NONLINEAR CONTROL with LIMITED INFORMATIONliberzon.csl.illinois.edu/research/ICC-plenary.pdf ·...
Transcript of NONLINEAR CONTROL with LIMITED INFORMATIONliberzon.csl.illinois.edu/research/ICC-plenary.pdf ·...
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NONLINEAR CONTROL with
LIMITED INFORMATION
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
Plenary talk, 2nd Indian Control Conference, Hyderabad, Jan 5, 2015 1 of 21
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Plant
Controller
INFORMATION FLOW in CONTROL SYSTEMS
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INFORMATION FLOW in CONTROL SYSTEMS
• Limited communication capacity• many control loops share network cable or wireless medium• microsystems with many sensors/actuators on one chip
• Need to minimize information transmission (security)• Event-driven actuators
• Coarse sensing
• Theoretical interest3 of 21
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[Brockett,Delchamps, Elia, Mitter,Nair, Savkin, Tatikonda, Wong,…]
• Deterministic & stochastic models
• Tools from information theory
• Mostly for linear plant dynamics
BACKGROUNDPrevious work:
• Unified framework for• quantization• time delays• disturbances
Our goals:• Handle nonlinear dynamics
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Caveat:This doesn’t work in general, need robustness from controller
OUR APPROACH
(Goal: treat nonlinear systems; handle quantization, delays, etc.)
• Model these effects via deterministic error signals,
• Design a control law ignoring these errors,
• “Certainty equivalence”: apply control,combined with estimation to reduce to zero
Technical tools:
• Input-to-state stability (ISS)• Lyapunov functions
• Small-gain theorems• Hybrid systems
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QUANTIZATION
Encoder Decoder
QUANTIZER
finite subset of
is partitioned into quantization regions
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QUANTIZATION and ISS
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– assume glob. asymp. stable (GAS)
QUANTIZATION and ISS
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QUANTIZATION and ISS
no longer GAS
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QUANTIZATION and ISS
quantization error
Assume
class
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Solutions that start inenter and remain there
This is input-to-state stability (ISS) w.r.t. measurement errors
In time domain:
QUANTIZATION and ISS
quantization error
Assume
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LINEAR SYSTEMS
Quantized control law:
∃ feedback gain & Lyapunov function
Closed-loop:
(automatically ISS w.r.t. )
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DYNAMIC QUANTIZATION
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DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
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DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
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Zoom out to overcome saturation
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
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After ultimate bound is achieved,recompute partition for smaller region
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
ISS from to ISS from to small-gain conditionProof:
Can recover global asymptotic stability
[L–Nešić, ’05, ’06, L–Nešić–Teel ’14]9 of 21
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QUANTIZATION and DELAY
Architecture-independent approach
Based on the work of Teel
Delays possibly large
QUANTIZER DELAY
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QUANTIZATION and DELAY
where
Can write
hence
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SMALL – GAIN ARGUMENT
if
then we recover ISS w.r.t. [Teel ’98]
Small gain:
Assuming ISS w.r.t. actuator errors:
In time domain:
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FINAL RESULT
Need:
small gain true
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FINAL RESULT
Need:
small gain true
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FINAL RESULT
solutions starting inenter and remain there
Can use “zooming” to improve convergence
Need:
small gain true
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EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
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EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
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Issue: disturbance forces the state outside quantizer range
Must switch repeatedly between zooming-in and zooming-out
Result: for linear plant, can achieve ISS w.r.t. disturbance
(ISS gains are nonlinear although plant is linear; cf. [Martins])
EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
After zoom-in:
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NETWORKED CONTROL SYSTEMS [Nešić–L]
NCS: Transmit only some variables according to time scheduling protocol
Examples: round-robin, TOD (try-once-discard)
QCS: Transmit quantized versions of all variables
NQCS: Unified framework combining time scheduling and quantization
Basic design/analysis steps:• Design controller ignoring network effects
• Apply small-gain theorem to compute upper bound onmaximal allowed transmission interval (MATI)
• Prove discrete protocol stability via Lyapunov function
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ACTIVE PROBING for INFORMATION
dynamic
dynamic
(changes at sampling times)
(time-varying)
PLANT
QUANTIZER
CONTROLLER
Encoder Decoder
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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(dependent on )
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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
If this is ISS w.r.t. as before, then 18 of 21
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ROBUSTNESS of the CONTROLLER
ISS w.r.t.
Same condition as before (restrictive, hard to check)
checkable sufficient conditions ([Hespanha-L-Teel])
ISS w.r.t.
Option 1.
Option 2. Look at the evolution of
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LINEAR SYSTEMS
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LINEAR SYSTEMS
[Baillieul, Brockett-L, Hespanha, Nair-Evans, Petersen-Savkin,Tatikonda]
Between sampling times,
where is Hurwitz0
• divided by 3 at each sampling time
• grows at most by in one period
amount of static infoprovided by quantizer
sampling frequency vs. open-loop instability
global quantity:
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OTHER RESEARCH DIRECTIONS
• Modeling uncertainty (with L. Vu)
• Disturbances and coarse quantizers (with Y. Sharon)
• Multi-agent coordination (with S. LaValle and J. Yu)
• Quantized output feedback and observers (with H. Shim)
• Vision-based control (with Y. Ma and Y. Sharon)
• Performance-based design (with F. Bullo)
• Quantized control of switched systems
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