Performance-driven Design of Engine Control Tasks

Transcription

1 Performance-driven Deign of Engine Control Tak Aleandro Biondi, Marco Di Natale, Giorgio Buttazzo Scuola Superiore Sant Anna, Pia, Italy {aleandro.biondi, marco, Abtract Engine control tak include computational activitie triggered at pecific rotation angle of the crankhaft, making the computational load increae with the engine peed. To avoid overload at high peed, implified control implementation are ued, defining different operational mode at different peed interval. The deign of a et of adaptive variable rate tak i an optimization problem, coniting in determining the rotation peed at which mode change hould occur to optimize the ytem performance while guaranteeing the chedulability. Thi paper preent three method for tackling the optimization problem under a et of aumption about the performance metric and the problem contraint. Two are heuritic and one i a branch and bound that i guaranteed, when it terminate, to find the optimum within a given granularity. In addition, a imple method to compute a performance upper bound i preented. The analyi of the problem reveal everal inight for the deign and the heuritic are hown to be quite cloe to the performance upper bound and the optimum with finite granularity. I. INTRODUCTION Engine control i one the mot challenging example of Cyber-Phyical Sytem (CPS), where the oftware that control injection and combution mut be deigned to take everal phyical characteritic of the engine into account. From a timing perpective, engine control oftware require the execution of different type of computation, ome cyclicly activated at fixed interval (periodic tak) ranging from a few milliecond up to 00 m, and ome triggered at predefined rotation angle of the crankhaft (angular tak) [2]. Such angular tak generate a dynamic workload trictly dependent on the actual engine peed. To avoid overloading the proceor at high engine peed, angular tak are typically implemented a a et of operational mode, each characterized by a different computational demand and acting on a different peed interval [9]. Since angular tak adapt their functionality with the peed and are activated at non contant rate, they are alo referred to a adaptive variable-rate (AVR). The chedulability analyi of AVR tak ha received ignificant attention from the reearch community. A model for decribing an AVR tak wa firt propoed by Kim, Lakhmanan, and Rajkumar [3], who alo derived a chedulability analyi under very retrictive aumption, conidering a ingle AVR tak running at the highet priority with a period alway maller than thoe of the other tak. In addition, relative deadline were aumed to be equal to period and prioritie were aigned baed on the Rate-Monotonic algorithm. Pollex et al. [4] derived a ufficient feaibility tet for fixed priority cheduling, but auming a contant engine peed. Davi et al. [0] ued an Integer Linear Programming (ILP) formulation to derive a ufficient chedulability tet of AVR tak under fixed-prioritie, alo taking acceleration into account, but conidering a finite et of (dicretized) initial engine peed. The exact characterization of the interference produced by an AVR tak under fixed prioritie ha been preented by Biondi et al. [5] a a earch approach in the peed domain, where the concept of dominant peed i ued to reduce the complexity and avoid peed quantization. Such a method ha then been extended to derive an exact repone time analyi of fixed prioritie AVR tak [6]. Other work addreed the analyi of AVR tak under the Earliet Deadline Firt (EDF) cheduling algorithm. Guo and Baruah [] propoed a peedup factor analyi and ufficient chedulability tet for AVR tak cheduled with EDF. Biondi et al. [4] propoed a precie workload characterization generated by an AVR tak that i ued to derive a feaibility analyi under EDF cheduling. The implementation of an AVR tak alo require determining the precie engine peed at which mode change hould occur. Thi problem ha been addreed under EDF cheduling by Buttazzo, Bini, and Buttle [8], who propoed a method for identifying the highet witching peed that bound the utilization of an AVR tak to a deired value. In reality, the election of the tranition peed i not driven by chedulability contraint alone, but ha to optimize a et of performance indexe, related to power, fuel conumption and emiion (among other). The control implementation are deigned to achieve the bet poible combination of all the performance indexe within a given computation complexity and for a given et of engine peed. Thi proce require the tuning of a ignificant number of configuration parameter and i performed at the tet bench, where each implementation i teted for different peed recording the performance parameter of interet. Intuitively, the mot ophiticated control implementation have the bet performance but they require a higher computational demand that cannot be afforded (without incurring in deadline mie) when they are executed more frequently (i.e., at high engine peed). Converely, impler control implementation have lower computational requirement and tend to work better at higher rotational peed, where the engine i more table. The reulting behavior of engine-control tak i a modechange among a et of different control implementation, each one executed in a given interval of engine peed. In our framework, the mode change iue i formulated a a deign optimization problem, coniting in finding the witching peed that optimize the overall ytem performance while guaranteeing the chedulability of the tak et. We aume the knowledge of the performance function aociated to each control implementation (they can be derived by fitting the

