( 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 - Universit´ e de Toulouse, FRANCE CNRS-RAS cooperative research project ”Robust and adaptive control of complex systems: Theory and applications”

Introduction CNRS-RAS cooperation objectives ➙ Investigate robustness issues of adaptive algorithms for control both theoretically and through experiments ➙ Adaptive Identification (CCA’07, ALCOSP’07) ➙ Direct adaptive control (ROCOND’06, ALCOSP’07, ACC’07) This presentation ➙ State-estimation in limited-band communication channel with adaptive tuning of coder/decoder & 1 IFAC ACA’07, June 2007, Toulouse

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) y(t) y(t) u(t) + − Observer Controller System Observer X(t) e(kT) e(kT) e(t) Adaptive Adaptive Decoder Coder Decoder tuning tuning Communication channel q(kT) ➙ Experiments with an ”Helicopter” benchmark & 2 IFAC ACA’07, June 2007, Toulouse

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 ) + a 1 ˙ ¨ θ ( t ) + a 0 sin( θ ( t ) − θ 0 ) = bu ( t ) a 1 , a 0 , θ 0 and b have been identified and model validated on experiments. & 3 IFAC ACA’07, June 2007, Toulouse

System, Observer and Controller PID with estimates of the state � t u ( t ) = k P ǫ ( t ) + k I 0 ǫ ( τ ) dτ + k D ˆ ω ( t ) where ǫ ( t ) = θ ref ( t ) − ˆ θ ( t ) and (ˆ ω ) are estimates of ( θ, ω = ˙ θ, ˆ θ ) . ( k P , k I , k D ) designed by loop shaping for the linearized system. Dynamic observer for estimation ˙       ˆ θ ( t ) ω ( t ) ˆ  l 1  =  +  ¯ e ( t )       ˙ ω ( t ) − a 0 sin(ˆ   ω ( t ) ˆ − a 1 ˆ θ ( t ) − θ 0 ) + bu ( t ) l 2 where ¯ e ( t ) = ¯ e ( kT ) for t ∈ [ kT ( k + 1) T [ . ( l 1 , l 2 ) designed by pole placement for the linearized system. & 4 IFAC ACA’07, June 2007, Toulouse

Coder / Decoder u(t) y(t) y(t) u(t) + − Observer Controller System Observer X(t) e(kT) e(kT) e(t) Adaptive Adaptive Decoder Coder Decoder tuning tuning Communication channel q(kT) Communication channel characteristics ➙ T sampling period ➙ µ symbols in the coding alphabet. ➙ Channel data rate: R = log 2 ( µ ) bit per interval, ¯ R = R/T bit/s ➙ [Nair, Evans Automatica 2003] closed loop α -stability cannot hold if � R < log 2 | λ i /α | | λ i | >α where λ i are the poles of the sampled open-loop discrete time system. ➙ For the considered system and T = 0 . 1 s one needs R > 0 . 4 bit per interval. & 5 IFAC ACA’07, June 2007, Toulouse

Coder / Decoder Coder characteristics ➙ Uniform scaled saturated quantization q ( kT ) = sat M k ( β round ( β − 1 δ ( kT ))) q(kT) +M δ (kT) − M ➙ One-step memory centering: δ ( kT ) = e ( kT ) − c k with c k +1 = c k + q ( kT ) . ➙ One-step memory adaptive zooming ( ρ > 1 ): M k +1 = m + ρM k | q ( kT ) + q (( k + 1) T ) | ≥ 2 M k if M k +1 = m + ρ − 1 M k otherwise where m = (1 − ρ − 1 ) M min & 6 IFAC ACA’07, June 2007, Toulouse

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 M k +1 = m + ρM k | σ ( kT ) + σ (( k + 1) T ) | � = 0 if M k +1 = m + ρ − 1 M k otherwise ➙ Decoding is ¯ e ( kT ) = M k σ ( kT ) . & 7 IFAC ACA’07, June 2007, Toulouse

Implementation requirements u(t) y(t) y(t) u(t) + − Observer Controller System Observer X(t) e(kT) e(kT) e(t) Adaptive Adaptive Decoder Coder Decoder tuning tuning Communication channel q(kT) ➙ Both Observers and Adaptive algorithms have same initial conditions ➙ If control signal u ( t ) not available to the sensor then duplicate controller Controller X(t) u(t) y(t) y(t) u(t) + − Observer Controller System Observer X(t) e(kT) e(kT) e(t) Adaptive Adaptive Decoder Coder Decoder tuning tuning Communication channel q(kT) ➙ Needs to synchronize both controllers with same reference signal: other coder/decoder pair for reference signal transmission. & 8 IFAC ACA’07, June 2007, Toulouse

Experimental results ”Ideal” control using sensor data ¯ R = 300 bit/s Same experimental conditions with adaptive coder/decoder at T = 0 . 05 s. & 9 IFAC ACA’07, June 2007, Toulouse

Experimental results Same experimental conditions with adaptive coder/decoder at T = 0 . 1 s. & 10 IFAC ACA’07, June 2007, Toulouse

Conclusions Control with band limited communication channel ➚ Smart sensors with observers and adaptive coder/decoder ➚ Relatively low transmission rate validated on experiments Future experiments ➙ 3-D control of the benchmark ➙ Transmission channel with delays ➙ Transmission channel with errors & 11 IFAC ACA’07, June 2007, Toulouse