Multiple Loop Self-Triggered Model Predictive Control for Network Scheduling and Control

Transcription

1 Mutipe Loop Sef-Triggered Mode Predictive Contro for Networ Scheduing and Contro Eri Henrisson, Danie E. Quevedo, Edwin G.W. Peters, Henri Sandberg, Kar Henri Johansson arxiv:5.38v [cs.sy] Feb 5 Abstract We present an agorithm for controing and scheduing mutipe inear time-invariant processes on a shared bandwidth imited communication networ using adaptive samping intervas. The controer is centraized and computes at every samping instant not ony the new contro command for a process, but aso decides the time interva to wait unti taing the next sampe. The approach reies on mode predictive contro ideas, where the cost function penaizes the state and contro effort as we as the time interva unti the next sampe is taen. The atter is introduced in order to generate an adaptive samping scheme for the overa system such that the samping time increases as the norm of the system state goes to zero. The paper presents a method for synthesizing such a predictive controer and gives expicit sufficient conditions for when it is stabiizing. Further expicit conditions are given which guarantee confict free transmissions on the networ. It is shown that the optimization probem may be soved off-ine and that the controer can be impemented as a ooup tabe of state feedbac gains. Simuation studies which compare the proposed agorithm to periodic samping iustrate potentia performance gains. Index Terms Predictive Contro; Networed Contro Systems; Process Contro; Stabiity; Scheduing; Sef-triggered Contro. I. INTRODUCTION Wireess sensing and contro systems have received increased attention in the process industry over the ast years. Emerging technoogies in ow-power wae-up radio enabes engineering of a new type of industria automation systems where sensors, controers and actuators communicate over a wireess channe. The introduction of a wireess medium in the contro oop gives rise to new chaenges which need to be handed []. The aim of this paper is to address the probem of how the medium access to the wireess channe coud be divided between the oops, taing the process dynamics into consideration. We investigate possibiities to design a sef-triggered controer, that adaptivey chooses the samping period for mutipe contro oops. The aim is to reduce the amount of generated networ traffic, whie maintaining a guaranteed eve of performance in respect of driving the initia system states to zero and the contro effort needed. Consider the networed contro system in Fig., which shows how the sensors and the controer are connected E. Henrisson, H. Sandberg and K.H. Johansson are with ACCESS Linnaeus Centre, Schoo of Eectrica Engineering, Roya Institute of Technoogy, 44 Stochom, Sweden. e-mai: {erie, hsan, aej}@ee.th.se D.E. Quevedo and E.G.W. Peters are with The University of Newcaste, NSW 38, Austraia. e-mai: dquevedo@ieee.org, edwin.g.w.peters@gmai.com. This wor was supported by the Swedish Governmenta Agency for Innovation Systems through the WiComPi project, the Swedish Research Counci under Grants and , the Knut and Aice Waenberg Foundation, and the Austraian Research Councis Discovery Projects funding scheme project number DP9886. A s A P s P S Networ Manager C Networ Fig.. Actuators A and processes P are wired to the controer C whie the sensors S communicate over a wireess networ, which in turn is coordinated by the Networ Manager. through a wireess networ. The wireess networ is controed by a Networ Manager which aocates medium access to the sensors and triggers their transmissions. This setup is motivated by current industry standards based on the IEEE standard, e.g., [], [3], which utiizes this structure for wireess contro in process industry. Here the triggering is in turn generated by the controer which, in addition to computing the appropriate contro action, dynamicay determines the time of the next sampe by a sef-triggering approach [4]. In doing so the controer gives varying attention to the oops depending on their state, whie trying to communicate ony few sampes. To achieve this the controer must, for every oop, trade contro performance against inter samping time and give a quantitative measure of the resuting performance. The main contribution of the paper is to show that a seftriggering controer can be derived using a receding horizon contro formuation where the predicted cost is used to jointy determine what contro signa to be appied as we as the time of the next samping instant. Using this formuation we can guarantee a minimum and a maximum time between sampes. We wi initiay consider a singe-oop system. We wi then extend the approach to the mutipe-oop case which can be anayzed with additiona constraints on the communication pattern. The resuts presented herein are extensions to the authors previous wor presented in [5]. These resuts have been extended to hande mutipe contro oops on the same networ whie maintaining contro performance and simutaneousy guarantee confict free transmissions on the networ. The deveopment of contro strategies for wireess automation has become a arge area of research in which, up unti recenty, most efforts have been made under the assumption of periodic communication [6]. However the idea of adaptive S s

