Fast Model Predictive Control for Magnetic Plasma Control MPC using fast online 1st-order QP methods
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- Olivia Gilbert
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1 AWP15-ENR01/JSI-02 Fast Model Predictive Control for Magnetic Plasma Control MPC using fast online 1st-order QP methods Matija Perne Jožef Stefan Institute
2 Linear dynamic system, optimum Discrete-time model: Matija Perne FMPCFMPC PM Ljubljana Page 2
3 Linear dynamic system, optimum Discrete-time model: Matija Perne FMPCFMPC PM Ljubljana Page 3
4 Linear dynamic system, optimum Discrete-time model: Cost minimization: Matija Perne FMPCFMPC PM Ljubljana Page 4
5 Linear dynamic system, optimum Discrete-time model: Cost minimization: Constraints: Matija Perne FMPCFMPC PM Ljubljana Page 5
6 Linear dynamic system, optimum Discrete-time model: Cost minimization: Constraints: Matija Perne FMPCFMPC PM Ljubljana Page 6
7 Planning the future Matija Perne FMPCFMPC PM Ljubljana Page 7
8 Planning the future Matija Perne FMPCFMPC PM Ljubljana Page 8
9 Planning the future Matija Perne FMPCFMPC PM Ljubljana Page 9
10 Quadratic program Matija Perne FMPCFMPC PM Ljubljana Page 10
11 Quadratic program Hessian Matija Perne FMPCFMPC PM Ljubljana Page 11
12 Quadratic program Hessian Gradient Matija Perne FMPCFMPC PM Ljubljana Page 12
13 Quadratic program Hessian Affine set Gradient Matija Perne FMPCFMPC PM Ljubljana Page 13
14 Quadratic program Hessian Affine set Gradient Polyhedron Matija Perne FMPCFMPC PM Ljubljana Page 14
15 Quadratic program Hessian Affine set Gradient Polyhedron Dual method Matija Perne FMPCFMPC PM Ljubljana Page 15
16 Solving the QP Matija Perne FMPCFMPC PM Ljubljana Page 16
17 Solving the QP Explicit MPC: solving the problem in advance Matija Perne FMPCFMPC PM Ljubljana Page 17
18 Solving the QP Explicit MPC: solving the problem in advance Solution piecewise linear Matija Perne FMPCFMPC PM Ljubljana Page 18
19 Solving the QP Explicit MPC: solving the problem in advance Solution piecewise linear ( Matija Perne FMPCFMPC PM Ljubljana Page 19
20 Solving the QP Explicit MPC: solving the problem in advance Solution piecewise linear Active and inactive constraints: ( Matija Perne FMPCFMPC PM Ljubljana Page 20
21 Solving the QP Explicit MPC: solving the problem in advance Solution piecewise linear Active and inactive constraints: ( Matija Perne FMPCFMPC PM Ljubljana Page 21
22 Solving the QP Explicit MPC: solving the problem in advance Solution piecewise linear Active and inactive constraints: ( Matija Perne FMPCFMPC PM Ljubljana Page 22
23 Solving the QP Explicit MPC: solving the problem in advance Solution piecewise linear Active and inactive constraints: Active set methods Matija Perne FMPCFMPC PM Ljubljana Page 23
24 Solving the QP Explicit MPC: solving the problem in advance Solution piecewise linear Active and inactive constraints: Active set methods Online search for active constraints Matija Perne FMPCFMPC PM Ljubljana Page 24
25 Solving the QP Explicit MPC: solving the problem in advance Solution piecewise linear Active and inactive constraints: Active set methods Online search for active constraints Iterative Matija Perne FMPCFMPC PM Ljubljana Page 25
26 Solving the QP Interior point methods Matija Perne FMPCFMPC PM Ljubljana Page 26
27 Solving the QP Interior point methods Cost is modified with a steep smooth function close to polyhedral constraints Matija Perne FMPCFMPC PM Ljubljana Page 27
28 Solving the QP Interior point methods Cost is modified with a steep smooth function close to polyhedral constraints Modified problem solved with unconstrained optimization method Matija Perne FMPCFMPC PM Ljubljana Page 28
29 Solving the QP 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 Matija Perne FMPCFMPC PM Ljubljana Page 29
30 Solving the QP 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 Matija Perne FMPCFMPC PM Ljubljana Page 30
31 Solving the QP 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 Matija Perne FMPCFMPC PM Ljubljana Page 31
32 Solving the QP 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 Matija Perne FMPCFMPC PM Ljubljana Page 32
33 Existence of a solution Matija Perne FMPCFMPC PM Ljubljana Page 33
