Parallel Explicit Tube Model Predictive Control · Parallel Explicit Tube MPC Controller (Recursive...

Parallel ExplicitTube Model Predictive Control

Kai Wang1, Yuning Jiang1, Juraj Oravec2, Mario E. Villanueva1,

Boris Houska1

1ShanghaiTech University

2Slovak University of Technology in Bratislava

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Overview

Background on Tube MPC

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Model Predictive Control (MPC)

Rigid Tube MPC

Contribution

Parallel Real-time Optimization Algorithm

Numerical Example

Robust MPC

Parallel Explicit Tube MPC Controller


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GoalObstacles

Uncertain Control System:


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Minimize distance to the dotted line

Subject to: System dynamics

State and input constraints

Uncertain Control System:

Certainty equivalent MPC:


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wait for new measurement

re-optimize the trajectory

Repeat:

apply optimal control law


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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

design tailored feedback policies to react to uncertainties

Limitations:

exponential number of scenarios

much more expensive than certainty equivalent MPC

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Main idea:

take all possible uncertainty scenarios into account

design tailored feedback policies to react to uncertainties

What is Robust MPC?

Limitations:

exponential number of scenarios

much more expensive than certainty equivalent MPC

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Main idea:

enclose all trajectories by a tube of constant polytopic cross-sections

optimize the central path of the tube

Rigid Tube MPC

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Mathematical Formulation of Rigid Tube MPC

Uncertain control system:

Constraints:

Linear feedback law:

Nominal system: (disturbance-free)

and

denotes a precomputed robust invariant set, w.r.t.,

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Mathematical Formulation of Rigid Tube MPC

Optimal Control Problem (OCP):

stage cost:

terminal cost:

and are symmetric positivedefinite

Standard assumptions:

terminal constraint set

Overview

Background on Tube MPC

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Robust Model Predictive Control (MPC)

Rigid Tube MPC

Contribution


Numerical Example

Tube-Based Robust MPC

Parallel Explicit Tube MPC Scheme

Goal: develop a real-time Tube MPC algorithm with limited memory footprint

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Rewrite the OCP of the Rigid Tube MPC

Stacked vectors and matrices

Constraint sets

Cost function

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Rewrite the OCP of the Rigid Tube MPC

Constraint sets

Cost function

Stacked vectors and matrices

equivalent

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Parallel Real-time Optimization Algorithm (Coupled QP)

Main idea:

problem approximates without needing inequality constraints

for problems and are equivalent

problem can be solved efficiently using a sparse linear algebra solver

Unconstrained auxiliary optimization problem:

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Parallel Real-time Optimization Algorithm (Decoupled QPs)

Augmented Lagrangian optimization problem:

strictly convex quadraticprograms

current approximations, and

Notice that has a separable structure ：


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using multi-parametric quadratic programming

pre-compute threesolution maps

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The memory requirements do not depend on the prediction horizon


Step1: input

Step2: compute using decoupled QPs

Step3: set

Step4: update next iterate using coupled QP:

End

For

Optimization Algorithm:

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The algorithm converges linearly:

Step1: input

Step2: compute using decoupled QPs

Step3: set

Step4: update next iterate using coupled QP:

End

For

Optimization Algorithm:

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update the OCPcurrent state measurement

proposedoptimization algorithm

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initial guess

re-scaling


update the OCP

uncertain dynamic system

current state measurement


sub-optimal solution

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initial guess

re-scaling


update the OCP




shift horizon

update


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initial guess

re-scaling

update the OCP




shift horizon

update


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initial guess

re-scaling

Recursive Feasibility

we assume that the state constant set to be robust control invariant, the subop-timal solution preserves the recursive feasibility

Parallel Explicit Tube MPC Controller (Recursive Feasibility)

Parallel Explicit Tube MPC Controller (Stability)

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Theorem 1

If the number of inner loops in the proposed optimization algorithm satisfies

the proposed scheme yields an asymptotically stable closed-loop controller.

linear convergence rate

satisfy the inequality

satisfies the inequality

Numerical Example

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Double integrator example:

Disturbance set:

State and input constraint:

Matrices:

Numerical Example (Memory Requirements)

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Explicit Tube MPCParallel Explicit Tube MPC

VS

Numerical Example (Runtime Performance)

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CPU Time [s]

On-line Centralized Tube MPC (qpOASES)

Parallel Explicit Tube MPC

Summary

Derived a real-time and parallelizable Tube MPC scheme

Conclusions:

Maintained recursive feasibility and stability

Future work:

Implementation on the embedded applications

Extension to nonlinear systems

19/19

Reduced the storage of Explicit MPC by orders of magnitude

Summary

Derived a real-time and parallelizable Tube MPC scheme

Conclusions:

Maintained recursive feasibility and stability

Future work:

Implementation on the embedded applications

Extension to nonlinear systems

Contact: [email protected]

Thank you! Questions?

19/19

Reduced the storage of Explicit MPC by orders of magnitude

Parallel Explicit Tube Model Predictive Control · Parallel Explicit Tube MPC Controller (Recursive...

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