgn mutpe actuator nk schedung and contro sgnas. There have aso been deveopments n usng MPC under event-based sampng. In [30], a method for tradng contro performance and transmsson rate n systems wth mutpe sensors s gven. In [31], an event-based MPC s proposed where the decson to recacuate the contro aw s based on the dfference between predcted and measured states. The probem addressed n ths paper, namey, the jont desgn of a sef-trggerng rue and the approprate contro sgna usng MPC has been ess studed than ts event-trggered counterpart. In [32], an approach reyng on an exhaustve search that utzes sub-optma soutons gvng the contro pocy and a correspondng sef-trggerng pocy s presented. In [33], t s suggested that a porton of the open-oop trajectory produced by the MPC shoud be apped to the process. The tme between reoptmzatons s then decded va a sef-trggerng approach. The approach taken n ths paper dffers from the above two n that the open oop cost we propose the MPC to sove s desgned to be used n an adaptve sampng context. Further, our extenson to hande mutpe oops usng a sef-trggered MPC s new and so s the guarantee of confct free transmssons. The outne of ths paper s as foows. In Secton II, the sef-trggered network schedung and contro probem s defned and formuated as a recedng horzon contro probem. Secton III presents the open-oop optma contro probem for a snge oop, to be soved by the recedng horzon controer. The optma souton s presented n Secton IV. Secton V presents the recedng horzon contro agorthm for a snge oop n further deta and gves condtons for when t s stabzng. The resuts are then extended to the mutpeoop case n Secton VII, where condtons for stabty and confct free transmssons are gven. The proposed method s expaned and evauated on the smuated exampes n Secton VIII. The concudng dscusson s made n Secton IX. II. SELF-TRIGGERED NETWORKED CONTROL ARCHITECTURE We consder the probem of controng s 1 processes P 1 through P s over a shared communcaton network as n Fg. 1. The processes are controed by the controer C that computes the approprate contro acton and schedue for each process. Each process P s gven by a near tme-nvarant LTI system x k + 1 = A x k + B u k x k R n, u k R m. 1 The controer works n the foowng way: at tme k = k, sensor S transmts a sampe x k to the controer that then computes the contro sgna u k and sends t to the actuator A. Here, {1, 2,...,s} s the process ndex. The actuator n turn w appy ths contro sgna to the process unt a new vaue s receved from the controer. Jonty wth decdng u k the controer aso decdes how many dscrete tme steps, say I k N + {1, 2,...}, t w wat before t needs to change the contro sgna the next tme. Ths vaue I k s sent to the network manager, whch w schedue the sensor S to send a new sampe at tme k = k + I k. To guarantee confct-free transmssons on the network, ony one sensor s aowed to transmt at every tme nstance. Hence, when decdng the tme to wat I k, the controer must make sure that no other sensor S q, q =, aready s schedued for transmsson at tme k = k + I k. We propose that the controer C shoud be mpemented as a recedng horzon controer that for an ndvdua oop at every sampng nstant k = k soves an open-oop optma contro probem. It does so by mnmzng the nfnte-horzon quadratc cost functon x k + 2 Q + u k + 2 R =0

3 HENRIKSSON et a.: MPC FOR NETWORK SCHEDULING AND CONTROL 2169 Fg. 2. Predcton horzon. Here, typca sgna predctons are shown wth I k = 3andp = 5. subject to the user defned weghts Q and R, whe consderng system dynamcs. In Secton III, we w embesh ths cost functon such that contro performance, nter sampng tme, and overa network scheduabty aso are taken nto consderaton. Remark 1: The focus s on a networked system where the network manager and controer are ntegrated n the same unt. Ths means that the controer can send the schedues that contan the transmsson tmes of the sensors drecty to the network manager. Ths nformaton needs to be transmtted to the sensor node such that t knows when to sampe and transmt. For exampe, usng the IEEE superframe, descrbed n [3], ths can be done wthout creatng addtona overhead. Here, the network manager broadcasts a beacon n the begnnng of each superframe, whch occurs at a fxed tme nterva. Ths beacon s receved by a devces on the network and ncudes a schedue of the sensors that are aowed to transmt at gven tmes. III. ADAPTIVE SAMPLING STRATEGY For pedagogca ease, we w n ths secton study the case when we contro a snge process on the network aowng us to drop the oop ndex. The process we contro has dynamcs xk + 1 = Axk + Buk, xk R n, uk R m 2 and the open-oop cost functon we propose the controer to mnmze at every sampng nstant s Jxk,, U = + xk + 2 Q + uk + 2 R 3 =0 where R + s a desgn varabe that s used to trade off the cost of sampng aganst the cost of contro and = I k {1, 2,..., p} s the number of dscrete tme unts to wat before takng the next sampe, where p N + s the maxmum number of tme unts to wat unt takng the next sampe. Further, the desgn varabes 0 < Q and 0 < R are symmetrc matrces of approprate dmensons. We optmze ths cost over the constrant that the contro sequence U {uk, uk + 1,...} shoud foow the specfc shape shown n Fg. 2, for some fxed perod p. The perod p s the maxmum amount of tme steps the sensor s aowed to wat before takng the next sampe. If p = 1, a sampe s taken at every tme nstance and a reguar recedng horzon cost s obtaned. If one, on the other hand, woud seect a arge vaue of p, very ong sampng ntervas can be obtaned. Ths can though affect the contro performance sgnfcanty n case there are dsturbances n the system that make the sze of the state ncrease. Thus, the number of dscrete tme unts = I k {1, 2,...,p} to wat before takng the next sampe xk + as we as the eves n the contro sequence U are free varabes over whch we optmze. Note that nether the state nor the contro vaues have a constraned magntude. The constrant on the shape of the contro trajectory U s motvated by the dea that ths sequence s apped to the actuator from tme k unt tme nstant k +. At tme xk +, we take a new sampe and redo the optmzaton. By ths method, we get a jont optmzaton of the contro sgna to be apped as we as the number of tme steps to the next sampng nstant. The reason for ettng the system be controed by a contro sgna wth perod p after ths s that we hope for the recedng horzon agorthm to converge to ths sampng rate. We w provde methods for choosng p so that ths happens n Secton V. The reason for wantng convergence to a down samped contro s that we want the system to be samped at a sow rate when t has reached steady state, whe we want t to be samped faster durng the transents. If one ncudes the above-mentoned constrants, then 3 can be rewrtten as Jxk,, U = xk + 2 Q + uk 2 R =0 p 1 xk + + r p + 2 Q r=0 =0 + uk + + r p 2 R 4 where and U {uk, uk +, uk + + η p,...