2 samping is receiving increased attention. The efforts within this area may coarsey be divided into the two paradigms of event- and sef-triggered contro. In event-triggered contro, e.g., [7], [8], [9], [], [], [], [3], [4], [5] the sensor continuousy monitors the process state and generates a sampe when the state vioates some predefined condition. Sef-triggered contro, e.g., [4], [6], [7], [8], utiizes a mode of the system to predict when a new sampe needs to be taen in order to fufi some pre-defined condition. A possibe advantage of event- over sef-triggered contro is that the continuous monitoring of the state guarantees that a sampe wi be drawn as soon as the design condition is vioated, thus resuting in an appropriate contro action. The sef-triggered controer wi instead operate in open-oop between sampes. This coud potentiay be a probem as disturbances to the process between sampes cannot be attenuated. This probem may however be avoided by good choices of the inter-samping times. The possibe advantage of sef- over event-triggered contro is that the transmission time of sensor pacets is nown a-priori and hence we may schedue them, enabing sensors and transmitters to be put to seep in-between sampes and thereby save energy. The research area of joint design of contro and communication is currenty very active, especiay in the context of eventtriggered contro. In [9] a joint optimization of contro and communication is soved using dynamic programming, pacing a communication scheduer in the sensor. In [], [] and [] the contro aw and event-condition are co-designed to match performance of periodic contro using a ower communication rate. In [3] this idea is extended to decentraized systems. The use of predictive contro is aso gaining popuarity within the networed contro community [4], [5], [6], [7]. In [8] predictive methods and vector quantization are used to reduce the controer to actuator communication in mutipe input systems. In [9] mode predictive contro MPC is used to design mutipe actuator in scheduing and contro signas. There have aso been deveopments in using MPC under event-based samping. In [3] a method for trading contro performance and transmission rate in systems with mutipe sensors is given. In [3] an event-based MPC is proposed where the decision to re-cacuate the contro aw is based on the difference between predicted and measured states. The probem addressed in the present paper, namey the joint design of a sef-triggering rue and the appropriate contro signa using MPC has been ess studied than its event-triggered counterpart. In [3] an approach reying on an exhaustive search which utiizes sub-optima soutions giving the contro poicy and a corresponding sef-triggering poicy is presented. In [33] it is suggested that a portion of the open oop trajectory produced by the MPC shoud be appied to the process. The time between re-optimizations is then decided via a seftriggering approach. The approach taen in this paper differs from the above two in that the open oop cost we propose the MPC to sove is designed to be used in an adaptive samping context. Further our extension to hande mutipe oops using a sef-triggered MPC is new. So is the guarantee of confict free transmissions. The outine of the paper is as foows. In Section II the sef-triggered networ scheduing and contro probem is defined and formuated as a receding horizon contro probem. Section III presents the open-oop optima contro probem for a singe oop, to be soved by the receding horizon controer. The optima soution is presented in Section IV. Section V presents the receding horizon contro agorithm for a singe oop in further detai and gives conditions for when it is stabiizing. The resuts are then extended to the mutipe oop case in Section VII where conditions for stabiity and confict free transmissions are given. The proposed method is expained and evauated on simuated exampes in Section VIII. Concuding discussions are made in Section IX. II. SELF-TRIGGERED NETWORKED CONTROL ARCHITECTURE We consider the probem of controing s processes P through P s over a shared communication networ as in Fig.. The processes are controed by the controer C which computes the appropriate contro action and schedue for each process. Each process P is given by a inear time-invariant LTI system x + = A x +B u, x R n, u R m. The controer wors in the foowing way: At time =, sensor S transmits a sampe x to the controer, which then computes the contro signa u and sends it to the actuator A. Here {,,...,s} is the process index. The actuator in turn wi appy this contro signa to the process unti a new vaue is received from the controer. Jointy with deciding u the controer aso decides how many discrete time steps, say I N + {,,...}, it wi wait before it needs to change the contro signa the next time. This vaue I is sent to the Networ Manager which wi schedue the sensor S to send a new sampe at time = +I. To guarantee confict free transmissions on the networ, ony one sensor is aowed to transmit at every time instance. Hence, when deciding the time to wait I, the controer must mae sure that no other sensor S q, q, aready is schedued for transmission at time = +I. We propose that the controer C shoud be impemented as a receding horizon controer which for an individua oop at every samping instant = soves an open-oop optima contro probem. It does so by minimizing the infinite-horizon quadratic cost function x + Q + u + R = subject to the user defined weights Q and R, whie taing system dynamics into account. In Section III we wi embeish this cost function, such that contro performance, inter samping time and overa networ scheduabiity aso are taen into consideration. Remar. The focus is on a networed system where the networ manager and controer are integrated in the same unit. This means, that the controer can send the schedues, that contain the transmission times of the sensors, directy to

