1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

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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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. Simulations and Results
The system response

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

30. 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

31. Simulations and Results
The system response

32. Simulations and Results
The steady state results

33. 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

34. 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

35. 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