2 tet bench performance data to a family of relatively imple analytical function). The conideration of uch performance function in the deign optimization proce together with the chedulability contraint i the main contribution of thi work. Thi paper preent three method for tackling uch an optimization problem auming a given performance metric and a et of contraint. Two heuritic approache are firt propoed to reduce the computational complexity and a performance upper bound i computed. Then, a branch and bound method i preented to find the optimum within a given peed granularity. Simulation reult how that the performance achieved by the heuritic i quite cloe to the performance upper bound and the optimum with a finite granularity. Paper tructure. The remainder of the paper i tructured a follow: Section II preent the model and the notation ued throughout the paper. Section III formally tate the problem conidered in the paper. Section V preent the two heuritic approache. Section VI decribe the branch and bound algorithm. Section VII illutrate a et of experimental reult aimed at comparing the propoed approache. Finally, Section VIII tate our concluion and future work. II. SYSTEM MODEL Thi paper conider application coniting of a et n realtime preemptive tak Γ = {τ,τ 2,...,τ n }. Each tak can be a regular periodic tak, or an AVR tak, activated at pecific crankhaft rotation angle. Whenever needed, an AVR tak may alo be denoted a τi. The rotation ource triggering the AVR tak i characterized by the following tate variable: θ the current rotation angle of the crankhaft; ω the current angular peed of the crankhaft; α the current angular acceleration of the crankhaft. The rotation peed ω i aumed to be limited within a range [ω min, ω max ] and the acceleration α i aumed to be limited within a range [α, α + ]. Both periodic and AVR tak are characterized by a wortcae execution time (WCET)C i, an interarrival time (or period) T i, and a relative deadline D i. However, while for regular periodic tak uch parameter are fixed, for angular tak they depend on the engine rotation peed ω. An AVR tak τi i characterized by an angular period Θ i and an angular phae Φ i, o that it i activated at the following angle:θ i = Φ i +kθ i, for k = 0,,2,... Thi mean that the inter-arrival of an AVR tak i inverely proportional to the engine peed ω and can be expreed a T i (ω) = Θ i ω. () An angular tak τ i i alo characterized by a relative angular deadline i expreed a a fraction δ i of the angular period (δ i [0,]). In the following, i = δ i Θ i repreent the relative angular deadline. All angular phae Φ i are relative to a reference poition called Top Dead Center (TDC) correponding to the crankhaft angle for which at leat one piton i at the highet poition in it cylinder. Without lo of generality, the TDC poition i aumed to be at θ = 0. A explained in the introduction, an AVR tak τ i i typically implemented a a et M i of M i execution mode. Each mode m ha a different WCET Ci m and operate in a predetermined range of engine peed (ωi m+,ωi m ], where ωmi+ i = ω min and ωi = ω max. Hence, the et of mode of tak τi can be expreed a M i = {(C m i,ωm i ),m =,2,...,M i}. The vector of the et of witching peed i denoted a ω. We aume that the wort-cae execution time of an AVR tak τ i can be expreed a a non-increaing tep function C i (ω) of the intantaneou peed ω at it releae, that i, C i (ω) {C i,...,c Mi i }. (2) An example of C(ω) function i illutrated in Figure. C i (ω) C M i i C m i Ci 2 Ci ω min i ω M i i ω m i ω 2 i ω i Figure : Computation time of an AVR tak a a function of the peed at the job activation. In the following, when a ingle AVR tak i addreed, the tak index i removed by the AVR tak parameter for the ake of readability. To upport the preentation of the derived reult it i convenient to define the teady-tate utilization of the j th execution mode a a function of a generic witching peed w, that i U j (ω) = Cj ω. (3) Θ Moreover, U j = U j (ω j ) denote the teady-tate utilization of the j th execution mode at it current witching peed ω j. A. Schedulability Analyi of AVR Tak In thi paper, the chedulability analyi of mixed tak et coniting of periodic tak and AVR tak i performed uing the tet preented in [6]. Conidering the peed domain a a continuum, a chedulability tet for tak et including AVR tak mut take into account all poible peed evolution of the rotation ource to cope with potential wort-cae ituation reulting in deadline mie. The adaptive behaviour of AVR tak cauing mode change a a function of the engine peed further complicate the identification of wort-cae repone time, preventing the treatment of uch tak a claical poradic tak. The analyi in [6] i baed on the computation of the interference caued by higherpriority tak, which i then ued to compute the tak repone time. By retricting to a finite et of dominant peed [5], it i poible to limit the number of peed evolution pattern that have to be examined for deriving the wort-cae interference. In [6] the repone time computation ha been approached a ω

3 a earch problem in the peed domain: the notion of dominant peed i firt ued to reduce the number of critical intant (i.e., the ituation in which all tak are releaed imultaneouly) and then the earch pace i reduced to dominant peed only. III. PROBLEM DEFINITION The analyi of thi paper i retricted to a repreentative cla of automotive application coniting of a ingle AVR tak and a et of periodic tak, cheduled by fixed-priority on a ingle core. Since there i only one AVR tak, we drop it index in the definition of it parameter. The addreed problem i firt tated uing a general formulation and then retricted to a number of pecific cae under a et of aumption. The general formulation conit of the definition of the input parameter, the optimization variable, the et of contraint function, and the performance function to be optimized. Input parameter. We conider an AVR tak coniting of Q implementation Λ j (j =,...,Q) of the ame functionality at different complexity. Note that we ditinguih between implementation and mode, ince, after the deign proce, one or more implementation can be merged into a ingle mode, to be executed in a range of peed (to be