An improved distributed version of Han s method for distributed MPC of canal systems

Transcription

1 Deft University of Technoogy Deft Center for Systems and Contro Technica report An improved distributed version of Han s method for distributed MPC of cana systems M.D. Doan, T. Keviczky, and B. De Schutter If you want to cite this report, pease use the foowing reference instead: M.D. Doan, T. Keviczky, and B. De Schutter, An improved distributed version of Han s method for distributed MPC of cana systems, Proceedings of the 12th IFAC Symposium on Large Scae Systems: Theory and Appications, Vieneuve d Ascq, France, 6 pp., Juy Deft Center for Systems and Contro Deft University of Technoogy Mekeweg 2, 2628 CD Deft The Netherands phone: (secretary) fax: URL: This report can aso be downoaded viahttp://pub.deschutter.info/abs/10_013.htm

2 An improved distributed version of Han s method for distributed MPC of cana systems Minh Dang Doan Tamás Keviczky Bart De Schutter Deft University of Technoogy, Deft, The Netherands (e-mai: {m.d.doan,t.keviczky,b.deschutter}@tudeft.n) Abstract: Recenty, we have introduced a distributed version of Han s method that can be used for distributed mode predictive contro (DMPC) of dynamicay couped inear systems, under couping constraints (Doan et a., 2009). Some DMPC probems of water networks can be cast into this type. In this paper, we propose an improved version of this method and appy it to a cana system. The simuation resuts show that the modifications ead to faster convergence of the method, thus making it more practica in contro of water networks. Keywords: distributed optimization, mode predictive contro, water networks, dua decomposition, decentraized and cooperative contro 1. INTRODUCTION Optimization techniques have payed a fundamenta roe in designing automatic contro systems for most part of the past haf century. This dependence is even more obvious in today s wide-spread use of onine optimizationbased contro methods, such as Mode Predictive Contro (MPC) (Maciejowski, 2002; Rawings and Mayne, 2009). The abiity to express important process constraints and characterize comprehensive economic objective functions has made MPC the industry standard for controing argescae systems ranging from chemica processes to basic infrastructure. For contro of arge-scae networked systems, centraized MPC may be considered impractica, infexibe, and unsuitabe due to information exchange requirements and computationa aspects. The subsystems in the network may beong to different authorities that prevent sending a necessary information to one processing center. Moreover, the optimization probem yieded by centraized MPC can be excessivey arge for rea-time computation. In order to dea with these imitations, distributed mode predictive contro (DMPC) has been proposed for contro of such arge-scae systems, by decomposing the overa system into sma subsystems (Jia and Krogh, 2001; Camponogara et a., 2002; Rawings and Stewart, 2008). The subsystems then empoy distinct MPC controers that ony sove oca optimization probems, use oca information from neighboring subsystems, and coaborate to achieve gobay attractive soutions. Approaches to DMPC design differ from each other in the probem setup. For systems with decouped dynamics, Dunbar and Murray (2006) proposed a DMPC scheme focusing on mutipe vehices with couped cost functions, and utiizing predicted trajectories of the neighbors in each subsystem s optimization. A DMPC scheme with a sufficient stabiity test for dynamicay decouped systems was proposed by Keviczky et a. (2006), in which each subsystem optimizes aso over the behaviors of its neighbors. Richards and How (2007) proposed a robust DMPC method for decouped systems with couped constraints, based on constraint tightening and a seria soution approach. For systems with couped dynamics and decouped constraints Venkat et a. (2008) proposed a distributed MPC scheme, based on a Jacobi agorithm that deas with the prima probem, using a convex combination of new and od soutions. Other research reated to the DMPC fied is reported by Jia and Krogh (2002); Du et a. (2001); Li et a. (2005); Camponogara and Taukdar (2007); Mercangoz and Doye III (2007); Aessio and Bemporad (2007, 2008); Necoara et a. (2008). A recent survey on DMPC can be found in (Scattoini, 2009). Recenty, we have deveoped a distributed version of Han s parae method for convex optimization (Doan et a., 2009). The method aims to define oca controers for dynamicay couped subsystems, which share couping constraints and minimize a separabe objective function. Reying on a decomposition of the dua optimization probem such that oca probems have anaytica soutions, the agorithm has an iterative update procedure which converges asymptoticay to the goba optimizer of the prima probem. At each iteration, the controers exchange