OBSERVER-BASED QUANTIZED
OUTPUT FEEDBACK CONTROL of
NONLINEAR SYSTEMS
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
1 of 11 IFAC World Congress, Seoul, Korea, July 2008
QUANTIZED OUTPUT FEEDBACK
PLANT
QUANTIZER
CONTROLLER
Motivation:
• limited communication between sensor and actuator• trade-off between communication and computation
Objectives:
• analyze effect of quantization on system stability
• design controllers robust to quantization errors2 of 11
QUANTIZER
is the range, is the quantization error bound
Assume such that:
1.
2.
Encoder Decoderfinite setQUANTIZER
Output space is divided into quantization regions
For , the quantizer saturates3 of 11
LINEAR SYSTEM
Plant:
quantizationerror
Closed-loop system:
Luenberger observer-based controller:
or in short
where is Hurwitz if and are Hurwitz
[Brockett-L]
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LINEAR SYSTEM (continued)
level sets of V
Solutions go from the larger level set to the smaller one
Recall:
Hurwitz
For we have
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is of class if
• for each fixed
• as for each
INPUT-TO-STATE STABILITY (ISS) [Sontag]
class function
where
ISS:
Equivalent Lyapunov characterization:
when for some
Example:
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NONLINEAR SYSTEM
Plant:
Dynamic controller:
Assume: this is ISS w.r.t. quantization error
Closed-loop system:
quantization error
or in short
(so in particular, should have GAS when )7 of 11
NONLINEAR SYSTEM (continued)
level sets of V
Solutions go from the larger level set to the smaller one
ISS
Can recover GAS using dynamic quantization
Lyap. function and class function s.t.
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ISS ASSUMPTION: CLOSER LOOK
Closed-loop system
Reason: cascade argument
if for some we have
1.
2.
and
is ISS
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Can extend this via a small-gain argument (need )
ISS CONTROLLER DESIGN
Closed-loop system:
1.
• Not always possible to achieve [Freeman ’95, Fah ’99]
• Results exist for classes of systems [Freeman & Kokotovic ’93, ’96, Freeman ’97, Fah ’99, Jiang et al. ’99, Sanfelice & Teel ’05, Ebenbauer, Raff & Allgower ’07, ’08]
• ISS assumption is fundamental in quantized control of nonlinear systems [L ’03]
This is ISS property of control law w.r.t. observation errors:
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ISS OBSERVER DESIGN
Closed-loop system:
2.
This is ISS property of observer w.r.t. additive output errors
• This property can be achieved for
with detectable and globally Lipschitz, very restrictive
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• Almost no results on design of such ISS observers exist,
except recent work of H. Shim, J.H. Seo, A.R. Teel, J.S. Kim
More research on this problem is needed
