Nonlinear and robust MPC with applications in robotics Boris Houska, Mario Villanueva, Benoît Chachuat ShanghaiTech, Texas A&M, Imperial College London 1
Overview Introduction to Robust MPC Min-Max Differential Inequalities 2
Model Predictive Control (MPC) Certainty equivalent MPC: minimize distance to dotted line subject to: system dynamics and constraints 3
Model Predictive Control (MPC) Certainty equivalent MPC: minimize distance to dotted line subject to: system dynamics and constraints 4
Model Predictive Control (MPC) Repeat: wait for new measurement re-optimize the trajectory 5
Model Predictive Control (MPC) Problem: certainty equivalent prediction is optimistic infeasible (worst-case) scenarios possible 6
What is Robust MPC? Main idea: take all possible uncertainty scenarios into account important: we can react to uncertainties 7
What is Robust MPC? Main idea: take all possible uncertainty scenarios into account important: we can react to uncertainties 8
What is Robust MPC? Main idea: take all possible uncertainty scenarios into account important: we can react to uncertainties 9
What is Robust MPC? Main idea: take all possible uncertainty scenarios into account important: we can react to uncertainties 10
What is Robust MPC? Problem: exponentially exploding amount of scenarios possible much more expensive than certainty equivalent MPC 11
Tube-based Robust MPC [Langson 04, Rakovic 05,...] Idea: optimize set-valued tube that encloses all possible scenarios no exponential scenario tree, but set enclosures needed 12
Tube-based Robust MPC [Langson 04, Rakovic 05,...] Idea: optimize set-valued tube that encloses all possible scenarios no exponential scenario tree, but set enclosures needed 13
Notation 1 Closed-loop system dynamics: ẋ(t) = f (x(t), µ(t, x(t)), w(t)) 14
Notation 2 Constraints: µ(t, x(t)) U, x(t) X, w(t) W (all compact sets) 15
Notation 3 Set-valued tubes: X µ (t) R nx denotes robust forward invariant tube: all solutions of ẋ(t) = f (x(t), µ(t, x(t)), w(t)) with x(t ) X µ (t) satisfy x(t) X µ (t) for all t t and all w with w(t) W. 16
Mathematical Formulation of Robust MPC Optimize over future feedback policy µ: inf µ:r X U s.t. t+t t l(x µ (τ)) dτ X µ (t) = {ˆx t }, X µ (τ) X optional terminal constraints Function l denotes scalar performance criterion ˆx t denotes current measurement T denotes finite prediction horizon 17
Overview Introduction to Robust MPC Min-Max Differential Inequalities 18
Differential Inequalities Scalar case: uncertain scalar ODE without controls: ẋ(t) = f (x(t), w(t)) with x(0) = x 0 19
Differential Inequalities Scalar case: Interval X(t) = [ x L (t), x U (t) ] is robust forward invariant if ẋ L (t) min w W f (x L (t), w) ẋ U (t) max w W f (x U (t), w) (Differential Inequalities) 20
Min-Max Differential Inequalities Scalar case with controls: Interval X(t) = [ x L (t), x U (t) ] is robust forward invariant if ẋ L (t) max u U min w W f (x L (t), u, w) ẋ U (t) min u U max w W f (x U (t), u, w) x L (t) x U (t) 21
Generalized Differential Inequalities General case: The state vector x(t) may have more than one component, ẋ(t) = f (x(t), u(t), w(t)) with x(0) = x 0 22
Generalized Differential Inequalities Definition: The support function of a compact set X is denoted by V [X](c) = max x X ct x 23
Generalized Differential Inequalities Theorem [Villanueva et al., 2016]: If f Lipschitz, X(t) X convex and compact, and x X(t) V [X(t)](c) min max c T f (x, u, w) c u U x,w T x = V [X(t)](c) w W for a.e. (t, c), then X(t) is a robust forward invariant tube. 24
Application to Robust MPC Conservative reformulation: inf Y t+t t l(y (τ)) dτ X(t) = {ˆx t }, s.t. X(τ) X V [X(t)](c) min max c T f (x, u, w) u U x,w optional terminal constraints x X(t) c T x = V [X(t)](c) w W Parameterize set X(t); not the feedback law µ! 25
Affine Parameterization: Affine Set Parameterizations X(t) = {A(t)ξ + b(t) ξ E m } Basis set: E m, domain constraint: [A(t), b(t)] D m,n 26
Numerical Example Spring-mass-damper system: ẋ1(t) = ẋ 2 (t) x 2 (t) + w 1 (t) k0 exp ( x1)x1(t) M h dx 2(t) M + u(t) M + w2(t) M 27
Numerical Example Spring-mass-damper system: ẋ1(t) = ẋ 2 (t) x 2 (t) + w 1 (t) k0 exp ( x1)x1(t) M h dx 2(t) M + u(t) M + w2(t) M 28
Numerical Example Spring-mass-damper system: ẋ1(t) = ẋ 2 (t) x 2 (t) + w 1 (t) k0 exp ( x1)x1(t) M h dx 2(t) M + u(t) M + w2(t) M 29
Conclusions Introduction to Robust MPC Needs accurate model of the system and uncertainties Useful whenever certainty equivalent MPC is too optimistic Apply if primary objective is safety (rather than CPU-time), example: human-robot cooperation Min-Max Differential Inequalities Min-Max DI leads to conservative reformulation, but parameterizes sets rather than feedback laws Boundary feedback law is minimizer of the RHS of min-max DI 30
Conclusions Introduction to Robust MPC Needs accurate model of the system and uncertainties Useful whenever certainty equivalent MPC is too optimistic Apply if primary objective is safety (rather than CPU-time), example: human-robot cooperation Min-Max Differential Inequalities Min-Max DI leads to conservative reformulation, but parameterizes sets rather than feedback laws Boundary feedback law is minimizer of the RHS of min-max DI 31
References M.E. Villanueva, B. Houska, B. Chachuat. Unified Framework for the Propagation of Continuous-Time Enclosures for Parametric Nonlinear ODEs. JOGO, 2015. B. Houska, M.E. Villanueva, B. Chachuat. Stable Set-Valued Integration of Nonlinear Dynamic Systems using Affine Set Parameterizations. SINUM, 2015. M.E. Villanueva, R. Quirynen, M. Diehl, B. Chachuat, B. Houska. Robust MPC via Min-Max Differential Inequalities. (submitted: April 2016) 32