Approximate nonlinear explicit MPC based on reachability analysis

Transcription

1 Approximate nonlinear explicit MPC based on reachability analysis D.M. Raimondo 1, M. Schulze Darup 2, M. Mönnigmann 2 Università degli studi di Pavia Ruhr-Universität Bochum

2 The system Introduction: Nonlinear MPC

3 Introduction: Nonlinear MPC The optimization problem

4 Introduction: Nonlinear MPC The optimization problem

5 Introduction: Nonlinear MPC The Receding Horizon approach Closed-loop control law

6 Objectives Controller requirements Fast online computation Suitable for inexpensive hardware Feasibility and stability guarantees

7 Objectives Controller requirements Fast online computation Suitable for inexpensive hardware Feasibility and stability guarantees Explicit MPC

8 Explicit MPC Linear Explicit MPC multi-parametric program can be solved exactly feasible set is closed and convex continuous polyhedral piecewise affine control law

9 Explicit MPC Linear Explicit MPC multi-parametric program can be solved exactly feasible set is closed and convex continuous polyhedral piecewise affine control law Drawbacks?

10 Explicit MPC Complexity Strongest dependence on the number of constraints Strong dependence on the number of free moves Weak dependence on the number of states

11 Approximate Explicit MPC One approach to alleviate complexity (will be used in the nonlinear case..) Solve iteratively 1-step optimization problems with varying terminal set constraint (P. Grieder and M. Morari 2003) where

12 Approximate Explicit MPC One approach to alleviate complexity (will be used in the nonlinear case..) Solve iteratively 1-step optimization problems with varying terminal set constraint (P. Grieder and M. Morari 2003) One-step set Set of states which can be steered to a target set within one time step (Bertsekas, 1971) T

13 Approximate Explicit MPC One approach to alleviate complexity (will be used in the nonlinear case..) Solve iteratively 1-step optimization problems with varying terminal set constraint (P. Grieder and M. Morari 2003) One-step set Set of states which can be steered to a target set within one time step (Bertsekas, 1971) T 0 Q(T 0 )

14 Approximate Explicit MPC One approach to alleviate complexity (will be used in the nonlinear case..) Solve iteratively 1-step optimization problems with varying terminal set constraint (P. Grieder and M. Morari 2003) One-step set Set of states which can be steered to a target set within one time step (Bertsekas, 1971) T 0 T 1

15 Approximate Explicit MPC One approach to alleviate complexity (will be used in the nonlinear case..) Solve iteratively 1-step optimization problems with varying terminal set constraint (P. Grieder and M. Morari 2003) Feasible set of the original OP T 0 N times T N

16 Image from P. Grieder and M. Morari CDC 2003 Approximate Explicit MPC One approach to alleviate complexity (will be used in the nonlinear case..) Solve iteratively 1-step optimization problems with varying terminal set constraint (P. Grieder and M. Morari 2003) i=0 Online: since the regions of the 1-step multiparametric programs may overlap apply the feedback control computed at the smallest iteration number i.

17 Image from P. Grieder and M. Morari CDC 2003 Approximate Explicit MPC One approach to alleviate complexity (will be used in the nonlinear case..) Solve iteratively 1-step optimization problems with varying terminal set constraint (P. Grieder and M. Morari 2003) i=1 Online: since the regions of the 1-step multiparametric programs may overlap apply the feedback control computed at the smallest iteration number i.

18 Image from P. Grieder and M. Morari CDC 2003 Approximate Explicit MPC One approach to alleviate complexity (will be used in the nonlinear case..) Solve iteratively 1-step optimization problems with varying terminal set constraint (P. Grieder and M. Morari 2003) i=2 Online: since the regions of the 1-step multiparametric programs may overlap apply the feedback control computed at the smallest iteration number i.

