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

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.