determined) and characterized by a known WCET C j. The Λ j are indexed uch that their execution time are trictly increaing, that i, C j < C j+, j =,...,Q. Optimization variable. The objective of the propoed optimization algorithm i to find the et of witching peed between mode of the AVR tak and to aign a fixed priority to the AVR tak and the periodic tak to optimize the engine performance while guaranteeing the tak et chedulability. Note that in the preence of AVR tak the Rate-Monotonic priority aignment i not guaranteed to be optimal due to the large period variation of the AVR tak [4]. The et of witching peed i labeled a ω j [ω min,ω max ]. Contraint. The contraint we conider in our optimization i the chedulability of the tak et, that i, the condition under which all the tak in the ytem meet their deadline. More pecifically, the following notion of chedulability i conidered in thi paper: a tak et i aid to be chedulable if there exit at leat a fixed priority aignment uch that all the tak meet their deadline, otherwie the tak et i aid to be unchedulable. The chedulability tet conidered in thi paper i the one preented in [6] and ummarized in Section II-A. Performance Metric. The effectivene of a deign olution i evaluated by aigning each implementation a performance function f j (ω) which i monotonically increaing with C j, that i, ω, f j (ω) > f j (ω) C j > C j. The rationale behind uch a retriction i that a more complex control implementation only make ene if it improve the performance. The overall ytem performance over the entire peed range, conidering all the poible execution mode i defined a P(ω,...,ω Q ) = Q j= ω j ω j+ f j (ω) dω (4) where j pan over all the mode j =,...,Q, and, by definition, ω = ω max and ω Q+ = ω min. In thi paper we conider two familie of performance function. In the firt, each implementation bring a contribution to the performance that doe not depend on the rotation peed at which it i applied. Each Λ j ha an aociated performance k j and the overall ytem performance i P(ω,...,ω Q ) = Q k j (ω j ω j+ ). (5) j= The econd et of function repreent control implementation in which the performance depend on the control law that i implemented, but alo on the rotation peed at which it i applied. The elected function (for each mode) i f j (ω) = k j, e kj,2 ω (6) and the ytem performance i computed a P(ω,...,ω Q ) = Q j= ω j j,2 k j, e k ω dω. (7) ω j+ The rationale for electing an exponential function of thi type i the following. The mot ophiticated control function (j = Q, with poibly multiple injection during the cycle) ha the bet performance, which i kept unchanged acro the peed range (f Q (ω) =, with k Q,2 = 0), and i conidered a a baeline for the other mode. Simplified control function tend to work better at higher rotational peed, where the engine i more table, but become le effective at ome cutoff peed, repreented by the parameter k i,2. The parameter k i, i ued a an additional degree of freedom to improve fitting the performance curve to the actual experimental data. Figure 2 how the typical hape of our exponential performance function for different value of k j,,k j,2, and Figure 3 how technical data (from the web) expreing the engine output power a a function of the engine peed, in upport of a poible fit with a family of exponential function a in Equation (6). Figure 2: A et of exponential performance function for elected value of k j,,k j,2. Formally, the firt type of function i a pecial cae of the

4 from the EU INTERESTED project []) coniting of 4 periodic tak with overall utilization U = 0.825, and an AVR tak with 6 mode. The period and the computation time of the periodic tak are reported in Table I, wherea the computation time of the 6 mode of the AVR tak are reported in Table II. To better explore the input pace, AVR computation time are expreed a a function of a caling factor. In the following, two ubcae are generated, uing = 6.0 and = 8.0 (all time in µ). Tak Tak2 Tak3 Tak4 C T=D Table I: Parameter of the tak et ued in the example. Figure 3: The power output curve of a the Evinrude E-TEC-250 engine (from continuouwave.com). exponential one where k j,2 = 0 for all j. However, in our experiment they are handled in a different way, becaue the optimization algorithm leverage the contant gradient of the contant function to avoid recomputing it at every tep, thu peeding up the computation. Although thee two clae of function are ued in the experiment, the propoed method i not limited to their conideration. The only requirement i that the ued function i monotonic with the mode index (and hence with the mode WCET) for all ω and it i integrable in the peed domain ω. The integral in Equation (7) cannot be computed analytically, thu i computed numerically through the exponential integral function Ei(x), o obtaining P(ω,...,ω Q ) = Q Y j (ω j ) Y j (ω j+ ), (8) j= where ( ) ) k Y j (ω) = k (k j, j,2 j,2 Ei +ω e kj,2 ω. (9) ω For notational convenience, we introduce a hortcut p j ( ω) for the partial derivative of the performance function with repect to the witching peed, defined a p j ( ω) = P( ω) ω j. Deign objective. The deign goal i to compute the et of witching peed {ω j,j =,...,Q} and the multidimenional optimization problem can be tated a follow: Definition (Optimization problem). A. Running example maxp(ω,...,ω Q ) ω > ω 2 >... > ω Q uch that the ytem i chedulable. To illutrate the outcome of each algorithm and the optimization procedure, we ue a running example (uing a configuration Λ Λ 2 Λ 3 Λ 4 Λ 5 Λ 6 C j k j Table II: Computation time of the AVR tak in the example. The running example ha the only purpoe of explaining the typical outcome of the propoed algorithm and i only conidered for the firt et of performance function (contant over the entire peed range). The experiment in the experimental ection ha been conducted alo for exponential performance function. In our running example, the coefficient k j of