19 Image from P. Grieder and M. Morari CDC 2003 Approximate Explicit MPC One approach to alleviate complexity (will be used in the nonlinear case..) Solve iteratively 1-step optimization problems with varying terminal set constraint (P. Grieder and M. Morari 2003) i=3 Online: since the regions of the 1-step multiparametric programs may overlap apply the feedback control computed at the smallest iteration number i.

20 Approximate Explicit MPC One approach to alleviate complexity (will be used in the nonlinear case..) Solve iteratively 1-step optimization problems with varying terminal set constraint (P. Grieder and M. Morari 2003) Online: since the regions of the 1-step multiparametric programs may overlap apply the feedback control computed at the smallest iteration number i. Guaranteed feasibility and stability of the approach

21 Approximate Explicit NMPC Making use of the previous approach in the nonlinear case Solving mp-nlp is difficult Exact solutions cannot be found in the general nonlinear case Feasible set non convex in general

22 Approximate Explicit NMPC One-step set in the nonlinear case T 0 The exact computation of this set is not possible in general The objective is to obtain a tight inner approximation Q(T 0 )

23 Approximate Explicit NMPC One-step set in the nonlinear case T 0 The exact computation of this set is not possible in general The objective is to obtain a tight inner approximation Q(T 0 ) Reachability analysis Overestimate one-step ahead reachable set for overestimation Methods: interval-arithmetic, DC-programming nonlinear mapping

24 Approximate Explicit NMPC One-step set in the nonlinear case The exact computation of this set is not possible in general The objective is to obtain a tight inner approximation

25 Approximate Explicit NMPC One-step set in the nonlinear case The exact computation of this set is not possible in general The objective is to obtain a tight inner approximation Partitions Choose a partitioning that results in fast online computation Hyperrectangles + binary tree structure

26 Bisection, binary tree How to choose u(x) for each hyperrectangle? root set f ( B, u) u?

27 Bisection, binary tree How to choose u(x) for each hyperrectangle? Entire U set root set u? f ( B, u)

28 Bisection, binary tree How to choose u(x) for each hyperrectangle? Set of hyperrectangles root set u?

29 Bisection, binary tree How to choose u(x) for each hyperrectangle? Bisect the input space root set u?

30 Bisection, binary tree How to choose u(x) for each hyperrectangle? Bisect the input space root set u 1 u 2 u1 u2

31 Bisection, binary tree How to choose u(x) for each hyperrectangle? Solve the NLP at the vertices and interpolate root set u? e.g. Hierarchical function approximation with interpolets

32 Bisection, binary tree How to choose u(x) for each hyperrectangle? Input space gridding root set u?

33 Bisection, binary tree How to choose u(x) for each hyperrectangle? Input space gridding root set u?

34 Bisection, binary tree How to organize hyperrectangular state space representation? Use bisection and binary tree can not be steered towards uncertain can be steered towards

35 Bisection, binary tree How to organize hyperrectangular state space representation? Use bisection and binary tree If f(b, u) bisect B intersects the target set to find the subsets entering T can not be steered towards uncertain can be steered towards

36 Bisection, binary tree How to organize hyperrectangular state space representation? Use bisection and binary tree root node can not be steered towards uncertain can be steered towards

37 Bisection, binary tree How to organize hyperrectangular state space representation? Use bisection and binary tree can not be steered towards uncertain can be steered towards

38 Bisection, binary tree How to organize hyperrectangular state space representation? Use bisection and binary tree can not be steered towards uncertain can be steered towards

39 Bisection, binary tree How to organize hyperrectangular state space representation? Use bisection and binary tree Which u do we choose? The one that minimizes the upper bound of the interval

40 Bisection, binary tree How to organize hyperrectangular state space representation? Use bisection and binary tree can not be steered towards uncertain can be steered towards

41 Bisection, binary tree How to organize hyperrectangular state space representation? Use bisection and binary tree can not be steered towards uncertain can be steered towards

42 Bisection, binary tree How to organize hyperrectangular state space representation? Use bisection and binary tree can not be steered towards uncertain can be steered towards

