On Event-Triggered and Self-Triggered Control over Sensor/Actuator Networks

Transcription

1 Proceeings of the th IEEE Conference on Decision an Control Cancun, Mexico, Dec. 9-11, 8 TuA1. On Event-Triggere an Self-Triggere Control over Sensor/Actuator Networks Manuel Mazo Jr. an Paulo Tabuaa Abstract Event-triggere an self-triggere control have been recently propose as an alternative to the more traitional perioic execution of control tasks. The possibility of reucing the number of executions while guaranteeing esire levels of performance makes event-triggere an self-triggere control very appealing in the context of sensor/actuator networks. In this setting, reucing the number of times that a feeback control law is execute implies a reuction in transmissions an thus a reuction in energy expenitures. In this paper we introuce two novel istribute implementations of event-triggere an self-triggere policies over sensor/actuator networks an iscuss their performance in terms of energy expeniture. I. INTRODUCTION Sensor networks research has extensively ealt with the extraction of information from the physical worl. Many of the evelope applications concentrate on how to obtain this information for posterior off-line analysis [1], [] or on-line processing of this information for tracking [], [], istribute optimization [5] or mapping []. In all of these applications there is a common esire for small power consumption, which woul exten the life span of the network. Many of the approaches use to reuce the power consumption concentrate on the communication requirements, as the works on compressive sensing [], network throughput optimization [8], [9], message-passing algorithms [1] or sleep-scheuling of the noes [11]. Still, most of these stuies are performe only for on- or off-line analysis on the information gathere. We aress the problem of minimizing energy consumption for applications in which actuation plays a major role, namely control applications. In this context, energy expenitures can be irectly relate to the frequency at which measurements are being taken an transmitte through the network. Although the choice of the sampling frequency is a problem as ol as the use of microprocessors for control, this question never receive a efinitive answer an engineers still rely on rules of thumb [1], [1], [1] such as sampling with a frequency times the system banwith. Moreover, the choice of perioic sampling is not even a natural one since the behavior of a control system can be quite ifferent in ifferent regions of the state space. In this paper, we show how to minimize energy consumption by resorting to event-triggere an self-triggere This work has been partially fune by the Spanish Ministry of Science an Eucation/UCLA fellowship LA- an by the NSF CSR-EHS 15 grant. M. Mazo Jr. an P. Tabuaa are with the Department of Electrical Engineering, University of California at Los Angeles, Los Angeles, CA , USA mmazo,tabuaa@ee.ucla.eu. sampling strategies over sensor/actuator networks (SAN). The event-triggere iea has been previously explore in the literature [15]. It has been shown in [1] that by sampling an computing the controller only when a certain threshol conition on the state is violate, one can sample less frequently than with a perioic scheme while guaranteeing the same performance. In [1] an [18] the self-triggere sampling paraigm was explore, for linear an non-linear systems respectively, in an attempt to eliminate the requirement of continuous state measurement. Self-triggere strategies compute the next sampling time using the last state measurement an thus o not require knowlege of the current state. The contribution of this paper is to show how the techniques introuce in [1], an later refine in [19], [1] for H controllers, can be implemente over sensor-actuator networks to consierably reuce the number of network transmissions. We propose an event-triggere strategy in which each noe uses its local information to etermine when to make a transmission an a self-triggere strategy in which the actuator noe etermines for how long shoul the sensing noes sleep before collecting an transmitting fresh measurements. The paper is organize in the following way: Section II introuces the notation an states the problem. Then we procee to escribe the propose ecentralize self-triggere algorithm in Section III. Next we propose a istribute eventtriggere protocol in Section IV, followe by Section V where we formalize how to obtain the times between upates in the previously escribe algorithms. The propose techniques are illustrate with two examples in Section VI. The paper conclues with a brief iscussion in Section VII. II. NOTATION AND PROBLEM STATEMENT We consier a linear control system: ẋ = Ax + Bu, x R n, u R m (1) stabilize by a linear controller: u = Kx () In what follows, we assume that K was obtaine through a LQR esign an consier its implementation over a sensor/actuator network. Therefore, K is given by K = B T P where P is the solution of the matrix Riccati equation: A T P + P A Q + L = () with Q = P BB T P an for some L an P = P T >. Furthermore, we assume m=1, for simplicity of presentation, an also the existence of a istinguishe noe collocate /8/$5. 8 IEEE 5

