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CONTROL of NONLINEAR SYSTEMS
with LIMITED INFORMATION
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
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
LIMITED INFORMATION SCENARIO
– partition of D
– points in D,
Quantizer/encoder:
Control:
for
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
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)
QUANTIZATION ERROR and RANGE
is the range, is the quantization error bound
For , the quantizer saturates
Assume such that:
1.
2.
OBSTRUCTION to STABILIZATION
Assume: fixed,M
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?
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?
STATE QUANTIZATION: LINEAR SYSTEMS
Quantized control law:
where is quantization error
Closed-loop system:
is asymptotically stable
9 Lyapunov function
LINEAR SYSTEMS (continued)
Recall:
Previous slide:
Lemma: solutions
that start in
enter in
finite time
Combine:
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. :
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
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:
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?
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
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
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
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?
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
sampling times
Example:
Zoom out to get initial bound
Between samplings
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. Look at the evolution of
Easier to verify (e.g., GES & glob. Lip.)
RESEARCH DIRECTIONS
• ISS control design
• Locational optimization
• Performance and robustness
• Applications
REFERENCES
Brockett & L, 2000 (IEEE TAC)Bullo & L, 2003, L & Hespanha, 2004(http://decision.csl.uiuc.edu/~liberzon)
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