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![Page 1: CONTROL of NONLINEAR SYSTEMS with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.](https://reader035.fdocuments.us/reader035/viewer/2022062716/56649dbd5503460f94ab0395/html5/thumbnails/1.jpg)
CONTROL of NONLINEAR SYSTEMS
with LIMITED INFORMATION
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
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0
Control objectives: stabilize to 0 or to a desired set
containing 0, exit D through a specified facet, etc.
CONSTRAINED CONTROL
Constraint: – given
control commands
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LIMITED INFORMATION SCENARIO
– partition of D
– points in D,
Quantizer/encoder:
Control:
for
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MOTIVATION
• Limited communication capacity
• many systems/tasks share network cable or wireless medium
• microsystems with many sensors/actuators on one chip
• Need to minimize information transmission (security)
• Event-driven actuators
• PWM amplifier
• manual car transmission
• stepping motor
Encoder Decoder
QUANTIZER
finite subset
of
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QUANTIZER GEOMETRY
is partitioned into quantization regions
uniform logarithmic arbitrary
Dynamics change at boundaries => hybrid closed-loop system
Chattering on the boundaries is possible (sliding mode)
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QUANTIZATION ERROR and RANGE
is the range, is the quantization error bound
For , the quantizer saturates
Assume such that:
1.
2.
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OBSTRUCTION to STABILIZATION
Assume: fixed,M
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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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?• What are the difficulties for nonlinear systems?
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STATE QUANTIZATION: LINEAR SYSTEMS
Quantized control law:
where is quantization error
Closed-loop system:
is asymptotically stable
9 Lyapunov function
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LINEAR SYSTEMS (continued)
Recall:
Previous slide:
Lemma: solutions
that start in
enter in
finite time
Combine:
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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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SUMMARY: PERTURBATION APPROACH
1. Design ignoring constraint
2. View as approximation
3. Prove that this still solves the problem
(in a weaker sense)
Issue:
error
Need to give ISS w.r.t. measurement errors
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INPUT QUANTIZATION
where
Control law:
Closed-loop system:
Analysis – same as before
Control law:
where
Need ISS with respect to actuator errors
Closed-loop system:
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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?• What are the difficulties for nonlinear systems?
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LOCATIONAL OPTIMIZATION: NAIVE APPROACH
This leads to the problem:
for Also true for nonlinear systemsISS w.r.t. measurement errors
Smaller => smaller
Compare: mailboxes in a city, cellular base stations in a region
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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
Zoom out to overcome saturation
Can recover global asymptotic stability
(also applies to input and output quantization)
– zooming variable
Hybrid quantized control: is discrete state
zoom out
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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?• What are the difficulties for nonlinear systems?
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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
sampling times
Example:
Zoom out to get initial bound
Between samplings
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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. Look at the evolution of
Easier to verify (e.g., GES & glob. Lip.)
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RESEARCH DIRECTIONS
• ISS control design
• Locational optimization
• Performance and robustness
• Applications
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REFERENCES
Brockett & L, 2000 (IEEE TAC)Bullo & L, 2003, L & Hespanha, 2004(http://decision.csl.uiuc.edu/~liberzon)