Self-triggered sampling selection based on quadratic programming

Self-triggered sampling selection based on quadratic programming ⋆ P.Mill´ an, L. Orihuela, D.Mu˜ noz de la Pe˜ na, C. Vivas, F.R. Rubio Dpto. Ingenier´ ıa de Sistemas yAutom´ atica, Universidad de Sevilla, Camino de los Descubrimientos s/n, Isla de la Cartuja, Seville,Spain {pmillan,orihuela,davidmps,vivas,rubio}@cartuja.us.es Abstract: This paper proposes amodel-based control strategy for networked control systems subject to norm bounded disturbances. The communication channel is supposed to be shared with several processes and, therefore, the access to the network needs to be minimized to avoid collisions and packet losses. Wepropose to use avariable sample rate scheme in which the controller operates in open-loop between successivestate measurements. The sampling time is decided on-line solving asequence of quadratic optimization problems in order to minimize the access to the common network while guaranteeing closed-loop practical stability.Both discrete and continuous time schemes are considered. 1. INTRODUCTION Networked control systems (NCSs) are those in which ashared communication network links the sensors, controllers and actuators of several control loops, Zampieri [2008]. In some cases the use of networks is motivated by the very nature of the problem, while in others, the network is introduced to exploit the advantages that this architecture provides. In general, introducing acommunication network in the control loop may reduce the costs, increase the flexibility and maintenance of the system and facilitate system diagnosis. However,in many practical applications the inclusion of anetwork to interchange real time control information introduces anumber of shortcomings that must be addressed. When acertain number of devices are sharing acommunication channel, which in general is not intended for processes with real time requirements, different effects, such as data losses or time varying delays, may appear degrading the control performance and even unstabilizing the plant. Therefore, NCS involves anumber of challenging control problems that havebeen studied during the last decade. Some works address the problem of controlling aplant subject to these undesired network induced dynamics. Forinstance, the effects of the quantization phenomenon and newadvanced quantization strategies havebeen investigated in Brockett and Liberzon [2000], Canudas-de Wit etal. [2006] and Elia and Mitter [2001]. Another important research direction deals with NCS in which the network induced delays, that are in general time-varying, becomes important in the control performance or stability,see e.g. Yue etal. [2005], Jiang et al. [2008] or Mill´an et al. [2009]. The problem of the network package dropouts has also receivedgreat attention and has been addressed using predictivecontrol and buffering techniques in Mu˜noz de la Pe˜na and Christofides [2008] and Mill´an et al. [2008]. Another approach consists on controlling the plan while minimizing the access to the network by using avariable sampling ⋆The authors would liketoacknowledge CICYT (Grant DPI2010-19154), and the European Commission (EC) (FeedNetBack Project, grant agreement 223866), for funding this work. rate in which measurements are only sent when theyare indeed necessary.Minimizing the network load is critical in large scale systems in which the amount of data transmitted may be very large. In this way,instead of using aconstant sampling period, network access is scheduled and employed only when necessary. There can be found in the literature two different frameworks for the problem of selecting the sampling times: event-based and self-triggered control. Under the former framework (Arz´en [1999], Tabuada [2007], Hetel et al. [2008], Cogill [2009], Lunze and Lehmann [2010]) the controller execution is triggered according to the state or output of the plant, which requires acontinuous monitoring of the state of the plant. This drawback does not appear in the latter approach (Anta and Tabuada [2008], Cogill [2009], Anta and Tabuada [2010]). Selftriggered systems try to emulate the event-based ideas, but avoiding the continuous measuring of the state and, hence, the implementation problems this incurs. It is worth mentioning the difference existing between these approaches and other control schemes in