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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
LIMITED INFORMATION SCENARIO
– partition of D
– points in D,
Quantizer:
Control:
for
OBSTRUCTION to STABILIZATION
Asymptotic stabilization is usually lost
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?
STATE QUANTIZATION: LINEAR SYSTEMS
Quantized control law:
Closed-loop:
9 feedback gain & Lyapunov function
quantization error
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. :
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]
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
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
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
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
ACTIVE PROBING for INFORMATION
PLANT
QUANTIZER
CONTROLLER
dynamic
dynamic
(changes at sampling times)
(time-varying)
Encoder Decoder
very small
LINEAR SYSTEMS
(Baillieul, Brockett-L, Hespanha et. al., Nair-Evans,
Petersen-Savkin, Tatikonda, and others)
LINEAR SYSTEMS
sampling times
Zoom out to get initial bound
Example:
Between sampling times, let
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
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
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
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 ?
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.)
SOME RESEARCH DIRECTIONS
• ISS control design
• ISS of impulsive systems (work with Hespanha, Teel)
• Performance and robustness (work with Nesic)
• Applications
• Other?