A Framework for Distributed Model Predictive Control March 2005 A Framework for Distributed Model Predictive Control Leonardo Giovanini Industrial Control Centre University of Strathclyde Glasgow Industrial Control Center

IEEE Colloquium on MPC University of Sheffield March 2005 Outline Background and motivation Problem Formulation Communication based MPC Properties (Stability, Convergence, Performance) Cooperative based MPC Simulations Conclusions and Future directions April 2005 IEEE Colloquium on MPC University of Sheffield Industrial Control Center

IEEE Colloquium on MPC University of Sheffield Background Decentralized control and the traditional design mindset. Several small units rather than a single monolithic unit Significant literature in the late seventies and early nineties focused on improved decentralized control. Limited focus on handling constraints, optimality stability of control methods. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield March 2005 Background Since nineties linear model predictive control became a dominant advanced control technology. Properties of single linear MPC are well established (stability, performance, feasibility). Potential benefits of integrating several MPCs (system resilience, reconfigurability). April 2005 IEEE Colloquium on MPC University of Sheffield Industrial Control Center

IEEE Colloquium on MPC University of Sheffield Background Example: Heat-exchanger networks The objective are: Control the output temperatures of streams C1, C2, H1 and H2 in the whole operating space Minimize the use of services S1, S2 and S3 Let consider a heat-exchanger network. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Background Example: Heat-exchanger networks The control inputs are constrained The network needs to employ the extra services (S2 and S3 ) to achieve some output targets and reject some disturbances. Therefore, a control strategy capable of make the sub systems work cooperatively and optimize the system behavior. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Background Why not centralized MPC? The main reasons are organizational and computational. Different sections of the networked systems may be owned by different organizations (power systems, integrated chemical plants, communication networks, manufacturing supply chain). Large centralized controllers are inflexible, hard to maintain and modify. Need of control methods for effective collaboration between the different subsystems. April 2005 IEEE Colloquium on MPC University of Sheffield

The Problem Formulation Complex high dimensional systems need efficient control architectures and algorithms. Conventional approach is based on the decomposition of the system (to reduce the computational burden) and coordination of the subsystems (to tackle with interactions). The development of distributed control systems brings new requirements and potential benefits to control field. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield The Problem Formulation The requirements for a game approach to decentralized MPC are Modeling Optimal partitioning of the system Models for the decentralized and interaction models Assumptions All cost functions are positive definite Linear models Convex inequalities Synchronous communication Levels of collaboration April 2005 IEEE Colloquium on MPC University of Sheffield

