parallel explicit tube model predictive control · parallel explicit tube mpc controller (recursive...
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Parallel ExplicitTube Model Predictive Control
Kai Wang1, Yuning Jiang1, Juraj Oravec2, Mario E. Villanueva1,
Boris Houska1
1ShanghaiTech University
2Slovak University of Technology in Bratislava
1/19
Overview
Background on Tube MPC
2/19
Model Predictive Control (MPC)
Rigid Tube MPC
Contribution
Parallel Real-time Optimization Algorithm
Numerical Example
Robust MPC
Parallel Explicit Tube MPC Controller
Model Predictive Control (MPC)
3/19
GoalObstacles
Uncertain Control System:
Model Predictive Control (MPC)
3/19
Minimize distance to the dotted line
Subject to: System dynamics
State and input constraints
Uncertain Control System:
Certainty equivalent MPC:
Model Predictive Control (MPC)
3/19
wait for new measurement
re-optimize the trajectory
Repeat:
apply optimal control law
Model Predictive Control (MPC)
3/19
Problem:
certainty equivalent prediction is optimistic
infeasible (worst-case) scenarios possible
4/19
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
4/19
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
5/19
Main idea:
enclose all trajectories by a tube of constant polytopic cross-sections
optimize the central path of the tube
Rigid Tube MPC
5/19
Main idea:
enclose all trajectories by a tube of constant polytopic cross-sections
optimize the central path of the tube
Rigid Tube MPC
5/19
Main idea:
enclose all trajectories by a tube of constant polytopic cross-sections
optimize the central path of the tube
Rigid Tube MPC
5/19
Main idea:
enclose all trajectories by a tube of constant polytopic cross-sections
optimize the central path of the tube
Rigid Tube MPC
6/19
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.,
6/19
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.,
7/19
Mathematical Formulation of Rigid Tube MPC
Optimal Control Problem (OCP):
stage cost:
terminal cost:
and are symmetric positivedefinite
Standard assumptions:
terminal constraint set
7/19
Mathematical Formulation of Rigid Tube MPC
Optimal Control Problem (OCP):
stage cost:
terminal cost:
and are symmetric positivedefinite
Standard assumptions:
terminal constraint set
7/19
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
8/19
Robust Model Predictive Control (MPC)
Rigid Tube MPC
Contribution
Parallel Real-time Optimization Algorithm
Numerical Example
Tube-Based Robust MPC
Parallel Explicit Tube MPC Scheme
Goal: develop a real-time Tube MPC algorithm with limited memory footprint
Overview
Background on Tube MPC
8/19
Robust Model Predictive Control (MPC)
Rigid Tube MPC
Contribution
Parallel Real-time Optimization Algorithm
Numerical Example
Tube-Based Robust MPC
Parallel Explicit Tube MPC Scheme
Goal: develop a real-time Tube MPC algorithm with limited memory footprint
9/19
Rewrite the OCP of the Rigid Tube MPC
Stacked vectors and matrices
Constraint sets
Cost function
9/19
Rewrite the OCP of the Rigid Tube MPC
Stacked vectors and matrices
Constraint sets
Cost function
9/19
Rewrite the OCP of the Rigid Tube MPC
Stacked vectors and matrices
Constraint sets
Cost function
9/19
Rewrite the OCP of the Rigid Tube MPC
Stacked vectors and matrices
Constraint sets
Cost function
9/19
Rewrite the OCP of the Rigid Tube MPC
Constraint sets
Cost function
Stacked vectors and matrices
equivalent
10/19
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:
10/19
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:
11/19
Parallel Real-time Optimization Algorithm (Decoupled QPs)
Augmented Lagrangian optimization problem:
strictly convex quadraticprograms
current approximations, and
Notice that has a separable structure :
11/19
Parallel Real-time Optimization Algorithm (Decoupled QPs)
Augmented Lagrangian optimization problem:
strictly convex quadraticprograms
current approximations, and
Notice that has a separable structure :
Parallel Real-time Optimization Algorithm (Decoupled QPs)
12/19
Parallel Real-time Optimization Algorithm (Decoupled QPs)
using multi-parametric quadratic programming
pre-compute threesolution maps
12/19
The memory requirements do not depend on the prediction horizon
Parallel Real-time Optimization Algorithm (Decoupled QPs)
using multi-parametric quadratic programming
pre-compute threesolution maps
12/19
The memory requirements do not depend on the prediction horizon
Parallel Real-time Optimization Algorithm
Step1: input
Step2: compute using decoupled QPs
Step3: set
Step4: update next iterate using coupled QP:
End
For
Optimization Algorithm:
13/19
Parallel Real-time Optimization Algorithm
Step1: input
Step2: compute using decoupled QPs
Step3: set
Step4: update next iterate using coupled QP:
End
For
Optimization Algorithm:
13/19
Parallel Real-time Optimization Algorithm
Step1: input
Step2: compute using decoupled QPs
Step3: set
Step4: update next iterate using coupled QP:
End
For
Optimization Algorithm:
13/19
Parallel Real-time Optimization Algorithm
The algorithm converges linearly:
Step1: input
Step2: compute using decoupled QPs
Step3: set
Step4: update next iterate using coupled QP:
End
For
Optimization Algorithm:
13/19
Parallel Explicit Tube MPC Controller
update the OCPcurrent state measurement
proposedoptimization algorithm
14/19
initial guess
re-scaling
Parallel Explicit Tube MPC Controller
update the OCP
uncertain dynamic system
current state measurement
proposedoptimization algorithm
sub-optimal solution
14/19
initial guess
re-scaling
Parallel Explicit Tube MPC Controller
update the OCP
uncertain dynamic system
current state measurement
proposedoptimization algorithm
shift horizon
update
sub-optimal solution
14/19
initial guess
re-scaling
update the OCP
uncertain dynamic system
current state measurement
proposedoptimization algorithm
shift horizon
update
sub-optimal solution
14/19
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)
15/19
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
16/19
Double integrator example:
Disturbance set:
State and input constraint:
Matrices:
Numerical Example (Memory Requirements)
17/19
Explicit Tube MPCParallel Explicit Tube MPC
VS
Numerical Example (Runtime Performance)
18/19
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
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