the performance function have been et a k j {2,3,4,5,7,0}. IV. REDUCING THE DESIGN SPACE Exploring the deign pace it ha been experimentally oberved that the chedulability contraint determine a nonconvex region in the pace of the witching peed ω j. Thi reult wa not urpriing conidering that the chedulability region for claical periodic tak in the pace of interarrival time i known to be non convex [3] and, to the bet of our knowledge, there i no known convex upper or lower bound of good quality for checking the chedulability of AVR tak under fixed-priority cheduling. Although an analytical decription of the feaibility region cannot be derived in a cloed form, we provide a numerical method for computing (with an arbitrary accuracy) an upperbound ω j (ub) of the maximum peed at which each control implementation can be executed, thu limiting the deign pace. The upper bound i uch that no feaible olution exit for the optimization problem for peedω j ω j (ub). The determination of thi upper bound alo lead to an upper-bound P (ub) on the maximum ytem performance. The propoed algorithm probe the chedulability region in the mot favorable condition for each poible control implementation j, that i when (i) no more complex control implementation are active and (ii) the AVR tak perform a direct mode change to the implet control implementation having WCET C, for any ω > ω j. An example of uch upper bound i hown in Figure 4, in which the mode configuration for determining ω j (ub) i hown a a dahed line. Under thi

5 C(ω) ω j+ (ub) A. Running example The peed upper bound for our running example are hown in Table III. C j C ω j (ub) ω j (ub) ω ω (ub) ω 2 (ub) ω 3 (ub) ω 4 (ub) ω 5 (ub) ω 6 (ub) P (ub) = = Table III: Upper bound on the tranition peed for the conidered example. ω min ω maxω Figure 4: Computing the tranition peed upper bound. condition, the algorithm compute the maximum peed ω j (ub) at which each control implementation Λ j can be executed. In Figure 4, the (typically unfeaible) ytem mode configuration obtained by conidering all the upper bound tranition peed i hown a a thick line. The approach i guaranteed to be correct becaue of the utainability property of uniproceor fixed-priority cheduling [7], which tate that chedulability never improve when computation time are increaed. : procedure COMPUTEUBS(C,...,C Q ) 2: w(ub) = ωmax ; 3: for j = 2 to j = Q do 4: M {(C,ω max ),(C j,ω min )}; 5: ω j (ub) = MAXBINSEARCH(j,M); 6: end for 7: return {ω j (ub),j =,...,Q}; 8: end procedure Figure 5: Procedure for computing the upper-bound for the witching peed. The algorithm to compute the witching peed upper bound i hown in Figure 5. The firt tep determine whether there are mode that can be afely avoided. Thi i done by finding the implementation with the larget C j that can afely be executed in ω min and dicarding thoe with higher index, and the implementation with the larget C that i feaible in ω M and dicarding thoe with lower index. For implicity, in the following, we aume that the only control implementation feaible at ω max i the firt one (j = ) and that the control implementation with the larget WCET C Q i feaible at ω min (that i, the number of mode are equal to the number of available implementation M = Q). The firt control implementation ha maximum peed w (ub) = ωmax, ince it i not poible to exceed the allowed peed range (ee line 2). The maximum peed are then determined for each mode j > by a imple binary earch (line 5) on the feaibility condition, by eeking a peed ω j (ub) uch that mode C j i executed in [ω min,ω j (ub)] and mode C i active in (ω j (ub),ωmax ]. V. HEURISTIC APPROACHES The non-convexity of the problem together with the lack of an analytical (cloed-form) characterization of the chedulability contraint, prevent the application of tandard method for computing the optimal olution. A number of poible heuritic approache can be devied to olve the problem. However, ome of them are not immediately applicable, a explained below. A. Gradient-baed earch A firt poible approach i to earch the pace of the witching peed ω j tarting from a known feaible point and then increaing all the candidate tranition peed according to the gradient of the performance function P. Thi i done by electing a tep δ and increaing each peed ω j by δ p j ( ω) (δ =50 in our experiment) until the boundary of the feaibility region i encountered. From that point on, the algorithm may proceed along the chedulability boundary with a local earch. A local earch conit in trying to improve a much a poible (uing a binary earch on the chedulability condition) all the tranition peed one by one, in order of their performance gradient p j ( ω). Thi intuitive approach doe not work in many cae. Thi i becaue in mot cae, the gradient of the performance function ha higher component for lower tranition peed and the problem definition require that ω < ω 2 <... < ω M. By projecting the deired direction onto thi contraint, all the ω j are increaed by the ame amount until we obtain the olution that compute the highet poible ω for the firt mode. Thi i often uboptimal (ee Figure 6, for a pace with only two peed, with the uboptimal olution ω o, the gray curve line join the et of point with the ame performance value, the black line the feaibility boundary). Indeed, thi policy i imilar to two other heuritic: a topdown or bottom-up greedy earch. A bottom-up earch (the topdown i imilar) conit of finding the larget poible value of ω that i feaible. Once ω i determined, the algorithm trie to increae ω 2 to the larget poible amount that till reult in a feaible ytem configuration and o on. Unfortunately, thee method are too greedy and often reult in olution in which mot mode are imply not uable at all, and compute olution that are far from optimality. For example in our running example, a top-down algorithm would produce ω 2 = w(ub) 2 and prevent the execution of the other control implementation Λ j,j > 2.

