nonlinear control with limited informationliberzon.csl.illinois.edu/research/icc-plenary.pdf ·...
TRANSCRIPT
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
Plant
Controller
INFORMATION FLOW in CONTROL SYSTEMS
2 of 21
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
[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
4 of 21
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
5 of 21
QUANTIZATION
Encoder Decoder
QUANTIZER
finite subset of
is partitioned into quantization regions
6 of 21
QUANTIZATION and ISS
7 of 21
– assume glob. asymp. stable (GAS)
QUANTIZATION and ISS
7 of 21
QUANTIZATION and ISS
no longer GAS
7 of 21
QUANTIZATION and ISS
quantization error
Assume
class
7 of 21
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
7 of 21
LINEAR SYSTEMS
Quantized control law:
∃ feedback gain & Lyapunov function
Closed-loop:
(automatically ISS w.r.t. )
8 of 21
DYNAMIC QUANTIZATION
9 of 21
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
9 of 21
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
9 of 21
Zoom out to overcome saturation
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
9 of 21
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
QUANTIZATION and DELAY
Architecture-independent approach
Based on the work of Teel
Delays possibly large
QUANTIZER DELAY
10 of 21
QUANTIZATION and DELAY
where
Can write
hence
11 of 21
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:
12 of 21
FINAL RESULT
Need:
small gain true
13 of 21
FINAL RESULT
Need:
small gain true
13 of 21
FINAL RESULT
solutions starting inenter and remain there
Can use “zooming” to improve convergence
Need:
small gain true
13 of 21
EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
14 of 21
EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
14 of 21
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:
14 of 21
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
15 of 21
ACTIVE PROBING for INFORMATION
dynamic
dynamic
(changes at sampling times)
(time-varying)
PLANT
QUANTIZER
CONTROLLER
Encoder Decoder
very small16 of 21
NONLINEAR SYSTEMS
sampling times
Example:
Zoom out to get initial bound
Between samplings
17 of 21
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 )
17 of 21
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
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
19 of 21
LINEAR SYSTEMS
20 of 21
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:
20 of 21
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
21 of 21