Robust Constrained Model Predictive Control Arthur George - MIT

209 Pages·2004·5.02 MB·English

Robust Constrained Model Predictive Control
by
Arthur George Richards
Master of Science
Massachusetts Institute of Technology, 2002
Master of Engineering
University of Cambridge, 2000
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2005
(cid:13)c Massachusetts Institute of Technology 2005. All rights reserved.
Author ..............................................................
Department of Aeronautics and Astronautics
November 22, 2004
Accepted by.........................................................
Jaime Peraire
Professor of Aeronautics and Astronautics
Chair, Committee on Graduate Students
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Robust Constrained Model Predictive Control
by
Arthur George Richards
Accepted by.......................................................................
Jonathan P. How
Associate Professor of Aeronautics and Astronautics
Thesis Supervisor
Accepted by.......................................................................
Eric M. Feron
Associate Professor of Aeronautics and Astronautics
Accepted by.......................................................................
John J. Deyst Jr.
Associate Professor of Aeronautics and Astronautics
Accepted by.......................................................................
Dr. Jorge Tierno
ALPHATECH Inc.
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Robust Constrained Model Predictive Control
by
Arthur George Richards
Submitted to the Department of Aeronautics and Astronautics
on November 22, 2004, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
This thesis extends Model Predictive Control (MPC) for constrained linear systems
subject to uncertainty, including persistent disturbances, estimation error and the
effects of delay. Previous work has shown that feasibility and constraint satisfaction
can be guaranteed by tightening the constraints in a suitable, monotonic sequence.
Thisthesisextendsthatworkinseveralways, includingmoreflexibleconstrainttight-
ening, applied within the prediction horizon, and more general terminal constraints,
applied to ensure feasible evolution beyond the horizon. These modifications reduce
the conservatism associated with the constraint tightening approach.
Modifications to account for estimation error, enabling output feedback control,
are presented, and we show that the effects of time delay can be handled in a similar
manner. A further extension combines robust MPC with a novel uncertainty estima-
tion algorithm, providing an adaptive MPC that adjusts the optimization constraints
to suit the level of uncertainty detected. This adaptive control replaces the need for
accurate a priori knowledge of uncertainty bounds. An approximate algorithm is de-
veloped for the prediction of the closed-loop performance using the new robust MPC
formulation, enabling rapid trade studies on the effect of controller parameters.
The constraint tightening concept is applied to develop a novel algorithm for
Decentralized MPC (DMPC) for teams of cooperating subsystems with coupled con-
straints. The centralized MPC optimization is divided into smaller subproblems, each
solving for the future actions of a single subsystem. Each subproblem is solved only
once per time step, without iteration, and is guaranteed to be feasible. Simulation
examples involving multiple Uninhabited Aerial Vehicles (UAVs) demonstrate that
the new DMPC algorithm offers significant computational improvement compared to
its centralized counterpart.
The controllers developed in this thesis are demonstrated throughout in simulated
examples related to vehicle control. Also, some of the controllers have been imple-
mented on vehicle testbeds to verify their operation. The tools developed in this
thesis improve the applicability of MPC to problems involving uncertainty and high
complexity, for example, the control of a team of cooperating UAVs.
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Acknowledgements
I’dliketothankmyadvisor, Prof.JonathanHow, forallhisguidance, supportandfor
letting me loose on some interesting problems. I also thank my committee members,
Prof. Eric Feron, Prof. John Deyst and Dr. Jorge Tierno for their input and oversight,
and Prof. Steve Hall for participating in the defense.
Many fellow students have shaped my MIT experience, at work and beyond, in-
cluding the members of the Aerospace Controls Lab, the Space Systems Lab and
the various students of the Laboratory for Random Graduate Study in Room 409.
Special thanks to the other founding members of Jon How’s MIT group for making
settling in such a pleasure and for their continuing friendship. Yoshiaki Kuwata and
Nick Pohlman were good enough to read this hefty document and help its preparation
with their feedback. Simon Nolet generously gave much time and effort to help me
with the SPHERES experiments.
I thank my mother for her constant support throughout this endeavour and all
that came before. Finally, I dedicate this thesis to two other family members who
both had a hand in this somehow.
For AER and GTB
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Contents
1 Introduction 19
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.1.1 Robust feasibility and constraint satisfaction . . . . . . . . . 21
1.1.2 Decentralization/Scalability . . . . . . . . . . . . . . . . . . . 21
1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2.1 Model Predictive Control . . . . . . . . . . . . . . . . . . . . 22
1.2.2 Decentralized Control . . . . . . . . . . . . . . . . . . . . . . 23
1.3 Outline and Summary of Contributions . . . . . . . . . . . . . . . . 24
2 Robust MPC 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Robustly-Feasible MPC . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Robust Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5.1 Robust Feasibility Demonstration . . . . . . . . . . . . . . . 47
2.5.2 Spacecraft Control . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5.3 Graphical Proof of Robustness . . . . . . . . . . . . . . . . . 54
2.5.4 Robust Convergence . . . . . . . . . . . . . . . . . . . . . . . 60
2.6 Variable Horizon MPC . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.6.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . 62
2.6.2 Variable Horizon Controller . . . . . . . . . . . . . . . . . . . 63
2.6.3 MILP Implementation . . . . . . . . . . . . . . . . . . . . . . 66
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2.7 Variable Horizon Examples . . . . . . . . . . . . . . . . . . . . . . . 69
2.7.1 Space Station Rendezvous . . . . . . . . . . . . . . . . . . . . 69
2.7.2 UAV Obstacle Avoidance . . . . . . . . . . . . . . . . . . . . 72
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.A Proof of Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.A.1 Proof of Proposition 2.1 . . . . . . . . . . . . . . . . . . . . . 75
2.A.2 Proof of Proposition 2.2 . . . . . . . . . . . . . . . . . . . . . 77
2.A.3 Proof of Proposition 2.3 . . . . . . . . . . . . . . . . . . . . . 77
3 MPC with Imperfect Information 79
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2 Handling Imperfect State Information . . . . . . . . . . . . . . . . . 82
3.2.1 MPC with Estimation Error . . . . . . . . . . . . . . . . . . 82
3.2.2 Finite Memory Estimator . . . . . . . . . . . . . . . . . . . . 84
3.2.3 Combining Finite Memory Estimator with MPC . . . . . . . 86
3.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2.5 Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.2.6 Time Delay Example . . . . . . . . . . . . . . . . . . . . . . 91
3.3 Adaptive Prediction Error Bounds . . . . . . . . . . . . . . . . . . . 94
3.3.1 Bounding Set Estimation for Vector Signals . . . . . . . . . . 94
3.3.2 Adaptive MPC . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3.3 Adaptive MPC Examples . . . . . . . . . . . . . . . . . . . . 97
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.A Bound Estimation for Scalar Signals . . . . . . . . . . . . . . . . . . 100
4 Decentralized MPC 107
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3 Centralized Robust MPC Problem . . . . . . . . . . . . . . . . . . . 111
4.4 Decentralized MPC Algorithm . . . . . . . . . . . . . . . . . . . . . 114
4.5 DMPC Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
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