34 Existence of a solution Matija Perne FMPCFMPC PM Ljubljana Page 34
35 Existence of a solution Matija Perne FMPCFMPC PM Ljubljana Page 35
36 Existence of a solution Soft constraints Matija Perne FMPCFMPC PM Ljubljana Page 36
37 Making the QP convenient Matija Perne FMPCFMPC PM Ljubljana Page 37
38 Simplifying the QP Move blocking, constraint reduction Matija Perne FMPCFMPC PM Ljubljana Page 38
39 Simplifying the QP Move blocking, constraint reduction Matija Perne FMPCFMPC PM Ljubljana Page 39
40 Simplifying the QP Move blocking, constraint reduction Matija Perne FMPCFMPC PM Ljubljana Page 40
41 Simplifying the QP Move blocking, constraint reduction Matija Perne FMPCFMPC PM Ljubljana Page 41
42 Simplifying the QP Move blocking, constraint reduction Matija Perne FMPCFMPC PM Ljubljana Page 42
43 Simplifying the QP Move blocking, constraint reduction Matija Perne FMPCFMPC PM Ljubljana Page 43
44 Cat (
45 Gradient method (ordinary) (F. Borrelli, A. Bemporad, M. Morari, Predictive Control for linear and hybrid systems, 2015.) unconstrained constrained Matija Perne FMPCFMPC PM Ljubljana Page 45
46 Lipschitz constant Matija Perne FMPCFMPC PM Ljubljana Page 46
47 Gradient method (ordinary) (F. Borrelli, A. Bemporad, M. Morari, Predictive Control for linear and hybrid systems, 2015.) unconstrained constrained Matija Perne FMPCFMPC PM Ljubljana Page 47
48 Gradient method (fast) (F. Borrelli, A. Bemporad, M. Morari, Predictive Control for linear and hybrid systems, 2015.) unconstrained constrained Matija Perne FMPCFMPC PM Ljubljana Page 48
49 Lagrange multipliers ("Lagrange multiplier." Wikipedia, The Free Encyclopedia, 2015.) Matija Perne FMPCFMPC PM Ljubljana Page 49
50 Lagrange multipliers ("Lagrange multiplier." Wikipedia, The Free Encyclopedia, 2015.) Matija Perne FMPCFMPC PM Ljubljana Page 50
51 Lagrange multipliers ("Lagrange multiplier." Wikipedia, The Free Encyclopedia, 2015.) Matija Perne FMPCFMPC PM Ljubljana Page 51
52 Lagrange and dual functions (Stephen Boyd & Lieven Vandenberghe, Convex Optimization, Cambridge University Press 2004) Matija Perne FMPCFMPC PM Ljubljana Page 52
53 Lagrange and dual functions Matija Perne FMPCFMPC PM Ljubljana Page 53
54 Lagrange and dual functions Matija Perne FMPCFMPC PM Ljubljana Page 54
55 Lagrange and dual functions Matija Perne FMPCFMPC PM Ljubljana Page 55
56 Lagrange and dual functions Matija Perne FMPCFMPC PM Ljubljana Page 56
57 Lagrange and dual functions Matija Perne FMPCFMPC PM Ljubljana Page 57
58 FGMdual (generalized fast dual gradient method) Matija Perne FMPCFMPC PM Ljubljana Page 58
59 FGMdual (generalized fast dual gradient method) Matija Perne FMPCFMPC PM Ljubljana Page 59
60 FGMdual (generalized fast dual gradient method) Matija Perne FMPCFMPC PM Ljubljana Page 60
61 QPgen Matija Perne FMPCFMPC PM Ljubljana Page 61
62 QPgen Code generator: Matija Perne FMPCFMPC PM Ljubljana Page 62
63 QPgen Code generator: Problem described in MATLAB Matija Perne FMPCFMPC PM Ljubljana Page 63
64 QPgen Code generator: Problem described in MATLAB MATLAB does the offline calculations Matija Perne FMPCFMPC PM Ljubljana Page 64
65 QPgen Code generator: Problem described in MATLAB MATLAB does the offline calculations MATLAB outputs the algorithm code in C Matija Perne FMPCFMPC PM Ljubljana Page 65
66 QPgen 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 Matija Perne FMPCFMPC PM Ljubljana Page 66
67 QPgen 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 Matija Perne FMPCFMPC PM Ljubljana Page 67
68 QPgen 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: Matija Perne FMPCFMPC PM Ljubljana Page 68
69 QPgen 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 Matija Perne FMPCFMPC PM Ljubljana Page 69
70 QPgen 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 Matija Perne FMPCFMPC PM Ljubljana Page 70
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76 Restart (B. O Donoghue, E. Candès, Adaptive Restart for Accelerated Gradient Schemes, Found Comput Math 2013.) Matija Perne FMPCFMPC PM Ljubljana Page 76
77 Complexity certification (S. Richter, Computational Complexity Certification of Gradient Methods for Real-Time Model Predictive Control, ETH Zurich 2012) Upper bound on the required number of floating point operations that in turn stems from an upper bound on the iteration count Matija Perne FMPCFMPC PM Ljubljana Page 77