3 3 x x +i x +i+p x +i+ p u u +i u +i+p... u +i+ p Fig.. The prediction horizon. Here typica signa predictions are shown with I = 3 and p = 5. the networ manager. This information needs to be transmitted to the sensor node such that it nows when to sampe and transmit. For exampe, using the IEEE superframe, described in [3], this can be done without creating additiona overhead. Here the networ manager broadcasts a beacon in the beginning of each superframe, which occurs at a fixed time interva. This beacon is received by a devices on the networ and incudes a schedue of the sensors that are aowed to transmit at given times. III. ADAPTIVE SAMPLING STRATEGY For pedagogica ease we wi in this section study the case when we contro a singe process on the networ aowing us to drop the oop index. The process we contro has dynamics x + = Ax+Bu, x R n, u R m and the open-oop cost function we propose the controer to minimize at every samping instant is Jx,i,U = α i + x+ Q+ u+ R 3 = where α R + is a design variabe that is used to trade off the cost of samping against the cost of contro and i = I {,,...,p} is the number of discrete time units to wait before taing the next sampe, where p N + is the maximum number of time units to wait unti taing the next sampe. Further, the design variabes < Q and < R are symmetric matrices of appropriate dimensions. We optimize this cost over the constraint that the contro sequence U {u,u +,...} shoud foow the specific shape iustrated in Fig., for some fixed period p. The period p is the maximum amount of time steps the sensor is aowed to wait before taing the next sampe. If p =, a sampe is taen at every time instance and a reguar receding horizon cost is obtained. If one on the other hand woud seect a arge vaue of p, very ong samping intervas can be obtained. This can though affect the contro performance significanty in case there are disturbances in the system that mae the size of the state increase. Thus, the number of discrete time units i = I {,,...,p} to wait before taing the next sampe x +i, as we as the eves in the contro sequence U are free variabes over which we optimize. Note that neither the state nor the contro vaues have a constrained magnitude.... The constraint on the shape of the contro trajectory U is motivated by the idea, that this sequence is appied to the actuator from timeunti time instant+i. At timex+i we tae a new sampe and redo the optimization. By this method we get a joint optimization of the contro signa to be appied as we as the number of time steps to the next samping instant. The reason for etting the system be controed by a contro signa with period p after this is that we hope for the receding horizon agorithm to converge to this samping-rate. We wi provide methods for choosing p so that this happens in Section V. The reason for wanting convergence to a down samped contro is that we want the system to be samped at a sow rate when it has reached steady state, whie we want it to be samped faster during the transients. If one incudes the above mentioned constraints, then 3 can be rewritten as Jx,i,Ui = α i i + x + Q + u R + r= = p = x +i+r p+ Q + u +i+r p R where i and Ui {u,u + i,u + i + η p,...}, η N +, are the decision variabes over which we optimize. The term α/i refects the cost of samping. We use this cost to weight the cost of samping against the cassica quadratic contro performance cost. For a givenx, choosing a arge α wi force i to be arger and hence give onger inter samping times. By the construction of the cost, we may tune Q, R and α via simuations to get the desired samping behaviour whie maintaining the desired contro performance. One coud imagine a more genera cost of samping. Here however we found α/i sufficient to be abe to trade contro performance for communication cost, see simuations in Section VIII-B. IV. COST FUNCTION MINIMIZATION Having defined the open-oop cost 4 we proceed by computing its optima vaue. We start by noticing that, even though we have a joint optimization probem, we may state it as minimize minimize Jx,i,Ui. 5 i Ui We wi use this separation and start by soving the inner probem, that of minimizing Jx, i, Ui for a given vaue of i. In order to derive the soution, and for future reference, we need to define some variabes. Definition. We define notation for the ifted mode as i A i = A i, B i = A q B. q= and notation for the generaized weighting matrices associated 4

4 4 to 4 as Q i = Q i +A i T QA i R i = R i +B i T QB i +R N i = N i +A i T QB i where i {,,...,p}, Q = Q, R = R and N =. Using Definition it is straightforward to show the foowing emma. Lemma. It hods that i x + Q + u R = = x T Q i x+u T R i u+x T N i u and x +i = A i x+b i u. Lemma. Assume that < Q, < R and that the pair A p,b p is controabe. Then min Ui r= p = x +i+r p+ Q + u +i+r p R = x +i P p and the minimizing contro signa characterizing Ui is given by u +i+r p = L p x +i+r p. In the above, 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 A pt P p B p +N p T. Proof. The proof is given in Appendix A. Using the above resuts we may formuate the main resut of this section as foows. Theorem Cosed Form Soution. Assume that < Q, < R and that the pair A p,b p is controabe. Then where 6 min Ui Jx,i,Ui = α i + x P i 7 P i = Q i +A it P p A i A it P p B i +N i L i L i = R i +B it P p B i A it P p B i +N i T and P p is given by Lemma. Denoting the vector of a ones in R n as n, the minimizing contro signa sequence is given by U = { L i x T i, Lp x +i+r p T p,...