using mpc in mpc tim robinson. using mehrotra’s predictor-corrector scheme in model predictive...
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![Page 1: Using MPC in MPC Tim Robinson. Using Mehrotra’s Predictor-Corrector Scheme in Model Predictive Control Tim Robinson](https://reader035.vdocument.in/reader035/viewer/2022062516/56649d2a5503460f949feae1/html5/thumbnails/1.jpg)
Using MPC in MPCUsing MPC in MPC
Tim RobinsonTim Robinson
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Using Mehrotra’s Using Mehrotra’s Predictor-Corrector Predictor-Corrector Scheme in Model Scheme in Model Predictive ControlPredictive Control
Tim RobinsonTim Robinson
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What is Model Predicitive Control?What is Model Predicitive Control?
PlantModel
Predictive Controller
Set Point
Control
Action
Output
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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
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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)
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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
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State Space Model of PlantState Space Model of Plant
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Velocity Form of State Space Velocity Form of State Space ModelModel
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Formulation of Control Signal Formulation of Control Signal TrajectoryTrajectory
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Formulation of Control Signal Formulation of Control Signal TrajectoryTrajectory
k
u
. . .
c1
c2
c3
c4
c5
c6
c7
c8
cNc - 2
cNc - 1
cNc
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Formulation of Control Signal Formulation of Control Signal TrajectoryTrajectory
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Classical Laguerre PolynomialsClassical Laguerre Polynomials
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Discrete Laguerre FunctionsDiscrete Laguerre Functions
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Discrete Laguerre FunctionsDiscrete Laguerre Functions
The first five discrete Laguerre functions
10 20 30 40 50
-0.4
-0.2
0.2
0.4
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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
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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
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Formulation of Cost FunctionFormulation of Cost Function
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Unconstrained ControlUnconstrained Control
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Constrained ControlConstrained Control
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Model Predictive ControlModel Predictive Control
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Formulation of Primal-Dual Formulation of Primal-Dual Quadratic ProgramQuadratic Program
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Formulation of Primal-Dual Formulation of Primal-Dual Quadratic ProgramQuadratic Program
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KKT Conditions for OptimalityKKT Conditions for Optimality
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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
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Mehrotra’s Predictor-Corrector Mehrotra’s Predictor-Corrector AlgorithmAlgorithm
central path
t11
t22
infeasible region
affine-scaling step
centering-corrector step
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Mehrotra’s Predictor-Corrector Mehrotra’s Predictor-Corrector AlgorithmAlgorithm
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Affine Scaling DirectionAffine Scaling Direction
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Centering-Corrector DirectionCentering-Corrector Direction
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Combined Search DirectionCombined Search Direction
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Step Length HeuristicStep Length Heuristic
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Optimal TrajectoryOptimal Trajectory
y(k)
u(k)
u(k)
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Early Termination (MPC)Early Termination (MPC)
y(k)
u(k)
u(k)
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Early TerminationEarly Termination(Active Set Strategy)(Active Set Strategy)
y(k)
u(k)
u(k)
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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.