Optimal Integrated Control and Scheduling of Systems with Communication Constraints

Transcription

1 Otimal Integrated Control and Scheduling of Systems with Communication Constraints Mohamed El Mongi Ben Gaid, Arben Çela and Yskandar Hamam Abstract This aer addresses the roblem of the otimal control and scheduling of Networked Control Systems over limited bandwidth time-slotted networks Multivariable linear systems subject to communication constraints are modeled in the Mixed Logical Dynamical (MLD) framework The translation of the MLD model into the Mixed Integer Quadratic Programming (MIQP) formulation is described This formulation allows the solving of the otimal control and scheduling roblem using efficient branch and bound algorithms Advantages and drawbacks of on-line and off-line scheduling algorithms are discussed Based on this discussion, an efficient on-line scheduling algorithm, which can be seen as a comromise, is resented and its erformance is evaluated I INTRODUCTION Shared communication networks are increasingly being used to suort the information exchange in distributed control systems Using a control network has many advantages, such as a higher reliability, an easier deloyment and maintenance However, many communication networks are subject to bandwidth constraints (for examle Underwater Acoustic Networks [] or Wireless Networks [2]) Reasons behind these resource constraints are multile Guaranteeing deterministic real-time communications induces imortant restrictions on the available bandwidth, esecially when the communication channel is noisy In other situations, some nodes of the communication network may be autonomous and battery-owered The lifetime of the batteries limits the energy used in the transmission and thus limits the bandwidth of the communications On the other hand, in the automotive industry, an increasing number of alications are being develoed in order to imrove the driving safety and comfort These alications need to share and to exchange an imortant amount of information As a consequence, the local bus is becoming more and more loaded Using a more exensive technology can solve these roblems, but comonents rice is strongly balanced by roduction cost requirements Many research works addressed the roblems stemming from the use of communication networks in control loos from different oints of view, see for examle [3] [6] Recently, it was shown that considering jointly control and real-time scheduling leads to an imrovement of the control erformance [7], [8] In [9], the roblem of the otimal off-line control and scheduling over TDMA networks is M-M Ben Gaid and A Çela are with the COSI Lab, ESIEE, Cité Descartes, BP 99, 9362 Noisy-Le-Grand Cedex, FRANCE ( {bengaidm,celaa}@esieefr) Y Hamam is with the A 2 SI Lab, ESIEE, Cité Descartes, BP 99, 9362 Noisy-Le-Grand Cedex, FRANCE ( hamamy@esieefr) addressed Network scheduling is modeled using the notion of communication sequence [] This modeling leads to a comlex combinatorial otimization roblem, whose solution is obtained by a heuristic search In [], a model redictive controller is used to calculate on-line both the control and the scheduling of a time-slotted network, assuming that control commands that cannot be udated are set to zero Using this assumtion, as well as a cost function based on the infinity norm, the otimization roblem was shown to be equivalent to the Generalized Linear Comlementarity Problem In this aer, the roblem of the distributed control over deterministic real-time networks is addressed The corresonding otimal control and scheduling roblem is formulated An efficient aroach for the solving of this roblem is roosed An on-line scheduling algorithm, which can be seen as a comromise between off-line and online scheduling algorithms, is resented and its erformance illustrated This aer is organized as follows In section II, the modeling of a multivariable control system with communication constraints in the Mixed Logical Dynamical framework is described Section III addresses the translation of this model into the Mixed Integer Quadratic Programming Formulation The finite-time otimal control and scheduling roblem based on this formulation is then illustrated Finally, in section IV, an efficient on-line scheduling algorithm (the Otimal Pointer Placement scheduling algorithm) is roosed and its effectiveness is shown in simulation examles II PROBLEM FORMULATION Consider a collection of N continuous-time LTI systems (S j ) j N described by the state equations: ẋ j (t)=a j c x j (t)+b j c u j (t) () where u j (t) is a column vector with m j entries, j N Assume that all the systems are controllable and that their state vectors are measurable Suose that system S j contains m j distinct distributed actuators which are connected to a main controller through a bus with a limited bandwidth Let A c = Diag ( A c,,a N ) c and Bc = Diag ( B c,,b N ) c, then systems (S j ) j N can be described using the state equation: ẋ(t)=a c x(t)+b c u(t) (2) where x(t)= x (t) x N (t) and u(t)= u (t) u N (t)