}, r N 8 where aso L p is given by Lemma. Proof. The proof is given in Appendix B. Now, getting bac to the origina probem 5. Provided that the assumptions of Theorem hod, we may appy it giving that { α } min Jx,i,Ui = min i,ui i i + x P. i Unfortunatey the authors are not aware of any method to sove this probem in genera. If however i is restricted to a nown finite set I N + we may find the optima vaue within this set for a given vaue of x by simpy evauating α i + x P i i I and by this obtaining the i which gives the owest vaue of the cost. This procedure gives the optimum of 5. Note that the computationa compexity of finding the optimum is not necessariy high, as we may compute P i i I off-ine prior to execution. V. SINGLE LOOP SELF-TRIGGERED MPC Having presented the receding horizon cost, we now continue with formuating its impementation in further detai. We wi assume that < Q, < R and that the down-samped pair A p,b p is controabe. Let us aso assume that the finite and non-empty set I N + is given and et γ = maxi. 9 From this we may use Theorem to compute the pairs P i,l i i I and formuate our proposed receding horizon contro agorithm for a singe oop: Agorithm. Singe Loop Sef-Triggered MPC At time = the sampe x is transmitted by the sensor to the controer. Using x the controer computes I α = argmin i I i + x P i u = L I x 3 a The controer sends u to the actuator which appies u = u to unti = +I. b The networ manager is requested to schedue the sensor to transmit a new sampe at time = + I. Remar. Note that the contro signa vaue is sent to the actuator at the same time as the controer requests the scheduing of the next sampe by the sensor. Remar 3. The proposed agorithm guarantees a minimum and a maximum inter-samping time. The minimum time is time step in the time scae of the underying process and the maximum inter samping time is γ time steps. This impies that there is some minimum attention to every oop independent of the predicted evoution of the process. Remar 4. Even though we are woring in uniformy samped discrete time the state is not samped at every time instant. Instead the set of sampes of the state actuay taen is given

5 5 by the set D, which assuming that the first sampe is taen at =, is given by D = {x,xi,xii,...}. VI. ANALYSIS Having estabished and detaied our receding horizon contro aw for a singe oop we continue with giving conditions for when it is stabiizing. Letting λa denote the set of eigenvaues of A we first reca the foowing controabiity conditions. Lemma 3. [34] The system A i,b i is controabe if and ony if the paira,b is controabe andahas no eigenvaue λ λa such that λ and λ i =. Using the above, and the resuts of Section IV, we may now give conditions for when the proposed receding horizon contro agorithm is stabiizing. Theorem Practica stabiity [35]. Assume < Q, < R and that A,B is controabe. If we choose i I N + and p = p given by p = max{i i I, λ λa λ i if λ } and appy Agorithm, then the system state is utimatey bounded as per: α α γ im min i I i + x P α i ǫ p ǫ. γ In the above, γ is as in 9 and ǫ is the argest vaue in the interva,] which i I fufis A i B i L i T P p A i B i L i ǫp i and is guaranteed to exist. Proof. The proof is given in Appendix C. Remar 5. The bound given in Theorem scaes ineary with the choice of α. Assumption. Assume that λ λa except possiby λ = such that λ = and the compex argument λ = π γ n for some n N +. Lemma 4. Let Assumption hod, then p = γ. Proof. The proof is given in Appendix D. Coroary Asymptotic stabiity of the origin. Assume < Q, < R and that A,B is controabe. Further assume that either Assumption hods orα =. If we choosei I N + and p = p given by and appy Agorithm, then im x = Proof. The proof is given in Appendix E. From the above resuts we may note the foowing. Remar 6. If the assumptions of Theorem hod, then Coroary can be used to ensure asymptotic stabiity of the origin, except in the extremey rare case that the underying system A, B becomes uncontroabe under down samping by a factor γ, see Lemma 3. Otherwise one may use Lemma 4 to re-design I giving a new vaue of γ which recovers the case p = γ. Remar 7. It is worth noticing that the deveoped framewor is not imited to just varying the time to the next sampe i as in Fig.. One coud expand this by using a mutistep approach wherein the inter samping time is optimized over severa frames before reverting to constant samping with period p. VII. EXTENSION TO MULTIPLE LOOPS Having detaied the controer for the case with a singe oop on the networ and given conditions for when it is stabiizing we now continue with extending to the case when we contro mutipe oops on the networ, as described in Fig.. The idea is that the controer C now wi run s such singe oop controers described in Agorithm in parae, one for each process P, L = {,,...,s}, controed over the networ. To guarantee confict free communication on the networ the controer C wi, at the same time, centray coordinate the transmissions of the different oops. A. Cost Function We start by extending the resuts in Section III to the case when we have mutipe oops. The cost function we propose the controer to minimize at every samping instant for oop is then J x,i,u i = α i i + x + Q + u R + r= = p x +i+r p + Q = + u +i+r p R derived in the same way as 4 now with α R +, < Q < R and period p N + specific for the contro of process P given by. From this we can state the foowing. Definition. We define the notation in the mutipe oop case foowing Definition. For a matrix E, e.g., A and Q, we denote E i by E i. Theorem 3 Cosed Form Soution. Assume that < Q, < R and that the pair A p,b p is controabe. Then where min J x,i,u i = α + x U i i P i P i = Q i +A i L i = R i +B i T p P A i T p P B i A i i A T p P B i +N i T p P B i +N i L i T