information with other neighboring subsystems, with which they are connected in terms of dynamics or constraint couping. In this paper, we present an improved distributed version of Han s parae agorithm for a cass of convex optimization probems (Han and Lou, 1988; Doan et a., 2009) and show that it is appicabe for DMPC of water networks. The improvements are iustrated in a simuation of the new DMPC scheme for a 4-reach cana system. The paper is organized as foows. The probem setup is described in Section 2. In Section 3, we summarize the origina Han s method and the distributed version, foowed by the new modified distributed version to speed up the convergence of the agorithm. The simuation resuts in Section 4 iustrate the properties of the DMPC scheme for the exampe

3 setup of the 4-reach cana. Section 5 concudes the paper and indicates some directions for future research. 2.1 The cana system 2. PROBLEM SETUP In this paper we iustrate the appication of the nove DMPC approach to the contro of a system of irrigation canas. Irrigation canas are arge systems, consisting of many interacting components, and spanning vast geographica areas. For the most safe and efficient operation of these canas, maintaining the eves of the water fows cose to pre-specified reference vaues is crucia, both under norma operating conditions as we as in extreme situations. Manipuation of the water fows in irrigation canas is done using devices such as pumps and gates. The exampe irrigation cana to be considered is a 4- reach cana system as iustrated in Figure 1. In this system, water fows from an upstream reservoir through the reaches, under the contro of 4 gates and a pump at the end of the cana system that discharges water. The contro design is based on the master-save contro paradigm, in which the master controers compute the fows through the gates, whie each save controer uses the oca contro actuators to guarantee the fow set by master controer (Schuurmans et a., 1999). We wi use the new DMPC method to design the master controers. upstream reservoir gate 1 reach 1 gate 2 reach 2 gate 3 Fig. 1. The exampe cana system 2.2 Modeing the cana reach 3 gate 4 reach 4 pump Subsystem modeing The cana system is divided into 4 subsystems, each of which corresponds to a reach and aso incudes the oca controer at the upstream gate of the reach. The 4 th subsystem has one more controer, corresponding to the pump at its downstream end. We first use a simpified mode for each subsystem as iustrated in Figure 2, and then obtain an overa mode by connecting subsystem modes. A subsystem is approximatey modeed by a reservoir with upstream in-fow and downstream out-fow. The discrete-time mode of reach i is represented by: h i k+1 h i k = T s [( ) Q i ( in Q i ) ] k out A i s where superscript i represents the subsystem index, subscript k is for the time index, T s is the samping time, h is the downstream water eve of the reach (zero eve k (1) is set at the autonomous steady state), A s is the water surface (voume of reservoir = h A s ), Q in and Q out are in-fow and out-fow of the cana which are measured at the upstream and downstream ends, respectivey. Denote the fow passing i th gate by q i, and the fow passing the pump by p 4. Due to the mass conservation aw, we have Q i out = Q i+1 in = q i+1, for i = 1,2,3, and Q 4 out = p 4. Q in Fig. 2. Mode of a reach A s h Q out In order to derive oca dynamics, we choose input and state vectors of subsystem i as x i k = h i k qk i, i = 1,2,3 u i [ ] k = q i k, i = 4 p i k The dynamics of each subsystem can be represented by a discrete-time, inear time-invariant mode of the form: x i k+1 = A ij x j k + Bij u j k, (2) j=1,,4 with the state-space matrices: A ii = 1, i = 1,,4; A ij = 0, i j B ii = T s /A i s, i = 1,2,3; B44 = [ ] T s /A 4 s T s /A 4 s B i(i+1) = T s /A i s, i = 1,2; B34 = [ T s /A 4 s 0 ] B ij = 0, j {i,i + 1} Centraized MPC probem The centraized MPC probem makes use of a quadratic cost function: 4 N 1 ( (u ) i TRi J = k u i k + ( x i ) TQi k+1 xk+1) i (3) i=1 k=0 in which N is the prediction horizon, and {Q i,r i } i=1,,4 are given positive definite weights. It is easy to verify that this cost function can be rewritten as J = x T Hx where H is a bock-diagona, positive definite matrix. The constraints of the optimization probem incude dynamica constraints (i.e. the mode equations), initia state constraint, termina constraint x i N = 0,i = 1,,4, and oca state and input constraints: u i k u i max, x i k x i max Foowing the method of Doan et a. (2009), the optimization probem to be soved by the centraized MPC controer at each samping interva can be represented in a compact form as min x x T Hx (4) s.t. a T x = b, = 1,...,n eq a T x b, = n eq + 1,...,s with s = n eq + n ineq, where n eq and n ineq are the number of scaar equaity and inequaity constraints, respectivey.