43 max. depth Bisection, binary tree How to organize hyperrectangular state space representation? Use bisection and binary tree can not be steered towards uncertain can be steered towards

44 max. depth Bisection, binary tree How to organize hyperrectangular state space representation? Use bisection and binary tree control law can not be steered towards uncertain can be steered towards

45 Numerical example Terminal set / state constraints Example bilinear system (Chen and Allgöwer, 1998) T 0 f1( x, u) x1 0.1x2 0.1 ( x1 ) u f2( x, u) x2 0.1x1 0.1 ( x2) u state and input constraints X X U x u R 2 R u x 2 2

46 Numerical example Feasible set Example bilinear system (Chen and Allgöwer, 1998) f ( x, u) x 0.1x 0.1 ( x ) u f ( x, u) x 0.1x 0.1 ( x ) u state and input constraints X U x u R 2 R u x 2 2 N=1

47 Numerical example Feasible set Example bilinear system (Chen and Allgöwer, 1998) f ( x, u) x 0.1x 0.1 ( x ) u f ( x, u) x 0.1x 0.1 ( x ) u state and input constraints X U x u R 2 R u x 2 2 N=2

48 Numerical example Feasible set Example bilinear system (Chen and Allgöwer, 1998) f ( x, u) x 0.1x 0.1 ( x ) u f ( x, u) x 0.1x 0.1 ( x ) u state and input constraints X U x u R 2 R u x 2 2 N=3

49 Numerical example Feasible set Example bilinear system (Chen and Allgöwer, 1998) f ( x, u) x 0.1x 0.1 ( x ) u f ( x, u) x 0.1x 0.1 ( x ) u state and input constraints X U x u R 2 R u x 2 2 N=5

50 Numerical example Feasible set Example bilinear system (Chen and Allgöwer, 1998) f ( x, u) x 0.1x 0.1 ( x ) u f ( x, u) x 0.1x 0.1 ( x ) u state and input constraints X U x u R 2 R u x 2 2 N=10

51 Numerical example Feasible set Example bilinear system (Chen and Allgöwer, 1998) f ( x, u) x 0.1x 0.1 ( x ) u f ( x, u) x 0.1x 0.1 ( x ) u state and input constraints X U x u R 2 R u x 2 2 N=12

52 Numerical example Feasible set Example bilinear system (Chen and Allgöwer, 1998) f ( x, u) x 0.1x 0.1 ( x ) u f ( x, u) x 0.1x 0.1 ( x ) u state and input constraints X U x u R 2 R u x 2 2 N=15

53 Numerical example Feasible set PWC-controller

54 Numerical example Feasible set Considerations Stability is guaranteed Once the terminal set is attained we switch to the auxiliary controller k (x) Differently from the original method, the use of hyperrectangles provides non-overlapping regions f

55 Numerical example Feasible set Considerations In order to reduce suboptimality Candidate regions can be further split according to the cost function and stop when the gap between keeping the same controller or differentiating it for each subset is smaller than a threshold ε.

56 Numerical example Feasible set Considerations How conservative is the inner approximation of the feasible set?

57 Numerical example Feasible set Considerations Inner approximation of infeasible set u U: T 0 0 u U: R n \ X X Compute the regions for which all U n R \ X with N lead to

58 Numerical example Feasible set Considerations Why the gap?

59 Numerical example Feasible set Considerations Finite horizon for the green region

60 Numerical example Feasible set Considerations Not feasible also when T is not attained in N steps

61 Numerical example Feasible set Considerations What about solving directly the N-step problem? Stability: difficult to get a suboptimality gap from the optimal solution.

62 Numerical example Feasible set Considerations Fast online evaluation Minimal storage requirements Hash Map Representation: Memory saving Search Tree Representation Fast online evaluation time The control law can be evaluated in 31ns (Hash) or 0.5µs (Search Tree) 3240 regions