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( t ) + a 1 ( t ) + a 0 sin( ( t ) 0 ) = bu ( t ) a 1 , a 0 , - - PowerPoint PPT Presentation
( t ) + a 1 ( t ) + a 0 sin( ( t ) 0 ) = bu ( t ) a 1 , a 0 , - - PowerPoint PPT Presentation
State estimation over the limited-band communication channel for pitch motion control of LAAS helicopter benchmark Boris ANDRIEVSKY & Alexander FRADKOV & Dimitri PEAUCELLE IPME-RAS - St Petersburg, RUSSIA LAAS-CNRS -
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Introduction ➙ Sensor data transmission over band limited communication channels:
navigation systems, distributed sensor networks, remote surveillance systems...
➙ Hybrid control:
continuous-time system ad control / discrete-time communication channel
➙ Use of smart sensors/coders: transmit condensed information ➙ Observer-based synchronization [Fradkov et. al., Physical Review 2006]
u(t) u(t)
− +
q(kT) e(kT) e(kT) Communication channel Observer Controller System Observer Decoder Adaptive tuning Coder Adaptive tuning Decoder X(t) e(t) y(t) y(t)
➙ Experiments with an ”Helicopter” benchmark
& 2 IFAC ACA’07, June 2007, Toulouse
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System, Observer and Controller
”Helicopter” Benchmark by Quanser at LAAS-CNRS ”Pitch” motion under assumptions (neglected gyroscopic torques, airspeed depen-
dence, aerodynamical forces, dry friction, travel and elevation coupling, motor dynamics)
¨ θ(t) + a1 ˙ θ(t) + a0 sin(θ(t) − θ0) = bu(t) a1, a0, θ0 and b have been identified and model validated on experiments.
& 3 IFAC ACA’07, June 2007, Toulouse
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System, Observer and Controller
PID with estimates of the state
u(t) = kPǫ(t) + kI
t
0 ǫ(τ)dτ + kDˆ
ω(t)
where ǫ(t) = θref(t) − ˆ
θ(t) and (ˆ θ, ˆ ω) are estimates of (θ, ω = ˙ θ). (kP, kI, kD) designed by loop shaping for the linearized system.
Dynamic observer for estimation
˙ ˆ θ(t) ˙ ˆ ω(t)
=
ˆ ω(t) −a1ˆ ω(t) − a0 sin(ˆ θ(t) − θ0) + bu(t)
+ l1
l2
¯
e(t)
where ¯
e(t) = ¯ e(kT) for t ∈ [ kT (k + 1)T[. (l1, l2) designed by pole placement for the linearized system.
& 4 IFAC ACA’07, June 2007, Toulouse
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Coder / Decoder
u(t) u(t)
− +
q(kT) e(kT) e(kT) Communication channel Observer Controller System Observer Decoder Adaptive tuning Coder Adaptive tuning Decoder X(t) e(t) y(t) y(t)
Communication channel characteristics
➙ T sampling period ➙ µ symbols in the coding alphabet. ➙ Channel data rate: R = log2(µ) bit per interval, ¯ R = R/T bit/s ➙ [Nair, Evans Automatica 2003] closed loop α-stability cannot hold if R <
- |λi|>α
log2 |λi/α|
where λi are the poles of the sampled open-loop discrete time system.
➙ For the considered system and T = 0.1s one needs R > 0.4 bit per interval.
& 5 IFAC ACA’07, June 2007, Toulouse
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Coder / Decoder
Coder characteristics
➙ Uniform scaled saturated quantization q(kT) = satMk(βround(β−1δ(kT)))
q(kT) δ(kT) −M +M
➙ One-step memory centering: δ(kT) = e(kT) − ck with ck+1 = ck + q(kT). ➙ One-step memory adaptive zooming (ρ > 1): Mk+1 = m + ρMk
if
|q(kT) + q((k + 1)T)| ≥ 2Mk Mk+1 = m + ρ−1Mk
- therwise
where m = (1 − ρ−1)Mmin
& 6 IFAC ACA’07, June 2007, Toulouse
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Coder / Decoder
Binary coding
➙ Binary coding (2 symbols in the coding alphabet, {−1, 1})
is optimal w.r.t. channel data rate
➙ Centering not needed (coding of error signal) ➙ Coding resumes to σ(kT) = sign(e(kT)) ➙ Adaptive zooming is given as Mk+1 = m + ρMk
if
|σ(kT) + σ((k + 1)T)| = 0 Mk+1 = m + ρ−1Mk
- therwise
➙ Decoding is ¯ e(kT) = Mkσ(kT).
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Implementation requirements
u(t) u(t)
− +
q(kT) e(kT) e(kT) Communication channel Observer Controller System Observer Decoder Adaptive tuning Coder Adaptive tuning Decoder X(t) e(t) y(t) y(t)
➙ Both Observers and Adaptive algorithms have same initial conditions ➙ If control signal u(t) not available to the sensor then duplicate controller
− +
e(kT) e(kT) q(kT) Communication channel Decoder Observer Controller Adaptive tuning Coder System Controller Observer Adaptive tuning Decoder
e(t) X(t) y(t) y(t) X(t) u(t) u(t)
➙ Needs to synchronize both controllers with same reference signal:
- ther coder/decoder pair for reference signal transmission.
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Experimental results
”Ideal” control using sensor data ¯
R = 300 bit/s
Same experimental conditions with adaptive coder/decoder at T = 0.05s.
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Experimental results
Same experimental conditions with adaptive coder/decoder at T = 0.1s.
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