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SAMPLING PERIOD ASSIGNMENT FOR NETWORKED CONTROL SYSTEMS BASED ON THE PLANT OPERATION MODE

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SAMPLING PERIOD ASSIGNMENT FOR NETWORKED CONTROL SYSTEMS BASED ON THE PLANT OPERATION MODE
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  SAMPLING PERIOD ASSIGNMENT FOR NETWORKED CONTROL SYSTEMSBASED ON THE PLANT OPERATION MODE Daniel A. Perez ∗ , Ubirajara F. Moreno ∗ , Carlos B. Montez ∗ , Tito L. M. Santos ∗∗ PGEAS - Programa de P´ os-Gradua¸c˜ ao em Engenharia de Automa¸c˜ ao e Sistemas DAS - Departamento de Automa¸c˜ ao e Sistemas, UFSC - Universidade Federal de Santa Catarina Caixa Postal 476, CEP 88040-900, Florian´ opolis, Santa Catarina, Brasil  Emails: dap@das.ufsc.br, moreno@das.ufsc.br, montez@das.ufsc.br, tito@das.ufsc.br Abstract— In this paper, a co-design methodology for networked control systems (NCS), based on the feedbackscheduling theory is proposed. In the proposed approach, the run-time information of the controlled process inemployed to dynamically reassign the computational resources. The policy is to assign a lower bound of resourcesfor each control loop for plants operating in the steady state, and allocate the exceeding resources to controlloops of plants that are in a transient behavior. Keywords— Dynamic Resource Allocation, Networked Control System Resumo— No presente artigo, uma metodologia de co-design para um sistema de controle via redes (NCS),baseada na teoria de escalonamento via realimenta¸c˜ao ´e proposta. Na abordagem sugerida, s˜ao utilizadas in- forma¸c˜oes do processo controlado em tempo de opera¸c˜ao para realocar dinamicamente recursos computacionais (largura de banda). A pol´ıtica se d´a com a determina¸c˜ao de um limite inferior das necessidades de recursos para cada malha de controle para as plantas que est˜ao operando em regime permanente. Assim os resursos excedentesnas malhas em regime permanente s˜ao alocadas para as malhas em regime transit´orio. Palavras-chave— Aloca¸c˜ao Dinˆamica de Recurso, Sistemas de Controle via Redes 1 Introduction Many approaches for resource allocation in net-worked control systems are based on fixed param-eters of the network’s load. They are defined andsettled at the initialization stage, and they arekept fixed during all the operation time of the sys-tem. At execution time, in general, the resourcesare shared among the control loops according tostatic specifications.As discussed (Mart´ı et al., 2002), it is notnecessary to assign the same amount of resourcesthat is demanded to reject a perturbation or a setpoint change in order to maintain a plant in steadystate. This statement suggests that to keep thesame distribution of resources during all the timemay be seen as a waste of resource in a networkedcontrol system (NCS).In this paper, a control-scheduling co-designmethodology that regards the plant output behav-ior is proposed. The proposed approach employsrun-time information from the controlled processto dynamically reassign computational resources(sampling period). The principle of the procedureis to allocate the exceeding bandwidth to thosecontrol loops that have their respective plants intransient response.This paper is organized as follows. In Section2 the general problem is described and the conceptof feedback scheduling and the procedure used toevaluate the control quality degeneration are pre-sented . The co-design methodology of NCS isexposed in section 3. In the Section 4, an illus-trative example of the methodology is presented.Finally, this paper is concluded in the Section 5. 2 Problem and Concepts 2.1 Problem  The problem studied in this paper is the real-timecontrol of a set of processes with controllers im-plemented in a remote computer, interconnectedthrough a computer network with limited band-width. There is a set of  m continuous plants tobe controlled. Associated to each process i , where i = 1 , 2 ,...,m , there are two devices physicallyconnected, the sensor i and the actuator i , and aremote component, the controller i .It is considered the situation which executiontime is not an accessible parameter, hence jittercompensation techniques are not considered. Theplant is described by the continuous-time linearsystem P  i ( s ), the plant output is sampled peri-odically with interval h by the sensor S  i . Thecontroller is represented by the discrete-time lin-ear system K  i ( z ), followed by the actuator A i thatincludes