The algebraic approach to state and constant parameter estimation: - PDF document

The algebraic approach to state and constant parameter estimation: Some experimental results H. Sira-Ram ´ ırez CINVESTAV-IPN Departamento de Ing. El´ ectrica Secci´ on de Mecatr´ onica M´ exico, D.F., M´ exico Ciudad Real, March 2006

Contents of the Presentation ◦ Introduction ◦ Algebraic state estimation via time deriva- tives calculation ◦ Control of a mass position system ◦ Sinusoid frequency estimation ◦ Conclusions 1

The fundamental mathematical developments supporting the approaches to state estimation and constant parameter identification, to be presented in this talk, are due primarily to M. Fliess and his highly professional mathematical vision of real life engineering problems. M. Fliess and H. Sira-Ram ´ ırez, “An algebraic framework for linear identification”, ESAIM COCV, Vol. 9, pp 151- 168, 2003. M. Fliess and H. Sira-Ram ´ ırez, “Reconstructeurs d’etat”, C.R. Acad. Sci. Paris t332, (1), pp. 91-96, 2004. M. Fliess, M. Mboup, H. Mounier, and H. Sira-Ram ´ ırez, “Questioning some paradigms of signal processing via concrete examples”. Ch 1 of Algebraic methods in flat- ness, signal processing and state estimation . Editorial Lagares, Mexico. M. Fliess, “Anayse non standard du bruit”. C.R. Acad. Sci. Paris, Ser. I 342, 2006. 2

Algebraic state estimation via time derivatives calculation 3

Algebraic state estimation via time deriv- atives calculation • In an observable system, the state estima- tion problem is intimately related to the problem of computing the successive time derivatives of the output and input signals in a sufficiently large number (See Diop y Fliess 1991). • In this work, we revisit a recently proposed non-asymptotic algebraic means for the ap- proximate estimation of the system states from the calculation of a finite number of time derivatives of the output signal. The method furnishes some specific formulae for the time derivatives of a measurable signal. 4

• The estimation method to be proposed may be combined with the notion of differential flatness to complete a feedback loop with a desirable closed loop dynamics which is based on output feedback alone of a dif- ferentially flat system. • There are some other interesting contribu- tions in the existing literature that propose non asymptotic approaches to state esti- mation in dynamical control systems – Diop, Grizzle, Moral, Stephanopoulou (1994), – Diop, Grizzle y Chaplais (2000), – Ibir (2004). 5

Derivative calculations Given a signal y ( t ), we want to compute a cer- tain number of its time derivatives, with the following restrictions: • The information on y ( t ) is obtained “on line” and it is required to on-line compute y ( t ) , · · · , y ( k ) ( t ). the time derivatives ˙ y ( t ) , ¨ • The signal y ( t ) is contaminated by noises whose statistics are unknown. • It is desired that the estimation process does not depend on the model of the dy- namic system that generates the output signal y ( t ). 6

Derivatives calculation In a laboratory, one of the most popular me- thods for obtaining the first order time deriv- ative of a signal y ( t ), is the so called finite difference method. It consists in the following approximation to the derivative ˙ y ( t ): y e ( t i ) ≈ y ( t ) − y ( t i ) ˙ , t ≥ t i t − t i where t − t i = ǫ > 0, is a known scalar and t is an arbitrary instant of time close to t i . 7

Derivatives calculation We note: • The method is quite general and it does not depend on the system model. • The quality of the approximation depends on ǫ = t − t i . • The estimation process is not asymptotic and the estimate is available “instantaneously” immediately after the instant t . • The method is quite sensitive to the pre- sence of noise perturbations in the signal to be processed. 8

Derivatives calculation The finite difference approximation of the deriv- ative is evidently based on a truncated expan- sion of the Taylor series of the underlying sig- nal, ˜ y ( t ) = y ( t i ) + [ ˙ y ( t i )]( t − t i ) , t ≥ t i We insist upon the fact that, the model is that of the signal and not of the system producing it. d 2 � y dt 2 = 0 The local problem is reduced to compute the unmeasured state of a homogeneous, second order, linear time-invariant system. 9

Derivatives calculation A higher order model for the approximation of the output signal y ( t ), at t = t i , may be proposed to be: N − 1 � 1 k ! y ( k ) ( t i )( t − t i ) k , y ( t ) = t ≥ t i � k =0 This approximation satisfies the homogeneous linear, time-invariant, differential equation: d N � y dt N = 0 The local problem is then reduced to compute the states of a time-invariant, linear, homoge- neous system of order N , from unknown initial conditions. 10

Derivatives calculation y ( N ) ( t ) = 0 The linear approximation adopted: ˜ satisfies, in terms of operational transforms, the following relation: d N � � s N ˜ Y ( s ) = 0 ds N The expressions given by s − k d N � � s N Y ( s ) = 0 , k = N − 1 , N − 2 , · · · , N − k ds N contain, respectively, implicit information on the first, second, etc., k -th derivatives of y ( t ) in an approximate manner. 11

Example Consider a fifth order approximation around t i = 0 of a sufficiently differentiable signal, y ( t ): 5 � 1 k ! t k y ( k ) (0) ˜ y ( t ) = k =0 We have d 6 ˜ y dt 6 = 0 and hence: d 6 � � s 6 ˜ y − s 5 y (0) − ... − y (5) (0) = 0 ds 6 or d 6 � � s 6 ˜ y = 0 ds 6 12