4 3. DISTRIBUTED MODEL PREDICTIVE CONTROL METHOD The optimization probem (4) wi be soved by a distributed agorithm that is based on Han s parae method for convex programs (Han and Lou, 1988). In the foowing we wi give a summary of Han s method for convex quadratic programs, and then describe the distributed version of Han s method for quadratic programs in the form (4). Then we wi proceed by describing the modified distributed version, which is the main contribution in this paper. The main idea behind the distributed version of Han s method is iustrated in Figures 3 and 4, with a simpe system consisting of 4 subsystems and the couping matrix that shows how subsystems are couped via their variabes (boxes on the same row iustrate the variabes that are couped in one constraint). In Han s method using goba variabes, a subsystem has to communicate with a other subsystems in order to compute the updates of the goba variabes. For the distributed version of Han s method, each subsystem ony communicates with the other subsystems of which the variabes are necessary for computing the updates of its oca variabes. 3.1 Han s parae method for convex quadratic programs The origina Han s method considers genera convex optimization probems where the constraint is an intersection of many convex sets. The agorithm is based on Fenche s duaity to perform a dua decomposition, and iterativey projects the dua variabes onto oca constraint sets. The sum of dua variabes can be shown to converge to the minimizer of the dua probem (Han and Lou, 1988). A simpified version of Han s method for the quadratic optimization probem (4) is summarized in Agorithm 1. Agorithm 1. Han s method for convex programs Choose parameter α big enough 1. For p = 1,2,...: 1) For = 1,...,s, find z (p) z 1 that soves min z + αy(p 1) x (p 1) s.t. a T z = b or a T z b ( ) 2) Assign y (p) = y (p 1) + (1/α) z (p) x (p 1) 3) Set y (p) = y (p) y s (p) 4) Compute: x (p) = H 1 y (p) In this representation, each vector y is a dua variabe corresponding to th constraint. For probem (4), Han s method was proved to converge to the goba optimum if the cost function is strongy convex, or equivaenty if H is positive definite (Han and Lou, 1988). An interesting property of this method is that the number of parae processes is equa to the number of constraints (as opposed to other dua decomposition methods where the number of parae processes often equas the number of variabes). 3.2 Distributed version of Han s method Han s agorithm invoves cacuation of the goba variabes, therefore a goba coordination method is required. A distributed version of Han s method was proposed by Doan et a. (2009), and it makes use of the expicit soutions in Step 1 of Agorithm 1, and expoits the structure of (4) to decompose the computations, hence avoiding goba communications. 1 Han and Lou (1988) recommended α = α 0 s/ρ, where s is the number of constraints and ρ is one haf of the smaest eigenvaue of H Fig. 3. Communication inks of the 2 nd subsystem in the centraized coordination version of Han s agorithm for an exampe 4-subsystem probem. An update for a goba variabe requires the 2 nd subsystem to communicate with a the others Fig. 4. Communication ink of the 2 nd subsystem in the distributed coordination version of Han s agorithm for an exampe 4-subsystem probem. The 2 nd subsystem ony cares about its oca variabe, therefore it does not need to communicate with the others that do not coupe with it. The distributed version of Han s method was proved to achieve the same convergence property as the origina method of Han (Doan et a., 2009). 3.3 Modifications of Han s method to speed up convergence A disadvantage of Han s method (and its distributed version) is the sow convergence rate, due to the fact that it is essentiay a projection method to sove the dua probem of (4). Therefore, we need to modify the method to achieve better convergence rate. In this paper, we present 2 modifications of the distributed version of Han s method:

5 Scaing of the step sizes reated to dua variabes by using different α vaues for the update of each dua variabe instead of the same α for a dua variabes. Use of nonzero initia guesses, which aows taking the current MPC soution as the start for the next sampe step. We wi use the same notations as in Doan et a. (2009, Section VI), which are briefy summarized beow: L i : the set of indices of constraints that subsystem i is responsibe for updating their dua variabes throughout the agorithm. N i : the neighborhood of subsystem i, consisting of i itsef and other subsystems that have direct dynamica or constraint coupings with subsystem i. L N i: the set of indices of constraints within responsibiity of a subsystems in N i. x (p) i : the sef image of the goba variabe vector x (p) made by subsystem i; this vector has the same size as x (p), containing a variabes of subsystem i at the right positions, and zeros for the other entries. x (p) N i : the neighborhood image of x (p) made by subsystem i, using variabes of a subsystems inside N i at the right positions, and zeros for the other entries. x (p) N i assumed : the assumed neighborhood image x(p) made by subsystem i. The difference between x (p) N i assumed and x (p) N i is that ony the vaues of variabes beonging to subsystem i are correct, whie for the variabes of other neighboring subsystems j {N i \i}, the vaues coud be different from the rea ones. I i : index matrix of subsystem i; it is the mask for the goba variabe x such that ony variabes of subsystem i are kept, i.e. x (p) i = I i x (p). We present the improved distributed version of Han s method in the foowing agorithm: Agorithm 2. Improved distributed agorithm for the MPC optimization probem Pre-computed parameters: Each subsystem i computes and stores the foowing parameters throughout the contro scheme: For each L i : α = ( k α ) α 0, where k α is the scaing vector. α acts as oca step size regarding th dua variabe, and therefore k α shoud be chosen such that the convergence rates of a s dua variabes are improved. The method to choose k α wi be discussed in this section. For each L i : c = 1 a T a H 1 a. We can see that c can be computed ocay by a oca controer with a priori knowedge of the parameter a and the weighting bocks on the diagona of H that correspond to the non-zero eements of a. MPC step: At the beginning of the MPC step, the current states of a subsystems are measured. The sequences of predicted states and inputs generated in the previous MPC step are shifted forward one step, then we add zero states and zero inputs to the end of the shifted sequences. The new sequences are then used as the initia guess for soving the optimization probem in the current MPC step. The initia guess for each subsystem can be defined ocay. For subsystem i, denote the initia guess as x (0) i. At the first MPC step, we have x (0) i = 0, i. The idea of using previousy predicted states and inputs for initiaization is a popuar technique in MPC (Rawings and Mayne, 2009). Especiay with Han s method, whose convergence rate is sow, an initia guess that is cose to the optima soution wi be very hepfu to reduce the number of iterations. The current state is pugged into the MPC probem, then we get an optimization probem of the form (4). This probem wi be soved in a distributed way by the foowing iterative procedure. Distributed iterative procedure to sove the optimization probem: Initiaize with p = 0. Each subsystem i communicates with the neighbors j N i to get x (0) j, then constructs x (0) N i = j N i x(0) j. Subsystem i computes its oca dua variabe y (0) N i = Hx (0) N i, and then computes initia intermediate variabes: γ (0) = max{a T (x (0) N i y (0) N i ) b,0}, L i Next, for p = 1,2,..., the foowing steps are executed: 1) Communications to get the updated main variabes Each controer i communicates with its neighbors j N i to get updated vaues of their variabes, contained in x (p 1) j. Vice versa, i aso sends its updated variabes in x (p 1) i to its neighbors as requested. After getting information from the neighbors, controer i constructs the neighborhood image x (p 1) N i as: x (p) N i = j N i x (p) j 2) Update intermediate variabes γ in parae In this step, the oca controers update γ corresponding to each constraint under their responsibiity. More specificay, each oca controer i updates γ for each L i in the foowing manner: If constraint is an equaity constraint ( {1,...,n eq }), then γ (p) = a T x(p 1) N i +γ (p 1) b. If constraint is an inequaity constraint ( {n eq +1,...,s}), then γ (p) = max{a T x(p 1) N i + γ (p 1) b,0}. 3) Communications to get the updated intermediate variabes Each oca controer i communicates with its neighbors to get updated γ (p) vaues that the neighbors just computed in Step 2). 4) Update main variabes in parae Loca controer i uses a γ (p) vaues that it has (by communications and those computed by itsef) to compute an assumed neighborhood image of x:

6 x (p) N i assumed = L N i 1 α γ (p) c (5) Then controer i seects the vaues of its variabes in x (p) N i assumed to construct the new sef image: x (p) i = I i x (p) N i assumed (6) which contains u i,(p) 0,...,u i,(p) N 1,xi,(p) 1,...,x i,(p) N. After updating their variabes, each oca controer checks the oca termination criteria. When a oca controers have converged 2, the agorithm stops and the oca contro actions are impemented, otherwise the controers proceed to Step 1) to start a new iteration. Impement MPC input: When the iterative procedure finishes, each subsystem appies the first input u i,(p) 0, then waits for the next state measurement to start a new MPC step. Method to choose the scaing vector: In the modified version of distributed Han s method, a good choice of the scaing vector heps to dramaticay improve the convergence speed. We have observed that the convergence rate of some dua variabes under the responsibiity of a subsystem i wi affect the convergence rate of dua variabes under the responsibiity of its neighbors in N i. Therefore the choice of scaing vector shoud focus on improving the convergence rate of sower convergent dua variabes. In our simuation, we rey on the Hessian to find the scaing vector. Specificay, for a subsystem i whose variabes have the average weight h i (e.g. average of entries reated to i s states and inputs in the diagona of the Hessian), we choose the scae factor ( k α ) = 1/ h i, with a L i. We aso mutipy the scaing vector k α with a factor θ < 0 for enarging the step sizes of a dua variabes; this θ is tuned in the first MPC step. The choice of the scaing vector depends on the structure of the centraized optimization probem, thus we ony need to choose it once in the first MPC step. Then for the next MPC steps, we can reuse the same scaing vector. 4. SIMULATION RESULTS AND DISCUSSION DMPC methods are appied to the reguation probem of the simuated cana system of Section 2, which has a perturbed initia state. We use distributed Han s method with and without the modifications described in Section 3.3 for the same setup, and compare the resuts. Figure 5 shows that the distributed Han s method with modifications achieves better convergence rate, aowing the distributed optimization to converge within an acceptabe number of iterations. A simuation of cosed-oop MPC is performed for 20 sampe steps. Figure 6 shows that the distributed soutions converge to the centraized soutions in every sampe step. 2 Checking the termination criteria in a distributed fashion requires a dedicated ogic scheme, the description of which is omitted for brevity. x * cent x(p) dist 2 / x* cent Od distributed agorithm using common α Improved method using scaed α p Fig. 5. Comparison of convergence rates of the former and the new distributed versions of Han s method in the first samping time (k=1) x * cent x(p) dist 2 / x* cent p Fig. 6. Normaized norm of difference between the centraized and the distributed soutions versus the iteration step p and sampe step k Athough the new scheme is verified by this simuation, there are sti severa theoretica issues that need to be addressed: Firsty, there is no convergence proof for the modified distributed version of Han s method yet. We observe that in setups that are more compicated, the method to choose scaing vector proposed in this simuation does not aways work we (sometimes after severa sampe steps, the agorithm does not converge in the next sampe steps). Note that with this method we aim to sove the dua probem, therefore the prima iterate woud be infeasibe uness the agorithm converges. Secondy, in the MPC formuation we keep both inputs and states as variabes of the centraized optimization probem. This formuation is advantageous in distributed MPC because the Hessian wi have a diagona structure, and the neighborhood of each subsystem wi ony contain its direct neighbors (the neighborhood woud be greaty extended if we eiminate the states in the optimization probem). However, using states as variabes requires considering the dynamica equations as equaity constraints of the optimization probem, and the existence of equaity constraints typicay requires an exact soution in order 10 k 20