a zero order hold. 2.2 Problem Modeling  A control cycle can be modeled by a real-timeend-to-end task segmented in subtasks with prece-dence constraints (Sun, 1997). The segmenta-tion of a control cycle T  cci could be off-line ana-lyzed, and it is divided in three subtasks: sensor-controller T  cci, 1 message; computation of the con-trol law T  cci, 2 and controller-actuator T  cci, 1 message,as presented in Figure 1).In (Cervin and Eker, 2000) the feedback scheduling  was proposed. The main idea is to dis-tribute computational resources to optimize the  cc T  1,1 cc T  2,1 cc T  3,1 cc T  1,2 cc T  2,2 cc T  3,2 ccm T  1, ccm T  2, ccm T  3, Figure 1: Modeling of a set with m NCSs.global performance of the control, in a processorsusceptible to overloads.In the feedback scheduling approach, thescheduler feedbacks the consumption of the re-sources of the system (e.g. execution times of tasks) to determine the load of the system. Sys-tem parameters, such as periods and priorities of tasks, are reconfigured to lead the utilization toa specific level of reference. If some task comesto overrun, the scheduler may detect the overloadand reconfigure the tasks to deal with the over-load.In previous works a scheduling algorithmcalled Feedback Control EDF  is presented(Stankovic et al., 1999; Lu et al., 1999). This ap-proach consists in an implementation of a PIDcontroller in the scheduler, that regulates thedeadline miss rate for soft real-time tasks withvariable execution time, through the adjustmentof the processor’s utilization. Another approachis used in (Beccari et al., 1999) where samplingintervals are assigned during run-time to preventoverload of the processor. In this same con-text (Henriksson and Cervin, 2005) can be cited,where tasks reassignment in overload conditionswas used. However in the majority of previouswork, the plant operation mode was not evaluatedfor a NCS. 2.3 Estimation of the Control Performance De-generation  The use of a shared network among the compo-nents of a control loop introduces variable delays(delay jitter) in the execution of the control cycle.This uncertainty leads the studied control loopsinto time-varying systems, disallowing the directuse of linear systems criteria to evaluate the de-generation of the system’s stability margins.On the other hand, there are some criteria tomeasure the degeneration of the stability marginsby implementation factors in linear time-invariantsystems. Equations (1) and (2), evaluate thephase lag due to the controller discretization andto the constant delay in the control cycle. ∆ ϕ m isthe sum of the degeneration factors sum. ∆ ϕ m ( d ) = ω c h 2(1)∆ ϕ m ( a ) = ω c L (2)∆ ϕ m = ∆ ϕ m ( a ) + ∆ ϕ m ( d ) (3) An approach to deals with NCS is to turn itinto a time-invariant problem, using buffers in thecontroller and actuator nodes to reduce delay jit-ters (Luck and Ray, 1990). The system becomestime-invariant, when the release time in buffer islonger than the worst-case response time of trans-mission and computation of the message betweenthe nodes of the control loop. The main drawbackof changing a NCS in a time-invariant system isthe unnecessary increase of the control delay, be-cause the average of the delays becomes equal theworst-case response time. 3 The co-design methodology Using the feedback scheduling concept, adaptedto the NCS context, it is possible to assign morecomputational resources for the control loops thatare demanding a better quality of control in eachmoment of the system operation.Figure 2: Dynamic allocation of resources throughthe feedback scheduling.Differently of the majority of the works car-ried through on feedback scheduling, the controlvariable in this proposal is the operation mode of the plants. Thus, the idea is to apply feedbackin two levels of the real-time control system, aspresented in the Figure 2. There is a standardfeedback used by controllers, and a second levelthat represents the feedback inside of the real-timesystem to assign dynamically the computationalresources between the control loops. The distri-bution of the resources is based on the currentsituation of operation of the controlled plants.The principle of the adopted resources al-location in the methodology comes from (Mart´ıet al., 2002), that suggests that the plant needsdifferent degrees of computational resources to becontrolled. The process needs less amount of re-sources when it is in steady state than in the situa-tions that the control system is rejecting a distur-bance or is responding to change in the reference.  