Example Consider: s − 6 d 6 ds 6 [ s 6 ˜ y ( s )] = 0 We obtain the following time-varying realiza- tion of the underlying approximating system: z 2 + 36 t 5 y ( t ) ˙ = z 1 z 3 − 450 t 4 y ( t ) z 2 ˙ = z 4 + 2400 t 3 y ( t ) ˙ = z 3 z 5 − 5400 t 2 y ( t ) z 4 ˙ = z 5 ˙ = z 6 + 4320 t y ( t ) ˙ = − 720 y ( t ) z 6 t 6 y ( t ) z 1 = The approximate derivatives of y ( t ) are lin- ear combinations of the unstable, time-varying, linear filter states. 13

Indeed: y ( t ) ≈ t − 7 [ tz 2 − z 1 ] ˙ y ( t ) ≈ t − 8 � � − 624 z 1 − 12 tz 2 + t 2 z 3 ¨ y (3) ( t ) ≈ t − 9 � � − 9672 z 1 + 540 tz 2 − 18 t 2 z 3 + t 3 z 4 The obtained formulae present a singularity at t = 0, which disappears for any t = ǫ > 0. The computation is feasible, provided the math processor yields an accurate quotient at t = ǫ . 14

Example We propose the following estimate of the first order time derivative of y ( t ) with respect to time:   arbitrary 0 ≤ t < ǫ y e ( t ) = ˙  t − 7 [ tz 2 − z 1 ] t ≥ ǫ z 2 + 36 t 5 y ( t ) z 1 ˙ = z 3 − 450 t 4 y ( t ) z 2 ˙ = z 4 + 2400 t 3 y ( t ) z 3 ˙ = z 5 − 5400 t 2 y ( t ) z 4 ˙ = z 5 ˙ = z 6 + 4320 t y ( t ) z 6 ˙ = − 720 y ( t ) t 6 y ( t ) z 1 = 15

Example Similarly, we propose the following estimate of the second order time derivative of y ( t ) with respect to time:  arbitrary 0 ≤ t < ǫ  y e ( t ) = ¨ t − 8 � �  − 624 z 1 − 12 tz 2 + t 2 z 3 t ≥ ǫ z 2 + 36 t 5 y ( t ) z 1 ˙ = z 3 − 450 t 4 y ( t ) ˙ = z 2 z 4 + 2400 t 3 y ( t ) ˙ = z 3 z 5 − 5400 t 2 y ( t ) ˙ = z 4 z 5 ˙ = z 6 + 4320 t y ( t ) z 6 ˙ = − 720 y ( t ) t 6 y ( t ) z 1 = etc. 16

Example 17

Remark The validity of the formulae for the calculation of the time derivatives is limited in the time horizon. For this reason, it becomes necessary to re-initialize the computations at some time t r > 0. As the derivatives drift from their actual val- ues; so will the estimated signal, computed on the basis of the truncated Taylor series approx- imation, from the measured signal values. 18

Remark An automatic resetting of the calculations can be devised on the basis of a weighted integral square error of the reconstructed signal devi- ation and a pre-specified threshold value for such a reconstruction error, � t W | y ( σ ) − y e ( σ ) | 2 dσ, e = W > 0 t r + ǫ y ( t r + ǫ )]( t − t r − ǫ )+1 y ( t r + ǫ )]( t − t r − ǫ ) 2 + · · · y e ( t ) = y ( t r + ǫ )+[ ˙ 2[¨ 19

On-line time derivative calculation  i )( t − t r ) 2 + ¨ 2 y (3) ( t − 1  y e ( t − y e ( t − r )( t − t r ) + ˙ r ) ,  e  t ∈ [ t r , t r + ǫ ) y e ( t ) = ˙ n 1 ( t )   t > t r + ǫ  d ( t ) , where n 1 ( t ) = 30( t − t r ) 5 y ( t ) + z 1 , d ( t ) = ( t − t r ) 6  y (3) ( t − y e ( t −  r )( t − t r ) + ¨ r ) , t ∈ [ t r , t r + ǫ ) e ¨ y e ( t ) = n 2 ( t )  d ( t ) , t > t r + ǫ where n 2 ( t ) = − 300( t − t r ) 4 y ( t ) + 24( t − t r ) 5 ˙ y e ( t ) + z 2 ,  y (3) ( t −  r ) t ∈ [ t r , t r + ǫ ) ... e y e ( t ) = n 3 ( t )  d ( t ) , t > t r + ǫ where 1200( t − t r ) 3 y ( t ) − 180( t − t r ) 4 ˙ n 3 ( t ) = y e ( t ) +18( t − t r ) 5 ¨ y e ( t ) + z 3 20

Recall that z 3 , z 2 , z 1 are given by: z 2 + 36 t 5 y ( t ) z 1 ˙ = z 3 − 450 t 4 y ( t ) z 2 ˙ = z 4 + 2400 t 3 y ( t ) z 3 ˙ = z 5 − 5400 t 2 y ( t ) z 4 ˙ = ˙ = z 6 + 4320 t y ( t ) z 5 ˙ = − 720 y ( t ) z 6 t 6 y ( t ) z 1 = t r is a calculation resetting instant decided upon by either an integral square error criterion, com- paring y ( t ) and y e ( t ), or taken to be periodical 21

ǫ = 0 . 02 , T = 0 . 05 22

ǫ = 0 . 02 , T = 0 . 2 23

ǫ = 0 . 02 , T = 0 . 5 24