2 th IEEE CDC, Cancun, Mexico, Dec. 9-11, 8 TuA1. with the actuator calle the actuator noe. The remaining noes are sensor noes an further escribe below. The control signal is obtaine from zero-orer sample an hol measurements so that: u(t) = Kx(t k ), t [t k, t k+1 ) where t k an t k+1 are two consecutive sampling instants. For the sake of reaability, we rop the explicit epenence of t but enote with the subinex k those quantities that are constant an equal to the value at the upate time t k, i.e. x k = x(t k ). With this kin of sample feeback the close loop is escribe by: ẋ = Ax + BKx k = (A + BK)x + BKe where e represents a measurement error efine by: t [t k, t k+1 ) = e(t) = x(t k ) x(t) Treating the measurement error e as a new variable we can rewrite the close loop ynamics as: [ ] [ ] [ ] ẋ A + BK BK x = ė A BK BK e since ė = ẋ. Moreover, the matrix P, obtaine as the solution of (), efines a quaratic Lyapunov function V = x T P x whose erivative along () satisfies: V = x T Lx + e T Qe u T k u k The esire performance for the close loop system can be escribe by a matrix S > for which the inequality: V x T Sx () hols. The matrix S woul then efine the esire rate of ecay for V. In orer to enforce () we nee to select the sampling instants t k so that x T Lx + e T Qe u T k u k x T Sx. This can be achieve by sampling when (5) hols. x T (L S)x + e T Qe u T k u k = (5) At t = t k the execution of the controller reners x k = x an e = x k x =, forcing x T k (Q+L S)x k. Now we can appreciate how (Q + L S) captures the trae-off between sampling an performance. In a typical esign one woul pick L to esign the continuous optimal controller, Q will be etermine by the LQR esign, an we will get to pick again S. The closer S an L are, the better the performance of the system will be (closer to the prescribe continuous esign) at the expense of more frequent sampling. In a practical setting the measurements x k will actually be receive after a certain elay which shoul be incorporate into the efinition of e. For the sake of simplicity we ignore this elay by assuming = although we can incorporate the elay in the propose framework by appropriately selecting S in the rule () by following the approach escribe in [1]. We are intereste in the use of event-triggere an selftriggere control techniques as they require fewer measurements than perioic techniques. In particular, in the context of SAN s, this also means that less transmissions are require for control an thus less energy usage. We will assume without loss of generality that each sensor measures one entry of the full state of the system. With all of the above in min, the problem we solve in this paper is: Problem.1: Given a Linear Time Invariant system as in (1), a controller () renering the close-loop system asymptotically stable, esign a istribute algorithm enforcing the controller upate rule () over a SAN. We can now see that the funamental ifficulty when implementing an event-triggere or self-triggere strategy in a istribute-sensing setting is that the triggering conition () has to be evaluate continuously an it epens on full state information. A continuous centralization of local measurements to check () woul lea to extensive communication an unacceptably high energy consumption, thus justifying a istribute esign. In aition to a reuction in energy consumption, we will also require the istribute algorithms to be energy-balance, i.e. all the sensing noes shoul consume energy at the same pace, since as soon as one sensing noe exhausts its energy reserves the stability an performance of the close loop system becomes seriously compromise. In what follows will enote the usual Eucliean norm of a vector an σ m,m ( ) the minimum (m) or maximum (M) singular value of a given matrix. III. A SELF-TRIGGERED PROTOCOL The first istribute implementation follows the selftriggere paraigm for control [], [1], [18]. Accoring to this paraigm, in aition to use the current state to compute the controller, the current state is also use to compute the next execution time of the controller. In this section we escribe a particular protocol to implement this paraigm over a sensor/actuator network. A. Algorithm escription We start by assuming the availability of a routing tree with root at the actuation noe. The construction of a routing tree is typically one uring the iscovery phase of the network by resorting to several efficient algorithms available in the literature [1]. We also assume that the actuator noe has access to larger energy reserves since it rives one or several actuators that are typically more power consuming. Therefore, we assume that the actuation noe has the ability to broacast messages to all noes without seriously reucing its energy reserves. The propose ecentralize algorithm for self-triggere control consists of two phases: A measurement collection phase in which the network computes in a istribute manner, as explaine later in this section, two scalars: u k = Kx k an x k. The actuator noe uses this information to upate the actuator an to compute the next time instant at which new samples shoul be collecte an transmitte. A broacast phase uring which the actuator noe broacasts the next upate time to all the sensing noes.