6 ω 2 ω o ω opt P Figure 6: The modified gradient-baed earch. ω : procedure GRADIENTSEARCH(C,...,C Q ) 2: w (k) = ω max, k; 3: for j = 2 to j = Q do 4: ω j (0) = ω min ; 5: end for 6: k 0; 7: while SCHEDULABLE({(C j,ω j (k)), j}) do 8: for j = 2 to j = Q do 9: ω j (k +) ω j (k)+δt j ( ω(k)); 0: end for : k k +; 2: end while 3: ω (end) LOCALSEARCH( ω(k )); 4: return ω (end) ; 5: end procedure Figure 7: Peudo-code for the propoed corrected gradientbaed earch. To improve reult, the gradient-driven heuritic can be corrected by a penalty factor aociated to the gradient term for a given ω j when the algorithm approache it upper bound ω j (ub). An effective penalty function i: z j (ω) = e ( ω j ) 2 (ub) ω ω j (ub). (0) The trajectory component T j ( ω) for each witching peed w j i expreed by T j ( ω) = z j (ω j )+ pj (ω j ), () pmax where p max = max j p j ( ω). Pleae note that the econd term in T j (ω) correpond to the original gradient direction with a normalization tep of p max. Each witching peed ω j i progreively increaed with rate δ T j ( ω) (δ = 5rpm in our experiment), generating a equence of peed ω(0), ω(),..., ω(k),..., in which each component progree a: ω j (0) = ω min ω j (k +) = ω j (k)+δ T j ( ω(k)), ω (k) = ω max j = 2,...,Q; k. We holdω (k) = ω max for the reaon explained in Section IV. The effect of the corrected gradient trajectory i hown in Figure 6 (dahed line). From Equation (0), when approaching the upper-bound ω j (ub) for a mode j, the penalty term zj (ω) i lowered and reduce the increae rate for ω j. Thi correction attempt at ecaping from trivial local minima and improving the obtained olution. Figure 7 illutrate the peudo code for the corrected gradient-baed heuritic. The algorithm initialize the witching peed at ω min (except for the firt mode). Then, a loop (line 7) update the witching peed according to the gradient trajectory T j ( ω) until the boundary of the chedulability contraint i reached. Finally, a local earch i performed (ee line 3) tarting from the mode having the maximum gradient coefficient, until no further tep can be performed without violating the chedulability contraint. ) Running example: The application of the corrected gradient heuritic to our running example provide the reult reported in Table IV. ω ω 2 ω 3 ω 4 ω 5 ω 6 P % of P (ub) % of P (ub) Table IV: Reult of the application of the corrected gradient heuritic to the running example. Clearly, the reult in the econd cae are poor, ince the mode after the firt are all collaped into a very mall range and the final performance i far from the upper bound. B. Utilization and Performance-Driven Backward Search An additional inight help building a better heuritic: lowering a witching peed can give more freedom to the feaibility range of the other. In particular, analyzing the experimental reult of a more exhautive branch and bound earch (dicued in the next ection), it became clear (a expected) that reducing the witching peed of the mode with the larget teady-tate utilization (defined a U (ub) = U( ω ub )) provide more freedom for the other peed, that i, allow to keep them cloer to their upper bound. At the ame time, the reduction of any tranition peed ω j caue a correponding performance reduction that hould be traded off with the utilization gain. Both conideration are combined in the Utilization and Performance-Driven backward earch. The witching peed are iteratively lowered, generating a equence ω(0), ω(),..., ω(k),..., tarting from the upper bound ω(0) = ω (ub). At each iteration, each ω j i lowered by a quantity δ R j, that i ω j (k+) = ω j (k) δ R j ( ω(k)), j = 2,...,Q (2) until a feaible et of peed i found. The reduction tep R j ( ω)

7 i R j ( ω) = max(ûj + P j ( ω),r min ). (3) R min repreent a minimum tep to enure progreion (in our experiment R min = 0.2 RPM). Û j and P j ( ω) are normalized indexe that relate to the utilization and performance gradient at the current peed ω j (k). The index Ûj i defined a with Û j = Uj U min U max Umin, (4) U max = max{u j } and U min = min{u j }. (5) j j Similarly, Pj ( ω) i computed a P j ( ω) = pmax p j ( ω) p max, (6) pmin where p max = max j p j ( ω) and p min = min j p j ( ω). Figure 8 illutrate the peudo code for the propoed utilization and performance-driven backward earch. The algorithm lower each witching peed ω j until the ytem become chedulable (ee line 7), then it proceed with a local earch, a the gradient-baed earch of Section V-A. : procedure BACKWARDSSEARCH(C,...,C Q ) 2: w (k) = ω max, k; 3: for j = 2 to j = Q do 4: ω j (0) = ω j ub ; 5: end for 6: k 0; 7: while NOT SCHEDULABLE({(C j,ω j (k)), j}) do 8: for j = 2 to j = Q do 9: ω j (k +) ω j (k) δr j ( ω(k)); 0: end for : k k +; 2: end while 3: ω (end) LOCALSEARCH( ω(k)); 4: return ω (end) ; 5: end procedure Figure 8: Peudo-code for the propoed utilization and performance-driven backward earch. ) Running example: The application of the backward earch heuritic to our running example provide the reult reported in Table V. ω ω 2 ω 3 ω 4 ω 5 ω 6 P % of P (ub) % of P (ub) Table V: Reult of the application of the backward earch heuritic to the running example. The backward earch heuritic i very effective. The running example i not a pecial cae, a hown in our experimental reult. The computed value are alway very cloe to the optimum. Alo, compared with the previou heuritic, a et of higher tranition peed for lower index mode i traded for a lower tranition peed for the lat mode (which reult in a better performance anyway). VI. BRANCH AND BOUND In many cae, the optimum can be computed (within a given peed granularity) by performing an exhautive earch tarting from an initial feaible olution and attempting to extend each tranition peed toward it upper bound. The knowledge of performance upper and lower bound allow introducing effective pruning rule that top the