6 6 and P p = Q p +A p A p L p = R p +B p A p T p P A p T p P B p +N p T p P B p T P p B p +N p L p T. Denoting the vector of a ones in R n as n, the minimizing contro signa sequence is given by U = { L i x T i, L p x +i+r p T p,...}, r N. Proof. Appication of Theorem on the individua oops. B. Mutipe Loop Sef-Triggered MPC To formuate our mutipe oop receding horizon contro aw we wi re-use the resuts in Section V and appy them on a per oop basis. For each process P with dynamics given in et the weights α R +, < Q < R and period p N + specific to the process be defined. If we further define the finite set I N+ for oop we may appy Theorem 3 to compute the pairs P i, L i i I. Provided of course that the pair A p,a p is controabe. As discussed in Section II when choosing i the controer must tae into consideration what transmission times other oops have reserved as we as overa networ scheduabiity. Hence at time = oop is restricted to choose i I I where I contains the feasibe vaues of i which gives coisions free scheduing of the networ. Note that at time = the contro input u and time for the next sampe I ony have to be computed for a singe oop. How I shoud be constructed when mutipe oops are present on the networ is discussed further ater in this section. We may now continue with formuating our proposed agorithm for controing mutipe processes over the networ. The foowing agorithm is executed whenever a sampe is received by the controer. Agorithm. Mutipe Loop Sef-Triggered MPC At time = the sampe x of process P is transmitted by the sensor S to the controer C. The controer C constructs I. 3 Using x the controer C computes I = arg min i I u = L I x. α i + x P i 4 a The controer C sends u to the actuator A which appies u = u to unti = +I. b The Networ Manager is requested to schedue the sensor S to transmit a new sampe at time = +I. Note that at time step 3 ony needs to be performed for oop. Remar 8. When the controer is initiaized at time = it is assumed that the controer has nowedge of the state x for a processes P controed over the networ. It wi then execute Agorithm entering at step, in the order of increasing oop index. C. Scheduabiity What remains to be detaied in the mutipe oop receding horizon contro aw is a mechanism for oop to choosei to achieve coision free scheduing. We now continue with giving conditions for when this hods. First we note that when using Theorem 3 we mae the impicit assumption that it is possibe to appy the corresponding optima contro signa sequence U. For this to be possibe we must be abe to measure the state x at the future time instances S = { +I, +I +p, +I +p, +I +3p,...}. Hence this samping pattern must be reserved for use by sensor S. We state the foowing to give conditions for when this is possibe. Lemma 5. Let oop choose its set I of feasibe times to wait unti the next sampe to be I = {i I i next q +n p q m p, m,n N, q L\{}} where q next is the next transmission time of sensor S q. Then it is possibe to reserve the needed samping pattern S in at time =. Proof. The proof is given in Appendix F. Constructing I as above we are not guaranteed that I. To guarantee this we mae the foowing assumption. Assumption. Assume that for every oop on the networ I = I and p = p. Further assume that L I and maxl p. Theorem 4. Let Assumption hod. If every oop chooses I = {i I i next q +r p, r Z, q L\{}} a transmissions on the networ wi be confict free and it wi aways be possibe to reserve the needed samping pattern S in. Proof. The proof is given in Appendix G. Remar 9. The resut in Lemma 5 requires the reservation of an infinite sequence. This is no onger required in Theorem 4 as a oops cooperate when choosing the set of feasibe times to wait. In fact oop ony needs to now the current time, the period p and the times when the other oops wi transmit next q next q L\{} in order to find its own vaue I. Remar. If Assumption hods and every oop on the networ chooses I according to Theorem 4, then it is

7 7 guaranteed that at time we can reserve and that no other oop can mae conficting reservations. Hence at time +I the sequence S +I = { +I +p, +I +p, +I +3p,...}. is guaranteed to be avaiabe. Thus p I +I. TABLE I THE PRE-COMPUTED CONTROL LAWS WITH RELATED COST FUNCTIONS AND INTERMEDIATE VARIABLES. i A i B i Q i R i N i L i P i V i x.7.7 α/+p x α/+p x α/3+p 3 x α/4+p 4 x 5=p α/5+p 5 x D. Stabiity We continue with giving conditions for when the mutipe oop receding horizon contro aw described in Agorithm is stabiizing. Extending the theory deveoped Section V to the mutipe oop case we may state the foowing. Theorem 5. Assume < Q, < R and that A,B is controabe. Further et Assumption hod. If we then choose i I I N +, with I chosen as in Theorem 4, and p = p given by p = max{i i I, λ λa λ i if λ } 3 and appy Agorithm, then as α γ min α i I i + x α P i ǫ p ǫ γ where γ = maxi and ǫ is the argest vaue in the interva,] which i I fufis i A B i Li TP p Proof. The proof is given in Appendix H. i A B i Li ǫ P i. Coroary. Assume < Q, < R and that A,B is controabe. Further et Assumption hod. In addition et I be chosen so that the resuting γ = maxi guarantees that Assumption hods for every oopor aternativey et α = for every oop. If we then choose i I I N +, with I chosen as in Theorem 4, and p = p given by 3 and appy Agorithm it hods that im x =. Proof. The proof foows from the resuts in Theorem 5 anaogous to the