78 Complexity certification 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 Matija Perne FMPCFMPC PM Ljubljana Page 78
79 Complexity certification Explicit: fixed complexity Matija Perne FMPCFMPC PM Ljubljana Page 79
80 Complexity certification Explicit: fixed complexity Active set: Matija Perne FMPCFMPC PM Ljubljana Page 80
81 Complexity certification Explicit: fixed complexity Active set: Finite termination Matija Perne FMPCFMPC PM Ljubljana Page 81
82 Complexity certification Explicit: fixed complexity Active set: Finite termination Worst case number of iterations huge Matija Perne FMPCFMPC PM Ljubljana Page 82
83 Complexity certification Explicit: fixed complexity Active set: Finite termination Worst case number of iterations huge Early termination unexplored Matija Perne FMPCFMPC PM Ljubljana Page 83
84 Complexity certification Explicit: fixed complexity Active set: Finite termination Worst case number of iterations huge Early termination unexplored Interior point: Matija Perne FMPCFMPC PM Ljubljana Page 84
85 Complexity certification Explicit: fixed complexity Active set: Finite termination Worst case number of iterations huge Early termination unexplored Interior point: Certificates very conservative Matija Perne FMPCFMPC PM Ljubljana Page 85
86 Complexity certification Explicit: fixed complexity Active set: Finite termination Worst case number of iterations huge Early termination unexplored Interior point: Certificates very conservative First order: Matija Perne FMPCFMPC PM Ljubljana Page 86
87 Complexity certification 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 Matija Perne FMPCFMPC PM Ljubljana Page 87
88 Complexity certification State of the art for first order methods: Matija Perne FMPCFMPC PM Ljubljana Page 88
89 Complexity certification State of the art for first order methods: Derived for certain (simpler) algorithms Matija Perne FMPCFMPC PM Ljubljana Page 89
90 Complexity certification State of the art for first order methods: Derived for certain (simpler) algorithms (not generalized, no restarting) Matija Perne FMPCFMPC PM Ljubljana Page 90
91 Complexity certification State of the art for first order methods: Derived for certain (simpler) algorithms (not generalized, no restarting) Derived for certain (simpler) constraints Matija Perne FMPCFMPC PM Ljubljana Page 91
92 Complexity certification State of the art for first order methods: Derived for certain (simpler) algorithms (not generalized, no restarting) Derived for certain (simpler) constraints Matija Perne FMPCFMPC PM Ljubljana Page 92
93 Nesterov (Y. Nesterov, Introductory lectures on convex optimization : a basic course, Springer 2004) Convex optimization Matija Perne FMPCFMPC PM Ljubljana Page 93
94 Nesterov (Y. Nesterov, Introductory lectures on convex optimization : a basic course, Springer 2004) Convex optimization Does not mention duality Matija Perne FMPCFMPC PM Ljubljana Page 94
95 Richter (S. Richter, Computational Complexity Certification of Gradient Methods for Real-Time Model Predictive Control, ETH Zurich 2012) MPC with some QP Matija Perne FMPCFMPC PM Ljubljana Page 95
96 Richter (S. Richter, Computational Complexity Certification of Gradient Methods for Real-Time Model Predictive Control, ETH Zurich 2012) MPC with some QP Duality: partial Lagrange relaxation (simple state constraints), shows that complete is worse, but complete is what we need Matija Perne FMPCFMPC PM Ljubljana Page 96
97 Richter (S. Richter, Computational Complexity Certification of Gradient Methods for Real-Time Model Predictive Control, ETH Zurich 2012) 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 Matija Perne FMPCFMPC PM Ljubljana Page 97
98 Temporary conclusion What to do? Transform the MPC-obtained QP into dual space Matija Perne FMPCFMPC PM Ljubljana Page 98
99 Temporary conclusion What to do? Transform the MPC-obtained QP into dual space Follow either Richter or Nesterov with dual QP Matija Perne FMPCFMPC PM Ljubljana Page 99
100 Temporary conclusion What to do? Transform the MPC-obtained QP into dual space Follow either Richter or Nesterov with dual QP Preconditioning? Restarting? Matija Perne FMPCFMPC PM Ljubljana Page 100