algorithm when a olution with ufficient quality i obtained. A. Definition of the algorithm The algorithm make ue of peed upper bound to compute a performance upper bound P ub that i in general not feaible. Alo, one of the heuritic in the previou ection (the backward earch i the bet option) allow computing a lower bound on the optimum performance P lb. The algorithm require the definition of a peed reolution δ (in our experiment δ = 5 RPM) and a tarting feaible olution with a configuration of tranition peed that ha ω Q at the highet poible value that allow for the execution of the other mode. The tarting olution hould alo be maximal, meaning that any poible increae of any tranition peed would create a non-feaible olution. Given any olution, a maximal olution can be imply found by a local earch. Becaue of the monotonicity property of our performance function (on Λ i and C i ), a maximal olution i alway going to have higher performance than any olution for which the tranition peed are component-wie le than or equal. The earch algorithm i baed on the obervation that, given a maximal olution, any increae in a tranition peedω j can only be obtained by decreaing at leat one of the tranition peed ω k with k > j. The algorithm work iteratively, attempting to improve on an initial feaible olution ω with ω Q equal to the larget poible value that allow for the execution of the other mode (the algorithm to compute the initial olution i explained later). At each iteration with indexj (j goe from3toq), the value in the et {ω Q,ωQ,...,ω j+ } are left unchanged from ω (the et i empty when j = Q), the peed ω j i iteratively reduced by m j δ, with m j N + ; and for each value of m j the algorithm trie all the poible extenion, of integer multiple of δ, of the peed {ω j,...,ω}, 2 until it reache the feaibility boundary (i.e., a maximal olution within the δ reolution). A a reult, the algorithm perform a branch and bound earch on the tree of peed with index lower than j. Figure 9 how an intermediate tep of the algorithm with the correponding earch tree below the element with index j. Since the index j i progreively increaed up to Q, all the poible peed combination (with granularity δ) are tried. The index j (controlling the peed that i electively reduced) tart from 3 becaue it i alway ω = ω M and reducing ω 2 i pointle becaue ω cannot be further increaed. Alo, it i neceary that ω Q i the larget poible value that allow the execution of the other mode, becaue ω Q i the only tranition peed that i not increaed in the earch.

8 ω min ω j ω j (ub) ω j ω j (ub) ω j 2 ω j 2 (ub) ω ω max ω Figure 9: The optimization algorithm a a branch and bound earch in the domain of the ω. At any point in time, the earch algorithm keep track of the bet performance olution found until then (initialized with the performance of the olution found by the backward earch heuritic). At each iteration, the algorithm perform a pruning on the peed ubtree when, after a reduction of m j δ of ω, j the current bet performance value cannot be improved by any olution available in the ubtree. The performance of all the olution available in the current ubtree are upper bound by the performance of the et of peed {ω Q,...,ωj+,ω j m j δ,ω j (ub),...,ω2 (ub),ω } contructed by leveraging on the knowledge of the peed upper bound. Computing the initial olution ω. The olution ω i computed iteratively. Firt, the value of ω Q i computed by earching back from ω Q (ub) (uing a binary earch) until the larget value that allow the execution of all other mode with lower index j at tranition peed ω Q +ǫ (Q j) (with ǫ arbitrarily mall). Next, ω Q i imilarly computed a the larget peed that allow executing all other mode with index j < Q with a tranition in ω Q +ǫ (Q j) and o on. The execution of the algorithm how how the optimum performance often reult in a configuration in which the larget peed decreae i for the mode with highet local utilization. Thi wa the motivation for the deriving the utilization-driven backward earch heuritic. The branch and bound compute olution of very good quality at the expene of time. A et of experiment (ee Section VII-C) ha been performed to evaluate the execution time and how the branch and bound reult compare with repect to the reult from the heuritic. However, it hould be noted that the runtime of the branch and bound earch i heavily dependent on the performance lower bound P lb that i provided to prune the olution tree at the beginning. Thi value i obtained by the backward earch heuritic. Hence, even in thoe cae in which the problem can be olved to (almot) optimality by the branch and bound earch, a practically uable execution time can only be achieved thank to the availability of a very good (and fat) heuritic. B. Running example The availability of the branch and bound exhautive earch for the optimum (with finite granularity) allow an evaluation of the quality of the heuritic. Table VI how a ummary of the reult for the cae with = 8. Algo ω ω 2 ω 3 ω 4 ω 5 ω 6 P H % of P (ub) H BS % of P (ub) BB % P (ub) UB P (ub) = Table VI: Reult for the running example with = 8 applying all the algorithm preented in thi paper. The table how the typical reult found in our experiment. Not only the backward heuritic i very cloe to the upper bound, but it i alo extremely cloe to the value computed by the branch and bound earch. In reality, the branch and bound reult i much cloer to the heuritic than it i to the upper bound. VII. EXPERIMENTAL RESULTS Thi ection report a et of experimental reult aimed at evaluating and comparing the approache preented in thi paper. All the algorithm have been implemented in the C++ language and teted over ynthetic workload for meauring their effectivene. In the experiment, the peed limit of the engine have been et to ω min = 500 RPM and ω max = 6500 RPM, repectively (typical value