2 The bus is scheduled using a Time Division Multile Access (TDMA) scheme At each slot, only one control command can be sent to a secified actuator The duration T s of the time slot, which deends on the bit rate and on the used communication rotocol, defines the elementary time unit of the model A similar model was considered in [9] and [] Let m be the total number of actuators of the global system: m = N j= m j (3) Network scheduling can be described by m binary functions δ i verifying: δ i (k)= the k th slot is used to udate the i th actuator (4) At the k th time slot, only one message can transmitted on the bus This constraint can be modeled by the inequality: m i= δ i (k) (5) In order to derive a digital control law, a discrete-time model of system (2), at the samling eriod T s is considered: x(k + )= Ax(k)+Bu(k) (6) The digital-to-analog converters, which are located at the actuators, use zero-order-holders to maintain the last received control commands constant until new control values are received Consequently, if a control command is not udated at the k th slot, then it is held constant This assertion can be modeled by the logic formula: δ i (k)= = u i (k)=u i (k ) (7) where u(k)=[u (k) u m (k)] T Finally, suose that control signals are limited: L i u i U i (8) and that L i < and U i > Equations (5), (6), (7) and (8) constitute the considered model of a networked control system with limited communication resources This model can be addressed using the Mixed Logical Dynamical (MLD) systems framework [2] III FINITE-TIME OPTIMAL CONTROL AND SCHEDULING PROBLEM The finite-time otimal control and scheduling roblem can be formalized as follows : Find the otimal control sequence u N =(u(),,u(n )) and the otimal communication sequence δ N =(δ(),,δ(n )) which are solutions of the following otimization roblem: min xt (N)Sx(N)+ N T x(k) x(k) Q u N,δ N k= u(k) u(k) subject to: x(k + )=Ax(k)+Bu(k), k {,,N } m i= δ i (k), k {,,N } δ i (k)= = u i (k)=u i (k ), k {,,N } x() given (9) where Q and S are definite ositive matrices Problem (9) is a constrained control roblem In fact, the use of TDMA scheduling imoses to m comonents of u(k) to remain equal to the corresonding comonents of u(k ) The only comonent of u(k) whose value can change at the k th slot is that whose index i satisfies δ i (k)= In order to solve this roblem, it is necessary to translate the logical formula (7) into linear inequalities The connective = can be eliminated if (7) is rewritten in the equivalent form: u i (k) u i (k )=δ i (k)u i (k) δ i (k)u i (k ) () However, equation () contains terms which are the roduct of logical variables and continuous variables The use of the rocedure described in [2] allows the translation of this roduct into an equivalent conjunction of linear inequalities For examle, let : z i (k)=δ i (k)u i (k) () Then () can be rewritten in the equivalent form: z i (k) U i δ i (k) z i (k) L i δ i (k) z i (k) u i (k) L i ( δ i (k)) z i (k) u i (k) U i ( δ i (k)) (2) Note that the same rocedure can be alied to w i (k) = δ i (k)u i (k ) δ() u() x() Let = U = X = δ(n ) u(n ) x(n) z() w() U Z = W = and V = X, then z(n ) w(n ) Z W roblem (9) can be written : { min V 2 V T HV + f T V (3) AV B where H, f, A, and B can be easily deduced form the discussion above Problem (3) is a Mixed-Integer Quadratic Program The advantage of this formulation is the existence of many efficient academic and commercial solvers, based on the branch and bound algorithm Examle : Consider the collection of 3 continuous-time LTI systems defined by : A 3 c = 8 A 2 4 c = 25 2 A 3 c = B 3 c = 6 84 B c = 224 B 2 c = 62 8 (S ) (S 2 ) (S 3 )

3 delta2 delta3 delta x, x2, x3 and x x x2 x3 x Fig Otimal schedule Fig 2 Global system resonse - states x, x 2 (system S ) and x 3, x 4 (system S 2 ) from t = s to t = s Systems S and S 3 are oen-loo unstable in oosition to system S 2 which is oen-loo stable The control commands are sent to the actuators through a bus with a slot time of 2 ms A discrete-time model of the global system (comosed of S, S 2 and S 3 ) at the samling eriod of 2 ms is considered Design criteria for each system are defined by matrices Q = Diag(,), R =, Q 2 = Diag(,), R 2 =, Q 3 = Diag(,,,) and R 3 = Desired closed loo secifications for the global system are described by the closed loo weighting matrices Q : and S : Q = Diag(c Q,c 2 Q 2,c 3 Q 3,c R,c 2 R 2,c 3 R 3 ) S = Diag(c Q,c 2 Q 2,c 3 Q 3 ) where c, c 2 and c 3 are the weighting coefficients In this examle, c, c 2 and c 3 were chosen equal to the inverse of steady state erformance index of each searate system controlled through a bus having an infinite bandwidth (c = 29, c 2 = 3 and c 3 = 6) The length of the otimal control and scheduling sequences is N = An otimal solution with an error bound of 5 was required The global system is started from the initial state x()=[ ] T The otimal schedule is deicted