Following this approach, operation of the controlloops was divided in two modes, Transient Mode and Steady Mode . In Transient Mode all the sur-plus of resources available will be assigned to thecontrol loop with a settling process in the instantof the evaluation. On the other hand, in Steady Mode the minimum amount of resources allowed,kept a lower predefined stability margin, is as-signed the control loop with the plant in steadystate in the instant of the evaluation.Each control loop can be switched betweenthese two operation modes during the run-time of the system, in accordance with the actual stateof the plant. During the control procedure of the plant, the controller receives the samples of the plant’s outputs, which are computed, result-ing in the control law. In the other feedback loop,used by the feedback scheduler, at each new con-trol cycle the controller verifies the actual plantstate. If the states of the plant have been mod-ified, since the last evaluation, the controller de-tects this change and feeds the feedback scheduler.The change in the resources distribution isdone through the modification of the sampling pe-riods h i of each control loop, therefore the qualityof control degenerative factors are directly associ-ated to this parameter. 3.1 Definitions and Assumptions Some assumptions and definitions must be madebefore the procedure description. First, a tech-nique to transform a NCS in a time-invariant sys-tem is applied to obtain a capable metric to give avalue of the quality of the control degeneration ineach control loop of the system. Thus, (1) and (2)become valid. In order to facilitate the executionof the network and computer schedulability tests,it is also assumed that the maximum time of thecontrol delay R never is greater than the value of the sampling interval h . The assignment methodruns every time that a controller detects that itsrespective controlled plant changes its operationmode.Considering that, there is a set of  m NCS’s,each time that the feedback scheduler runs to re-configure the allocation of resources, the set of control loops is divided in two subsets: TM  and SM  . The subset denoted by TM  is composed bythe control loops operating in a transient state,and the subset denoted by SM  is composed bythe control loops operating in a steady state. Thenumber of elements of each subset is denoted re-spectively by tm and sm .The phase margin of the system after its im-plementation in a NCS and its relative degenera-tion λ are defined as: ϕ NCSm = ϕ m − ∆ ϕ m (4) λ = ϕ NCSm ϕ m (5) In accordance with the operation modes, λ tm and λ sm are defined as the relative degeneration of thephase margin of the control loops that belongs tothe subsets TM  and SM  respectively.By denoting the utilization of the processorsby the set of control loops that belong to the sys-tem as U  c , the utilizations related to the subsets TM  and SM  are denoted, respectively, by U  SM  and U  TM  . 3.2 Sampling Periods Assignment  The employed methodology consists in assignsampling intervals h such that the control loopswhich are operating in the same operation modehave the same relative degeneration of the phasemargin λ . The sampling interval h of a NCS canbe related to its relative degeneration of the phaseedge λ .In accordance with the assumptions, let L = h . Thus, the sum of the degeneration factors ∆ ϕ is expressed as in (6) and ω c is the cross-over fre-quency. ∆ ϕ m =32 · ω c · h (6) On the other hand, the sum of the degenera-tive factors ∆ ϕ m can be related with the relativedegeneration of the phase margin λ , hence from(4) and (5): ∆ ϕ m = ϕ m (1 − λ ) (7) From (6) and (7), the value of  h can be ob-tained. h =23 · ϕ m (1 − λ ) ω c (8) Thus, the sampling period assignment foreach control loop in the subset SM  is donethrough (8) and by the project parameter λ sm .The utilization U  SM  is computed through thesum of the individual utilizations ( e SM  /h SM  ) of each control loop in the subset SM  and e denotesthe execution time. By applying (8), U  SM  is givenby U  SM  =32 · sm  i =1 e SM,i ω c SM,i ϕ m SM,i (1 − λ SM  )(9) Once determined the computational resourcesconsumed by the control loops in steady state,by applying (9), the remaining computational re-sources are allocated to the control loops in thesubset TM  , thus, the utilization U  TM  is given by: U  TM  = U  c − U  SM  (10) By developing a similar derivation for the