proof of Coroary. VIII. SIMULATION STUDIES To iustrate the proposed theory we now continue with giving simuations. First we show how the contro aw wors when a singe oop is controed over the networ and focus on the oop specific mechanisms of the controer. Secondy we iustrate how the controer wors when mutipe oops are present on the networ and focus on how the controer aocates networ access to different oops. A. Singe Loop Let us exempify and discuss how the controer handes the contro performance versus communication rate trade-off in an individua oop. We do this by studying the case with a singe system on the networ. The system we study is the singe integrator system which we discretize using sampe and hod with samping time T s = s giving us x+ = Ax+Bu with A, B =,. Since we want the resuting seftriggered MPC described in Agorithm to be stabiizing we need to mae sure that our design fufis the conditions of Theorem. If we further want it to be asymptoticay stabiizing we in addition need it to fufi the conditions of Coroary. The design procedure is then as foows: First we note that the systema,b is controabe. The next step is to decide the weights < Q and < R in the quadratic cost function 3. This is done in the same way as in cassica inear quadratic contro, see e.g. [36]. Here we for simpicity choose Q = and R =. We note that the system ony has the eigenvaue λ =, fufiing Assumption, so that in Theorem gives p = maxi. Hence Coroary, and thus Theorem, wi hod for every choice of I. This means that we may choose the eements in I, i.e., the possibe down samping rates, freey. A natura way to choose them is to decide on a maximum aowed down samping rate and then choose I to contain a rates from up to this number. Let s say that we here want the system to be samped at east every 5 T s s, then a good choice isi = {,,3,4,5}, givingp = maxi = 5. Now having guaranteed that the conditions of Theorem and Coroary hod we have aso guaranteed that the conditions of Theorem are fufied. Hence we may use it to compute the state feedbac gains and cost function matrices that are used in Agorithm. The resuts from these computations are shown in Tabe I, together with some of the intermediate variabes from Definition. Here we see that the cost functions are quadratic functions in the state x where the coefficients P i are functions of Q and R. We aso see that the cost to sampe α/i enters ineary and as we change it we wi change the offset eve of the curves and thereby their vaues reated to each other. However it wi not affect the state feedbac gains. A graphica iustration of the cost functions in Tabe I, for the choice α =., is shown in Fig. 3 together with the curve I = argmin I V i x, i.e., the index of the cost function which has the owest vaue for a given state x. This is the partitioning of the state space that the sef-triggered MPC controer wi use to choose which of the state feedbac

8 8 9 8 V V 7 6 V 3 V 4 V 5 x x Fig. 3. The cost functions V i x dashed together with the partitioning of the state space and the time to wait I = argmin I V i x soid. gains to appy and how ong to wait before samping again. Appying our sef-triggered MPC described in Agorithm using the resuts in Tabe I to our integrator system when initiaized in x = we get the response shown in Fig. 4a. Note here that the system wi converge to the fixed samping rate p as the state converges. It may now appear as it is sufficient to use periodic contro and sampe the system every p T s s to get good contro performance. To compare the performance of this periodic samping strategy with the sef-triggered strategy above we appy the contro which minimizes the same cost function 3 as above with the exception that the system now may ony be samped every p T s s. This is in fact the same as using the receding horizon contro above whie restricting the controer to choose i = p every time. The resuting simuations are shown in Fig. 4b. As seen, there is a arge degradation of the performance in the transient whie the stationary behaviour is amost the same. By this we can concude that it is not sufficient to sampe the system every p T s s if we want to achieve the same transient performance as with the seftriggered samping. In the initia transient response the sef-triggered MPC controer samped after one time instant. This indicates that there is performance to gain by samping every time instant. To investigate this we appy the contro which minimizes the same cost function 3, now with the exception that the system may be samped everyt s s, i.e., cassica unconstrained inear quadratic contro. Now simuating the system we get the response shown in Fig. 4c. As expected we get sighty better transient performance in this case compared to our sef-triggered samping scheme, it is however comparabe. Note however that this improvement comes at the cost of a drasticay increased communication need, which may not be suitabe for systems where mutipe oops share the same wireess medium. From the above we may concude that our sef-triggered MPC combines the ow communication rate in stationarity of the sow periodic controer with the quic transient response of the fast periodic samping. In fact we may, using our u a System response of the integrator system when minimizing the cost by using our singe oop sef-triggered MPC x u b System response of the integrator system when minimizing the cost by samping every 5 th second x u c System response of the integrator system when minimizing the cost by samping every second. Fig. 4. Comparing contro performance for different samping poicies.