the context of robust stability of NCS subject to time-varying sampling instants (Suh [2008], Fujioka [2009], Fujioka et al. [2010]) in which, although intervals between sampling time are also time-varying, there is no freedom in the choice of the following sampling instant. Aiming at reducing communication rates, different control strategies resort to the idea of using aplant model in the controller side. This idea has provenits effectiveness not only for periodic sampling (Montestruque and Antsaklis [2003, 2004], Orihuela et al. [2009]), butalso for event-based control of linear stable systems (Lunze and Lehmann [2010]). In this work the problem of reducing the use of abandwidthlimited channel is tackled in adifferent manner.Ascenario with acommunication network in the sensor-to-controller path is considered. The system is alinear time invariant plant, subject to bounded additivedisturbances. Starting from the knowledge of astabilizing feedback controller and an associated Lyapunov function, amodel-based controller is proposed in which Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011 978-3-902661-93-7/11/$20.00 © 2011 IFAC 8896 10.3182/20110828-6-IT-1002.02837 … Network x(k+1)=Ax(k)+Bu(k)+ ω (k) u(k)=Kxc(k)xc(k)=Axc(k)+Bu(k) device device collisions x(ks) ks+1 Fig. 1. Networked control system the controller operates in open-loop between two consecutive samples. The following sampling time is decided by the controller,insuch away that practical stability is guaranteed while minimizing the number of access to the shared network, that is, while maximizing the time between successivesamples. In order to decide the following sampling time the controller must solveon-line several quadratic optimization problems (QP). This Lyapunov-based sampling policyresults in aself-triggered sampling strategy as in Mazo et al. [2009]. The main difference with that work is the use of amodel to minimize the access to the shared network. The problem is solved in discrete time, but it isalso included an extension for continuous systems with sampled state measurements. The following section presents adescription of the system, controller and network. Section 3describes the Lyapunovbased sampling policy.The extension for continuous plants is giveninSection 4. Asimulation example is shown in Section 5. Conclusions and future research proposal are summarized in Section 6. 2. PROBLEM FORMULATION Consider the following discrete time linear system givenby: x(k+1)=Ax(k)+Bu(k)+ ω (k),(1) where x(k)∈Rn,and u(k)∈Rmare the state vector and control input vector respectively.The process disturbance is ω (k)∈Rn, and satisfies ω (k)⊆W,where: W={ ω ∈Rn: ω (k)∞≤ γ , γ >0}.(2) It is assumed that afeedback local controller K,associated with adiscrete Lyapunovfunction V(x)=xTPx,has been designed for system (1) so that the control lawu(k)=Kx(k)ensures practical stability of the closed-loop system. Consider system (1) being controlled through anetwork. The inclusion of such anetwork in the control loop induces collisions and packet dropouts. This problem becomes more important as the number of devices connected to the network and the sampling frequency ofsuch devices grow.In order to control the system while minimizing the network traffic load, we resort to amodel-based controller given by the following equations: xc(k+1)=Axc(k)+Bu(k),(3) xc(ks)=x(ks),s=0,1,2... (4) u(k)=Kxc(k),(5) where ksare the discrete time instants in which the sensors measure the state of the plant and send it to the controller. Figure 1shows an scheme of the proposed control system. The model state is updated wheneveranewsample arrives. Then, the model evolves in open-loop until another measure reaches the controller.The main difference between this approach and the one of Montestruque and Antsaklis [2003] is that, here, the following sampling time is decided on-line by the controller.As Figure 1suggests, the controller is close to the plant, hence, the same control signal is being applied to the system and is being fed to the model. Acommunication protocol between the sensors and the controller is assumed to be operating, in such away that it is possible for the controller to decide the sampling instants. This could be performed, for instance, if the controller sends apacket to the sensors which contains the following sampling instant. The arrivalofthis packet triggers