sub-set TM  , as for the subset SM  , an expressionthat relates the utilization U  TM  and λ tm , couldbe given by  U  TM  =32 · tm  i =1 e TM,i ω c TM,i ϕ m TM,i (1 − λ TM  )(11) Differently of the sampling periods assign-ment in the subset SM  , the utilization U  TM  isthe known parameter and the relative degenera-tion of the phase margin λ tm is the parameterto be calculated. The value of  λ tm can be ob-tained iteratively, starting form an initial value λ tm = λ sm . As described in the proposal of themethod, admitting that the component loops of the subset TM  will have a smaller degenerationthat the ones which form the subset SM  , thus λ TM  > λ SM  for all the considered cases.Once the value of  λ tm is estimated, the as-signment of the sampling period for each NCS k in TM  is given by (8).Thus, the sampling periods of all the controlloops in the system are assigned every time thatthe feedback scheduler is executed. 3.3 Modeling of the System Reconfiguration Pro-cedure The system reconfiguration can be described inthe following way. In each activation, the feed-back scheduler determines a new sampling inter-val for each control loop. The feedback sched-uler, then, brings up to date the sampling ratesof each controller through the communication be-tween processes in the computer, and brings up todate the sensors and actuators of the NCS throughthe broadcast of a message containing the new setof sampling intervals, for all the remote devicesthat compose the system.The reconfiguration procedure can be mod-eled as a real-time end-to-end task T  r , subdividedin two subtasks. First subtask T  r 1 is composed bythe execution of a procedure that computes thenew values of the sampling intervals and by theupdate of these values in the controllers, executedin the computer. The second subtask T  r 2 is con-stituted by the sending of a message, through thecommunication network, containing the update tothe sensors and remote actuators that belong tothe system.Differently of the control cycles, the reconfig-uration task of the system is modeled as an aperi-odic real-time task. Thus, the release of the feed-back scheduler instances has a non periodic behav-ior. The activations are caused by the occurrenceof disturbances or changes in the reference signalthat, in general, are events that occur without apredetermined pattern.The incorporation of the system reconfigura-tion in the schedulability analysis, and the deter-mination of the worst-case response times of thecontrol cycles, can be made using aperiodic real-time tasks scheduling techniques for the subtaskexecuted in the computer. To consider the aperi-odic messages in the communication network, theapproach will vary in accordance with the imple-mented network. For the case of a CAN network,presented in the following topic, is possible touse the Deferrable Server  (Lehoczky et al., 1987)scheduling strategy. 3.4 Application in a CAN network  A CAN network can be scheduled as a fixed pri-ority non preemptable real-time processor. Thus,established scheduling techniques can be used tocompute the schedulability and the response timesof a set of real-time messages transmitted throughthe network.The modeling of the studied problem lookslike the presented in Figure 1, with control cycleend-to-end tasks and the reconfiguration end-to-end task.The utilization of the network that guaranteesthe schedulability is given by U  i + u s + e s + b i h i ≤ U  RM  ( i + 1) (12) where U  i and u s are the periodic tasks and thedeferrable server utilizations, respectively, and b i is the blocking time. 4 Simulations To illustrate the application of the approach, anexample is presented. The example is composedby three control loops, whose controllers are im-plemented in a remote computer and using thesame CAN network to exchange the necessary in-formation to the control. In the computer, theadopted scheduling strategy is rate monotonic.Considering a CAN network with the biggesttransmission rate for this technology, 1 Mbits/s ,and messages with constant and equal size 120 bits (average size of a CAN frame), the transmissiontime of a message is equal 120 µs . The priorityassignment of the network nodes is fixed. Thecomputer messages has the biggest priority of thenetwork, the sensors are organized by the decreas-ing transmission rates when all the NCSs operatein the Transient Mode. It is assumed that theimplementation execution times of each controllerin the computer are constant and equal 150 µs .The continuous-time plants used in the exampleare given by (13). P  1 ( s ) =900( s 2 + 42 s + 900) P  2 ( s ) =4 · 10 4 ( s − 50)( s + 50)(13) P  3 ( s ) =5 · 10 7 s ( s 2 + 100 s + 2 . 