Communication based MPC IEEE Colloquium on MPC University of Sheffield The global optimization problem has been decomposed into a number of small coupled local optimization problem. Each optimization problems only consider the local part of objective function. The optimization problems are solved using only the local decision variables and communicating the result to others problems at each iteration. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Communication based MPC The basic algorithm An iterative algorithm for solving this problem is Exchange state and inputs trajectories information between controllers, Solve each sub problem using the trajectories provided by other agents and exchange information till all trajectories converge, Once the problems converge, apply the first element of the inputs trajectories of each subsystem. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Communication based MPC The basic algorithm Definition: A group of control decision is called to be Nash optimal solution if the following relation is held If the Nash solution is achieved, each agent (i) does not change its control decision (ui) because it has achieved the local optimum, otherwise the local performance index Ji will degrade. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Communication based MPC Convergence The control actions are given by where The convergence of control action only depends on Therefore the algorithm will converge if April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Communication based MPC Convergence To model the effects of communication failure, the matrices Tr and Tc are introduced into the control law. Their elements only assume the values 1 or 0. Then, the control actions are given by Therefore the algorithm will converge if April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Communication based MPC Stability The system can be written as where Then, the closed-loop system is given by which is stable if April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Communication based MPC Performance The solutions are Nash equilibrium, they are generally not Paretto optimal when the number of interacting agents is finite (Ramachadran et al., 1992; Dubbey and Rogawski, 2002). For example, lets assume two subsystems, two inputs, one output each and the prediction horizon is 1. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Communication based MPC Recovering Performance Regulatory constraints: The presence of constraints can change the number and location of the equilibrium points. This can used to improve the solution, approaching N to the Paretto set P. Deference: If one of the agents allows the other to use both decision variables, under the condition that new decision are not worse than the equilibrium point, the agent can drag the attractor inside of the Paretto set. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Communication based MPC Properties Trajectories generated by this scheme at each iteration p Feasible Guarantee the closed-loop stability Then The cooperation based scheme can be terminated in any intermediate iteration. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Cooperative based MPC The global optimization problem has been decomposed into a number of small coupled optimization problem. Each optimization problems consider the global objective function. Each optimization problem is solved using only local decision variables and communicating the result to others problems at each iteration. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Cooperative based MPC The basic algorithm An iterative algorithm for solving this problem is Exchange state and inputs trajectories information between controllers, Solve each subproblem using the trajectories provided by other agents and exchange information till all trajectories converge, Once the problems converge, apply the first element of the inputs trajectories of each subsystem. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Cooperative based MPC Properties: feasibility and convergence All iterates are system wide feasible. The sequence of cost functions is a non- increasing function of the iteration number p The sequence of iterates converges to an optimal limit point (centralized MPC). April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Cooperative based MPC Performance Returning to the initial example, two subsystems with two inputs, one output each and the prediction horizon is 1. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Cooperative based MPC Properties: closed-loop stability Feasibility at initial time implies feasibility at all futures times. Additionally (Aii, Bii) i=1,2, … ,m stabilizable ( Decentralized model ) (Aij, Bij) i=1,2, … ,m stable ( Interaction model ) Stabilizing constraints ( ui(k+j) = 0 j > N ) Then The origin is an asymptotically stable equilibrium point for the closed loop system for all initial states x(k) and all iteration numbers p. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Cooperative based MPC Properties Trajectories generated by this scheme at each iteration p Feasible Guarantee the closed-loop stability Then The cooperation based scheme can be terminated in any intermediate iteration. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Simulations and Results The heat-exchanger network Let consider the following heat-exchanger network The objective are: Guarantee the controllability of the system for any operational condition, Provide an optimal response for steady-state, Provide an optimal performance for any change. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Simulations and Results The heat-exchanger network The parameters of the streams are April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Simulations and Results Set point changes The simulations consist in a sequence of set point and load disturbances changes that drive the network outside of the original operational space. The set point changes are: Firstly, TC1 goes from 80 °C to 70 °C at 5 min. then, TC2 goes from 40 °C to 45 °C at 15 min. finally, TH2 goes from 100 C to 90 C at 25 min. April 2005 IEEE Colloquium on MPC University of Sheffield

Simulations and Results The system response

Simulations and Results The steady state results March 2005 Simulations and Results The steady state results Industrial Control Center

IEEE Colloquium on MPC University of Sheffield Simulations and Results Load disturbances changes The simulations consist in a sequence of load disturbances changes that drive the network outside of the original operational space. The load changes are: Firstly, T inH1 goes from 90 °C to 80 °C at 5 min. then, T inH2 goes from 130 °C to 140 °C at 15 min. finally, T inC1 goes from 30 C to 40 C at 25 min. April 2005 IEEE Colloquium on MPC University of Sheffield

Simulations and Results The system response

Simulations and Results The steady state results

IEEE Colloquium on MPC University of Sheffield Conclusions A general framework for decentralized model predictive control has been presented. The proposed framework is able to take advantage of system flexibility to handle conflicting situations while maintain operation in the optimal conditions. The framework allows to explicit handle the trade off between optimality and performance with a minimum computational effort. The proposed framework has a modular design, that can be easily update and allow to take advantage of decentralize systems. April 2005 IEEE Colloquium on MPC University of Sheffield

IEEE Colloquium on MPC University of Sheffield Future Directions Extend these ideas to uncertain and nonlinear systems. Develop procedures and algorithms for the optimal partition of the controlled system. Explore the inclusion of adaptation and learning capabilities into the decentralized scheme. April 2005 IEEE Colloquium on MPC University of Sheffield

A Framework for Distributed Model Predictive Control March 2005 A Framework for Distributed Model Predictive Control Leonardo Giovanini Industrial Control Centre University of Strathclyde Glasgow Industrial Control Center