for a production car). The acceleration range allow the engine to reach the maximum peed tarting from the minimum in 35 revolution [0], reulting in α + = α = rev/mec 2. A. Workload generation In the evaluation we conider a tak et compoed of N periodic tak, with utilization U P, and an AVR tak τ with Q = 6 poible control implementation. The period of the periodic tak are {5, 0, 20, 50, 80, 00}m, conidered a typical value for engine control application [2]. The execution time of the periodic tak are generated by the UUnifat algorithm [2]. The WCET of the poible control implementation for the AVR tak are generated by randomly chooing (with a uniform ditribution and a minimum eparation c ep ) a et of eed value {c,c 2,...,c Q } from the range [c min,c max ]. The actual WCET are computed uing a cale factor a C j = c j. The cale factor i a parameter that allow tuning the computational requirement of the AVR tak implementation. When the witching peed and conequently the interarrival time of the AVR tak are unknown, it i not poible to define the AVR load with a imple utilization metric. B. Performance function generation The performance function conidered in thi work are: (i) contant function, a in Equation (5), and (ii) exponential function of the engine peed, a in Equation (7). In the firt cae, each control implementation Λ j i aigned a performance coefficient k j that i randomly generated with a uniform ditribution in the range[k min,k max ], with a minimum

9 eparation of k ep. In the cae of exponential function, the generation involve two parameter k j, and k j,2 for each control implementation Λ j. The performance i normalized with repect to Λ Q (i.e., the implementation with the larget WCET with contant performance), which ha k Q, = and k Q,2 = 0. To provide for a uniform ditribution of the exponential performance function (ee Equation (6)), the coefficient k j,2 are generated with a uniform ditribution in a logarithmic cale with range [logk 2,min,logk 2,max ]. Finally, we et k j, =,j =,...,Q for implicity. C. Contant performance function In thi experiment we conider the contant performance function and evaluated the performance of the heuritic with repect to the upper-bound P (ub) obtained with the method decribed in Section IV. We generate 500 tak et with N = 5 periodic tak and an AVR tak with a et of poible control implementation. For each tak et, we teted 30 different et of performance coefficient and a variable cale factor from to 0, trying different configuration. For each value of the performance of the heuritic and the upper-bound are computed. The, performance value obtained by the heuritic are normalized with repect to the value of the upper-bound. The normalized performance value with repect to the upper-bound are lower-bound of the performance value normalized with repect to the actual optimal performance. The normalized performance value were then averaged among all the configuration for a given value of. The range and eparation c min = 00, c max = 000, c ep = 00 are ued for generating the computation time eed and k min =, k max = 50 and k ep = are ued for the generation of the performance coefficient. Normalized Performance Upper-Bound Backward Heuritic Gradient-baed Heuritic Figure 0: Performance of the heuritic normalized to the performance upper-bound a a function of for U P = 0.5. Figure 0 how the reult for the cae of a periodic tak utilization U P = 0.5. A hown by the graph, the backward heuritic provide an extremely good performance, alway greater than 99% of the upper-bound, and extremely cloe to the optimum. Converely, the gradient-baed heuritic how a degradation for increaing value of reaching a value lower than the 90% of the upper-bound for = 0. In our experiment, the gradient-baed heuritic alway perform wore than the backward earch. To ave time in our experiment, we focued the remaining evaluation cae on the backward earch heuritic. Figure report the reult for the ame experiment when the utilization of the periodic tak i increaed to U P = The performance of the backward earch heuritic i lightly wore, reaching a value of approximately 93% for = 0. Normalized Performance Upper-Bound Backward Heuritic Figure : Performance of the heuritic normalized to the performance upper-bound a a function of for U P = However, a explained at the beginning of thi ection, the reult normalized with repect to the upper-bound are only lower-bound of the actual performance, expreed by the ratio with repect to the true optimum performance value (when computable). For thi reaon we performed another et of experiment including the reult of the Branch and Bound algorithm (with δ =5 RPM), to tudy the performance of the backward earch heuritic with repect to the actual optimal performance (or a value mot likely cloe to it). Due to the large run-time of the branch and bound algorithm, thi experiment ha been conducted on a mall et of configuration with 50 tak et and 5 et of performance coefficient. The reult are hown in Figure 2. A hown by the graph, the optimal performance tend to recede from the upper-bound for increaing value of, confirming the effectivene of the backward earch heuritic which remain around 99% of the performance value found by the branch and bound algorithm. The maximum oberved run-time for the Backward Search heuritic i 756 econd with an average run-time of 5.6 econd. For the Branch and Bound algorithm with preciion δ =5 RPM we meaured a maximum run-time of 6070 econd with an average run-time of about 600 econd. Such reult have been obtained executing the algorithm on a machine equipped with Intel i7 proceor running at 3.2 Ghz and 8Gb of RAM. Normalized Performance Upper-Bound Branch and Bound (5 RPM) Backward Heuritic Figure 2: Performance of the heuritic normalized to the performance upper-bound a a function of for U P = 0.75.