5 · 10 5 )  The continuous-time controllers are given by (14). K 1 ( s ) =500( s + 70)( s + 60) s ( s + 1500) K 2 ( s ) =8 · 10 3 ( s + 2 . 5 · 10 5 )( s + 90)( s + 2000)( s 2 + 1 . 645 · 10 4 s + 1 . 35 · 10 8 )(14) K 3 ( s ) =478( s + 2 · 10 5 )( s 2 + 160 . 6 s + 1 . 655 · 10 5 )( s + 2740)( s + 1000)( s 2 + 2494 s + 7 . 109 · 10 6 ) Beginning the co-design procedure, some param-eters must be defined before the execution of thesimulation. In accordance with rate monotonictheory, the respective utilization that guaranteesthe schedulability for the network is U  = 0 . 73.Consequently, in accord with 12, the reload inter-val of the deferrable server was chosen as h s =1 ms , and the size of the recharge is enough tosend a message per cycle, e s = 120 µs . The valueof the phase margin relative degeneration for theSteady Mode was assumed to be λ sm = 0 . 35, lead-ing to a phase margin after the implementation of  ϕ ncsm > 20 ◦ for each mesh.The proposal, presented in this paper, is char-acterized by the dynamic change of the resourcesof allocation, according to the operation mode of the set of NCS. To evaluate possible benefits of theproposal, three cases are proposed in order to ex-plore the operational limits of each plant and theevolution of scenarios in which the assignment of the periods are modified. Case 1 - The plants are initially at steady state,the reference of all plants are changed to 0.The goal is to observe the control loops oper-ating, together, in the Transient Mode. Case 2 - The same reference change of  Case 1 isapplied for NCS 2, while the control loops 1and 3 are kept operating at the steady state(Steady Mode). The objective in this simu-lation is to place the biggest amount of re-sources for the control loop 2. Case 3 - In this scenario, the goal is to changethe operation modes of all NCS to show somereconfiguration procedures of the system. Todo that, the same disturbance is applied, in t = 0 . 03 s and t = 0 . 14 s , in the control loop2; and a reference change is imposed to NCS3 at t = 0 . 16 s . Case 1 for NCS 1.After the simulation of  Case 1 , the resultsobtained are presented in the Figure 3. This isthe configuration which each NCS receives lessamount of resources when operating in the Tran-sient Mode. Consequently, the greatest closedloop performance degeneration occurs in this sit-uation.The impact in the plant 2 due to the use of the proposed methodology in Case 2 is displayedin Figure 4. In this case, the plant 2 holds thegreatest availability of resources, since it operatesalone in Transient Mode ( TM  ). The instant of commutation t com and the phase margins ϕ ncsm of  Case 1 and Case 2 are shown in the Table I. 0 0.05 0.1 0.15−1−0.500.51 Plant 1     O   u    t   p   u    t Time (s) 0 0.05 0.1 0.1500.511.52x 10 −3 Plant 1Time (s)     S   a   m   p    l    i   n   g    I . 0 0.05 0.1 0.15−1−0.500.51 Plant 2     O   u    t   p   u    t Time (s) 0 0.05 0.1 0.1500.511.52x 10 −3 Plant 2Time (s)     S   a   m   p    l    i   n   g    I . 0 0.05 0.1 0.15−1−0.500.51 Plant 3     O   u    t   p   u    t Time (s) 0 0.05 0.1 0.1500.511.5x 10 −3 Plant 3Time (s)     S   a   m   p    l    i   n   g    I . (a) (b)Figure 3: Simulation results to the case 1 , (a)Plants responses e (b) Sampling interval (in sec-onds) used during the simulation.The responses of the control loops of  Case 3 are presented in the Figure 5(a). The values of the sample periods for each NCS during the simu-lation are displayed in the Figure 5(b), which was 0 0.05 0.1 0.15−1−0.500.51 Plant 2     O   u    t   p   u    t Time (s) 0 0.05 0.1 0.1500.511.52x 10 −3 Plant 2Time (s)     S   a   m   p    l    i   n   g    I . (a) (b)Figure 4: Simulation results to the case 2 (a)Plants responses e (b) Sampling interval (in sec-onds) used during the simulation.Table 1: Switching time t com and implementation phasemargin ϕ NCSm (degrees).Case 1 Case 2 t com ( s ) ϕ NCSm t com ( s ) ϕ NCSm 0 . 110 28 . 4 − − 0 . 123 28 . 7 0 . 045 37 . 30 . 117 30 . 6 − − Table 2: Implementation phase margin ϕ NCSm (degrees) forthe case 3 . ϕ NCSm a b c d e f g h 37 . 1 34 . 2 21 . 8 21 . 8 21 . 8 21 . 8 21 . 8 21 . 822 . 0 34 . 4 37 . 3 22 . 0 37 . 3 32 . 8 37 . 3 22 . 023 . 6 23 . 6 23 . 6 23 . 6 23 . 6 32 . 1 23 . 6 23 . 6
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