Towards to dynamic optimal control for large-scale distributed systems

Shaoyuan LI

Department of Automation,Shanghai Jiao Tong University,Shanghai 200240,China

Towards to dynamic optimal control for large-scale distributed systems

Shaoyuan LI

Department of Automation,Shanghai Jiao Tong University,Shanghai 200240,China

1 Dynamic optimal control for large-scale distributed systems

There is a class of complex,large-scale systems that are composed of many interacted and spatially distributed subsystems such as the power grids,transportation systems,and large scale chemical processes.With the progress of industrial technology,field bus,integrated circuit technologies and network communication technology,this automation system has taken a totally new configuration in which each subsystem owns only one hardware controller for controlling the subsystem itself,and all hardware controllers are connected by networks for exchanging information with each other or with upper layer controllers.In this distributed(or decentralized)framework,the distributed control algorithms are usually adopted because the classical centralized control solutions are often impractical due to their heavy computational demands and the lack of fault tolerance.

Among the distributed optimization algorithms,the distributed model predictive control(DMPC)that was formally proposed in 2001 in ACC[1,2],where each MPC controls an individual subsystem and coordinates with each other by making use of exchanging information,is the most frequently studied method in recent years.DMPC not only inherits MPC’s abilities of good optimization performance,explicitly accommodating constraints,and strong applicability,but also possesses the advantages of the distributed framework of error-tolerance with less computational effort,and the flexibility to the system structure.Especially,with the development of industrial 4.0 and cyber-physical systems(CPS),the flexible distributed optimization becomes even more important and DMPCs have been applied in increasingly wide industrial fields[3,4].

Comparing with the centralized MPC,DMPC is more flexible in control system configuration with higher computational speed and better error tolerance characteristics.Even when the optimization performance of DMPC is lower than that of the centralized MPC,DMPC can improve its optimization performance by making use of the potential capability of interacted subsystem’s information regardless of the independent characteristics and flexibility of centralized MPC from the point of view of algorithms.

Thus,how to design the information structure of each subsystem-based controller,and how to fully use the exchanged information so as to obtain a perfect coordination strategy for improving the optimization performance of the entire closed-loop system with high structural flexibility and low error tolerance for different types of systems are always the major duties and goals in the research field related to DMPC.In addition,since there are errors existing between the predictive state trajectories and the presuming state sequences,which destroy the recursive feasibility of each local MPC,how to design DMPCs to guarantee the recursive feasibility and the asymptotic stability with weakly appended constraints becomes one of the most difficult problems in theoretical aspects of DMPC.

2 The development of DMPC

So far, many DMPC algorithms have been developed for different purposes. The following are the recent progresses around the major problems of DMPC mentioned above.

The simplest and most frequently adopted strategy is that each local controller minimizes its own subsystem’s cost and uses the state prediction of the previous time instant to approximate the state sequence at the current time instant in computing the optimal solution[1,2].Based on this strategy,we proposed an iterative algorithm for obtaining the “Nash Optimality”of the closed-loop system[5].

Another commonly used coordination strategy, called cooperative DMPC, to improve the optimization performance of the entire closed-loop system is that each subsystem-based MPC optimizes the cost of the overall system to improve the global performance[6].In computing the optimal solution,we also use the state prediction of the previous time instant to approximate the state sequence at the current time instant.If the iterative algorithm is used,the“Pareto Optimality”can be obtained.This strategy can achieve a good global performance in some cases,but it reduces the flexibility and increases the communication load.To increase the flexibility of the closed-loop system,we developed the“Neighborhood Optimization-based”DMPC where the subsystems are described by input interacted models[7],and the N-step adjacent region optimization-based DMPC where each subsystem is described by state interacted models.In these two methods,each subsystem optimizes all the subsystems it will impact during the optimization horizon.The solution provided by both methods is equivalent to the cooperative DMPC,but the communication effort is less than the cooperative DMPC.The “Pareto Optimality”is obtained by the iterative version algorithms.

In an effort to achieve a trade-off between the global performance of the entire system and the computational burden,recently,we proposed an intuitively appealing strategy in[8,9],where each subsystem-based MPC only considers the cost of its own subsystem and those of the subsystems it directly impacts.Such a design can be referred to as impacted-region cost-optimization-based DMPC(ICO-DMPC).In particular,we apply this design idea to a metallurgy system,see[10],and explains why this coordination strategy can improve the global performance.Numerical data and application experiments show that this coordination strategy can obtain a performance close to that of a classical centralized MPC.

