fast model predictive control for magnetic plasma control mpc … · 2016. 2. 29. · for magnetic...
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![Page 1: Fast Model Predictive Control for Magnetic Plasma Control MPC … · 2016. 2. 29. · for Magnetic Plasma Control MPC using fast online 1st-order QP methods Matija Perne Jožef Stefan](https://reader035.vdocuments.site/reader035/viewer/2022071108/5fe2f0f31ca75358da04d5b2/html5/thumbnails/1.jpg)
Fast Model Predictive Control for Magnetic Plasma Control MPC using fast online 1st-order QP methods
Matija Perne Jožef Stefan Institute
AWP15-ENR-01/JSI-02
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Linear dynamic system, optimum
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 2
● Discrete-time model:
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Linear dynamic system, optimum
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 3
● Discrete-time model:
![Page 4: Fast Model Predictive Control for Magnetic Plasma Control MPC … · 2016. 2. 29. · for Magnetic Plasma Control MPC using fast online 1st-order QP methods Matija Perne Jožef Stefan](https://reader035.vdocuments.site/reader035/viewer/2022071108/5fe2f0f31ca75358da04d5b2/html5/thumbnails/4.jpg)
Linear dynamic system, optimum
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 4
● Discrete-time model:
● Cost minimization:
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Linear dynamic system, optimum
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 5
● Discrete-time model:
● Cost minimization:
● Constraints:
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Linear dynamic system, optimum
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 6
● Discrete-time model:
● Cost minimization:
● Constraints:
![Page 7: Fast Model Predictive Control for Magnetic Plasma Control MPC … · 2016. 2. 29. · for Magnetic Plasma Control MPC using fast online 1st-order QP methods Matija Perne Jožef Stefan](https://reader035.vdocuments.site/reader035/viewer/2022071108/5fe2f0f31ca75358da04d5b2/html5/thumbnails/7.jpg)
Planning the future
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 7
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Planning the future
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 8
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Planning the future
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 9
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Quadratic program
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 10
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Quadratic program
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 11
Hessian
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Quadratic program
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 12
HessianGradient
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Quadratic program
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 13
Hessian
Affine setGradient
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Quadratic program
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 14
Hessian
Affine set
Polyhedron
Gradient
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Quadratic program
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 15
Hessian
Affine set
Polyhedron
Gradient
Dual method
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 16
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 17
● Explicit MPC: solving the problem in advance
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 18
● Explicit MPC: solving the problem in advance
– Solution piecewise linear
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 19
● Explicit MPC: solving the problem in advance
– Solution piecewise linear
(http://www.seas.upenn.edu/~ese680/papers/explicit_linear_mpc.pdf)
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 20
● Explicit MPC: solving the problem in advance
– Solution piecewise linear
– Active and inactive constraints:
(http://www.seas.upenn.edu/~ese680/papers/explicit_linear_mpc.pdf)
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 21
● Explicit MPC: solving the problem in advance
– Solution piecewise linear
– Active and inactive constraints:
(http://www.seas.upenn.edu/~ese680/papers/explicit_linear_mpc.pdf)
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 22
● Explicit MPC: solving the problem in advance
– Solution piecewise linear
– Active and inactive constraints:
(http://www.seas.upenn.edu/~ese680/papers/explicit_linear_mpc.pdf)
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 23
● Explicit MPC: solving the problem in advance
– Solution piecewise linear
– Active and inactive constraints:
● Active set methods
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 24
● Explicit MPC: solving the problem in advance
– Solution piecewise linear
– Active and inactive constraints:
● Active set methods
– Online search for active constraints
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 25
● Explicit MPC: solving the problem in advance
– Solution piecewise linear
– Active and inactive constraints:
● Active set methods
– Online search for active constraints
– Iterative
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 26
• Interior point methods
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 27
• Interior point methods– Cost is modified with a steep smooth function
close to polyhedral constraints
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 28
• Interior point methods– Cost is modified with a steep smooth function
close to polyhedral constraints
– Modified problem solved with unconstrained optimization method
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 29
• Interior point methods– Cost is modified with a steep smooth function
close to polyhedral constraints
– Modified problem solved with unconstrained optimization method
– Modification decreased iteratively
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 30
• Interior point methods– Cost is modified with a steep smooth function
close to polyhedral constraints
– Modified problem solved with unconstrained optimization method
– Modification decreased iteratively
• First order methods
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 31
• Interior point methods– Cost is modified with a steep smooth function
close to polyhedral constraints
– Modified problem solved with unconstrained optimization method
– Modification decreased iteratively
• First order methods
– Inspired by gradient method
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Solving the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 32
• Interior point methods– Cost is modified with a steep smooth function
close to polyhedral constraints
– Modified problem solved with unconstrained optimization method
– Modification decreased iteratively
• First order methods
– Inspired by gradient method
– Simple, low order of convergence
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Existence of a solution
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 33
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Existence of a solution
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 34
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Existence of a solution
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 35
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Existence of a solution
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 36
Soft constraints