}, η N +, are the decson varabes over whch we optmze. The term / refects the cost of sampng. We use ths cost to weght the cost of sampng aganst the cassca quadratc contro performance cost. For a gven xk, choosng a arge w force to be arger and hence gve onger ntersampng tmes. By the constructon of the cost, we may tune Q, R, and va smuatons to get the desred sampng behavor whe mantanng the desred contro performance. One coud magne a more genera cost of sampng. Here, however, we found that / s suffcent to be abe to trade contro performance for communcaton cost see smuatons n Secton VIII-B. IV. COST FUNCTION MINIMIZATION Havng defned the open-oop cost 4, we proceed by computng ts optma vaue. We start by notcng that even though we have a jont optmzaton probem, we may state t as mnmze mnmze Jxk,, U U. 5

4 2170 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER 2015 We w use ths separaton and start by sovng the nner probem that of mnmzng Jxk,, U for a gven vaue of. To derve the souton, and for future reference, we need to defne some varabes. Defnton 1: We defne notaton for the fted mode as A = A, B = 1 q=0 A q B. and notaton for the generazed weghtng matrces assocated wth 4 as Q = Q 1 + A 1T QA 1 R = R 1 + B 1T QB 1 + R N = N 1 + A 1T QB 1 where {1, 2,...,p}, Q 1 = Q, R 1 = R, andn 1 = 0. Usng Defnton 1, t s straghtforward to show the foowng emma. Lemma 1: It hods that 1 =0 xk + 2 Q + uk 2 R = xk T Q xk + uk T R uk + 2xk T N uk and xk + = A xk + B uk. Lemma 2: Assume that 0 < Q and 0 < R, and that the par A p, B p s controabe. Then mn U p 1 xk + + r p + 2 Q r=0 =0 + uk + + r p 2 R = xk + 2 P p and the mnmzng contro sgna characterzng U s gven by In the above uk + + r p = L p xk + + r p. P p = Q p + A pt P p A p A pt P p B p + N p L p L p = R p + B pt P p B p 1 A pt P p B p + N p T. 6 Proof: The proof s gven n Appendx A. Usng the above resuts, we may formuate the man resut of ths secton as foows. Theorem 1 Cosed Form Souton: Assume that 0 < Q and 0 < R, and that the par A p, B p s controabe. Then mn U Jxk,, U = + xk 2 P 7 where P = Q + A T P p A A T P p B + N L L = R + B T P p B 1 A T P p B + N T and P p s gven by Lemma 2. Denotng the vector of a ones n R n as 1 n, the mnmzng contro sgna sequence s gven by U = { L xk1 T, Lp xk + + r p1 T p,...}, 8 r N where aso L p s gven by Lemma 2. Proof: The proof s gven n Appendx B. Now, gettng back to the orgna probem 5, provded that the assumptons of Theorem 1 hod, we may appy t gvng that { mn Jxk,, U = mn + xk }. 2,U P Unfortunatey, the authors are not aware of any method to sove ths probem n genera. However, f s restrcted to a known fnte set I 0 N +, we may fnd the optma vaue wthn ths set for a gven vaue of xk by smpy evauatng + xk 2 I 0 and by ths obtanng the, whch gves P the owest vaue of the cost. Ths procedure gves the optmum of 5. Note that the computatona compexty of fndng the optmum s not necessary hgh, as we may compute P I 0 offne pror to executon. V. SINGLE-LOOP SELF-TRIGGERED MPC Havng presented the recedng horzon cost, we now contnue wth formuatng ts mpementaton n further deta. We w assume that 0 < Q and 0 < R, and that the down-samped par A p, B p s controabe. Let us aso assume that the fnte and nonempty set I 0 N + s gven and et γ = max I 0. 9 From ths, we may use Theorem 1 to compute the pars P, L I 0 and formuate our proposed recedng horzon contro agorthm for a snge oop. Remark 2: Note that the contro sgna vaue s sent to the actuator at the same tme as the controer requests the schedung of the next sampe by the sensor. Remark 3: The proposed agorthm guarantees a mnmum and a maxmum ntersampng tme. The mnmum tme s 1 tme step n the tme scae of the underyng process 2 and the maxmum nter sampng tme s γ tme steps. Ths mpes that there s some mnmum attenton to every oop ndependent of the predcted evouton of the process. Remark 4: Even though we are workng n unformy samped dscrete tme, the state s not samped at every tme nstant k. Instead, the set of sampes of the state actuay taken s gven by the set D, assumng that the frst sampe s taken at k = 0, s gven by D = {x0, xi 0, xi I 0,...}. 10

5 HENRIKSSON et a.: MPC FOR NETWORK SCHEDULING AND CONTROL 2171 Agorthm 1 Snge Loop Sef-Trggered MPC VI. ANALYSIS Havng estabshed and detaed our recedng horzon contro aw for a snge oop, we contnue wth gvng condtons for when t s stabzng. Lettng λa to denote the set of egenvaues of A, we frst reca the foowng controabty condtons. Lemma 3 [34]: The system A, B s controabe f and ony f the par A, B s controabe and A has no egenvaue λ λa such that λ = 1andλ = 1. Usng the above, and the resuts of Secton IV, we may now gve condtons for when the proposed recedng horzon contro agorthm s stabzng. Theorem 2 Practca Stabty [35]: Assume 0 < Q and 0 < R and that A, B s controabe. If we choose I 0 N + and p = p gven by p = max{ I 0, λ λa λ = 1fλ = 1} 11 and appy Agorthm 1, then the system state s utmatey bounded as per the foowng: γ m + xk 2 k I 0 P 1 ɛ p 1 ɛ 1. γ In the above, γ s as n 9 and ɛ s the argest vaue n the nterva 0, 1] whch I 0 fufs A B L T P p A B L 1 ɛp and s guaranteed to exst. Proof: The proof s gven n Appendx C. Remark 5: The bound gven n Theorem 2 scaes neary wth the choce of. Assumpton 1: Assume that λ λ A except possby λ = 1 such that λ = 1 and the compex argument λ = 2π/γ n for some n N +. Lemma 4: Let Assumpton 1 hod, then p = γ. Proof: The proof s gven n Appendx D. Coroary 1 Asymptotc Stabty of the Orgn: Assume 0 < Q and 0 < R, andthata, B s controabe. Further assume that ether Assumpton 1 hods or = 0. If we choose I 0 N + and p = p gven by 11 and appy Agorthm 1, then m xk = 0. k Proof: The proof s gven n Appendx E. From the above resuts, we may note the foowng. Remark 6: If the assumptons of Theorem 2 hod, then Coroary 1 can be used to ensure asymptotc stabty of the orgn, except n the extremey rare case that the underyng system A, B becomes uncontroabe under down sampng by a factor γ see Lemma 3. Otherwse, one may use Lemma 4 to redesgn I 0 gvng a new vaue of γ that recovers the case p = γ. Remark 7: It s worth notcng that the deveoped framework s not mted to just varyng the tme to the next sampe as n Fg. 2. One coud expand ths usng a mutstep approach wheren the ntersampng tme s optmzed over severa frames before revertng to constant sampng wth perod p. VII. EXTENSION TO MULTIPLE LOOPS Havng detaed the controer for the case wth a snge oop on the network and gven condtons for when t s stabzng, we now contnue wth