7 to guarantee feasibiity. In future research, we wi aso study MPC formuations in which a states are eiminated, so that the centraized optimization ony has inequaity constraints. Such formuation woud aow stopping the agorithm in a finite number of steps, and the fina iterate coud be feasibe (athough it may be suboptima). Another probem is that the proposed method is for quadratic programs ony. Athough many MPC probems for inear time-invariant systems are formuated as quadratic programs, there are other variants that use different objective functions, and noninear MPC woud aso yied more compicated optimization probems than quadratic programs. With such probems, we might not be abe to impement Han s parae method in a distributed fashion. This issue motivates the research for other decomposition methods that can hande more genera probems, e.g. convex probems with inear or decouped noninear constraints. Last but not east, the MPC formuation in this paper empoys the termina constraint x N = 0, which is conservative since it reduces the domain of attraction of MPC. An improvement coud be made by repacing this constraint with ess restrictive conditions (e.g. termina constraint set and termina controer). However, there is sti no distributed scheme to construct the termina constraint set and the termina controer (and aso the termina penaty matrix that is soution of the Riccati equation), other than assuming them to be competey decouped. 5. CONCLUSIONS The modified distributed version of Han s method has an improved convergence rate, thus it is more suitabe for DMPC of arge-scae water networks. Future research wi invove finding a way to construct the scaing vector of the modified distributed version of Han s method together with a theoretica proof of the convergence. We wi aso investigate different distributed optimization methods using dua decomposition techniques to address noninear MPC with more genera optimization probem. Another direction is to find distributed MPC schemes for suboptima MPC. ACKNOWLEDGEMENTS Research supported by the European 7th framework STREP project Hierarchica and distributed mode predictive contro, contract number INFSO-ICT REFERENCES Aessio, A. and Bemporad, A. (2007). Decentraized mode predictive contro of constrained inear systems. In European Contro Conference, Kos, Greece. Aessio, A. and Bemporad, A. (2008). Stabiity conditions for decentraized mode predictive contro under packet drop communication. In American Contro Conference, Camponogara, E., Jia, D., Krogh, B., and Taukdar, S. (2002). Distributed mode predictive contro. IEEE Contro Systems Magazine, 22(1), Camponogara, E. and Taukdar, S. (2007). Distributed mode predictive contro: Synchronous and asynchronous computation. IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, 37(5), Doan, D., Keviczky, T., Necoara, I., Dieh, M., and De Schutter, B. (2009). A distributed version of Han s method for DMPC using oca communications ony. Contro Engineering and Appied Informatics, 11(3), Du, X., Xi, Y., and Li, S. (2001). Distributed mode predictive contro for arge-scae systems. In American Contro Conference, voume 4, Dunbar, W.B. and Murray, R.M. (2006). Distributed receding horizon contro for muti-vehice formation stabiization. Automatica, 42, Han, S.P. and Lou, G. (1988). A parae agorithm for a cass of convex programs. SIAM Journa on Contro and Optimization, 26(2), Jia, D. and Krogh, B. (2002). Min-max feedback mode predictive contro for distributed contro with communication. In American Contro Conference, voume 6, Jia, D. and Krogh, B. (2001). Distributed mode predictive contro. In American Contro Conference, voume 4, Keviczky, T., Borrei, F., and Baas, G.J. (2006). Decentraized receding horizon contro for arge scae dynamicay decouped systems. Automatica, 42, Li, S., Zhang, Y., and Zhu, Q. (2005). Nash-optimization enhanced distributed mode predictive contro appied to the she benchmark probem. Information Sciences, 170(2-4), Maciejowski, J.M. (2002). Predictive Contro with Constraints. Prentice Ha, Harow, Engand. Mercangoz, M. and Doye III, F.J. (2007). Distributed mode predictive contro of an experimenta four-tank system. Journa of Process Contro, 17(3), Necoara, I., Doan, D., and Suykens, J. (2008). Appication of the proxima center decomposition method to distributed mode predictive contro. In IEEE Conference on Decision and Contro, Rawings, J.B. and Mayne, D.Q. (2009). Mode Predictive Contro: Theory and Design. Nob Hi Pubishing, Madison, Wisconsin. Rawings, J.B. and Stewart, B.T. (2008). Coordinating mutipe optimization-based controers: New opportunities and chaenges. Journa of Process Contro, 18(9), Richards, A. and How, J. (2007). Robust distributed mode predictive contro. Internationa Journa of Contro, 80(9), Scattoini, R. (2009). Architectures for distributed and hierarchica mode predictive contro - A review. Journa of Process Contro, 19(5), Schuurmans, J., Hof, A., Dijkstra, S., Bosgra, O.H., and Brouwer, R. (1999). Simpe water eve controer for irrigation and drainage canas. Journa of Irrigation and Drainage Engineering, 125(4), Venkat, A., Hiskens, I., Rawings, J., and Wright, S. (2008). Distributed MPC strategies with appication to power system automatic generation contro. IEEE Transactions on Contro Systems Technoogy, 16(6),