in figure In this schedule, δ i = means that the bus is dedicated to the transmission of control signal u i The resonse of the global system is illustrated in figures 2 and 3 The first network slots are mainly dedicated to system S until it is stabilized It can be observed that system S 2, which is oen-loo stable and whose resonse time is bigger than S, needs only three time slots to be stabilized After the stabilization of systems S and S 2, network resources are entirely dedicated to the stabilization of system S 3 When system S 3 is close to the equilibrium state (from t = 3 s), then its control signals changes are minor Consequently, the scheduling has no significant imact on control, because control signals of the three systems are relatively constant, which exlains the shae of the scheduling diagram, after t = 3 s However, this otimal schedule is deendent from initial conditions If the initial condition x() is modified, then the otimal control and schedule would be different It is x5, x6, x7 and x Fig 3 Global system resonse - states x 5 x 8 (system S 3 ) clear that such an oen-loo schedule, which is generated offline, cannot be alied at run time as static schedule In fact, assume that system S is disturbed at t = 24 s Observing the schedule, no slots are allocated to the transmission of the messages of system S between t = 24 s and t = 28 s, which would induce erformance degradations These observations show that in the same way as the otimal control, the otimal scheduling deends on the current state of the system Consequently, fixed schedules may not be otimal if the system is started from another initial state or if it is disturbed at runtime However, from a comuter science oint of view, off-line scheduling has many advantages, essentially because it consumes few comuting resources and does not induce execution overheads In order to obtain static schedules which are otimal from a certain oint of view, it is necessary to use erformance criteria which deend on the intrinsic characteristics of the system, not on a articular evolution, for examle, based on the H 2 or H erformance indices [3] IV OPTIMAL POINTER PLACEMENT SCHEDULING Oen-loo otimization roblems constitute the cornerstone of a successful control method: the model redictive control (MPC) MPC has strong theoretical foundations, and many interesting roerties which make it suitable to address constrained control roblems However, its main drawback x5 x6 x7 x8

4 is that it requires very imortant comuting resources, which make it only alicable to slow systems, like chemical rocesses The use of the MPC technique leads to an on-line solving of an oen-loo otimization roblem For the articular MLD model described in equations (5)(6)(7)(8), the alication of the MPC technique requires the finding of the otimal control sequence v N =(v(),,v(n )) and the otimal bus allocation sequence σ N =(σ(),,σ(n )) which are solutions of the following otimization roblem: T y(h) Q y(h) min yt (N)Sy(N)+ N v N,σ N h= subject to: y()=x(k) y(h + )=Ay(h)+B, h {,,N } m i= σ i (h),h {,,N } σ i ()= = v i ()=u i (k ) σ i (h)= = v i (h)=v i (h ), h {,,N } (4) The solution of this roblem is based on the redicted evolution of the system over a horizon N and deends on the current state x(k) The variables y(h), h [, N] reresent the redicted values of system states x(k + h), based on the nominal model and knowing x(k) Sequences (v(),,v(n )) and (σ(),,σ(n )) are resectively called virtual control sequence and virtual bus allocation sequence because they are used to redict a ossible evolution of the system, based on x(k) Let (v (),,v (N )) and (σ (),,σ (N )) be the otimal solutions of this roblem Then the actual control command is obtained by setting u(k)=v () and only sending the control command u i (k) whose index verifies δ i (k)= where δ(k)=σ () At the next samling eriod, the whole rocedure is reiterated, based on x(k + ) Although the alication of this technique gives very good resulats [4], its major drawback is that it requires very imortant comutational resources, which makes its alication to fast systems imracticable The motivation behind the Otimal Pointer Placement (OPP) scheduling algorithm resented in this section is to be a comromise between the advantages of the on-line scheduling (control erformance) and those of the off-line scheduling (a very limited usage of comuting resources) Assume that a eriodic off-line controller as well as a eriodic off-line schedule (both of eriod P) guaranteeing the asymtotic stability of the system exist The eriodic controller is defined by a eriodic sequence of state feedback control gains (corresonding to the global system) K P =(K(),,K(P )) and the schedule by the eriodic communication sequence γ P =(γ(),,γ(p )) Note that otimal K P and γ P selection, considering worst case initial conditions, is described in [9] At runtime, the execution of the eriodic off-line controller and schedule can be described using the notion