Lecture 8 & Tutorial 1: Experimental idention of low order model - PowerPoint PPT Presentation

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011 Lecture 8 & Tutorial 1: Experimental idention of low order model Jean-Luc Battaglia, Laboratory I2M, Department TREFLE , I2M Bordeaux

Measurement inversion 35 30 25 20 Τ m ( t ) Measured 15 10 measured 5 simulated with the identified system 0 identified -5 0 1 2 3 4 5 time (sec) Τ ext ϕ ( t ) h j 100 80 { } ( ) ( ) = M ϕ T t t m phi (W/m²) 60 ϕ = 40 0 known 20 0 0 1 2 3 4 5 time (sec) system

Classical requirements for a good resolution of the inverse problem • Implementing thermal sensors in the system at strategic locations and making measurements • Having a reliable and accurate model that describes well the experiment. The reliability of the direct model rests on the accuracy on two sets of data: – the thermal properties – the location of the sensor. Uncertainties on those data will lead to a very low confidence domain for the estimated heat flux

System identification 35 30 Measured 25 20 Τ m ( t ) 15 10 measured 5 simulated with the identified system 0 known -5 0 1 2 3 4 5 time (sec) Τ ext ϕ ( t ) h j 100 80 { } ( ) ( ) = M ϕ T t t m phi (W/m²) 60 ϕ = 40 0 identified 20 0 0 1 2 3 4 5 time (sec) system

Linear monovariable systems • The impulse response ∞ ( ) ( ) ( ) ( ) ( ) ∫ = ∗ ϕ = − τ ϕ τ τ d T t h t t h t m m m 0 • For monovariable linear systems, the impulse response fully characterizes the thermal behaviour. Therefore, any kind of inverse strategy can be based on the direct model expressed as the impulse response of the system

System identification approach • Advantages • The system identification approach will be first interesting to obtain a reliable and accurate low order model that will require less computational time for simulation. • There is no need to know the thermal properties of the system (thermal conductivity, density, specific heat, heat exchange coefficients, thermal resistances at the interfaces, parameters related to thermal radiation…). • It is not required to know the sensor location inside the system. • It is not required calibrating the sensor. The identification procedure is fast (this will be viewed later with the description • of the different techniques). Drawbacks • • The model identification must be achieved in the exactly same conditions as those encountered during the inversion (heat exchanges between the surrounding and the system must remain the same for the two configurations). • The prediction of the identified model rests on strong assumptions (in particular, it is better reaching the stationary behaviour during the system identification process). In general, the identified system is only valid for the time duration of the system identification process.

The deconvolution technique 1/2 k k ( ) ( ( ) ∑ ∑ ( ) ( ) ) ( ) ( ) ∆ = − ∆ ϕ ∆ = ∆ ϕ − ∆ T k t h k i t k t h k t k i t m m m = = i 0 i 0 ϕ       T h 0 0 m 0       ϕ ϕ T h       = 1 1 0 m 1       � � � � �       ϕ ϕ ϕ �       T h N N 1 0 mN Assuming an additive measurement error of normal distribution (zero mean and constant standard deviation) k ( ) ∑ ( ) ( ) ( ) ( ) ( ) ( ) ∆ = ∆ + ∆ = ∆ ϕ − ∆ + ∆ y k t T k t e k t h k t k i t e k t m m m = 0 i

The deconvolution technique 2/2 = lim h 0 →∞ k k ϕ       y e 0 0 0       ϕ ϕ   h y e       1 0 m 0 1 1         � � � � � h   = + m 1       ϕ ϕ ϕ   � � y e       Q 1 0 Q Q         � � � � � � h     mQ �     ϕ ϕ ϕ  �      y   e H − + − � N N Q 1 N Q � N ����� ����� � N Q Φ Y E N N N ( ) ( ) − 1 = Φ Φ Φ T T T H Y E E Q N N N N N N

The correlation technique ∞ ( ) ( ) ( ) ( ) ∫ = − τ ϕ τ τ + y t h t d e t m m 0 ∞ ∞ ∞ ∞ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫∫ ∫ ϕ − τ = − τ ϕ τ ϕ − τ τ + ϕ − τ y t t d t h t t d d t t e t d t m m 0 0 0 0 ∞ ( ) ( ) ( ) ( ) ( ) ∫ τ = − τ τ τ + − τ τ C h t C d h t C ϕ ϕ ϕ y m y m e m m 0 ( ) ( ) C ϕϕ τ = δ τ ( ) ( ) ϕ τ = τ C h y m