9 9 5 Average samping interva.5 Sef-triggered Periodic Average empiric cost α = 6.5 α = α = α =.5 α =.5 α =.3 α = 5 α = 5 α = Fig. 5. The average samping interva for different vaues of α for the integrator system. α 5 5 Average samping interva Fig. 6. Performance of the integrator using the sef-triggered agorithm compared to the simpe periodic agorithm for different samping intervas. Some vaues of α are mared with the bac crosses. method, recover the transient behaviour of fast periodic samping at the communication cost of one extra sampe compared to sow periodic samping. The reason for this is that the fast samping rate ony is needed in the transient whie we in stationarity can obtain sufficient performance with a ower rate. B. Comparison to periodic contro To mae a more extensive comparison between the seftriggered agorithm and periodic samping, we sweep the vaue of the design parameter α in the cost function, and run simuations of sampes for each vaue. To add randomness to the process, the mode is extended to incude disturbances such that for process x + = A x +B u +E ω, 4 where ω N,σ is zero-mean norma distributed with variance σ =. and E =. We increase the maximum aowed samping interva to p = maxi = 5. This though has the effect, that P =.83 whereas P =.74, such that it is aways more favourabe to have a sampe interva of sampes instead of, even when α =. We therefore changed the cost of contro to R =. such that we aow for arger contro signas and force the process to sampe at every time instance if α = is chosen. The initia state is randomy generated as x N,σx, where, σ x = 5., Fig. 5 shows the average samping interva during the simuations for different choices of α for the integrator system. The reation between the vaue of α and the average samping interva is in genera monotonic but highy affected by the system to be controed and the statistics of the disturbances in the system. Since we now have the range of vaues in which we want to sweep α, we present a simpe periodic MPC cost to which we compare our agorithm. This cost is given by J x,u = T s x ++rt s Q r= = + u ++rt s R, which using Definition and Lemmas and can be rewritten as J x,u = r= for samping period T s where x +T s = A Ts x +rt s + u Q Ts +rt s R Ts +x +rt s T N i u +rt s 5 x +B Ts u. When T s =, 5 reduces to cassica unconstrained inear quadratic contro. The empiric cost for each simuation is cacuated by T x T Q x +u T R u, 6 T = where T = is the ength of the simuation. Fig. 6 shows the average performance that is obtained for different averaged samping intervas obtained by sweeping α from to 6. Fig. 6 iustrates that the sef-triggered agorithm performs significanty better than periodic samping especiay when communication costs are medium to high. However, at samping intervas of and 3 time-steps the sef-triggered agorithm performs sighty worse than the periodic agorithm. This is caused by the fact, that if the state of the system is cose to zero when samped, the cost of samping is much higher than the cost of the state error and contro, hence the time unti the next sampe is taen is arge. To avoid this phenomenon, in future wor, one coud tae the statistics of the process noise into account in the cost function. It can further be noticed, that when α, the performance of the sef-triggered agorithm is very cose to the cost of the periodic agorithm. This is as expected, since the cost of α wi be greater than the cost of the state and contro, which wi resut in sampe intervas of i = p = 5. This reduces the sef-triggered agorithm to periodic contro. C. Mutipe Loops We now continue with performing a simuation study where we contro two systems over the same networ. We wi start by showing a simpe exampe foowed by a more extensive performance comparison of the Agorithm to periodic contro. We wi eep the integrator system from

10 Section VIII-A now denoting it process P with dynamics x + = A x + B u with A,B =, as before. In addition we wi the contro process P which is a doube integrator system which we discretize using sampe and hod with samping time T s = s giving x + x } {{ + = } x + x x }{{}}{{ +.5 }}{{} A x B u. We wish to contro these processes using our proposed mutipe oop sef-triggered MPC described in Agorithm. As we wish to stabiize these systems we start by checing the conditions of Theorem 5 and Coroary. First we may easiy verify that both the pairs A,B and A,B are controabe. To use the stabiity resuts we need Assumption to hod, impying that we must choose p = p = p, I = I = I and choosei such that{,} I and p. For reasons of performance we wish to guarantee that the systems are samped at east every5 T s s and therefore choose I = I = I = {,,3,4,5} fufiing the requirement above. We aso note that λa = {} and λa = {,} and that hence both system fufi Assumption for this choice of I, impying that 3 in Theorem 5 gives p = maxi = 5. Thus choosing p = p as stated in Theorem 5 resuts in that Assumption hods. What now remains to be decided are the weights α, Q and R. For the integrator process P we eep the same tuning as in Section VIII-A with Q = R =. Having decided Q, R, I and p we use Theorem 3 to compute the needed state feedbac gains and cost function matricesp i,li i I needed by Agorithm. We aso eep α =. as it gave a good communication versus performance trade-off. For the doube integrator process P the weights are chosen to be Q = I as we consider both states equay important and R = to favor contro performance and aow for arger contro signas. Having decided Q, R, I and p we may use Theorem 3 to compute the needed state feedbac gains and cost function matrices P i,li i I needed by Agorithm. The samping cost is chosen to be α =, as this gives a good tradeoff between contro performance and the number of sampes. We have now fufied a the assumptions of both Theorem 5 and Coroary. Hence appying Agorithm choosing I according to Theorem 4 wi asymptoticay stabiize both process P and P. ControingP andp using our mutipe oop sef-triggered MPC described in Agorithm with the above designed tuning we get the resut shown in Fig. 7. As