of ointer The ointer can be seen as a variable which contains the index of the control gain to use and the actuator to udate The ointer is incremented at each slot If it reaches the end of the sequence, its osition is reset More formally, if the ointer is started at the osition ( < P) its exression I (k) at the k th slot is : I (k)=(k + ) mod P (5) The idea behind the OPP scheduling heuristic is that instead of finding an otimal solution to roblem (4), the search is restricted to a sub-otimal solution, based on the off-line schedule, according to the following roblem: min J( x(k), )=y T (N)Sy(N)+ N h= T y(h) Q y(h) subject to: y()=x(k) v( )=u(k ) and for all h {,,N } = Γ(I (h))k(i (h))y(h)+(i m Γ(I (h))v(h ) y(h + )= Ay(h)+B (6) where Γ(l)=Diag(γ (l),,γ m (l)), I m is the m m identity x(k) matrix and x(k)= The horizon N is constrained u(k ) to be a multile of P The cost function is calculated according to a rediction of the future evolution of the system (described by y(h)) This evolution is calculated assuming that sequences K P and γ P are started from the osition The solution of roblem (6) is the ointer s osition which minimizes the cost function J( x(k), ) The control command u i (k) =v i () is sent to the actuator verifying γ i ( )= When the horizon N is infinite, a formal roof of the stability of the OPP scheduling algorithm can be given This result is stated in the following theorem: Theorem : If N is infinite, and if the asymtotic stability of system (5)(6)(7) is guarantied by the off-line control and scheduling using the control gains sequence K P and the network scheduling sequence γ P, than it is also ensured by the Otimal Pointer Placement scheduling algorithm Proof: Let x() be an arbitrary initial condition and an arbitrary initial ointer osition for the static scheduling algorithm Let x ss and u ss (resectively x o and u o )bethe state trajectory and the control of the system when scheduled using the static scheduling algorithm (resectively the OPP algorithm) Let J ss ( x(i),i, f, ) = f [ x ss ] T [ (h) x ss ] (h) h=i u ss Q (h) u ss (h) be be the cost function corresonding to an evolution from instant [ h = i to instant ] h = f starting from the extended state x(i) x(i) = where the static scheduling algorithm is u(i ) alied and where the ointer at instant i is laced at osition Let J o ( x(i),i, f )= f [ x o ] T [ (h) x o ] (h) h=i u o Q (h) u o be the (h) cost function corresonding to an evolution from instant k = i to instant k = f starting from the state x(i) where the OPP algorithm is alied According to the OPP strategy, the

5 ointer osition (l) at instant l is chosen such that: (l)=argminj ss ( x o (l),l,+, ) For l =, min J ss ( x(),,+, ) J ss ( x(),,+, ) and at each ste l >, the relation J o ( x(),,l ) + minj ss ( x o (l),l,+, ) J ss ( x(),,+, ) holds When l +, the last relation reduces to: J o ( x(),,+ ) J ss ( x(),,+, ) (7) Q is definite ositive Consequently, J ss ( x(),,+, ) (resectively J o ( x(),,+ )) is finite if and only if the system is asymtotically stable Knowing the asymtotic stability of the system scheduled using the static scheduling algorithm and using relation (7), the theorem is roved Examle 2: In order to illustrate the effectiveness of the OPP scheduling algorithm, examle of section III is reconsidered The control erformance corresonding to the use of a static scheduling (SS) algorithm, an OPP algorithm and the MPC algorithm is comared Static scheduling and OPP algorithms use the communication sequence: γ 6 = which together with the control gains sequence K 6 = (K,K 2,K,K 2,K,K 3 ) guarantees the asymtotic stability of the global system, where: K =( 4 4 ) K 2 =( ) K 3 =( ) The eriod P of the schedule is equal to 6 and the horizon N of the OPP and MPC algorithms is equal to 6 A sub-otimal solution with a relative error of 5 was required for the MPC algorithm System resonses corresonding to state variables x, x 3, x 5 and x 6 (the ositions) are deicted in figures 4 and 5 The continuous-time cost functions corresonding to these resonses are illustrated in figure 6 The global system is started from the initial state [ ] T The three systems S, S 2 and S 3 converge rogressively to the steady state At t = 4 s, the system S is disturbed This deviation is quickly corrected The best resonse is achieved by the MPC However, this algorithm cannot be imlemented in ractice, because the required comutation time is too long (a few seconds at each ste in this examle) For N = 6, the number of feasible communication sequences is 3 6 = The fact that the used branch and bound algorithm finishes in few seconds shows its efficiency to address this articular roblem The OPP algorithm significantly imroves the control erformance with resect to the static scheduling algorithm, requiring fewer comuting resources than the MPC In fact, x and x3 x5 and x x (SS) x3 (SS) x (OPP) x3 (OPP) x (MPC) x3 (MPC) Fig 4 System resonses - states x and x 3 x5 (SS) x6 (SS) x5 (OPP) x6 (OPP) x5 (MPC) x6 (MPC) Fig 5 System resonses - states x 5 and x 6 SS OPP MPC Fig 6 Continuous time cost functions using the OPP scheduling algorithm, the maximum number of ossible communication sequences is equal to P = 6 The schedule obtained by the OPP and MPC algorithms are resectively deicted in figures 7 and 8 At t = s, the OPP scheduling algorithm chooses to reserve the first slot to system S 3 This choice contributes to the imrovement of its erformance and that of the global system (shown in figure 5) MPC and OPP have the ability to change on-line the samling eriod of each system MPC can comensate for these changes Consequently, the scheduling attern is irregular Network slots are allocated in order to imrove the control erformance When a system is closer to the

6 delta3 6 5 SS OPP MPC 4 delta delta Fig 7 OPP schedule Fig 9 Cumulative continuous time cost functions delta delta2 delta Fig 8 MPC schedule equilibrium, then its control commands remains constant and they may be sent less frequently over the network That s why the MPC algorithm does not allocate network slots to systems in the equilibrium, when other systems are far form the steady state At t = 4 s, when the system S is disturbed, OPP and MPC algorithms allocates the network slots mainly for the transmission of the control command of system S, allowing to react earlier and quicker to the disturbance This contrasts with the static scheduling algorithm, where the allocation of the network resources is indeendent from the dynamical state of the systems Finally, the three systems are disturbed with a band limited white noise characterized by a noise ower of and a correlation time of 5 Simulation results are deicted in figure 9 Performance imrovements using the OPP and the MPC algorithms are similar to those observed in the revious simulations V CONCLUSIONS In this aer, the roblem of the integrated otimal control and scheduling was addressed theoretically A model of a control system and its limited communication network was described, allowing control and scheduling to be tightly couled in order to imrove the control erformance A new on-line scheduling algorithm for networked control systems was introduced Stability conditions for the systems scheduled using this algorithm were stated Simulation results show that erformance imrovements are significant and aroach those obtained using the model redictive control algorithm, which can be seen as an otimal on-line control and scheduling algorithm These imrovements are due to a more efficient use of the available network resources, which are dynamically allocated as a function of the states variables of the different systems REFERENCES [] E Sozer, M Stojanovic, and J Proakis, Underwater acoustic networks, IEEE Journal of Oceanic Engineering, vol 25, no, 72 83, January 2 [2] L Xiangheng and A Goldsmith, Wireless medium access control in networked control systems, in Proceedings of the 23rd American Control Conference, Boston, USA, June-July 24 [3] G Walsh and H Ye, Scheduling of networked control systems, IEEE Control Systems Magazine, vol 2, no, 57 65, February 2 [4] M Branicky, S Phillis, and W Zhang, Scheduling and feedback co-design for networked control systems, in Proceedings of the 4st IEEE Conference on Decision and Control, Las Vegas, USA, December 22 [5] J P Hesanha and Y Xu, Otimal communication logics for networked control systems, in Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, December 24 [6] D Hristu and K Morgansen, Limited communication control, System and Control Letters, vol 37, no 4, 93 25, July 999 [7] A Cervin, J Eker, B Bernhardsson, and K-E Årzén, Feedbackfeedforward scheduling of control tasks, Real-Time Systems, vol 23, no, 25 53, July 22 [8] D Simon, D Robert, and O Sename, Robust control/scheduling codesign: alication to robot control, in Proceedings of the th IEEE Real-Time and Embedded Technology and Alications Symosium, San Francisco, USA, March 25 [9] H Rehbinder and M Sanfridson, Scheduling of a limited communication channel for otimal control, Automatica, vol 4, no 3, 49 5, March 24 [] R Brockett, Stabilization of motor networks, in Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, Los Angeles, USA, December 995 [] L Palooli, A Bicchi, and A Sangiovanni Vincentelli, Numerically efficient control of systems with communication constraints, in Proceedings of the 4st IEEE Conference on Decision and Control, Las Vegas, USA, December 22 [2] A Bemorad and M Morari, Control of systems integrating logic, dynamics, and constraints, Automatica, vol 35, no 3, , 999 [3] L Lu, L Xie, and M Fu, Otimal control of networked systems with limited communication: a combined heuristic and convex otimization aroach, in Proceedings of the 42nd IEEE Conference on Decision and Control, Hawaii, USA, December 23 [4] M-M Ben Gaid and A Çela, Model redictive control of systems with communication constraints, in Proceedings of the 6th IFAC World Congress, Prague, Czech Reublic, July 25