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Adaptive Parameter Identification for Simplified 3D-Motion Model of LAAS Helicopter Benchmark Sylvain Le Gac & Dimitri PEAUCELLE & Boris ANDRIEVSKY SEDITEC LAAS-CNRS - Universit e de Toulouse, FRANCE IPME-RAS

  1. Adaptive Parameter Identification for Simplified 3D-Motion Model of ‘LAAS Helicopter Benchmark’ Sylvain Le Gac § & Dimitri PEAUCELLE † & Boris ANDRIEVSKY ‡ § SEDITEC † LAAS-CNRS - Universit´ e de Toulouse, FRANCE ‡ IPME-RAS - St Petersburg, RUSSIA CNRS-RAS cooperative research project ”Robust and adaptive control of complex systems: Theory and applications”

  2. 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, ACA’07) ➙ State-estimation in limited-band communication channel Other cooperations ➙ Also part of ECO-NET project ”Polynomial optimization for complex systems”, funded by French Ministry of Foreign Affairs, and handled by Egide. Concerned countries : Czech Republic, France, Russian Federation, Slovakia. ➙ Submitted a PICS project ”Robust and adaptive control of complex systems” (funded by CNRS and RFBR). & 1 IFAC ALCOSP’07, August 2007, St. Petersburg

  3. Introduction ”Helicopter” Benchmark by Quanser at LAAS-CNRS ➙ Purpose : demonstration of research results & educational ➙ Simplified model needed with identified parameters ➙ Identification via adaptive algorithms ➙ Outline : Theory / Experiments & 2 IFAC ALCOSP’07, August 2007, St. Petersburg

  4. MISO LTI systems LTI system: order n with m inputs m � b in u ( n ) y ( n ) ( t )+ . . . + a 1 ˙ y ( t )+ a 0 y ( t ) = i ( t )+ . . . + b i 1 ˙ u i ( t )+ b i 0 u i ( t ) . i =1 Define the following vectors     u ( n − 1) y ( n − 1) ( t ) ( t ) i     . .     . . X y ( t ) = , X ui ( t ) = ,     . .         y ( t ) u i ( t ) � � φ T ( t ) = u ( n ) u ( n ) X T X T X T y ( t ) 1 ( t ) u 1 ( t ) m ( t ) um ( t ) · · · � � Ω T = a n − 1 . . . a 0 b 1 n . . . b 10 . . . b mn . . . b m 0 System compact model: y ( n ) ( t ) = φ T ( t )Ω . Identification: least square estimation of Ω assumed constant. & 3 IFAC ALCOSP’07, August 2007, St. Petersburg

  5. Filters D ( s ) : avoid derivation of y ( t ) and u i ( t ) ➘ Only y ( t ) and u i ( t ) are measured, numerical time-derivatives amplify noise ➚ Let an order n Hurwitz polynomial D ( s ) = s n + . . . + d 1 s + d 0 then y n ( t ) = ˜ y ( n ) ( t ) = φ T ( t )Ω ⇒ ˜ φ T ( t )Ω y n ( t ) = D − 1 ( s ) y ( n ) ( s ) and ˜ φ ( s ) = D − 1 ( s ) φ ( s ) obtained by: where ˜ � � φ T = ˜ ˜ ˜ ˜ X T X T X T ˜ ˜ u 1 n · · · u mn u 1 y um and for all z = y, u 1 , . . . u m :     0 1 0 0     .  ... ...    .       . ˙ ˜     X z ( t )     ˜        = X z ( t ) + z ( t ) 0 0 1 0      z n ( t ) ˜         1 − d 0 − d 1 · · · − d n − 1             1 − d 0 − d 1 · · · − d n − 1 & 4 IFAC ALCOSP’07, August 2007, St. Petersburg

  6. y n ( t ) = ˜ φ T ( t )Ω Kalman filtering for ˜ Estimator of Estimate Ω ∗ = Ω( t → ∞ ) where Ω( t ) solution of adaptive algorithm � ˜ � Ω( t ) = − Γ( t )˜ ˙ φ T ( t )Ω( t ) − ˜ φ ( t ) y n ( t ) Γ( t ) = − Γ( t )˜ ˙ φ ( t )˜ φ T ( t )Γ( t )+ α Γ( t ) For α = 0 : guaranteed convergence if permanent excitation on u i ( t ) . α > 0 small: forgetting factor, to be used for slowly time varying parameters. & 5 IFAC ALCOSP’07, August 2007, St. Petersburg