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Making the QP convenient
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 37
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Simplifying the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 38
Move blocking, constraint reduction
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Simplifying the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 39
Move blocking, constraint reduction
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Simplifying the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 40
Move blocking, constraint reduction
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Simplifying the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 41
Move blocking, constraint reduction
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Simplifying the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 42
Move blocking, constraint reduction
![Page 43: Fast Model Predictive Control for Magnetic Plasma Control MPC … · 2016. 2. 29. · for Magnetic Plasma Control MPC using fast online 1st-order QP methods Matija Perne Jožef Stefan](https://reader035.vdocuments.site/reader035/viewer/2022071108/5fe2f0f31ca75358da04d5b2/html5/thumbnails/43.jpg)
Simplifying the QP
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 43
Move blocking, constraint reduction
![Page 44: Fast Model Predictive Control for Magnetic Plasma Control MPC … · 2016. 2. 29. · for Magnetic Plasma Control MPC using fast online 1st-order QP methods Matija Perne Jožef Stefan](https://reader035.vdocuments.site/reader035/viewer/2022071108/5fe2f0f31ca75358da04d5b2/html5/thumbnails/44.jpg)
(http://animalswalls.blogspot.si/2011/08/cat-wallpapers.html)
Cat
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Gradient method (ordinary)
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 45
(F. Borrelli, A. Bemporad, M. Morari, Predictive Control for linear and hybrid systems, 2015.)
unconstrained
constrained
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Lipschitz constant
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 46
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Gradient method (ordinary)
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 47
(F. Borrelli, A. Bemporad, M. Morari, Predictive Control for linear and hybrid systems, 2015.)
unconstrained
constrained
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Gradient method (fast)
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 48
(F. Borrelli, A. Bemporad, M. Morari, Predictive Control for linear and hybrid systems, 2015.)
unconstrained
constrained
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Lagrange multipliers
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 49
("Lagrange multiplier." Wikipedia, The Free Encyclopedia, 2015.)
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Lagrange multipliers
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 50
("Lagrange multiplier." Wikipedia, The Free Encyclopedia, 2015.)
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Lagrange multipliers
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 51
("Lagrange multiplier." Wikipedia, The Free Encyclopedia, 2015.)
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Lagrange and dual functions
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 52
(Stephen Boyd & Lieven Vandenberghe, Convex Optimization, Cambridge University Press 2004)
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Lagrange and dual functions
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 53
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Lagrange and dual functions
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 54
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Lagrange and dual functions
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 55
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Lagrange and dual functions
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 56
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Lagrange and dual functions
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 57
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FGMdual
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 58
(generalized fast dual gradient method)
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FGMdual
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 59
(generalized fast dual gradient method)
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FGMdual
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 60
(generalized fast dual gradient method)
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QPgen
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 61
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QPgen
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 62
● Code generator:
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QPgen
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 63
● Code generator:– Problem described in MATLAB
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QPgen
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 64
● Code generator:– Problem described in MATLAB
– MATLAB does the offline calculations
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QPgen
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 65
● Code generator:– Problem described in MATLAB
– MATLAB does the offline calculations
– MATLAB outputs the algorithm code in C
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QPgen
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 66
● Code generator:– Problem described in MATLAB
– MATLAB does the offline calculations
– MATLAB outputs the algorithm code in C
– Code compiled, ran, result returned to MATLAB
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QPgen
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 67
● Code generator:– Problem described in MATLAB
– MATLAB does the offline calculations
– MATLAB outputs the algorithm code in C
– Code compiled, ran, result returned to MATLAB
– FGMdual one of the supported algorithms
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QPgen
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 68
● Code generator:– Problem described in MATLAB
– MATLAB does the offline calculations
– MATLAB outputs the algorithm code in C
– Code compiled, ran, result returned to MATLAB
– FGMdual one of the supported algorithms
● Knowing the code:
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QPgen
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 69
● Code generator:– Problem described in MATLAB
– MATLAB does the offline calculations
– MATLAB outputs the algorithm code in C
– Code compiled, ran, result returned to MATLAB
– FGMdual one of the supported algorithms
● Knowing the code:– MATLAB code is long
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QPgen
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 70
● Code generator:– Problem described in MATLAB
– MATLAB does the offline calculations
– MATLAB outputs the algorithm code in C
– Code compiled, ran, result returned to MATLAB
– FGMdual one of the supported algorithms
● Knowing the code:– MATLAB code is long
– Generated code is confusing
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Restart
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 76
(B. O’Donoghue, E. Candès, Adaptive Restart for Accelerated Gradient Schemes, Found Comput Math 2013.)