extendng to the case when we contro mutpe oops on the network, as descrbed n Fg. 1. The dea s that the controer C now w run s such snge-oop controers descrbed n Agorthm 1 n parae, one for each process P, L ={1, 2,...,s}, controed over the network. To guarantee confct-free communcaton on the network, the controer C w, at the same tme, centray coordnate the transmssons of the dfferent oops. A. Cost Functon We start by extendng the resuts n Secton III to the case when we have mutpe oops. The cost functon we propose the controer to mnmze at every sampng nstant for oop s then J x k,, U = 1 + x k + 2 Q + u k 2 R =0 p 1 + x k + + r p + 2 Q r=0 =0 + u k + + r p 2 R derved n the same way as 4 now wth R +,0< Q 0 < R and perod p N + specfc for the contro of process P gven by 1. From ths, we can state the foowng. Defnton 2: We defne the notaton n the mutpe-oop case foowng Defnton 1. For a matrx E,e.g.,A and Q, we denote E by E. Theorem 3 Cosed-Form Souton: Assume that 0 < Q and 0 < R, and that the par A p, B p s controabe. Then mn J x k,, U = + x k 2 U P

6 2172 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER 2015 where P = Q L = R + A + B T p P A T P p B 1 A T p P B A T p P B + N L + N T Agorthm 2 Mutpe Loop Sef-Trggered MPC and P p = Q p + A p T p P A p A p T p P B p + N p L p = R p + B p T p P B p 1 A p T p P B p + N p L p T. Denotng the vector of a ones n R n as 1 n, the mnmzng contro sgna sequence s gven by { } U = L x k1 T, Lp x k + + r p 1 T p,..., r N. Proof: oops. Appcaton of Theorem 1 on the ndvdua B. Mutpe-Loop Sef-Trggered MPC To formuate our mutpe-oop recedng horzon contro aw, we w reuse the resuts n Secton V and appy them on a per oop bass. For each process P wth dynamcs gven n 1, et the weghts R +,0< Q 0 < R, and perod p N + specfc to the process be defned. If we further defne the fnte set I 0 N+ for oop, we may appy Theorem 3 to compute the pars P, L I0. Provded of course that the par A p, A p s controabe. As dscussed n Secton II, when choosng, the controer must take nto consderaton what transmsson tmes other oops have reserved as we as the overa network scheduabty. Hence, at tme k = k, oop s restrcted to choose I k I 0 where I k contans the feasbe vaues of that gves coson-free schedung of the network. Note that at tme k = k, the contro nput u and tme for the next sampe I ony have to be computed for a snge oop. How I k shoud be constructed when mutpe oops are present on the network s dscussed further ater n ths secton. We may now contnue wth formuatng our proposed agorthm for controng mutpe processes over the network. The foowng agorthm s executed whenever a sampe s receved by the controer. Remark 8: When the controer s ntazed at tme k = 0, t s assumed that the controer has knowedge of the state x 0 for a processes P controed over the network. It w then execute Agorthm 2 enterng at step 2, n the order of ncreasng oop ndex. C. Scheduabty What remans to be detaed n the mutpe-oop recedng horzon contro aw s a mechansm for oop to choose I k to acheve coson-free schedung. We now contnue wth gvng condtons for when ths hods. Frst, we note that when usng Theorem 3, we make the mpct assumpton that t s possbe to appy the correspondng optma contro sgna sequence U. For ths to be possbe, we must be abe to measure the state x k at the future tme nstances S k ={k + I k, k + I k + p, k + I k + 2 p, k + I k + 3 p,...}. 12 Hence, ths sampng pattern must be reserved for use by sensor S. We state the foowng to gve condtons for when ths s possbe. Lemma 5: Let oop choose ts set I k of feasbe tmes to wat unt the next sampe to be { I k = I 0 = knext q k + n p q m p, } m, n N, q L \{} where kq next s the next transmsson tme of sensor S q.then t s possbe to reserve the needed sampng pattern S k n 12 at tme k = k. Proof: The proof s gven n Appendx F. Constructng I k as above, we are not guaranteed that I k =. To guarantee ths, we make the foowng assumpton. Assumpton 2: Assume that for every oop on the network I 0 = I0 and p = p. Further assume that L I 0 and max L p. Theorem 4: Let Assumpton 2 hod. If every oop chooses I k ={ I 0 = kq next k + r p, r Z, q L\{}} a transmssons on the network w be confct free and

7 HENRIKSSON et a.: MPC FOR NETWORK SCHEDULING AND CONTROL 2173 t w aways be possbe to reserve the needed sampng pattern S k n 12. Proof: The proof s gven n Appendx G. Remark 9: The resut n Lemma 5 requres the reservaton of an nfnte sequence. Ths s no onger requred n Theorem 4 as a oops cooperate when choosng the set of feasbe tmes to wat. In fact, oop ony needs to know the current tme k, the perod p, and the tmes when the other oops w transmt next kq next q L \{} to fnd ts own vaue I k. Remark 10: If Assumpton 2 hods and every oop on the network chooses I k accordng to Theorem 4, then t s guaranteed that at tme k, we can reserve 12 and that no other oop can make confctng reservatons. Hence, at tme k + I k, the sequence S k + I k ={k + I k + p, k + I k + 2 p, k + I k + 3 p,...} s guaranteed to be avaabe. Thus, p I k + I k. D. Stabty We contnue wth gvng condtons for when the mutpe oop recedng horzon contro aw descrbed n Agorthm 2 s stabzng. Extendng the theory deveoped n Secton V to the mutpe oop case, we may state the foowng. Theorem 5: Assume 0 < Q and 0 < R,andthatA, B s controabe. Further et Assumpton 2 hod. If we then choose I k I 0 N +, wth I k chosen as n Theorem 4, and p = p gven by p = max{ I 0, λ λa λ = 1fλ = 1} 13 and appy Agorthm 2, then as k γ mn + x k 2 I k P 1 ɛ p 1 ɛ 1 γ where γ = max I 0 and ɛ s the argest vaue n the nterva 0, 1], whch I 0 fufs A B L T p P A B L 1 ɛ P. Proof: The proof s gven n Appendx H. Coroary 2: Assume 0 < Q and 0 < R, and that A, B s controabe. Further et Assumpton 2 hod. In addton, et I 0 be chosen so that the resutng γ = max I 0 guarantees that Assumpton 1 hods for every oop or aternatvey et = 0 for every oop. If we then choose I k I 0 N +, wth I k chosen as n Theorem 4, and p = p gven by 13 and appy Agorthm 2, t hods that m x k = 0. k Proof: The proof foows from the resuts n Theorem 5 anaogous to the proof of Coroary 1. TABLE I PRECOMPUTED CONTROL LAWS WITH RELATED COST FUNCTIONS AND INTERMEDIATE VARIABLES VIII. SIMULATION STUDIES To ustrate the proposed theory, we now contnue wth gvng smuatons. Frst, we show how the contro aw works when a snge oop s controed over the network and focus on the oop-specfc mechansms of the controer. Second, we ustrate how the controer works when mutpe oops are present on the network and focus on how the controer aocates network access to dfferent oops. A. Snge Loop Let us exempfy and dscuss how the controer handes the contro performance versus communcaton rate tradeoff n