asensor event-based protocol that samples and sends the state of the plant at the appropriate time. Under these considerations closed-loop equations of system (1) and controller (3)-(5) while the latter is evolving in open-loop are givenby: x(k+1)=Ax(k)+BKxc(k)+ ω (k),(6) xc(k+1)=(A+BK)xc(k),∀k∈[ks,ks+1),(7) xc(ks)=x(ks),s=1,2,... (8) ks+1=f(x(ks)),(9) where time ks,with s=1,2,...,are the time instants in which the controller receivesthe measurements form the sensor.The sampling instants ksare calculated by the controller based on the state measurements received. In the next section, we present amethod to decide the next sampling instant ks+1based on the system model, the controller gain K,the Lyapunovfunction Vand the latest state measurement x(ks)in order to minimize the access to the network while guaranteing closed-loop practical stability. 3. LYAPUNOV-BASED SAMPLING PROCEDURE This section describes the proposed procedure to minimize the access to the network while preserving closed-loop practical stability. In viewofequation (1) and equations (3)-(5), the model error δ (k)can be defined as: δ (k)x(k)−xc(k),(10) where δ (ks)=0,∀ks.Apossible evolution of the state of the system x(k),the controller xc(k)and the error δ (k)is drawn in Figure 2. The dynamics of the controller state and the model error between two consecutivesampling times can be written as follows: xc(ks+j)=(A+BK)jx(ks),∀j∈N:{ks+j<ks+1}(11) δ (ks+j)= j ∑ i=1 Ai−1 ω (ks+j−i),∀j∈N:{ks+j<ks+1}(12) Obviously,from equations (2) and (12), one can see that: 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011 8897 ( ) ( ) k x(k) xc(k) δ (k) ksks+1ks+2ks+3 Fig. 2. Possible evolution of the state and the model error  δ (ks+j)∞< γ j ∑ i=1 Ai−1∞,(13) for all ω (ks+i),with i=0,..., j−1. In what follows, the Lyapunov-based sampling procedure is developed starting from the available pair controller-Lyapunovfunction and taking into account the closed-loop equations (6)-(9). The forward difference of the Lyapunovfunction for k∈[ks,ks+1)yields: ∆V(ks,ks+j)=V(x(ks+j))−V(x(ks)) =xT(ks+j)Px(ks+j) −xT(ks)Px(ks),∀j∈N:{ks+j<ks+1}(14) Now,substituting x(k)from equation (10), ∆V(ks,ks+j)= δ T(ks+j)P δ (ks+j)+2xT c(ks+j)P δ (ks+j) +xT c(ks+j)Pxc(ks+j)−xT c(ks)Pxc(ks).(15) The controller’sgoal is to maximize the next sampling instant ks+1while guaranteeing that the forward difference is negative for all possible disturbances in order to ensure practical stability.Tothis end, the controller solves the following optimization problem. max ks+1(16) subject to: ∆V(ks,ks+j)≤0,∀j∈N:{ks+j<ks+1} ∀ ω (ks+i),i=0,..., j−1 Next, we present an algorithm which solves (16) to determine the next sampling time in an iterativemanner: Algorithm 1. (1) Set j=1. (2) Solvethe problem min δ −∆V(ks,ks+j)(17) subject to:  δ ∞< γ j ∑ i=1 Ai−1∞. (3) If ∆V(ks,ks+j)≤0, increase j=j+1and go to Step 2. Otherwise, choose ks+1=ks+j. Algorithm 1increases ks+1iteratively while aworst case bound on the difference between the value of the Lyapunovfunction of the current state and the state corresponding to the next sampling time is negative.Once this constraint does not hold, the algorithm stops and decides the next sampling time. This implies, that the V(x(ks)) is adecreasing sequence of values with alower bound (given by the size of the uncertainty), and hence that the closed-loop system is practically stable. Next, will provethat this optimization problem can be stated as aQPproblem. First of all, the standard QP problem is introduced, see Nocedal and Wright [2006]. Quadratic programming problem. Assume ξ belongs to Rp space. The p×pmatrix His symmetric, and fis anyf×1 vector.The QP problem is stated as min ξ g( ξ )=1 2 ξ TH ξ +fT ξ +c,(18) subject to D ξ ≤b(inequality constraint) (19) Weprovenext that problem (17) can be stated as aQP. Proposition 1. Problem (17) can be formulated as aQPifthe elements of equations (18)-(19) are chosen as ξ = δ , H=−2P, fT=−2(A+BK)jxc(ks)TP, c=−(A+BK)jxc(ks)TP(A+BK)jxc(ks) +xT c(ks)Pxc(ks),(20) and for the inequality constraint D=In −In,b= γ j ∑ i=1 Ai−1∞¯ 1n −¯ 1n.(21) where ¯ 1nis acolumn vector of dimension nwhose components are ones and Inis the identity matrix with dimension n. Proof.The proof is straightforward from equations (15) and (18)-(19). 