  7. Implementation for ’helicopter’ identification Simplified model of 3 D -Motion of ’helicopter’ benchmark ¨ 1 ˙ θ ( t ) + a θ θ ( t ) + a θ 0 sin( θ ( t ) − θ 0 ) = b θ 0 µ d ( t ) 0 sin( ǫ ( t ) − ǫ 0 ) + c λθ ˙ λ ( t ) ˙ ǫ ( t ) + a ǫ ǫ ( t ) + a ǫ θ ( t ) = b ǫ ¨ 1 ˙ 0 µ s ( t ) cos θ ( t ) ¨ 1 ˙ λ ( t ) + a λ λ ( t ) = b λ 0 µ s ( t ) sin θ ( t ) & 6 IFAC ALCOSP’07, August 2007, St. Petersburg

  8. Identification of the pitch motion MISO model of the non-linear dynamics ¨ 1 ˙ θ ( t ) + a θ θ ( t ) + a θ 0 sin( θ ( t ) − θ 0 ) = b θ 0 µ d ( t ) ⇓ ¨ 1 ˙ θ ( t ) + a θ θ ( t ) = − a θ + b θ s ( t ) 0 µ d ( t ) 0 ���� sin( θ ( t ) − θ 0 ) ➙ θ 0 = − 7 . 8 o measured as the equilibrium for µ d = 0 . ➙ D ( s ) = s 2 + 2 ω d ρ d s + ω 2 = s 2 + 1 . 4 s + s 2 1.4 1.2 1 0.8 ➙ Permanent excitation: square + chirp 0.6 0.4 0.2 0 0 5 10 15 20 25 30 & 7 IFAC ALCOSP’07, August 2007, St. Petersburg

  9. Pitch identification results ✪ α ∈ [0 , 0 . 001] : good convergence (else oscillations appear) &'( & !'( ✪ No major dependency w.r.t. initial guess Ω(0) ! ! !'( ! & ! &'( ! " # $ % &! &" &# &$ &% "! # "+& " !+& ✪ Γ(0) ≃ 10 3 1 for quicker convergence ! ! !+& ! " ! "+& ! # ! "! #! $! %! &! '! (! )! *! "!! & 8 IFAC ALCOSP’07, August 2007, St. Petersburg

  10. Pitch identification results ✪ For different experimental conditions (various choices of the excitation signal, disturbances...) the identified parameters are close but slightly different. ✪ Obtained values are uncertain in intervals b θ a θ a θ 0 ∈ [0 . 25 , 0 . 3] , 0 ∈ [0 . 58 , 0 . 67] , 1 ∈ [0 . 058 , 0 . 068] ✪ A PID controller is designed for the median values of identified parameters ✪ Error in closed-loop behavior of non-linear model and system is satisfying ( " ! ( ! # ! ' ! ! ! & !" !# !! !$ !% &" &# &! & 9 IFAC ALCOSP’07, August 2007, St. Petersburg

  11. Identification of elevation and travel axis ✪ Both axes identified simultaneously because ➙ Both excited by µ s ( t ) , the sum of propeller forces ➙ Have coupled dynamics ✪ Identification done with PID control on µ d ( t ) , the difference of propeller forces ➙ Identification for various references θ ref on the pitch ➙ θ ref � = 0 for travel to be exited ✪ Results give about 20% variation on parameter between experiments ➙ Median values are given by c λθ = 0 . 026 , b ǫ a ǫ a ǫ 0 = 0 . 16 , 0 = 2 . 59 , 1 = 0 . 032 b λ a λ 0 = − 0 . 112 , 1 = 0 . 114 & 10 IFAC ALCOSP’07, August 2007, St. Petersburg

  12. Work done since the final paper - Conclusions Closed-loop 3D-motion experiments ➚ Good behavior of the model for some simple and slow moves ➙ Instability for quick changes of reference signal ➘ Errors in transient behavior of the model for low propeller speed ➘ Need to improve the model Identification with other filter D ( s ) ➘ Algorithm converges to other values of parameters ➘ Need to clarify the dependency of results w.r.t. excitation signal and D ( s ) & 11 IFAC ALCOSP’07, August 2007, St. Petersburg

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