an ndvdua oop. We do ths by studyng the case wth a snge system on the network. The system we study s the snge ntegrator system, whch we dscretze usng sampe and hod wth sampng tme T s = 1 s gvng us xk + 1 = Axk + Buk wth A, B = 1, 1. Sncewe want the resutng sef-trggered MPC descrbed n Agorthm 1 to be stabzng, we need to make sure that our desgn fufs the condtons of Theorem 2. If we further want t to be asymptotcay stabzng, we n addton need t to fuf the condtons of Coroary 1. The desgn procedure s then as foows. Frst, we note that the system A, B s controabe. The next step s to decde the weghts 0 < Q and 0 < R n the quadratc cost functon 3. Ths s done n the same way as n cassca near quadratc contro [36]. Here, we for smpcty choose Q = 1andR = 1. We note that the system ony has the egenvaue λ = 1, fufng Assumpton 1, so that 11 n Theorem 2 gves p = max I 0. Hence, Coroary 1 and thus Theorem 2, w hod for every choce of I 0. Ths means that we may choose the eements n I 0,.e., the possbe down sampng rates freey. A natura way to choose them s to decde on a maxmum aowed down sampng rate and then choose I 0 to contan a rates from 1 up to ths number. Let us say that we here want the system to be samped at east every 5 T s s, then a good choce s I 0 ={1, 2, 3, 4, 5}, gvngp = max I 0 = 5. Now havng guaranteed that the condtons of Theorem 2 and Coroary 1 hod, we have aso guaranteed that the condtons of Theorem 1 are fufed. Hence, we may use t to compute the state feedback gans and cost functon matrces that are used n Agorthm 1. The resuts from these computatons are shown n Tabe I, together wth some of the ntermedate varabes from Defnton 1. Here, we see that the cost functons are quadratc functons n the state x, where the coeffcents P are functons of Q and R. Weasosee that the cost to sampe / enters neary and as we change t,

8 2174 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER 2015 Fg. 3. Cost functons V xk dashed ne together wth the parttonng of the state space and the tme to wat I k = arg mn I 0 V xk sod ne. we w change the offset eve of the curves and thereby ther vaues reated to each other. However, t w not affect the state feedback gans. A graphca ustraton of the cost functons n Tabe I, for the choce = 0.2, s shown n Fg. 3 together wth the curve I k = arg mn I 0 V xk,.e., the ndex of the cost functon whch has the owest vaue for a gven state xk. Ths s the parttonng of the state space that the sef-trggered MPC controer w use to choose whch of the state feedback gans to appy and how ong to wat before sampng agan. Appyng our sef-trggered MPC descrbed n Agorthm 1 usng the resuts n Tabe I to our ntegrator system when ntazed n x0 = 2, we get the response shown n Fg. 4a. Note here that the system w converge to the fxed sampng rate p as the state converges. It may now appear as t s suffcent to use perodc contro and sampe the system every p T s s to get good contro performance. To compare the performance of ths perodc sampng strategy wth the sef-trggered strategy above, we appy the contro that mnmzes the same cost functon 3 as above wth the excepton that the system now may ony be samped every p T s s. Ths s n fact the same as usng the recedng horzon contro above whe restrctng the controer to choose = p every tme. The resutng smuatons are shown n Fg. 4b. As seen, there s a arge degradaton of the performance n the transent whe the statonary behavor s amost the same. By ths, we can concude that t s not suffcent to sampe the system every p T s sfwewant to acheve the same transent performance as wth the sef-trggered sampng. In the nta transent response, the sef-trggered MPC controer samped after one tme nstant. Ths ndcates that there s performance to gan by sampng every tme nstant. To nvestgate ths, we appy the contro that mnmzes the same cost functon 3, now wth the excepton that the system may be samped every T s s,.e., cassca unconstraned near quadratc contro. Now smuatng the system, we get the response shown n Fg. 4c. As expected, we get sghty better transent performance n ths case compared wth our sef-trggered sampng scheme; however, t s comparabe. Fg. 4. Comparson of contro performance for dfferent sampng poces. a System response of the ntegrator system when mnmzng the cost usng our snge-oop sef-trggered MPC. b System response of the ntegrator system when mnmzng the cost by sampng every ffth second. c System response of the ntegrator system when mnmzng the cost by sampng every second. Note, however, that ths mprovement comes at the cost of a drastcay ncreased communcaton need, whch may not be sutabe for systems where mutpe oops share the same wreess medum.

9 HENRIKSSON et a.: MPC FOR NETWORK SCHEDULING AND CONTROL 2175 Fg. 5. Average sampng nterva for dfferent vaues of for the ntegrator system. From the above, we may concude that our sef-trggered MPC combnes the ow communcaton rate n statonarty of the sow perodc controer wth the quck transent response of the fast perodc sampng. In fact, we may, usng our method, recover the transent behavor of fast perodc sampng at the communcaton cost of one extra sampe compared wth sow perodc sampng. The reason for ths s that the fast sampng rate ony s needed n the transent whe we n statonarty can obtan suffcent performance wth aowerrate. B. Comparson Wth Perodc Contro To make a more extensve comparson between the sef-trggered agorthm and perodc sampng, we sweep the vaue of the desgn parameter n the cost functon and run 100 smuatons of sampes for each vaue. To add randomness to the process, the mode 1 s extended to ncude dsturbances such that for process x k + 1 = A x k + B u k + E ω k 14 where ω k N 0,σ 2 s zero-mean norma dstrbuted wth varance σ 2 = 0.1 ande = 1. We ncrease the maxmum aowed sampng nterva to p = max I 0 = 15. Ths though has the effect, that P 1 = 1.83 whereas P 2 = 1.74, such that t s aways more favorabe to have a sampe nterva of two sampes nstead of one, even when = 0. We therefore changed the cost of contro to R = 0.1 such that we aow for arger contro sgnas and force the process to sampe at every tme nstance f = 0 s chosen. The nta state s randomy generated as x 0 N 0,σx 2 0,, where σx 2 0, = Fg. 5 shows the average sampng nterva durng the smuatons for dfferent choces of for the ntegrator system. The reaton between the vaue of and the average sampng nterva s n genera monotonc but hghy affected by the system to be controed and the statstcs of the dsturbances n the system. Snce we now have the range of vaues n whch we want to sweep, we present a smpe perodc MPC cost to whch we compare our agorthm. Ths cost s gven by J x k, U T s = x k + + rt s 2 Q + u k + + rt s 2 R r=0 =0 Fg. 6. Performance