In the DMPC framework,the design of the stabilized controller that takes state and/or input constraints into account without losing its advantage of flexible for system structure is an important and challenging problem.[11]provides the weakly coupled nonlinear continuous systems a design method which uses the consistent constraint to limit the error between the future state sequences(or called presumed sequences)of upstream neighbors,which are calculated based on the solution in the previous time instant,and the predictive states calculated by the corresponding subsystem in the current time instant to guarantee the recursive feasibility and asymptotic stability.Using similar techniques,we proposed a general design of stabilized DMPC for linear systems in[8],which could handle input constraints and is suitable for a class of DMPCs where each subsystem’s performance index takes over other subsystems’cost functions.Through this method,both the stabilized cooperative DMPC and the classical DMPC can be designed.In addition,[12]provides another stabilized iterative cooperative DMPC method that takes the advances of those optimal solutions to converge to a fixed point in the method to avoid the consistency problem.However,this method leads to much more communication loads and requires global information,which violates the flexibility of DMPC.

3 Further investigations of DMPC

Although the DMPC has been developed since 2001,there are still many important problems to be solved.

1)With the rapid development of industrial 4.0 which requires the product process to automatically satisfy customer demands,there are continuously new requirements for controllers to meet.In the control layer,the controllers should automatically adapt to the changed system structures,process parameters,and production planning,etc.,with high efficiency and good quality.Thus,highly flexible dynamic optimal control with high performance and plug-and-play features is required.

2)In DMPC,the coordination strategy for improving the global optimization performance is heuristic.There is no theoretical analysis to qualify the optimization performance.Only some concepts,e.g.,the “Pareto Optimality”,are given.To evaluate the performance of DMPC is challenging and meaningful.

3)How to design a non-global information-based stabilized DMPC for a system with strong couplings existing among subsystems is still not developed.So far,it is still a difficulty to be solved in DMPC field.

4)From the practical aspect,the distributed systems are usually constituted by different types of subsystems.Thus,different subsystem-based controller should be designed according to the characteristics of the corresponding subsystem.This may give a result of many different types of MPCs or controllers existing in a given DMPC framework.Thus,how to coordinate these different types of subsystem-based controllers is a challenge in DMPC implementations.

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[2]X.Du,Y.Xi,S.Li.Distributed model predictive control for large-scale systems.Proceedings of the American Control Conference,Arlington:IEEE,2001:3142–3143.

[3]P.D.Christofides,R.Scattolini,D.Mu˜noz de la Pe˜na,et al.Distributed model predictive control:a tutorial review and future research directions.Computers&Chemical Engineering,2013,51(SI):21–41.

[4]S.Li,Y.Zheng.Distributed Model Predictive Control for Plant-wide Systems.Hoboken:John Wiley&Sons,2015.

[5]S. Li, Y. Zhang, Q. Zhu. Nash-optimization enhanced distributed model predictive control applied to the Shell benchmark problem. Information Sciences, 2005, 170 (2/4): 329 – 349.

[6]Y.Zheng,S.Li,H.Qiu.Networked coordination-based distributed model predictive control for large-scale system.IEEE Transactions on Control Systems Technology,2013,21(3):991–998.

[7]Y.Zhang,S.Li.Networked model predictive control based on neighborhood optimization for serially connected large-scale processes.Journal of Process Control,2007,17(1):37–50.

[8]S.Li,Y.Zheng,Z.Lin.Impacted-region optimization for distributed model predictive control systems with constraints.IEEE Transactions on Automation Science and Engineering,2015,12(4):1447–1460.

[9]Y.Zheng,S.Li,N.Li.Distributed model predictive control over network information exchange for large-scale systems.Control Engineering Practice.2011,19(7):757–769.

[10]Y.Zheng,N.Li,S.Li.Hot-rolled strip laminar cooling process plant-wide temperature monitoring and control.Control Engineering Practice,2013,21(1):23–30.

[11]W.Dunbar.Distributed receding horizon control of dynamically coupled nonlinear systems.IEEE Transactions on Automatic Control,2007,52(7):1249–1263.

[12]B.T.Stewart,A.N.Venkat,J.B.Rawlings,et al.Cooperative distributed model predictive control.Systems&Control Letters,2010,59(8):460–469.

DOI10.1007/s11768-017-6197-8

E-mail:syli@sjtu.edu.cn.

©2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Shaoyuan LIwas graduated from Hebei University of Technology,China,in 1987.He received his M.S.degree from Hebei University of Technology,China,in 1992,and Ph.D.degree from Nankai University,China,in 1997.He is currently a professor of the Institute of Automation,Shanghai Jiao Tong University,China.His research interest includes nonlinear system control,and fuzzy systems.E-mail:syli@sjtu.edu.cn.