Robust MPC using min-max differential inequalities Boris Houska, - - PowerPoint PPT Presentation

Robust MPC using min-max differential inequalities

Boris Houska, Mario Villanueva, Benoît Chachuat

ShanghaiTech, Texas A&M, Imperial College London

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Overview

Introduction to Robust MPC Min-Max Differential Inequalities

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Model Predictive Control (MPC)

Certainty equivalent MPC: minimize distance to dotted line subject to: system dynamics and constraints

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Model Predictive Control (MPC)

Certainty equivalent MPC: minimize distance to dotted line subject to: system dynamics and constraints

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Model Predictive Control (MPC)

Repeat: wait for new measurement re-optimize the trajectory

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Model Predictive Control (MPC)

Problem: certainty equivalent prediction is optimistic infeasible (worst-case) scenarios possible

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What is Robust MPC?

Main idea: take all possible uncertainty scenarios into account important: we can react to uncertainties

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What is Robust MPC?

Main idea: take all possible uncertainty scenarios into account important: we can react to uncertainties

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What is Robust MPC?

Main idea: take all possible uncertainty scenarios into account important: we can react to uncertainties

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What is Robust MPC?

Main idea: take all possible uncertainty scenarios into account important: we can react to uncertainties

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What is Robust MPC?

Problem: exponentially exploding amount of scenarios possible much more expensive than certainty equivalent MPC

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Tube-based Robust MPC [Langson’04, Rakovic’05,. . .]

Idea:

• ptimize set-valued tube that encloses all possible scenarios

no exponential scenario tree, but set enclosures needed

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Tube-based Robust MPC [Langson’04, Rakovic’05,. . .]

Idea:

• ptimize set-valued tube that encloses all possible scenarios

no exponential scenario tree, but set enclosures needed

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Notation 1

Closed-loop system dynamics: ˙ x(t) = f (x(t), µ(t, x(t)), w(t))

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Notation 2

Constraints: µ(t, x(t)) ∈ U , x(t) ∈ X , w(t) ∈ W (all compact sets)

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Notation 3

Set-valued tubes: Xµ(t) ⊆ Rnx denotes robust forward invariant tube: all solutions of ˙ x(t) = f (x(t), µ(t, x(t)), w(t)) with x(t′) ∈ Xµ(t) satisfy x(t) ∈ Xµ(t) for all t ≥ t′ and all w with w(t) ∈ W.

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Mathematical Formulation of Robust MPC

Optimize over future feedback policy µ: inf

µ:R×X→U

t+T

t

ℓ(Xµ(τ)) dτ s.t.          Xµ(t) = {ˆ xt} , Xµ(τ) ⊆ X

• ptional terminal constraints

Function ℓ denotes scalar performance criterion ˆ xt denotes current measurement T denotes finite prediction horizon

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Overview

Introduction to Robust MPC Min-Max Differential Inequalities

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Differential Inequalities

Scalar case: uncertain scalar ODE without controls: ˙ x(t) = f (x(t), w(t)) with x(0) = x0

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Differential Inequalities

Scalar case: Interval X(t) =

• xL(t), xU(t)
• is robust forward invariant if

˙ xL(t) ≤ minw∈W f (xL(t), w) ˙ xU(t) ≥ maxw∈W f (xU(t), w) (Differential Inequalities)

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Min-Max Differential Inequalities

Scalar case with controls: Interval X(t) =

• xL(t), xU(t)
• is robust forward invariant if

˙ xL(t) ≤ maxu∈U minw∈W f (xL(t), u, w) ˙ xU(t) ≥ minu∈U maxw∈W f (xU(t), u, w) xL(t) ≤ xU(t)

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Generalized Differential Inequalities

General case: The state vector x(t) may have more than one component, ˙ x(t) = f (x(t), u(t), w(t)) with x(0) = x0

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Generalized Differential Inequalities

Definition: The support function of a compact set X is denoted by V [X](c) = max

x∈X cTx

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Generalized Differential Inequalities

Theorem [Villanueva et al., 2016]: If f Lipschitz, X(t) ⊆ X convex and compact, and ˙ V [X(t)](c) ≥ min

u∈U max x,w

         cTf (x, u, w)

• x ∈ X(t)

cTx = V [X(t)](c) w ∈ W          for a.e. (t, c), then X(t) is a robust forward invariant tube.

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Application to Robust MPC

Conservative reformulation: inf

Y

t+T

t

ℓ(Y (τ)) dτ s.t.                            X(t) = {ˆ xt} , X(τ) ⊆ X ˙ V [X(t)](c) ≥ min

u∈U max x,w

         cTf (x, u, w)

• x ∈ X(t)

cTx = V [X(t)](c) w ∈ W         

• ptional terminal constraints

Parameterize set X(t); not the feedback law µ!

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Affine Set Parameterizations

Affine Parameterization: X(t) = {A(t)ξ + b(t) | ξ ∈ Em} Basis set: Em, domain constraint: [A(t), b(t)] ∈ Dm,n

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Numerical Example

Spring-mass-damper system:   ˙ x1(t) ˙ x2(t)   =   x2(t) + w1(t) − k0 exp (−x1)x1(t)

M

− hdx2(t)

M

+ u(t)

M + w2(t) M

 

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Numerical Example

Spring-mass-damper system:   ˙ x1(t) ˙ x2(t)   =   x2(t) + w1(t) − k0 exp (−x1)x1(t)

M

− hdx2(t)

M

+ u(t)

M + w2(t) M

 

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Numerical Example

Spring-mass-damper system:   ˙ x1(t) ˙ x2(t)   =   x2(t) + w1(t) − k0 exp (−x1)x1(t)

M

− hdx2(t)

M

+ u(t)

M + w2(t) M

 

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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

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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

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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. AUTOMATICA, (provisionally accepted).

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