of the ntegrator usng the sef-trggered agorthm compared wth the smpe perodc agorthm for dfferent sampng ntervas. Some vaues of are marked wth the back crosses. whch usng Defnton 1 and Lemmas 1 and 2 can be rewrtten as J x k, U = x k + rt s 2 Q Ts + u k + rt s 2 R Ts r=0 + 2x k + rt s T N u k + rt s 15 for sampng perod T s where x k + T s = A T s x k + B T s u k. When T s = 1, 15 reduces to cassca unconstraned near quadratc contro. The emprc cost for each smuaton s cacuated by T 1 1 x k T Q x k + u k T R u k 16 T k=0 where T = s the ength of the smuaton. Fg. 6 shows the average performance that s obtaned for dfferent averaged sampng ntervas obtaned by sweepng from 0 to Fg. 6 shows that the sef-trggered agorthm performs sgnfcanty better than perodc sampng, especay when communcaton costs are medum to hgh. However, at sampng ntervas of 2 and 3 tme steps, the sef-trggered agorthm performs sghty worse than the perodc agorthm. Ths s caused by the fact that f the state of the system s cose to zero when samped, the cost of sampng s much hgher than the cost of the state error and contro, hence the tme unt the next sampe s taken s arge. To avod ths phenomenon, n future work, one coud consder the statstcs of the process nose n the cost functon. It can further be noted that when, the performance of the sef-trggered agorthm s very cose to the cost of the perodc agorthm. Ths s as expected, snce the cost of w be greater than the cost of the state and contro, whch w resut n sampe ntervas of = p = 15. Ths reduces the sef-trggered agorthm to perodc contro. C. Mutpe Loops We now contnue wth performng a smuaton study, where we contro two systems over the same network. We w start by showng a smpe exampe foowed by a more extensve performance comparson of the Agorthm 2 wth perodc contro. We w keep the ntegrator system

10 2176 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER 2015 from Secton VIII-A now denotng t process P 1 wth dynamcs x 1 k + 1 = A 1 x 1 k + B 1 u 1 k wth A 1, B 1 = 1, 1 as before. In addton, we w the contro process P 2, whch s a doube ntegrator system, whch we dscretze usng sampe and hod wth sampng tme T s = 1 s gvng x 1 k x 1 x2 2 = k 1 k x2 2 }{{}}{{ k + u k. } x 2 k+1 x 2 k } {{ } A 2 }{{} B 2 We wsh to contro these processes usng our proposed mutpe-oop sef-trggered MPC descrbed n Agorthm 2. As we wsh to stabze these systems, we start by checkng the condtons of Theorem 5 and Coroary 2. Frst, we may easy verfy that both the pars A 1, B 1 and A 2, B 2 are controabe. To use the stabty resuts, we need Assumpton 2 to hod, mpyng that we must choose p 1 = p 2 = p, I1 0 = I0 2 = I0, and choose I 0 such that {1, 2} I 0 and 2 p. For reasons of performance, we wsh to guarantee that the systems are samped at east every 5 T s s and therefore choose I1 0 = I0 2 = I0 = {1, 2, 3, 4, 5} fufng the requrement above. We aso note that λa 1 ={1} and λa 2 ={1, 1} and that hence both system fuf Assumpton 1 for ths choce of I 0, mpyng that 13 n Theorem 5 gves p = max I 0 = 5. Thus, choosng p = p, as stated n Theorem 5, resuts n that Assumpton 2 hods. What now remans to be decded are the weghts, Q,andR. For the ntegrator process P 1, we keep the same tunng as n Secton VIII-A wth Q 1 = R 1 = 1. Havng decded Q 1, R 1, I 0,andp, we use Theorem 3 to compute the needed state feedback gans and cost functon matrces P 1, L 1 I 0 needed by Agorthm 2. We aso keep 1 = 0.2 ast gave a good communcaton versus performance tradeoff. For the doube ntegrator process P 2, the weghts are chosen to be Q 2 = I, as we consder both states equay mportant and R 2 = 1/10 to favor contro performance and aow for arger contro sgnas. Havng decded Q 2, R 2, I 0,andp, we may use Theorem 3 to compute the needed state feedback gans and cost functon matrces P 2, L 2 I0 needed by Agorthm 2. The sampng cost s chosen to be 2 = 1, as ths gves a good tradeoff between contro performance and the number of sampes. We have now fufed a the assumptons of both Theorem 5 and Coroary 2. Hence, appyng Agorthm 2, choosng I k accordng to Theorem 4 w asymptotcay stabze both process P 1 and P 2. Controng P 1 and P 2 usng our mutpe-oop sef-trggered MPC descrbed n Agorthm 2 wth the above desgned tunng, we get the resut shown n Fg. 7. As expected, the behavor of the controer ustrated n Secton VIII-A carres through aso to the case when we have mutpe oops on the network. In fact, comparng Fg. 4a by showng how the controer handes process P 1 when controng t by tsef on the network and Fg. 7a that shows how P 1 s handed n the mutpe-oop case, we see that they are the same. Further we see that, as expected, n statonarty, the two oops controng process P 1 and P 2 both converge to the sampng rate p. Fg. 7. Processes P 1 and P 2 controed and schedued on the same network usng our mutpe-oop sef-trggered MPC. a System response for process P 1. b System response for process P 2. As mentoned prevousy, the controer uses the mechansm n Theorem 4 to choose the set of feasbe tmes to wat unt the next sampe. In Fg. 8, we can observe how the resutng sets I k ook n deta. At tme k = 0, oop 1 gets to run Agorthm 2 frst. As sensor S 2 s not schedued for any transmssons, yet I 1 0 = I 0 from whch the controer chooses I 1 0 = 1. Then oop 2 gets to run Agorthm 2 at tme k = 0. As sensor S 1 now s schedued for transmsson at tme k = 0 + I 1 0 = 1, Theorem 4 gves I 2 0 = I 0 \{1} from whch the controer chooses I 2 0 = 2. The process s then repeated every tme, a sampe s transmtted to the controer, gvng the resut n Fg. 8. As seen, both the set I and the optma tme to wat I converges to some fxed vaue as the state of the correspondng process P converges to zero. D. Comparson Wth Perodc Contro For a more thorough performance comparson, we smuate the systems usng Agorthm 2 and compare them to the perodc agorthm that uses the cost functon 15. The snge and doube ntegrator processes P 1 and P 2 are smuated usng the parameters mentoned n Secton VIII. The varance of the dsturbances for both processes are set to σ 2 = 0.1,,