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 77
• Upper bound on the required number of floating point operations that in turn stems from an upper bound on the iteration count
(S. Richter, Computational Complexity Certification of Gradient Methods for Real-Time Model Predictive Control, ETH Zurich 2012)
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 78
• Upper bound on the required number of floating point operations that in turn stems from an upper bound on the iteration count
• accuracy => no. of iterations => no. of operations => computing time
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 79
• Explicit: fixed complexity
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 80
• Explicit: fixed complexity• Active set:
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 81
• Explicit: fixed complexity• Active set:
– Finite termination
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 82
• Explicit: fixed complexity• Active set:
– Finite termination
– Worst case number of iterations huge
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 83
• Explicit: fixed complexity• Active set:
– Finite termination
– Worst case number of iterations huge
– Early termination unexplored
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 84
• Explicit: fixed complexity• Active set:
– Finite termination
– Worst case number of iterations huge
– Early termination unexplored
• Interior point:
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 85
• Explicit: fixed complexity• Active set:
– Finite termination
– Worst case number of iterations huge
– Early termination unexplored
• Interior point:– Certificates very conservative
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 86
• Explicit: fixed complexity• Active set:
– Finite termination
– Worst case number of iterations huge
– Early termination unexplored
• Interior point:– Certificates very conservative
• First order:
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 87
• Explicit: fixed complexity• Active set:
– Finite termination
– Worst case number of iterations huge
– Early termination unexplored
• Interior point:– Certificates very conservative
• First order:– Certificates within few orders of magnitude
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 88
• State of the art for first order methods:
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 89
• State of the art for first order methods:– Derived for certain (simpler) algorithms
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 90
• State of the art for first order methods:– Derived for certain (simpler) algorithms
(not generalized, no restarting)
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 91
• State of the art for first order methods:– Derived for certain (simpler) algorithms
(not generalized, no restarting)– Derived for certain (simpler) constraints
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Complexity certification
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 92
• State of the art for first order methods:– Derived for certain (simpler) algorithms
(not generalized, no restarting)– Derived for certain (simpler) constraints
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Nesterov
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 93
• Convex optimization
(Y. Nesterov, Introductory lectures on convex optimization : a basic course, Springer 2004)
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Nesterov
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 94
• Convex optimization• Does not mention duality
(Y. Nesterov, Introductory lectures on convex optimization : a basic course, Springer 2004)
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Richter
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 95
• MPC with some QP
(S. Richter, Computational Complexity Certification of Gradient Methods for Real-Time Model Predictive Control, ETH Zurich 2012)
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Richter
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 96
• MPC with some QP• Duality: partial Lagrange relaxation
(simple state constraints), shows that complete is worse, but complete is what we need
(S. Richter, Computational Complexity Certification of Gradient Methods for Real-Time Model Predictive Control, ETH Zurich 2012)
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Richter
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 97
• MPC with some QP• Duality: partial Lagrange relaxation
(simple state constraints), shows that complete is worse, but complete is what we need
• Duality: dual problem not strongly concave
(S. Richter, Computational Complexity Certification of Gradient Methods for Real-Time Model Predictive Control, ETH Zurich 2012)
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Temporary conclusion
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 98
• What to do? Transform the MPC-obtained QP into dual space
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Temporary conclusion
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 99
• What to do? Transform the MPC-obtained QP into dual space
• Follow either Richter or Nesterov with dual QP
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Temporary conclusion
Matija Perne | FMPCFMPC PM | Ljubljana | 25.02.2016 | Page 100
• What to do? Transform the MPC-obtained QP into dual space
• Follow either Richter or Nesterov with dual QP
• Preconditioning?• Restarting?