3 th IEEE CDC, Cancun, Mexico, Dec. 9-11, 8 TuA1. In practice, in orer to avoi synchronization issues one woul rather sen t k+1 t k instea of t k+1. The sensing noes woul then sleep for t k+1 t k units of time before starting the next measurement collection phase. These t k+1 t k times will be compute in Section V. In any implementation of a control system the control signal u has to be elivere to the controller. In aition to u k we will also nee to transmit aitional information in orer to preict the sampling times. For the techniques that we will present x k is enough; nevertheless, more information will help to increase the time between upates. The istribute computation of u k = Kx k an x k = n 1 x i (t k) is base on the fact that both quantities are sums of quantities locally available to the noes of the network. The computation of these quantities can then be one by a wave algorithm for trees (Tree Wave-Algorithm) [1] in which each noe collects its local quantity (K i x i k an (xi k ) respectively), as it to the quantities that it receives from its chilren an transmits the resulting sum to its parent. Having the root at the actuation noe ensures that the computation ens at the actuation noe. The algorithm also ensures that each noe transmits packets of the same size. The number of transmissions are also limite to one per noe, ue to the Tree Wave-Algorithm implementation. Briefly, the Tree Wave-Algorithm limits the communication by passing to the parent noe a token or signal only when the noe has receive one from each of the links to its chilren, an it has locally generate its own token. In this way when the root receives a token, the root noe knows that all noes in the network generate locally a token. If the maximum elay introuce by the measuring collection phase can be compute, it can be regare as a computation elay an accommoate by moifying the triggering rule as in [1]. B. Energy utilization In orer to etermine the energy consume by the propose protocol, we nee to take into account the power consume in transmitting ata, in listening for transmissions, an in computing. In the case of a self-triggere strategy the noes only perform the computations neee in the wave algorithm computing u k an x k. We will enote this computing cost by D 1. It is a fixe unit of energy consume per upate. The raio moule can also be kept asleep most of the time as there is no nee for communication between upates, therefore the listening cost coul be neglecte. As for the power consume in transmitting information, the noes on the self-triggere protocol only sen two scalars per upate. Thus, we represent this cost by two parameters: C 1 an C. The first parameter escribes the cost associate to the packet overhea transmission (inepenent of the payloa) while the secon parameter escribes the cost per transmitte scalar. Therefore, in the self-triggere case there woul be an energy cost of C 1 + C, an the total energy consume at a noe at any given point in time will be given by: J 1 (t) = (C 1 + C + D 1 ) () k Γ(t) where Γ(t) = {k : t k t}. IV. AN EVENT-TRIGGERED PROTOCOL A ifferent strategy consists in checking in a istribute manner the conition impose on the ecay of the Lyapunov equation. This leas to a istribute computation of the controller upate times. Our goal is to reuce the consumption at the noes of the network, on which communication has the biggest impact. Therefore we cannot exchange information frequently between noes. Instea we propose to have the noes estimate in a conservative way when the triggering conition stops to hol. Each of the noes will be making use of its local measurements in orer to perform this estimation. When all the noes agree in that the triggering conition has been violate a new controller upate is force. A. Algorithm escription Once more we will nee to assume the existence of a routing tree with root at the actuation noe visiting all the noes in the network. The event-triggere algorithm can be ivie in several phases: Phase 1: The actuation noe broacasts a request for new measurements. Phase : The sensing noes take their local measurement an compute their part of u k an x k an forwar them up the routing tree (towars the root). Phase : The actuation noe finishes the computation of u k an x k an broacasts those quantities to the rest of the noes. Phase : The sensing noes use u k an x k to upate their local estimators accoringly. When a noe s local triggering conition stops to hol it will generate a local token, starting the computation of the Tree Wave- Algorithm, explaine in a previous section. When the actuation noe receives tokens from all of its chilren noes it will broacast a request for new measurements an the network woul be back in Phase 1. The local triggering (generation of these tokens) will be explaine in etail in Section V. As the measurements from the noes are synchronize, any elay introuce in phases an can be regare as a computation elay an again ealt with moifying the triggering rule as in [1]. Note also that the performance of neither the self-triggere nor the event-triggere protocol is affecte by the topology of the network if no elays are taken into account. However, the topology will have an effect on the maximum elay present in a practical implementation, an thus on its performance. B. Energy utilization The event-triggere protocol requires the noes to continuously compute estimates, an listen to possible communications from their chilren. Therefore, we have a constant