11 HENRIKSSON et a.: MPC FOR NETWORK SCHEDULING AND CONTROL 2177 Fg. 8. Sets I of feasbe tmes to wat unt the next sampe for oop 1 and oop 2. The optma tme to wat I s marked by red crces, whereas the other feasbe tmes are starred. Fg. 9. Performance for both processes for the sef-trggered agorthm compared wth a smpe perodc agorthm for dfferent sampng ntervas. The maxmum sampng nterva p = 5 done by varyng from 0 to and cacuatng the average amount of sampes for every vaue of. the varance of the nta state to σx 2 0, = 25,, and E 2 =[1, 1] T n 14. The vaue of s dentca for both processes n each smuaton, such that 1 = 2. Further, R 1 = 0.1. The smuaton for both agorthms s ntazed as descrbed n Remark 8. Fg 9 shows the performance for the processes, P 1 and P 2, cacuated by averagng 16 over 100 smuatons each of ength for dfferent vaues of when p1 = p 2 = 5. Fg. 10 shows the performance when p1 = p 2 = 15. The cost of each process s normazed wth respect to ts argest cost for easer vewng. Both fgures show that the sef-trggered agorthm n genera outperforms perodc sampng. The performance margn ncreases as the sampng nterva ncreases. Fg. 9 shows that P 1 when = 0 sampes every 1.8 tme steps on average, whereas P 2 ony sampes every 2.2 tme steps. When ncreases, both processes sampe amost every 2 tme steps. The reason for ths s that the cost of the state for the down samped systems n some cases s ower. The worse performance of the sef-trggered agorthm at ower sampng ntervas s more sgnfcant n Fg. 10, where the performance of process P 1 shows a smar performance as n Fg. 6. The owest average sampng nterva for process P 2 s 3.6 tme steps when = 0. The sef-trggered agorthm, though, sgnfcanty outperforms the perodc agorthm when the average sampng nterva ncreases. Fg. 10. Performance for both processes for the sef-trggered agorthm compared wth a smpe perodc agorthm for dfferent sampng ntervas. The maxmum sampng nterva p = 15 done by varyng from 0 to 10 9 and cacuatng the average amount of sampes for every vaue of. As, the cost of sampng forces the sef-trggered agorthm to behave smary to the perodc samped agorthm. Therefore, the performance gap between the perodc and sef-trggered agorthms narrows, as the average sampng nterva s cose to p. IX. CONCLUSION We have studed the jont desgn of contro and adaptve schedung of mutpe oops, and have presented a method that at every sampng nstant computes the optma contro sgna to be apped as we as the optma tme to wat before takng the next sampe. It s shown that ths contro aw may be reazed usng MPC and computed expcty. The controer s aso shown to be stabzng under md assumptons. The smuaton resuts show that the use of the presented contro aw n most cases may hep reducng the requred amount of communcaton wthout amost any oss of performance compared wth fast perodc sampng. In the mutpe-oop case, we have aso presented an agorthm for guaranteeng confct free transmssons. It s shown that under md assumptons, there aways exsts a feasbe schedue for the network. The compexty of the mutpe-oop sef-trggered MPC and the correspondng schedung agorthm scaes neary n the number of oops. An nterestng topc for future research s to further nvestgate the compexty and possbe performance ncrease for such an extended formuaton. Intutvey, addtona performance gans can be acheved when the cost functon consders process nose as we. Expotng ths coud be of nterest n future research. Another topc for future research woud be to appy the presented framework to constraned contro probems. APPENDIX A. Proof of Lemma 2 Proof: Foowng Lemma 1, the probem s equvaent to xk + + r p T Q p xk + + r p mn U r=0 + uk + + r p T R p uk + + r p + 2xk + + r p T N p uk + + r p

12 2178 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER 2015 wth xk + + r + 1 p = A p xk + + r p + B p uk + + r p. Ths probem has the known optma souton [37], xk+ 2,whereP p s gven by the Rccat equaton 6, P p whch has a souton provded that 0 < R p, mped by 0 < R, and0< Q p,mpedby0< Q, and that the par A p, B p s controabe. It s exacty what s stated n the emma. B. Proof of Theorem 1 Proof: From the Theorem, we have that 0 < Q and 0 < R, and that the par A p, B p s controabe. Thus, we may use Lemma 2 to express the cost 4 as Jxk,, U = + xk + 2 P p 1 + xk + 2 Q + uk 2 R. =0 Now appyng Lemma 1, we get Jxk,, U = + xk + 2 P p + xk T Q xk + uk T R uk + 2xk T N uk wth xk + = A xk + B uk. Mnmzng Jxk,, U now becomes a fnte-horzon optma contro probem wth one predcton step nto the future. Ths probem has the we-defned souton 7 [37] gven by teratng the Rccat equaton 8. C. Proof of Theorem 2 Proof: By assumpton, A, B s controabe. Together wth the choce of p ths, va Lemma 3, mpes that A p, B p s controabe. Further, et ˆxk k denote an estmate of xk, gven a avaabe measurements up unt tme k. Defnng 1 ˆxk k 2 S ˆxk + k 2 Q + ûk k 2 R =0 17 we may, snce by assumpton 0 < Q and 0 < R, use Lemma 2 and Theorem 1 to express V k, the optma vaue of the cost 4 at the current sampng nstant k, as V k mn I 0,ûk k Jxk,, ûk k = mn + ˆxk + k 2 I 0 P p + ˆxk k 2 S = mn + ˆxk k 2 I 0 P. 18 We w use V k as a Lyapunov-ke functon. Assume that V k+ s the optma cost at the next sampng nstant k +. Agan usng Theorem 1, we may express t as V k+ mn Jxk +, j, ûk + k + j I 0,ûk+ k+ mn Jxk +, j = ûk+ k+ p, ûk + k + = p + ˆxk + k + 2 P p = p + ˆxk + k 2 P p 19 where the nequaty comes from the fact that choosng j = p s suboptma. Takng the dfference, we get V k+ V k p ˆxk k 2 S whch n genera s not decreasng. However, we may use the foowng dea to bound ths dfference: assume that there ɛ 0, 1], andβ R + such that we may wrte V k+ V k ɛv k + β for a k, k+ D 10, 18. Thus, at sampng nstances nto the future, whch happens at, et us say, tme k +,wehave that V k+ 1 ɛ V k + β 1 ɛ. Snce ɛ 0, 1], ths s equvaent to r=0 V k+ 1 ɛ 1 1 ɛ V k + β 1 1 ɛ whch as gves us an upper bound on the cost functon, V k+ β/ɛ. Appyng ths dea on our setup, we shoud fuf p ˆxk k S ɛ ɛ ˆxk + k 2 P p ɛ ˆxk k 2 S + β. Choosng β /p 1 ɛ/γ, we have fufment f ɛ ˆxk + k 2P + ˆxk k 2 p S ˆxk k 2 S. 20 Ceary there ɛ 0, 1] such that the above reaton s fufed f 0 < ˆxk k 2. For the case ˆxk k 2 = 0, S S we must, foowng the defnton 17 and the assumpton 0 < Q, havethat ˆxk k = 0and ˆxk + k = 0, and hence the reaton s fufed aso n ths case. Usng the fna step n 18, we may express 20 n easy computabe quanttes gvng the condton ˆxk + k 2 P p 1 ɛ ˆxk k 2 P. As ths shoud hod x and I 0, we must fuf A B L T P p A B L 1 ɛp whch s stated n the theorem. Summng up, we have V k+ 1 ɛ p 1 ɛ 1 γ whch s mnmzed by maxmzng ɛ. From the defnton of the cost 3, we may aso concude that /γ V k+. Wth V k+ = mn + ˆxk + k + 2 I 0 P

