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Using MPC in MPCUsing MPC in MPC
Tim RobinsonTim Robinson
Using Mehrotra’s Using Mehrotra’s Predictor-Corrector Predictor-Corrector Scheme in Model Scheme in Model Predictive ControlPredictive Control
Tim RobinsonTim Robinson
What is Model Predicitive Control?What is Model Predicitive Control?
PlantModel
Predictive Controller
Set Point
Control
Action
Output
How Does MPC Work?How Does MPC Work?
Discrete Time MPC
1. Sample the state of the system
2. Use the computer model of the plant to predict the future state of the system for a given input
3. Formulate an optimisation problem in order to penalise any deviation of the output from the given setpoints
4. Solve the optimisation problem to find the control signal that will drive the system along the setpoints
5. Set the input equal to the starting value of the optimal control signal, discard the rest of the signal and start again
Discrete Time Model Predicitive Discrete Time Model Predicitive ControlControl
k
k
k
k0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
u0(k)
u2(k)
u1(k)
u3(k)
What’s So Good About MPC?What’s So Good About MPC?
Allows traditional safety margins to be Allows traditional safety margins to be scrapped, so the plant can be operated scrapped, so the plant can be operated near to the constraint boundariesnear to the constraint boundaries
Allows for much greater efficiencyAllows for much greater efficiency
Used extensively in the petro-chemical Used extensively in the petro-chemical industryindustry
Has been of great interest to field of Has been of great interest to field of optimisationoptimisation
State Space Model of PlantState Space Model of Plant
Velocity Form of State Space Velocity Form of State Space ModelModel
Formulation of Control Signal Formulation of Control Signal TrajectoryTrajectory
Formulation of Control Signal Formulation of Control Signal TrajectoryTrajectory
k
u
. . .
c1
c2
c3
c4
c5
c6
c7
c8
cNc - 2
cNc - 1
cNc
Formulation of Control Signal Formulation of Control Signal TrajectoryTrajectory
Classical Laguerre PolynomialsClassical Laguerre Polynomials
Discrete Laguerre FunctionsDiscrete Laguerre Functions
Discrete Laguerre FunctionsDiscrete Laguerre Functions
The first five discrete Laguerre functions
10 20 30 40 50
-0.4
-0.2
0.2
0.4
Discrete Laguerre FunctionsDiscrete Laguerre Functions
10 20 30 40 50
-0.4
-0.2
0.2
0.4
10 20 30 40 50
-0.4
-0.2
0.2
0.4
10 20 30 40 50
-0.3
-0.2
-0.1
0.1
0.2
10 20 30 40 50
-0.1
0.1
0.2
a=0.5
a=0.9a=0.7
a=0.3
Effect of the scaling parameter a
Why Use Discrete Laguerre Why Use Discrete Laguerre Functions?Functions?
Converge rapidly to a typical control signalConverge rapidly to a typical control signal
Need less coefficients to describe the Need less coefficients to describe the control trajectorycontrol trajectory
Less constraints neededLess constraints needed
Formulation of Cost FunctionFormulation of Cost Function
Unconstrained ControlUnconstrained Control
Constrained ControlConstrained Control
Model Predictive ControlModel Predictive Control
Formulation of Primal-Dual Formulation of Primal-Dual Quadratic ProgramQuadratic Program
Formulation of Primal-Dual Formulation of Primal-Dual Quadratic ProgramQuadratic Program
KKT Conditions for OptimalityKKT Conditions for Optimality
What is Mehrotra’s Predictor-What is Mehrotra’s Predictor-Corrector Algorithm?Corrector Algorithm?
Primal-dual interior-point method of solving Primal-dual interior-point method of solving optimisation problemsoptimisation problems
Produces a sequence of feasible iterates which Produces a sequence of feasible iterates which has the optimal solution as the limit pointhas the optimal solution as the limit point
Incorporates a few heuristics for high Incorporates a few heuristics for high performanceperformance
Only just over 10 years old, but has become an Only just over 10 years old, but has become an industry standard for solving linear programsindustry standard for solving linear programs
Also very successful for quadratic programsAlso very successful for quadratic programs
Mehrotra’s Predictor-Corrector Mehrotra’s Predictor-Corrector AlgorithmAlgorithm
central path
t11
t22
infeasible region
affine-scaling step
centering-corrector step
Mehrotra’s Predictor-Corrector Mehrotra’s Predictor-Corrector AlgorithmAlgorithm
Affine Scaling DirectionAffine Scaling Direction
Centering-Corrector DirectionCentering-Corrector Direction
Combined Search DirectionCombined Search Direction
Step Length HeuristicStep Length Heuristic
Optimal TrajectoryOptimal Trajectory
y(k)
u(k)
u(k)
Early Termination (MPC)Early Termination (MPC)
y(k)
u(k)
u(k)
Early TerminationEarly Termination(Active Set Strategy)(Active Set Strategy)
y(k)
u(k)
u(k)
ReferencesReferences
Goodwin, G.C., Graebe, S.F., Salgado, M.E. (2001). Goodwin, G.C., Graebe, S.F., Salgado, M.E. (2001). Control System Control System Design.Design. Prentice-Hall, Upper Saddle River, N.J. Prentice-Hall, Upper Saddle River, N.J.
Mayne, D.Q., Rawlings, J.B., Rao, C.V. (1998). Model Predicitive Control: A Mayne, D.Q., Rawlings, J.B., Rao, C.V. (1998). Model Predicitive Control: A Review. Review. Automatica.Automatica.
Rao, C.V., Wright, S.J., Rawlings, J.B. (1998). Application of Interior-Point Rao, C.V., Wright, S.J., Rawlings, J.B. (1998). Application of Interior-Point Methods to Model Predicitive Control. Methods to Model Predicitive Control. Journal of Optimization Theory and Journal of Optimization Theory and ApplicationsApplications, 99:723-757., 99:723-757.
Wang, L. (2003). Discrete Model Predicitive Control Using Laguerre Wang, L. (2003). Discrete Model Predicitive Control Using Laguerre Functions. Functions. Technical Report, School of Electrical and Computer Technical Report, School of Electrical and Computer Engineering, Royal Melbourne Institute of Technology, Australia.Engineering, Royal Melbourne Institute of Technology, Australia.
Wang, L. (2002). A Tutorial on Model Predictive Control – Using a Linear Wang, L. (2002). A Tutorial on Model Predictive Control – Using a Linear Velocity-Form Model. Velocity-Form Model. Technical Report, School of Electrical and Computer Technical Report, School of Electrical and Computer Engineering, Royal Melbourne Institute of Technology, Australia.Engineering, Royal Melbourne Institute of Technology, Australia.
Wright, S.J. (1997). Wright, S.J. (1997). Primal-Dual Interior-Point Methods.Primal-Dual Interior-Point Methods. Society for Society for Industrial and Applied Mathematics, Philadelphia.Industrial and Applied Mathematics, Philadelphia.