1. Using MPC in MPC
Tim Robinson

2. Using Mehrotra’s Predictor-Corrector Scheme in Model Predictive Control
Tim Robinson

3. What is Model Predicitive Control?
Set Point
Model Predictive Controller
Plant
Control
Output
Action

4. How Does MPC Work? Discrete Time MPC Sample the state of the system
Use the computer model of the plant to predict the future state of the system for a given input
Formulate an optimisation problem in order to penalise any deviation of the output from the given setpoints
Solve the optimisation problem to find the control signal that will drive the system along the setpoints
Set the input equal to the starting value of the optimal control signal, discard the rest of the signal and start again

5. Discrete Time Model Predicitive Control
u0(k)
u1(k) k k 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
u2(k)
u3(k) k k 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

6. What’s So Good About MPC?
Allows traditional safety margins to be scrapped, so the plant can be operated near to the constraint boundaries
Allows for much greater efficiency
Used extensively in the petro-chemical industry
Has been of great interest to field of optimisation

7. State Space Model of Plant

8. Velocity Form of State Space Model

9. Formulation of Control Signal Trajectory

10. Formulation of Control Signal TrajectoryDu c1 c2 c8 c7 c3
cNc - 2
cNc k
cNc - 1 c6 c4 c5

11. Formulation of Control Signal Trajectory

12. Classical Laguerre Polynomials

13. Discrete Laguerre Functions

14. Discrete Laguerre Functions
The first five discrete Laguerre functions

15. Discrete Laguerre Functions
Effect of the scaling parameter a

16. Why Use Discrete Laguerre Functions?
Converge rapidly to a typical control signal
Need less coefficients to describe the control trajectory
Less constraints needed

17. Formulation of Cost Function

18. Unconstrained Control

19. Constrained Control

20. Model Predictive Control

21. Formulation of Primal-Dual Quadratic Program

22. Formulation of Primal-Dual Quadratic Program

23. KKT Conditions for Optimality

24. What is Mehrotra’s Predictor-Corrector Algorithm?
Primal-dual interior-point method of solving optimisation problems
Produces a sequence of feasible iterates which has the optimal solution as the limit point
Incorporates a few heuristics for high performance
Only just over 10 years old, but has become an industry standard for solving linear programs
Also very successful for quadratic programs

25. Mehrotra’s Predictor-Corrector Algorithm
t2l2
central path
affine-scaling step
centering-corrector step
t1l1
infeasible region

26. Mehrotra’s Predictor-Corrector Algorithm

27. Affine Scaling Direction

28. Centering-Corrector Direction

29. Combined Search Direction

30. Step Length Heuristic

31. Optimal Trajectory
y(k)
u(k)
Du(k)

32. Early Termination (MPC)
y(k)
u(k)
Du(k)

33. Early Termination (Active Set Strategy)
y(k)
u(k)
Du(k)

34. References
Goodwin, G.C., Graebe, S.F., Salgado, M.E. (2001). Control System Design. Prentice-Hall, Upper Saddle River, N.J.
Mayne, D.Q., Rawlings, J.B., Rao, C.V. (1998). Model Predicitive Control: A Review. Automatica.
Rao, C.V., Wright, S.J., Rawlings, J.B. (1998). Application of Interior-Point Methods to Model Predicitive Control. Journal of Optimization Theory and Applications, 99:
Wang, L. (2003). Discrete Model Predicitive Control Using Laguerre Functions. Technical Report, School of Electrical and Computer Engineering, Royal Melbourne Institute of Technology, Australia.
Wang, L. (2002). A Tutorial on Model Predictive Control – Using a Linear Velocity-Form Model. Technical Report, School of Electrical and Computer Engineering, Royal Melbourne Institute of Technology, Australia.
Wright, S.J. (1997). Primal-Dual Interior-Point Methods. Society for Industrial and Applied Mathematics, Philadelphia.