10 D. Exponential performance function Another experiment ha been conducted with the exponential performance function decribed in Section III. We focu on the comparion of the backward earch heuritic reult againt the performance upper-bound P (ub). Figure 3 how the reult a a function of the cale factor for two different value of the coefficient k 2,max of the performance function (keeping k 2,min contant). The utilization of the periodic tak i U P = 0.75 and for each value of we try 500 tak et and 30 et of performance coefficient, hence teting different configuration. A hown by the graph, the backward earch heuritic ha a performance alway greater than 99% of the upper-bound for k 2,max = 50k 2,min. In the cae k 2,max = 200k 2,min there i a light degradation of the performance of the heuritic that reache 96% of the upper-bound for = 0. Normalized Performance Upper-Bound Backward Heuritic (k 2,max = 50k 2,min ) Backward Heuritic (k 2,max = 200k 2,min ) Figure 3: Performance of the heuritic normalized to the performance upper-bound a a function of for U P = 0.75 with exponential performance function. The ratio between k 2,max and k 2,min determine the ditribution of the exponential performance function in the peed domain. Intuitively, the higher k 2,max /k 2,min the more the performance function are far apart. For thi reaon we conduct another experiment by varying the ratio k 2,max /k 2,min while holding the cale factor = 7. For each value of the ratio we tet 500 tak et and 50 et of performance coefficient k 2,j, hence configuration. The reult are hown in Figure 4 and confirm the trend of Figure 3, howing a graceful and quite limited degradation for increaing value of the ratio k 2,max /k 2,min. Normalized Performance Upper-Bound Backward Heuritic k 2,max /k 2,min VIII. CONCLUSIONS The problem of performance oriented deign of tranition peed in a (fuel injection) ytem with adaptive variable rate tak i dicued by preenting a et of optimization algorithm that apply to a quite general cenario, in which the performance of each control implementation i expreed by an arbitrary function that ha the only requirement of being integrable and monotonically increaing with the complexity of the implemented algorithm. The experimental reult how that the propoed heuritic are extremely cloe to the actual optimum value and allow the computation of the optimum with finite reolution in many cae. Future work include the analyi of actual performance data and improving the chedulability contraint to allow for temporary overload condition. REFERENCES [] INTERESTED, European project, Cordi project decription. URL: [2] E. Bini and G. C. Buttazzo. Meauring the performance of chedulability tet. Real-Time Sytem, 30(-2), [3] E. Bini, M. D. Natale, and G. Buttazzo. Senitivity analyi for fixedpriority real-time ytem. Real-Time Sytem, 39(-3):5 30, Augut [4] A. Biondi, G. Buttazzo, and S. Simoncelli. Feaibility analyi of engine control tak under EDF cheduling. In Proc. of the 27th Euromicro Conference on Real-Time Sytem (ECRTS 205), Lund, Sweden, July 8-0, 205. [5] A. Biondi, A. Melani, M. Marinoni, M. D. Natale, and G. Buttazzo. Exact interference of adaptive variable-rate tak under fixed-priority cheduling. In Proceeding of the 26th Euromicro Conference on Real- Time Sytem (ECRTS 204), Madrid, Spain, July 8-, 204. [6] A. Biondi, M. D. Natale, and G. Buttazzo. Repone-time analyi for real-time tak in engine control application. In Proceeding of the 6th International Conference on Cyber-Phyical Sytem (ICCPS 205), Seattle, Wahington, USA, April 4-6, 205. [7] A. Burn and S. Baruah. Sutainability in real-time cheduling. Journal of Computing Science and Engineering, 2():74 97, [8] G. Buttazzo, E. Bini, and D. Buttle. Rate-adaptive tak: Model, analyi, and deign iue. In Proc. of the Int. Conference on Deign, Automation and Tet in Europe, Dreden, Germany, March 24-28, 204. [9] D. Buttle. Real-time in the prime-time. In Keynote peech at the 24th Euromicro Conference on Real-Time Sytem, Pia, Italy, July 2, 202. [0] R. I. Davi, T. Feld, V. Pollex, and F. Slomka. Schedulability tet for tak with variable rate-dependent behaviour under fixed priority cheduling. In Proc. 20th IEEE Real-Time and Embedded Technology and Application Sympoium, Berlin, Germany, April 204. [] Z. Guo and S. Baruah. Uniproceor EDF cheduling of avr tak ytem. In Proc. of the ACM/IEEE 6th International Conference on Cyber-Phyical Sytem (ICCPS 205), Seattle, USA, April 205. [2] L. Guzzella and C. H. Onder. Introduction to Modeling and Control of Internal Combution Engine Sytem. Springer-Verlag, 200. [3] J. Kim, K. Lakhmanan, and R. Rajkumar. Rhythmic tak: A new tak model with continually varying period for cyber-phyical ytem. In Proc. of the Third IEEE/ACM Int. Conference on Cyber-Phyical Sytem (ICCPS 202), page 28 38, Beijing, China, April 202. [4] V. Pollex, T. Feld, F. Slomka, U. Margull, R. Mader, and G. Wirrer. Sufficient real-time analyi for an engine control unit with contant angular velocitie. In Proc. of the Deign, Automation and Tet Conference in Europe, Grenoble, France, March 8-22, 203. Figure 4: Performance of the heuritic normalized to the performance upper-bound a a function of k2,max k for U P = ,min and = 7.