13 HENRIKSSON et a.: MPC FOR NETWORK SCHEDULING AND CONTROL 2179 we may concude that γ m mn + ˆxk k 2 k I 0 P ɛ 1 p 1 ɛ 1. γ D. Proof of Lemma 4 Proof: From 11, t s cear that p = γ f λ λa except λ = 1 such that λ γ = 1. In poar coordnates, we have that λ = λ exp j λ mpyng λ γ = λ γ exp j γ λ = 1 may ony be fufed f λ =1and λ = 2π/γ n for some n N +, whch contradcts Assumpton 1. E. Proof of Coroary 1 Proof: From Theorem 2, we have that as k γ mn + xk 2 I 0 P 1 ɛ p 1 ɛ 1 γ whch as p = γ, gven by Lemma 4, smpfes to γ mn + xk 2 I 0 P γ ndependent of ɛ. Impyng that as k,wehavethat = γ and xk 2 = 0, snce 0 < P, provded 0 < Q, P ths mpes that xk = 0 ndependent of. In the case = 0, the bound from Theorem 2 smpfes to that as k 0 mn I xk 2 0 P 0 and hence for the optma, wehave xk 2 = 0 mpyng P xk = 0 as above. F. Proof of Lemma 5 Proof: When oop q was ast samped at tme k q < k, t was optmzed over I q k q and found the optma feasbe tme unt the next sampe I q k q. The oop then reserved the nfnte sequence S q k q ={k q + I q k q, k q + I q k + p q, k q + I q k q + 2 p q, k q + I q k q + 3 p q,...}. When oop now shoud choose I k, t must be abe to reserve 12. To ensure that ths s true, t must choose I k so that S k S q k q =, q L \{}. Ths hods f we have that k + I k + m p = k q + I q k q + n p q m, n N q L \{}. Smpfyng the above condton, we get condtons on the feasbe vaues of I k I k = k q + I q k q k + n p q m p m, n N, q L \{}. Notng that k q + I q k q s the next transmsson of oop q, we denote t by k next q, gvng the statement n the emma. G. Proof of Theorem 4 Proof: Assumng p = p and I 0 = I0 for a oops, Lemma 5 gves I k, as stated n the theorem. Snce by assumpton L I 0, we know that I k { L = kq next k + r p, r Z, q L\{}}. Further, snce max L p, we know that the set { L = k next q k + r p, r Z } contans at most one eement for a gven oop q. Thus { L = kq next k + r p, r Z, q L\{}} contans at east one eement as q L \{} L. Hence, I k aways contans at east one eement k,and thus there aways exsts a feasbe tme to wat. H. Proof of Theorem 5 Proof: Drect appcaton of Theorem 2 on each oop s done. The correspondng proof carres through as the choce of I k together wth the remanng assumptons guarantees that we may appy Theorem 3 on the feasbe vaues of n every step, thus the expresson for V k n 18 aways exsts. The crtca step s that the upper bound on V k+ n 19 must exst,.e., the choce j = p must be feasbe. Ths s aso guaranteed by the choce of I k Remark 10. REFERENCES [1] A. Wg, Recent and emergng topcs n wreess ndustra communcatons: A seecton, IEEE Trans. Ind. Informat., vo. 4, no. 2, pp , May [2] HART Communcaton Foundaton WreessHART. [Onne]. Avaabe: [3] IEEE Standard for Loca and Metropotan Area Networks Part 15.4: Low-Rate Wreess Persona Area Networks LR-WPANs, IEEE Standard , [Onne]. Avaabe: eee.org/about/get/802/ htm [4] M. Veasco, P. Martý, and J. M. Fuertes, The sef trggered task mode for rea-tme contro systems, n Proc. 24th IEEE Work-Prog. Sesson Rea-Tme Syst. Symp. RTSS, Cancun, Mexco, [Onne]. Avaabe: [5] E. Henrksson, D. E. Quevedo, H. Sandberg, and K. H. Johansson, Seftrggered mode predctve contro for network schedung and contro, n Proc. 8th IFAC Int. Symp. Adv. Contro Chem. Process., Sngapore, 2012, pp [6] P. Antsaks and J. Baeu, Speca ssue on technoogy of networked contro systems, Proc. IEEE, vo. 95, no. 1, pp. 5 8, Jan [7] K. J. Åström and B. Bernhardsson, Comparson of perodc and event based sampng for frst-order stochastc systems, n Proc. 14th Word Congr. IFAC, Bejng, Chna, 1999, pp [8] K.-E. Årzén, A smpe event-based PID controer, n Proc. 14th Word Congr. IFAC, Bejng, Chna, 1999, pp [9] P. Tabuada, Event-trggered rea-tme schedung of stabzng contro tasks, IEEE Trans. Autom. Contro, vo. 52, no. 9, pp , Sep [10] W. P. M. H. Heemes, J. H. Sandee, and P. P. J. Van Den Bosch, Anayss of event-drven controers for near systems, Int. J. Contro, vo. 81, no. 4, pp , [11] A. Bemporad, S. D Carano, and J. Júvez, Event-based mode predctve contro and verfcaton of ntegra contnuous-tme hybrd automata, n Proc. 9th Int. Workshop Hybrd Syst., Comput. Contro HSCC, Santa Barbara, CA, USA, Mar. 2006, pp [Onne]. Avaabe: [12] P. Varutt, B. Kern, T. Fauwasser, and R. Fndesen, Event-based mode predctve contro for networked contro systems, n Proc. 48th IEEE Conf. Decson Contro, 28th Chn. Contro Conf. CDC/CCC, Dec. 2009, pp

14 2180 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 23, NO. 6, NOVEMBER 2015 [13] D. Gross and O. Stursberg, Dstrbuted predctve contro for a cass of hybrd systems wth event-based communcaton, n Proc. 4th IFAC Workshop Dstrb. Estmaton Contro Netw. Syst., 2013, pp [14] A. Mon and S. Hrche, A b-eve approach for the desgn of eventtrggered contro systems over a shared network, Dscrete Event Dyn. Syst., vo. 24, no. 2, pp , [15] R. Bnd and F. Agöwer, On tme-trggered and event-based contro of ntegrator systems over a shared communcaton system, Math. Contro, Sgnas, Syst., vo. 25, no. 4, pp , [16] X. Wang and M. D. Lemmon, Sef-trggered feedback contro systems wth fnte-gan L 2 stabty, IEEE Trans. Autom. Contro, vo. 45, no. 3, pp , Mar [17] A. Anta and P. Tabuada, To sampe or not to sampe: Sef-trggered contro for nonnear systems, IEEE Trans. Autom. Contro, vo. 55, no. 9, pp , Sep [18] M. Mazo, Jr., A. Anta, and P. Tabuada, An ISS sef-trggered mpementaton of near controers, Automatca, vo. 46, no. 8, pp , [19] A. Mon and S. Hrche, On LQG jont optma schedung and contro under communcaton constrants, n Proc. 48th IEEE Int. Conf. Decson Contro, Shangha, Chna, Dec. 2009, pp [20] D. J. Antunes, W. P. M. H. Heemes, J. P. Hespanha, and C. J. Svestre, Schedung measurements and contros over networks Part II: Roout strateges for smutaneous protoco and controer desgn, n Proc. Amer. Contro Conf., Montrea, QC, Canada, Jun. 2012, pp [21] D. Antunes, W. P. M. H. Heemes, and P. Tabuada, Dynamc programmng formuaton of perodc event-trggered contro: Performance guarantees and co-desgn, n Proc. IEEE 51st Annu. Conf. Decson Contro, Mau, HI, USA, Dec. 2012, pp [22] W. P. M. H. Heemes, M. C. F. Donkers, and A. R. Tee, Perodc event-trggered contro for near systems, IEEE Trans. Autom. Contro, vo. 58, no. 4, pp , Apr [23] W. P. M. H. Heemes and M. C. F. Donkers, Mode-based perodc event-trggered contro for near systems, Automatca, vo. 49, no. 3, pp , [24] A. Casavoa, E. Mosca, and M. Papn, Predctve teeoperaton of constraned dynamc systems va nternet-ke channes, IEEE Trans. Contro Syst. Techno., vo. 14, no. 4, pp , Ju [25] P. L. Tang and C. W. de Sva, Compensaton for transmsson deays n an ethernet-based contro network usng varabe-horzon predctve contro, IEEE Trans. Contro Syst. Techno., vo. 14, no. 4, pp , Ju [26] G. P. Lu, J. X. Mu, D. Rees, and S. C. Cha, Desgn and stabty anayss of networked contro systems wth random communcaton tme deay usng the modfed MPC, Int. J. Contro, vo. 79, no. 4, pp , [27] D. E. Quevedo and D. Nešć, Robust stabty of packetzed predctve contro of nonnear systems wth dsturbances and Markovan packet osses, Automatca, vo. 48, no. 8, pp , [28] D. E. Quevedo, G. C. Goodwn, and J. S. Wesh, Mnmzng downnk traffc n networked contro systems va optma contro technques, n Proc. 42nd IEEE Conf. Decson Contro, Dec. 2003, pp [29] M. Lješnjann, D. E. Quevedo, and D. Nešć, Packetzed MPC wth dynamc schedung constrants and bounded packet dropouts, Automatca, vo. 50, no. 3, pp , [30] D. Bernardn and A. Bemporad, Energy-aware robust mode predctve contro based on nosy wreess sensors, Automatca, vo. 48, no. 1, pp , [31] D. Lehmann, E. Henrksson, and K. H. Johansson, Event-trggered mode predctve contro of dscrete-tme near systems subject to dsturbances, n Proc. Eur. Contro Conf., Zürch, Swtzerand, Ju. 2013, pp [32] J. D. J. Barradas Bergnd, T. M. P. Gommans, and W. P. M. H. Heemes, Sef-trggered MPC for constraned near systems and quadratc costs, n Proc. 4th IFAC Conf. Nonnear Mode Predct. Contro, Noordwjkerhout, The Netherands, 2012, pp [33] A. Eqtam, S. Heshmat-Aamdar, D. V. Dmarogonas, and K. J. Kyrakopouos, Sef-trggered mode predctve contro for nonhoonomc systems, n Proc. Eur. Contro Conf., Zürch, Swtzerand, Ju. 2013, pp [34] P. Coaner, R. Scatton, and N. Schavon, Reguaton of mutrate samped-data systems, Contro-Theory Adv. Techno., vo. 7, no. 3, pp , [35] Z.