Exponentially Unstable Plants with Saturating Actuators

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Exponentially Unstable Plants with Saturating
Actuators
ONR Young Investigator Award
Program Manager Dr. Allen Moshfegh
Zongli Lin, Principal Investigator
Department of Electrical
and Computer Engineering
University of
Virginia Program Review, Berkeley, CA,
August 2001
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Outline of the Presentation

• Introduction to the Problem
• Previous Results and Motivation
• A Two Step Design Method for Performance and
  Stability
• Simulation Results - TRANS3
• Future Work

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Introduction to the Problem
System Description
Controllability
Null controllable region C the set of
all states that can be driven to the origin.
Stabilization
Find a feedback law uf (x) that stabilizes the
system on C.
Issues in Stabilization

• Large stability region
• Speedy convergence of the trajectories

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Previous Results and Motivation

• Characterization of null controllable region C
  
• Hu and Lin, Control Systems with
  Actuator Saturation, Birkhauser, 2001

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Previous Results and Motivation

• Semi-global stabilization on C

Hu and Lin, Birkhauser, 2001
For any compact set X ? C , construct a feedback
law,

• such that
• the closed-loop system is locally
  asymptotically stable at x 0
• X is contained in the basin of attraction.

C
Observations

• large stability region
• slow convergence rate

x(0)
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Previous Results and Motivation

• Piecewise Linear Control (PLC)

Wredenhagen and Belanger, Automatica,
1994 Miotto, Shewchun, Feron, Paduano, AIAA
GNC, 1996
Observations

• by choosing we can cause
• the trajectories to converge faster as
• they enter an inner ellipsoid
• discontinuous feedback law
• saturation avoidance conservative!

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Previous Results and Motivation

• Scheduled PLC Controller

Megretski, IFAC World Congress, 1996
Observations

• Q(x) increases as x decreases
• continuous controller
• computationally formidable
• saturation avoidance conservative!

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A Two Step Design Method
Step 1. Design for Regional Performance
(e.g., the continuous switching
method)
Step 2. Augmentation for Large Stability Region
(e.g., semi-global scheme)
?o
C
?0.9C
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A Two Step Design Method
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A Two Step Design Method
A Continuously Switching Controller
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A Two Step Design Method
A Continuously Switching Controller
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A Two Step Design Method
A Continuously Switching Controller
An Example. Inverted Pendulum
LQR design
PLC design
Continuously switching
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A Two Step Design Method
A Continuously Switching Controller
LQR design
PLC design
Continuously switching
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Simulation Results
TRANS3 Longitudinal Dynamics (Texas A M)
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Future Work

• Actuator rate saturation
• Nonlinear systems subject to actuator
  saturation
• Systems with time delay in state and/or control
• Decentralized control for large scale systems.