2 In viewofProposition 1, the controller needs to solveseveral QP problems in Algorithm 1tofind the next sampling time. Remark. As one can see, in Algorithm 1the minimum sampling time is one. It is not possible to ensure that the Lyapunov function decreases for all kbecause of the presence of bounded disturbances ω (k),which can make∆V(k,k+1)strictly positivein aneighborhood of the origin. However,it is worth reminding that, by assumption, the system practical stability is guaranteed for the controller Kwith sampling time equal to one. It is important to remark that the QP problem that needs to be solved in order to solve(17) is amulti-parametric QP problem (mpQP), for which the explicit solution can be obtained, see Bemporad et al. [2002]. In particular,the parameter θ ∈Rn+1 of the mpQP problem is θ (ks,j)=  xc(ks+j) j ∑ i=1 Ai−1∞ . This allows implementing the proposed variable sample control scheme efficiently. 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011 8898 4. EXTENSION TOCONTINUOUS-TIME SYSTEMS The control scheme presented in the previous section can be readily extended to continuous-time systems under the following assumptions. Consider the following continuous time linear system subject to bounded disturbances. ˙x(t)=Ax(t)+Bu(t)+ ω (t),(22) where x(t)∈Rnis the state vector,u(t)∈Rmis the control input vector and ω (t)∈W⊂Rnis the process disturbance where: W={ ω ∈Rn: ω (t)∞≤ γ , γ >0}.(23) Weassume that there exists alinear controller u(t)=Kx(t) that asymptotically stabilizes the nominal system (system (22) with ω (t)=0) with acorresponding LyapunovfunctionV(x)= xTPx. System (22) is controlled through anetwork with the same structure as that in Figure 1, which implements the following model-based controller (which is the continuous time version of (3)-(5)): ˙xc(t)=Axc(t)+Bu(t),∀t∈[tk,tk+1)(24) u(t)=Kxc(t),(25) xc(tk)=x(tk),k=0,1,2... (26) where tkis the sampling time (equivalent to ks)inwhich the sensors send newinformation to the controller.ALyapunovbased control design procedure can nowbefollowed similarly to that in Section 3. Wedefine the model error variable δ (t)as δ (t)x(t)−xc(t).(27) The dynamic of the error equation nowbecomes ˙ δ (t)=˙x(t)−˙xc(t) =Ax(t)+Bu(t)+ ω (t)−Axc(t)−Bu(t), =A δ (t)+ ω (t),∀t∈[tk,tk+1).(28) Thus, the dynamics of the controller state and the model error between two consecutivesampling times evolves as: xc(t)=e(A+BK)(t−tk)xc(tk),∀t∈[tk,tk+1)(29) δ (t)=eA(t−tk) δ (tk)+t tk eA(t− τ ) ω ( τ )d τ = =t tk eA(t− τ ) ω ( τ )d τ ,∀t∈[tk,tk+1).(30) The following proposition is needed for further developments. Proposition 2. If the dynamics of the error variable is given by (30), the error can be bounded as follows:  δ (t)∞≤ γφ (t,tk)(31) where φ (t,tk)= 1 A∞(eA∞(t−tk)−1)and A∞is the infinite norm of A. Proof.Taking into account equation (30), the norm of the error can be bounded as follows:  δ (t)∞=t tk eA(t− τ ) ω ( τ )d τ ∞≤t tk eA(t− τ )∞ ω ( τ )∞d τ ≤ γ t tk eA∞(t− τ )d τ = γ 1 A∞ (eA∞(t−tk)−1). 2 In what follows, the Lyapunov-based sampling procedure is developed. The controller’sgoal is to maximize the next sampling instant tk+1,while guaranteeing that the derivativeof the Lyapunov function is negativefor all possible disturbances. Taking the time derivativeof the Lyapunovfunction for t∈[tk,tk+1) yields d dtV(t)=xT(t)P˙x(t)+˙xT(t)Px(t)=2xT(t)P˙x(t).(32) Now,substituting x(t)from equation (27), ˙ V(x(t)) =2( δ T(t)+xT c(t))P(˙ δ (t)+˙xc(t)) =2( δ T(t)+xT c(t))P(A δ (t)+ ω (t)+Axc(t)+Bu(t)) = δ T(t)(PA +ATP) δ (t)+2 δ T(t)P ω (t)+2xT c(t)P ω (t) +2 δ T(t)(PA +ATP+PBK)xc(t)+ +xT c(t)P(A+BK)+(A+BK)TPxc(t),∀t∈[tk,tk+1).(33) In the following algorithm we will ensure the negativedefiniteness of an upper bound on the time derivativeof the Lyapunov function (33) at time tfor all possible uncertainty trajectories. The objectiveof the controller is to maximize tk+1while guaranteing that the time derivativeof the Lyapunovfunction is negativefor all possible disturbances; that is, max tk+1(34) subject to: d dtV(x(t)) ≤0,∀t∈[tk,tk+1) (22)−(26). This optimization problem is very difficult to solve. The parameter to be optimize, i.e. tk+1,isinvolved in anonlinear equation and there are an infinite number of constraints, because they must be satisfied for all t∈[tk,tk+1).Inorder to obtain the next sampling time, we propose to define tk+1=Tmin +n∆and find the maximum nsuch that the time derivativeof the Lyapunov function is negativeat those time instants for all possible disturbances. Wepresent next an iterativealgorithm that under mild assumptions provide an approximate solution to (34). Algorithm 2. (1) Set tk+1=tk+Tmin. (2) Solvethe problem min δ (tk+1), ω (tk+1)−d dtV(x(tk+1)) (35) subject to:  ω (tk+1)∞≤ γ  δ (tk+1)∞≤ γφ (tk+1,tk) (3) If ˙ V(x(tk+1)) ≤0, increase tk+1=tk+1+∆and go to Step 2. Otherwise, choose tk+1. 