4 th IEEE CDC, Cancun, Mexico, Dec. 9-11, 8 TuA1. power consumption ue to listening an computation, which we will enote by l an respectively. In the event-triggere protocol there are two transmissions per noe per upate. The first one is use just to ecie the next upate time an contains no payloa, therefore incurring on a cost of C 1. The secon transmission is use to compute the control signal u k an x k, therefore having two scalars as payloa an incurring on a cost of C 1 +C. Finally the total energy consumption at any noe at any point in time will be given by the following equation: J (t) = (C 1 + C ) + (l + ) t () k Γ(t) where once more Γ(t) = {k : t k t}. V. HOW TO COMPUTE SLEEPING TIMES We will nee to relax conition () in orer to make it possible to check in a ecentralize way. In this section we will present these relaxations an how to take avantage of them to perform a istribute or ecentralize computation of the times between upates of the controller. A. Self-triggere control One of these possible relaxations provies the triggering conition: e T (Q + L S)e 1 σ m(l S) x k + u T k u k (8) which implies the inequalities: e T (Q + L S)e 1 xt k (L S)x k + u T k u k V x T Sx with α = M 1 AM 1 (1) β = min{β 1, β } (11) β 1 = M 1 (A + BK) xk ( (1) β = M 1 (A + BK) x k + (1) M 1 Buk + u T k B T M 1 A xk Our boun estimator woul therefore be ɛ = αɛ+β, which by the Comparison Lemma [], woul provie with ɛ(t) e(t) M. Integrating ɛ an taking into account ɛ(t k ) = leas to: ɛ(t) = β ( ) e α(t tk) 1 (1) α Therefore the next upate time will be given by: ) 1 ɛ (t k+1 ) = σ m(l S) x k + u T k u k (15) which can be solve analytically: t k+1 = t k + 1 ( α log 1 + ακ ) k (1) β σm (L S) κ k = x k + u T k u k (1) Let us summarize these results in the following proposition: Proposition 5.1: The istribute implementation of a LQR controller (), esigne to stabilize the linear system (1), efine by the algorithm escribe in Section III with upate times given by (1) guarantees stability of the close loop system an enforces V x T Sx. Moreover, the algorithm is energy-balance. also hol, where we use: x T (L S)x 1 xt k (L S)x k e T (L S)e The right-han-sie of equation (8) is a linear combination of the constants u k an x k obtaine from a istribute computation involving only the last measurements taken by the sensing noes. In orer to check the triggering conition on-line we will nee to estimate an upper boun for the evolution of e T (Q + L S)e. Provie that L an S where esigne so that (Q + L S) > we can fin a ynamic estimator ɛ satisfying e(t) M ɛ(t) an bouning above e M = M 1 e = Q + L Se : t e M ė M M 1 AM 1 e M + (A+BK)x k M Which now can be moifie to be use as a boun estimator epening only on quantities computable over the network: t e M α e M + β(u k, x k ) (9) 8 B. Event-triggere control A ifferent alternative consists of esigning estimators in which local measurements can be irectly injecte. In this way the triggering conitions can be locally checke. In orer to be able to obtain such estimators we also nee to relax the triggering conition () in the local noes. σ M (Q) e ( σ m (L S) x + u T ) k u k (18) V x T Sx Now, as in the esign of the self-triggere estimator, we can obtain estimators with irect injection of local measurements for e an x. Let us first partition an reorer the state-space as e i = [ e i, ê i] T, with ei enoting the local error variable at noe i, an ê i the remaining entries of the error vector. With this new escription of the error vector at noe i, we can rewrite the ynamics for e as t [ ei ê i ] = [ A1 A ] [ ei ê i ] A A [ ] [ A1 A xi (t k ) A A ˆx i (t k ) ] [ ] B1 u B k

5 th IEEE CDC, Cancun, Mexico, Dec. 9-11, 8 TuA1. Then, t ê i A e i + A ê i B u k [A A ]xk i A ê i + ( A e i B u k + [A A ] x k + (A e i B u k ) T [A A ] x k ) 1 an hence we can use as boun estimator: e T e e i + (ε i ) (19) t εi = A ε i + ` A e i B u k + [A A ] x k + (A e i B u k ) T [A A ] x k 1 () In an analogous way we can erive a lower boun estimator for x resulting in: x T x x i + (χ i ) (1) t χi = A χ i A x i + B u k () Therefore, local triggering conitions of the form σ M (Q) e i + (ε i ) σ m(l S) x i + (χ i ) + u T k u k () coul be use to ecie the upate times. The istribute algorithm propose in section IV-A will enforce a new upate only when all noes have their local rules violate. Therefore, the istribute algorithm will ensure that the controller follows the noe with least conservative estimates, an thus the system waits as long as possible before triggering a new upate. Now we can summarize