Distributed predictive control: A non-cooperative algorithm with neighbor-to-neighbor communication for linear systems☆
Introduction
Many large scale and spatially distributed systems, such as power networks, transport networks and hydro power plants (Negenborn et al., 2009, Šiljac, 1978), motivate the development of distributed control structures with limited local computation and transmission requirements. In the context of Model Predictive Control (MPC), many distributed control algorithms have been proposed. Specifically, completely decentralized architectures (Barcelli and Bemporad, 2009, Magni and Scattolini, 2006, Raimondo et al., 2006), distributed schemes (see, e.g., Dunbar (2007), Liu et al., 2010, Liu et al., 2009, Venkat, Hiskens, Rawlings, and Wright (2008) and Stewart, Venkat, Rawlings, Wright, and Pannocchia (2010)) and coordinated control techniques for dynamically decoupled systems (Dunbar and Murray, 2006, Ferrari-Trecate et al., 2009, Keviczki et al., 2006, Richards and How, 2007, Trodden and Richards, 2010) have been proposed.
Distributed MPC techniques can be classified according to the information exchange protocol needed (i.e., non-iterative or iterative algorithms), to the type of cost function which is optimized (i.e., cooperative or non-cooperative algorithms), and to the topology of the transmission network (i.e., fully connected or partially connected networks), see Scattolini (2009). Specifically, if we define the neighborhood of a subsystem as the set of the subsystems which directly affect its dynamics, algorithms based on a partially connected communication network generally require that data are transmitted to each subsystem from its neighbors solely (i.e., neighbor-to-neighbor information exchange). In Dunbar (2007) a non-iterative, non-cooperative distributed MPC technique is proposed for continuous-time systems based on neighbor-to-neighbor information exchange. In Liu et al., 2010, Liu et al., 2009 a non-iterative sequential (partially connected) algorithm and a novel iterative fully connected one are proposed. In Venkat et al. (2008) and Stewart et al. (2010) a cooperative fully connected output-feedback MPC algorithm for discrete time systems is discussed, where only input constraints can be assigned and full knowledge on the system dynamics is required for all the subsystems.
In this work a novel Distributed Predictive Control (DPC) scheme for linear discrete-time systems is proposed. Namely, DPC is a non-iterative, non-cooperative algorithm where a neighbor-to-neighbor (i.e., partially connected) communication network and partial (regional) structural information are needed. The rationale of the proposed technique is that, at each sampling time, each subsystem sends to its neighbors information about its future reference trajectory, and guarantees that the actual trajectory lies within a certain bound in the neighborhood of the reference one. Then, a robust MPC approach inspired by Mayne, Seron, and Raković (2005) provides a tool for the statement of the optimization problems solved by each subsystem.
The DPC algorithm handles state and input constraints and, under an assumption on the existence of a suitable decentralized auxiliary control law, convergence of the closed loop control system is proven. In the algorithm described here, the state of each subsystem is assumed to be available and the control algorithm is state-feedback; its extension to the output-feedback case is described in Farina and Scattolini (2011).
The highlights of DPC are: (i) it is not necessary for each subsystem to know the dynamical models governing the trajectories of the other subsystems (not even the ones of the neighbors); (ii) the transmission of information is limited, in that each subsystem needs the reference trajectories only of the variables of one’s neighbors, i.e. the subsystems which actually affect its dynamics and the ones which share one’s constraints; (iii) its rationale is very similar to the MPC algorithms presently employed in industry, where reference trajectories tailored on the dynamics of the system under control are used.
The paper is structured as follows. In Section 2 we introduce partitioned systems, while the DPC algorithm is defined in Section 3. The main convergence results of the proposed control scheme are illustrated in Section 4. In Section 5 the choice of the DPC tuning parameters is discussed, while Section 6 presents an application example. In Section 7 some conclusions are drawn. For clarity of presentation, all the proofs are postponed to an Appendix A Proof of, Appendix B Proof of the propositions.
Notation. A matrix is Schur if all its eigenvalues lie in the interior the unit circle. The short-hand denotes a column vector with (not necessarily scalar) components . The symbol denotes the Minkowski sum and . For a discrete-time signal and , , is denoted with . A continuous function is a function iff , it is strictly increasing and as . Finally, and are the maximum and the minimum eigenvalue of a matrix, respectively.