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011 8899 where Tmin is lower bound for the following sampling time. Next proposition shows that problem (35) can be stated as aQP. Proposition 3. Problem (35) can be formulated as aQPifthe elements of equations (18)-(19) are chosen as ξ = δ (t) ω (t), H=−2PA +ATPP P0, fT=−2xT c(tk+1)PA +ATP+KTBTPP , c=−xT c(tk+1)P(A+BK)+(A+BK)TPxc(tk+1), and for the inequality constraint D=   In0 −In0 0In 0−In   ,b=   γφ (tk+1,tk)¯ 1n γφ (tk+1,tk)¯ 1n γ ¯ 1n γ ¯ 1n   .(36) where ¯ 1nis acolumn vector of dimension nwhose components are ones and Inis the identity matrix with dimension n. Proof.The proof is straightforward from equations (33) and (18)-(19). 2 The value of ∆must be chosen small enough in away such that the dynamics of the controller state, and hence of the Lyapunov function, are smooth between two consecutivesampling, avoiding unexpected sign changes of the derivativeof the Lyapunov function from tkto tk+1.Ingeneral Tmin is chosen according with the minimum sampling time of the sensors. As in the discrete time case, the resulting QP problem is a mpQP,and hence, an explicit solution can be obtained. In this case the parameter of the QP problem is: θ (tk,j)=xc(tk+j∆) φ (tk+j∆,tk). Assuming that the sign of the time derivativeof the Lyapunov function does not change between two consecutivetimes, Algorithm 1provides asuboptimal solution to problem (34). Note that this assumption will be satisfied for asufficiently small ∆. The constraint on the upper bound of the time derivativeof the Lyapunovfunction is more restrictivethan the constraint on the difference of the Lyapunovfunction imposed in the discrete time controller studied in the previous section. Note that, in continuous time, aconstraint on the difference of the Lyapunovfunction does not yield aQPproblem and hence is more difficult to implement in real time. 5. NUMERICAL EXAMPLE In this section, we are going to apply the previous result to an unstable plant in order to showhowthe controller manages to reduce the traffic load maintaining the practical stability of the system. Consider the following discrete time LTIsystem: x(k+1)=2.72 2.70 02.69 x(k)+1 1.7u(k)+ ω (k),(37) 0 5 10 15 20 −200 −150 −100 −50 0 50 100 150 System’sstates k Fig. 3. Evolution of the system’sstates 0 2 4 6 8 10 12 14 0 0.5 1 1.5 2 2.5 3 3.5 4 Sampling intervals k Fig. 4. Evolution of the sampling intervals The initial condition for the system and the controller is x(0)T= [100 −20 ]. The following stabilizing controller has been designed for the discrete system: K=[−1.1868 −2.0415 ]. The Lyapunovfunction is defined by P=1061.67 0.67 0.67 0.60 . Suppose that this system is controlled using ashared network so it is interesting to reduce the number of access to the shared medium. Supposing that the disturbances are bounded  ω (k)∞≤1, the evolution of the system is illustrated in Figure 3, where the sampling instant are indicated with circles. In Figure 4the sampling instants obtained using the proposed method are shown. One can see that when the system is far of the equilibrium point it is possible to enlarge the sampling period still assuring asymptotic stability. Finally,the evolution of the Lyapunovfunction is drawn in Figure 5. 6. CONCLUSIONS Wehavepresented anovelmodel-based controller for networked systems, aimed at reducing the number of accesses to a shared network. It has been shown that, using predictions based on anominal model of the system, adequate asynchronous sampling times can be found by solving several QP problems which take explicitly into account the disturbances of the model. We 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011 8900 0 5 10 15 20 0 1 2 3 4 5 6 7 8 9 10 x 10 9 Value of V(k) k Fig. 5. Evolution of the Lyapunovfunction haveconsidered both discrete and continuous time controllers. The results havebeen demonstrated through simulation. REFERENCES A. Anta and P.Tabuada. Self-triggered stabilization of homogeneous control systems. 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