expected, the behavior of the controer iustrated in Section VIII-A carries through aso to the case when we have mutipe oops on the networ. In fact comparing Fig. 4a showing how the controer handes process P when controing it by itsef on the networ and Fig. 7a which shows how P is handed in the mutipe oop case we see that they are the same. Further we see that, as expected, in stationarity the two oops controing process P and P both converge to the samping rate p. As mentioned previousy the controer uses the mechanism in Theorem 4 to choose the set of feasibe times to wait x u a System response for process P x u b System response for process P Fig. 7. The processes P and P controed and schedued on the same networ using our mutipe oop sef-triggered MPC. unti the next sampe. In Fig. 8 we can see how the resuting sets I oo in detai. At time = oop gets to run Agorithm first. As sensor S is not schedued for any transmissions yet I = I from which the controer choosesi =. Then oopgets to run Agorithm at time =. As sensor S now is schedued for transmission at time = +I =, Theorem 4 gives I = I \{}. From which the controer chooses I =. The process is then repeated every time a sampe is transmitted to the controer, giving the resut in Fig. 8. As seen both the set I and the optima time to wait I converges to some fixed vaue as the state of the corresponding process P converges to zero. D. Comparison to periodic contro For a more thorough performance comparison, we simuate the systems using Agorithm and compare them to the periodic agorithm that uses the cost function 5. The singe and doube integrator processes P and P are simuated using the parameters mentioned in Section VIII. The variance of the disturbances for both processes are set to σ =.,, the variance of the initia state to σx = 5, ande, = [, ] T in 4. The vaue of α is identica for both processes in

11 5 4 3 I I I I Normaized average empiric cost Sef-triggered P Sef-triggered P Periodic P Periodic P 5 5 Average samping interva Fig.. Performance for both processes for the sef-triggered agorithm compared to a simpe periodic agorithm for different samping intervas. The max samping interva p = 5. This is done by varying α from to 9 and cacuating the average amount of sampes for every vaue of α. Fig. 8. The sets I of feasibe times to wait unti the next sampe for oop and oop. The optima time to wait I is mared by red circes whereas the other feasibe times are starred Normaized average empiric cost Sef-triggered P Sef-triggered P Periodic P Periodic P Average samping interva Fig. 9. Performance for both processes for the sef-triggered agorithm compared to a simpe periodic agorithm for different samping intervas. The max samping interva p = 5. This is done by varying α from to 5 4 and cacuating the average amount of sampes for every vaue of α. each simuation, such that α = α. Further R =.. The simuation for both agorithms is initiaized as described in Remar 8. Fig 9 shows the performance for the processes, P and P, cacuated by averaging 6 over simuations each of ength for different vaues of α when p = p = 5. Fig. shows the performance when p = p = 5. The cost of each process is normaized with respect to its argest cost for easier viewing. Both figures show that the sef-triggered agorithm in genera outperforms periodic samping. The performance margin increases as the samping interva increases. Fig. 9 shows that P when α = sampes every.8 time steps on average, whereas P ony sampes every. time steps. When α increases both processes sampe amost every time steps. The reason for this is that the cost of the state for the down samped systems in some cases is ower. The worse performance of the sef-triggered agorithm at ower samping intervas is more significant in Fig. where the performance of process P shows a simiar performance as in Fig. 6. The owest average samping interva for process P is 3.6 time steps when α =. The sef-triggered agorithm though significanty outperforms the periodic agorithm when the average samping interva increases. As α the cost of samping forces the sef-triggered agorithm to behave simiar to the periodic samped agorithm. Therefore the performance gap between the periodic and seftriggered agorithms narrows, as the average samping interva is cose to p. IX. CONCLUSIONS We have studied joint design of contro and adaptive scheduing of mutipe oops, and have presented a method which at every samping instant computes the optima contro signa to be appied as we as the optima time to wait before taing the next sampe. It is shown that this contro aw may be reaized using MPC and computed expicity. The controer is aso shown to be stabiizing under mid assumptions. Simuation resuts show that the use of the presented contro aw in most cases may hep reducing the required amount of communication without amost any oss of performance compared to fast periodic samping. In the mutipe oop case we have aso presented an agorithm for guaranteeing confict free transmissions. It is shown that under mid assumptions there aways exists a feasibe schedue for the networ. The compexity of the mutipe oop sef-triggered MPC and the corresponding scheduing agorithm scaes ineary in the number of oops. An interesting topic for future research is to further investigate the compexity and possibe performance increase for such an extended formuation. Intuitivey, additiona performance gains can be achieved when the cost function taes process noise into account as we. Expoiting this coud be of interest in future research. Another topic for future research woud be to appy the presented framewor to constrained contro probems. REFERENCES [] A. Wiig, Recent and emerging topics in wireess industria communications: A seection, IEEE Transactions on Industria Informatics, vo. 4, no., pp. 4, May 8. [] HART Communication Foundation, WireessHART, HART Communication Foundation, 3, [3] I. S. Association, IEEE standard for oca and metropoitan area networs - part 5.4: Networs ow-rate wireess LR-WPANs, IEEE Standards Association,,

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