the event-triggere protocol in the following proposition: Proposition 5.: The istribute implementation of a LQR controller (), esigne to stabilize the linear system (1), efine by the algorithm escribe in Section IV with upate times given by (),(), an () guarantees stability of the close loop system an enforces V x T Sx. Moreover, the algorithm is energy-balance. VI. EXAMPLE We illustrate these techniques in two small examples. Let us start with the system efine by A = , B = 1 1 5, xo = 1 The controller was esigne solving the Riccati equation (), with L =.5I an setting K = B T P. The S matrix was set to S =.I. For comparison we inclue plots of simulations using the centralize event-triggere rule e T Qe u T k u k x T (L S)x, which oes not involve any approximation or relaxation from (). This protocol woul be giving us the exact times when the controller nees to be upate. Obviously our two protocols for networks will give us more upates, but not necessarily always smaller times. In figure 1 we can see how the three strategies achieve the esire performance as measure by the ecay of the 5 x T Px Centralize Event Triggere Distribute Event Triggere Distribute Self Triggere Fig. 1. Lyapunov function ecay for centralize event-triggere, istribute event-triggere an ecentralize self-triggere. First example. t k+1 t k [s] Centralize Event Triggere Distribute Event Triggere Decentralize Self Triggere Fig.. Inter-sample times for centralize event-triggere, istribute eventtriggere an ecentralize self-triggere strategies for the first example. Lyapunov function. But looking at figure, while the times obtaine with the self-triggere strategy are much shorter than the centralize event-triggere controller, the istribute event-triggere strategy prouces times much closer to those from the centralize controller. This effect is explaine by the fact that in an event-triggere protocol measurements from the plant at a local level are taken continuously, while the self-triggere protocol is completely open loop between upates. Finally we present a comparison of the energy consumption evolution at a noe (an therefore at all of them as our algorithms are balance). The values for the ifferent costs use in the energy computation were: C 1 = 8.mJ, C =.mj, l = mw, = mw an D 1 = mj negligible. These values are approximations obtaine from the actual power consumption values for a MicaZ [] using ZigBee [] for wireless communication. Figure epicts the energy consumption using the self-triggere an the eventtriggere protocols, as given by () an (). Observe how the istribute event-triggere algorithm consumes less energy than the self-triggere. In this example the event-triggere strategy becomes more energy-efficient because of the great ifference on number of upates between event an selftriggere. But in general that oes not have to be the case. Even if the event-triggere algorithm prouces fewer upates than the self-triggere, the first algorithm coul consume more energy. The next example tries to portray this situation. 9

6 th IEEE CDC, Cancun, Mexico, Dec. 9-11, 8 TuA1. J 1, [mj] 1 x Distribute Event Triggere Distribute Self Triggere Hybri Strategy Fig.. Energy consumption for istribute event-triggere an ecentralize self-triggere strategies for the first example. J 1, [mj] Distribute Event Triggere Distribute Self Triggere Hybri Strategy Fig.. Energy consumption for istribute event-triggere an ecentralize self-triggere strategies for the secon example. The system is escribe by: A = , B = x o = [ ] T , an we use again L =.5I an S =.I. In figure we can appreciate how the self-triggere protocol again prouces more frequent upates, but if we look at the energy consumption we can see how the istribute event-triggere algorithm consumes more energy all the time. In this example the evolution of the system is quite slow, an therefore the listening an computing energy plays a bigger role, making the istribute event-triggere strategy less efficient. A hybri strategy in which the event-triggere protocol also uses the self-triggere preicte times to keep the raio-moule asleep is also presente in both figure an figure. This algorithm improves the performance of the event-triggere algorithm but oes not always outperform the self-triggere algorithm. VII. DISCUSSION We have propose two istribute implementations for controllers over SAN s base on the event-triggere an selftriggere control paraigms. The use of event-triggere an self-triggere control is justifie in terms of the reuction in energy consumption obtaine by reucing the number of messages exchange over the network while achieving the prescribe control performance. Both strategies were illustrate through two examples which show that no algorithm outperforms the other in terms of energy expeniture. A hybri strategy benefiting from both approaches was teste. This hybri protocol always outperforms the event-triggere approach, but that is not the case with the self-triggere protocol. 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