control of nonlinear systems under communication constraints daniel liberzon coordinated science...
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CONTROL of NONLINEAR SYSTEMS under
COMMUNICATION CONSTRAINTS
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
Caltech, Apr 1, 2005
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LIMITED INFORMATION SCENARIO
– partition of D
– points in D,
Quantizer:
Control:
for
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OBSTRUCTION to STABILIZATION
Asymptotic stabilization is usually lost
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BASIC QUESTIONS
• What can we say about a given quantized system?
• How can we design the “best” quantizer for stability?
• What can we do with very coarse quantization?
• What are the difficulties for nonlinear systems?
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STATE QUANTIZATION: LINEAR SYSTEMS
Quantized control law:
Closed-loop:
9 feedback gain & Lyapunov function
quantization error
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NONLINEAR SYSTEMS
For nonlinear systems, GAS such robustness
For linear systems, we saw that if
gives then
automatically gives
when
This is robustness to measurement errors
This is input-to-state stability (ISS) for measurement errors
when
To have the same result, need to assume pos.def. incr. :
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LOCATIONAL OPTIMIZATION
This leads to the problem:
for
Compare: mailboxes in a city, cellular base stations in a region
Also true for nonlinear systemsISS w.r.t. measurement errors
Small => small
[Bullo-L]
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MULTICENTER PROBLEM
Critical points of satisfy
1. is the Voronoi partition :
2.
This is the
center of enclosing sphere of smallest radius
Lloyd algorithm:
Each is the Chebyshev center
(solution of the 1-center problem).
iterate
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LOCATIONAL OPTIMIZATION: REFINED APPROACH
only need thisratio to be smallRevised problem:
. .. ..
.
.
...
.
. ..Logarithmic quantization:
Lower precision far away, higher precision close to 0
Only applicable to linear systems
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WEIGHTED MULTICENTER PROBLEM
This is the center of sphere enclosing
with smallest
Critical points of satisfy
1. is the Voronoi partition as before
2.
Lloyd algorithm – as before
Each is the weighted center
(solution of the weighted 1-center problem)
on not containing 0 (annulus)
Gives 25% decrease in for 2-D example
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DYNAMIC QUANTIZATION
zoom in
After ultimate bound is achieved,recompute partition for smaller region
Can recover global asymptotic stability
– zooming variable
Hybrid quantized control: is discrete state
Zoom out to overcome saturation
zoom out
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ACTIVE PROBING for INFORMATION
PLANT
QUANTIZER
CONTROLLER
dynamic
dynamic
(changes at sampling times)
(time-varying)
Encoder Decoder
very small
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LINEAR SYSTEMS
(Baillieul, Brockett-L, Hespanha et. al., Nair-Evans,
Petersen-Savkin, Tatikonda, and others)
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LINEAR SYSTEMS
sampling times
Zoom out to get initial bound
Example:
Between sampling times, let
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LINEAR SYSTEMS
Consider
• is divided by 3 at the sampling time
Example:
Between sampling times, let
• grows at most by the factor in one period
The norm
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where is stable
0
LINEAR SYSTEMS (continued)
Pick small enough s.t.
sampling frequency vs. open-loop instability
amount of static infoprovided by quantizer
• grows at most by the factor in one period
• is divided by 3 at each sampling time
The norm
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NONLINEAR SYSTEMS
• is divided by 3 at the sampling time
Let
Example:
Between samplings
• grows at most by the factor in one period
The norm
on a suitable compact region
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Pick small enough s.t.
NONLINEAR SYSTEMS (continued)
• grows at most by the factor in one period
• is divided by 3 at each sampling time
The norm
What properties of guarantee GAS ?
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ROBUSTNESS of the CONTROLLER
ISS w.r.t.
ISS w.r.t. measurement errors – quite restrictive...
ISS w.r.t.
Option 1.
Option 2. [Hespanha-L] Look at the evolution of
Easier to verify (e.g., GES & glob. Lip.)
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SOME RESEARCH DIRECTIONS
• ISS control design
• ISS of impulsive systems (work with Hespanha, Teel)
• Performance and robustness (work with Nesic)
• Applications
• Other?