-P. Jang and Y. Wang, Input-to-state stabty for dscrete-tme nonnear systems, Automatca, vo. 37, no. 6, pp , Jun [36] J. M. Macejowsk, Predctve Contro Wth Constrants. Engewood Cffs, NJ, USA: Prentce-Ha, [37] D. P. Bertsekas, Dynamc Programmng and Optma Contro, vo. 1. Bemont, MA, USA: Athena Scentfc, Erk Henrksson receved the M.Sc. degree n eectrca engneerng and the Ph.D. degree n automatc contro from the Roya Insttute of Technoogy, Stockhom, Sweden, n 2007 and 2014, respectvey. He was a Vstng Researcher wth the Unversta deg Stud d Sena, Sena, Itay, n 2007, and The Unversty of Newcaste, Caaghan, NSW, Austraa, n He s currenty wth Öhns Racng AB, Uppands Väsby, Sweden, where he s nvoved n semactve suspenson for hypersport motorcyces. Dane E. Quevedo S 97 M 05 SM 14 receved the Ingenería Cv Eectrónca and Magster en Ingenería Eectrónca degrees from the Unversdad Técnca Federco Santa María, Vaparaíso, Che, n 2000, and the Ph.D. degree from The Unversty of Newcaste, Caaghan, NSW, Austraa, n He s currenty a Professor of Automatc Contro at the Unversty of Paderborn, Germany. He has been a Vstng Researcher at varous nsttutons, ncudng Uppsaa Unversty, Uppsaa, Sweden, the Roya Insttute of Technoogy, Stockhom, Sweden, Kyoto Unversty, Kyoto, Japan, the Karsruher Insttut für Technooge, Karsruhe, Germany, the Unversty of Notre Dame, Notre Dame, IN, USA, INRIA, Grenobe, France, the Hong Kong Unversty of Scence and Technoogy, Hong Kong, Aaborg Unversty, Aaborg, Denmark, and Nanyang Technoogca Unversty, Sngapore. Hs current research nterests ncude automatc contro, sgna processng, and power eectroncs. Prof. Quevedo was a recpent of the IEEE Conference on Decson and Contro Best Student Paper Award n 2003, and was a fnast n He receved a fve-year Austraan Research Feowshp n He was supported by the Fu Schoarshp from the Aumn Assocaton durng hs tme wth the Unversdad Técnca Federco Santa María, and receved severa unversty-wde przes upon graduatng. He serves as an Edtor of the Internatona Journa of Robust and Nonnear Contro. Edwn G. W. Peters receved the B.Sc. degree n eectrca engneerng and the M.Sc. degree n sgna processng from Aaborg Unversty, Aaborg, Denmark, n 2011 and 2013, respectvey. He s currenty pursung the Ph.D. degree n contro systems wth The Unversty of Newcaste, Caaghan, NSW, Austraa. Hs current research nterests ncude networked contro, stochastc systems, and optmzaton.

15 HENRIKSSON et a.: MPC FOR NETWORK SCHEDULING AND CONTROL 2181 Henrk Sandberg receved the M.Sc. degree n engneerng physcs and the Ph.D. degree n automatc contro from Lund Unversty, Lund, Sweden, n 1999 and 2004, respectvey. He was a Post-Doctora Schoar wth the Caforna Insttute of Technoogy, Pasadena, CA, USA, from 2005 to In 2013, he was a Vstng Schoar wth the Laboratory for Informaton and Decson Systems, Massachusetts Insttute of Technoogy, Cambrdge, MA, USA. He has hed vstng appontments wth Austraan Natona Unversty, Canberra, ACT, Austraa, and the Unversty of Mebourne, Mebourne, NSW, Austraa. He s currenty an Assocate Professor wth the Department of Automatc Contro, Roya Insttute of Technoogy, Stockhom, Sweden. Hs current research nterests ncude secure networked contro, power systems, mode reducton, and fundamenta mtatons n contro. Dr. Sandberg was a recpent of the Best Student Paper Award from the IEEE Conference on Decson and Contro n 2004 and the Ingvar Carsson Award from the Swedsh Foundaton for Strategc Research n He s currenty an Assocate Edtor of IFAC Automatca journa. Kar Henrk Johansson F 13 receved the M.Sc. and Ph.D. degrees n eectrca engneerng from Lund Unversty, Lund, Sweden. He hed vstng postons wth the Unversty of Caforna at Berkeey, Berkeey, CA, USA, from 1998 to 2000, and the Caforna Insttute of Technoogy, Pasadena, CA, USA, from 2006 to He s currenty the Drector of the KTH ACCESS Lnnaeus Centre, and a Professor wth the Schoo of Eectrca Engneerng, Roya Insttute of Technoogy, Stockhom, Sweden. He s a Waenberg Schoar, and has been a sx-year Senor Researcher wth the Swedsh Research Counc. He s the Drector of the Stockhom Strategc Research Area ICT The Next Generaton. Hs current research nterests ncude networked contro systems, hybrd and embedded system, and appcatons n smart transportaton, energy, and automaton systems. Dr. Johansson was a recpent of the best paper award at the IEEE Internatona Conference on Mobe Ad-hoc and Sensor Systems n 2009, the Indvdua Grant for the Advancement of Research Leaders from the Swedsh Foundaton for Strategc Research n 2005, the Trenna Young Author Prze from IFAC n 1996, the Pecce Award from the Internatona Insttute of System Anayss, Austra, n 1993, and the Young Researcher Awards from Scana n 1996 and from Ercsson n 1998 and He has been a member of the IEEE Contro Systems Socety Board of Governors and the Char of the IFAC Technca Commttee on Networked Systems. He has been on the Edtora Boards of severa journas, ncudng Automatca, the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, andiet Contro Theory and Appcatons. He s currenty on the Edtora Board of IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS and the European Journa of Contro. He has been a Guest Edtor for speca ssues, ncudng two ssues of IEEE TRANSACTIONS ON AUTOMATIC CONTROL: one on wreess sensor and actuator networks and one on contro of cyber-physca systems. He was the Genera Char of the ACM/IEEE Cyber-Physca Systems Week n Stockhom n 2010 and the IPC Char of many conferences. He has served on the Executve Commttees of severa European research projects n the area of networked embedded systems. He receved the Waenberg Schoar, as one of the frst 10 schoars from a scences, by the Knut and Ace Waenberg Foundaton.