Spectral technique 1/2  ∞  ( ) ( ) ( ) ( ) ( ) ( ) ∫   = τ − τ τ τ = Φ =   FFT C FFT h t C d Y f f S f   ϕ ϕϕ ϕ y m m y m   m 0  ∞  ( ) ( ) ( ) ( ) ( ) ∫  τ  = ϕ − τ ϕ τ τ = Φ 2 =   FFT C FFT t d f S f   ϕϕ ϕϕ   0 ( ) ( ) ( ) ( ) = + S f H f S f S f ϕ ϕϕ ϕ y e m ( ) S f ( ) ϕ = y H f m ( ) S f ϕϕ ( ) ( ) ( ) ϕ = ϕ Π t t t Π τ

Spectral technique 2/2 ( )  π τ  sin f ( ) ( ) Φ = Φ ∗  τ  f f Π π τ   f ( ) ( ) ( ) ϕ = ϕ t t g t Π τ  π    2 ( ) t = − g t 0.5 1  cos    τ τ    

The parametric approach ( ) ( ) ( ) ( ) ϕ ϕ 2 2 d T t d T t d t d t ( ) ( ) + α + α + = β ϕ + β + β + m m � � T t t 1 2 0 1 2 m 2 2 d t d t d t d t ( ) ( ) ∂ ∂ 2 T x t , T x t , = < < > a 0 x e t , 0 ∂ ∂ 2 t x ( ) ∂ T x t , ( ) = > − = ϕ x 0, t 0 k t ∂ x ( ) ( ) = > ∂ T x t = , 0 x e t T x t , , 0 = 0 ∂ x

The parametric approach { } 1 { } 1 ( ) ( ) ( ) ( ) = θ = ϕ = Φ β = L T t s L t s s a ( ) ( ) β β β β m m k sinh e k sinh e 1 1 ( ) ( ) ( ) + ∞ θ = Φ = Φ 2 n 1 ∑ z s s s ( ) = ∀ ≥ ( ) + ∞ + + m β sinh z , z 0 ∞ 2 n 1 2 n 1 n 1 e s ( ) ∑ e ∑ + β 2 n 1 ! k ( ) = k n 0 ( ) + + n 1 + a 2 n 1 ! 2 n 1 ! = = n 0 n 0 ( ) ( )   = n − k d f t n 1 d f 0 ∑ ( ) − − −   n n k 1 L s F s s n k  d t  d t = 0 k ( ) + ∞ n 1 d T t ∑ ( ) α = ϕ m t n d t = n 0

Output error method ( ) ( ) ( ) ( ) ( ) ( ) = ϕ + ϕ − + ϕ − + − − − − − � � T k b k b k 1 b k 2 a T k 1 a T k 2 m 0 1 2 1 m 2 m y m ( k ) system ε ( k ) ϕ ( k ) output error T m ( k ) model ( ) , ∂ ( ) , T k ( ) ∂ = = T k m � ( ) 0, , S k i n = = m � S k i 1, , n ∂ b i b ∂ a i a i i ( ) ( ) ( ) ( ) + − + + − = − − = � � S k a S k 1 a S k n T k i , i 1, , n a 1 a n a m i i i

Output error method ( ) ( ) ( ) ( ) ( ) ( ) ( ) + − + + − = − − = = = = − = � � � S k a S k 1 a S k n T k i , i 1, , n S 0 S 1 S n 1 0 a 1 a n a m a a a i i i i i i ( ) ( ) ( ) = = = − = � S 0 S 1 S n 1 0 b b b i i i ( ) ( ) ( ) ( ) + − + + − = ϕ − = � � b S k b S k 1 b S k n k i , i 0, , n 0 b 1 b n b i i i n n ∑ ∑ ( ) ( ) ( ) ( ) ( ) ε = − = ∆ + ∆ k y k T k S k a S k b m m a i b i i i = = i 1 i 0 ∆   a 1   ( ) ( ) ( ) ( ) ( )  ε  �   n � � S n S n S n S n     a a b b   ( ) 1 n 0 n ε +  ∆  S n 1 a =    � � � �  E = = S = S ∆Θ n     ∆  ( ) ( ) ( ) ( )  � b   � � S N S N S N S N   0   a a b b ( )   ε 1 n 0 n �  N    ∆  b  n ( ) − 1 ∆Θ = S S S E Θ = Θ + ∆Θ T T ν ν − ν − 1 1

Predicton error method � ( ) ( ) ( ) ( ) ( ) ( ) = ϕ + ϕ − + ϕ − + − − − − − � � T k b k b k 1 b k 2 a y k 1 a y k 2 m 0 1 2 1 m 2 m y m ( k ) system ϕ ( k ) e ( k ) predictive model T m ( k ) ( ) ( ) ( ) = H Θ + y k k e k m [ ] ( ) ( ) ( ) ( ) ( ) Θ = H = −  − − − ϕ ϕ −  T � � a a b b � � k y k 1 y k n k k n   1 n 0 n m m