Section snippets
Partitioned systems
Consider a process which obeys to the linear dynamics where is the state vector and is the input signal. System (1) can be partitioned in low order interconnected non overlapping subsystems, where the -th sub-model has as state vector, i.e., and . According to this decomposition, the state transition matrices of the subsystems are the diagonal blocks of , whereas the non-diagonal blocks of (i.e.,
The distributed predictive control (DPC) algorithm
In order to design the distributed control scheme, assume that at any time instant , each subsystem transmits to the other subsystems its future state and input reference trajectories (to be later specified) and , , respectively. Moreover, by adding suitable constraints to its MPC formulation, each subsystem is able to guarantee that, for all , these trajectories lie in specified time-invariant neighborhoods of the reference trajectories, i.e, and
Convergence results
In order to state the main theoretical contribution of the paper, define the set of admissible initial conditions and initial reference trajectories , , for all as follows. Definition 1 Letting , denote by
DPC: implementation
In order to apply the DPC algorithm, some design parameters must be properly selected off-line, while the on-line phase reduces to solve the set of standard MPC problems (7), (8), (9), (10), (11), (12), (13), (14). Specifically, in the off-line design, one has to define:
- (1)
the stage and final costs and satisfying Assumption 2;
- (2)
the sets , , and satisfying Assumption 3;
- (3)
initial feasible reference trajectories , .
Example
The DPC algorithm has been used for control of the reactor–separator process already considered in Liu et al. (2009), Stewart et al. (2010) and shown in Fig. 1. The plant consists of three subsystems, i.e. two reactors and a separator. The reactant is inserted in the two reactors, where it is converted to product , with a side product ; a significant recirculation from the separator to the first reactor makes the system heavily coupled. The model, whose equations can be found in Liu et al.
Conclusions and future works
The distributed DPC algorithm presented in this paper enjoys the following properties: (i) it has been designed for controlling a wide class of large scale systems, in fact it can be used for dynamically coupled subsystems, as well as for subsystems with coupled constraints; (ii) each subsystem is required to know only the model governing its dynamics and how it is influenced by the inputs and outputs of the neighboring subsystems; (iii) it requires neighbor-to-neighbor communication among
Acknowledgments
The authors would like to greatly thank Giulio Betti and Davide Melzi for fruitful discussions.
Marcello Farina received the “Laurea” degree in Electronic Engineering in 2003 and the Ph.D. degree in Information Engineering in 2007, both from the Politecnico di Milano. In 2005 he was a visiting student at the Institute for Systems Theory and Automatic Control, Stuttgart, Germany. He is presently an Assistant Professor at the Dipartimento di Elettronica e Informazione, Politecnico di Milano. His research interests include modeling and identification of biological systems, distributed
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Marcello Farina received the “Laurea” degree in Electronic Engineering in 2003 and the Ph.D. degree in Information Engineering in 2007, both from the Politecnico di Milano. In 2005 he was a visiting student at the Institute for Systems Theory and Automatic Control, Stuttgart, Germany. He is presently an Assistant Professor at the Dipartimento di Elettronica e Informazione, Politecnico di Milano. His research interests include modeling and identification of biological systems, distributed control and state estimation, modeling and control of energy supply systems.
Riccardo Scattolini was born in Milano, Italy, in 1956. He received the “Laurea” degree in Electrical Engineering from the Politecnico di Milano, Milano, Italy, in 1979. He is presently Professor of automatic control with the Politecnico di Milano. From 1984 to 1985, he was a visiting researcher at the Department of Engineering Science, Oxford University, Oxford, UK. He also spent one year working in industry on the simulation and control of chemical plants. His current research interests include modeling, identification, simulation and control of industrial plants, with emphasis on model predictive control of large-scale systems. Dr. Scattolini received the Heaviside Premium from the Institution of Electrical Engineers, UK, in 1991.
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This research has been supported by the European 7th framework STREP project “Hierarchical and Distributed Model Predictive Control (HD-MPC)”, contract number INFSO-ICT-223854. The material in this paper was partially presented at the 18th IFAC World Conference. August 28–September 2, 2011, Milan, Italy. This paper was recommended for publication in revised form by Associate Editor Martin Guay under the direction of Editor Frank Allgöwer.
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