On the numerical differentiation problem of noisy signal

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1 On the numerical dierentiation problem o noisy signal O. Dhaou, L. Sidhom, I. Chihi, A. Abdelkrim National Engineering School O Tunis Research Laboratory LA.R.A in Automatic control, National Engineering School o Tunis (ENIT) University o Tunis El Manar BP 37, Le Belvédère, 2 Tunis, Tunisia dhaouola@gmail.com lilia.sidhom@gmail.com Abstract This paper discusses the relevant theoretical problem o the numerical derivative estimation o noisy signals. In this paper, a comparative study o some dierent schemes o the dierentiators is given: Kalman ilter, the well-known Super Twisting algorithm, Super Twisting with dynamic gains and Euler backward dierence method. The analysis o the study results can ocus on the strengths and weaknesses o each algorithm with some chosen criteria. Keywords numerical derivative, higher order sliding modes, Kalman ilter; simulations. I. INTRODUCTION Numerical dierentiation o measured signals is an old problem which is still considered as an attracted topic by many scientiic communities and also by some industrial applications. The major problem o noisy dierentiation signal is the ampliication o the noises on the signal estimation where it is so hard to discern between the noise and the basic signal. This problem becomes more important in practice, where the measurement noises are not obvious to model. Moreover, the estimation o any derivative can be destroyed even or a small high-requency noises which is the usual assumption considered in most cases. Indeed, the practical dierentiation consists to have a trade-o between exact dierentiation and robustness with respect to noises. In the literature, dierent approaches are proposed to deal with the problem o inding suitable linear or nonlinear algorithms able to reduce the noise eect while trying to leave the based signal with any phase shit. In signal processing, many researchers have been developed. to cast the problem in terms o digital ilter design requency domain. Another standard approach is to invert the transer unction o a digital integrator well adapted to obtain a digital dierentiator. One o the others traditional method is to use the inite dierence method such as the Euler backward dierence method. This one is simple to implement. However, with the sensors noises, this approach gives alse results. Although a low-pass ilter can attenuate the noise on the estimation signals, some phase delay can be inevitable introduced. Other approaches are directed towards an observer design problem where the knowledge o the system/noise model is necessary. In [], the proposed dierentiator is a high gain observer. The major drawback o this one is its sensitivity to measurement noises or perturbations which due to the ininity value o its gain. In the case o random noises/perturbations, the Kalman ilter [8] [9] can be used to estimate a derivative o some noisy signal which is optimal with the Gaussian white noises central is addressed by Kalman observers whose gains are computed by the resolution o an algebraic Ricatti equation [8]. In many cases the structure o the signal is unknown except rom some dierential inequalities, so, the observer approach can t be elaborated. An alternative approach is then addressed which consist o the design o dierentiator scheme. In such case, an algebraic dierentiator [5] [6] [7] are based on truncated Taylor series o the signal to be dierentiated are potentially interesting. Other possible method is proposed which is relies on the higher order sliding mode [] []. In [6], Levant proposes a st-order sliding mode dierentiator which is well-known by Super-Twisting. This last one is widely applied in the control context. Such algorithm has a simple orm and is thereore easy to be implemented. However, its perormance depends on the choice o the parameter values which are depended on the Lipschitz constant o the signal derivative. This constant is usually not known accurately beorehand, especially in the presence o noises. To avoid this problem, some research works were proposed a new scheme o the classical Super- Twisting, see or instance, [-4]. In this paper, three kinds o dierentiators are studied: Kalman algorithm [re], Super-Twisting [6] and a new scheme o Super Twisting which have dynamic gains. By using an academic example, a detailed comparative study is given. This paper is organized as ollows. Section 2 introduces the basic principle o the three dierentiators. In section 3 presented an academic example. Section 4 is dedicated to discuss and to synthesize the obtained results. II. STUDIED DIFFERENTIATORS A. Kalman Filter Started with the ollowing state system: x( t) Ax( t) Bu( t) Mw( t) y( t) Cx( t) Du( t) v( t) ()

2 Where x is the state vector, u and y are respectively the input and the output o the system. A, B, C and D called respectively state matrix, input matrix, output matrix and eed-orward matrix o some system. Where x ; u ; y n m q A ; B ; C ; D nn nm pn pm w(t) and v(t) are stationary random signals and their covariance is zero. Note that W and V are respectively the Power Spectral Density o w(t) and v(t). In order to have an optimal behavior, the Kalman ilter presents an assumption on W and V which must be Gaussian white noises central. So the Kalman ilter equations are as ollow: xˆ ( t) Axˆ ( t) Bu( t) K ( ( ) ˆ Cx t y( t)) (2) yˆ( t) Cxˆ( t) With K is the gain o the ilter and ˆx is the estimate state. ( t) x( t) xˆ ( t) The error expression can be taken as. To minimize the error () t, the gain K is so computed with using a deined ollows criterion: T T J ( t) E ( t) ( t) tracee ( t) ( t) (3) With E is an expected value o T ( t ) ( t ) quantity. Then with using the Riccati equation, the value o K is obtained as: ( ) T T K E t ( t) C V (4) B. Super TwistingDierentiator Consider () t a measurable locally bounded unction deined on, as ollowing: ( t) ( t) ( t) (5) Where () t is an unknown base signal with ( n)th derivative having a known Lipschitz constant C and () t is a bounded Lebesgue-measurable noise with unknown eatures, deined by: with is suiciently small. t In [6], Levant deines an ininite number o dierentiator th schemes which makes possible to estimate the n derivative o the considered signal. So, rom these schemes and or n, a Super-Twisting dierentiator (ST) can be deined by the ollowing equations: z v v 2 z z sgn( z ) z v v sgn( z v ) (6) Where z and z are the states o the algorithm and v is its output., are the Under suicient conditions given in [6], positive gains ensuring a inite-time convergence o the algorithm by satisying these inequalities: C 2 C (7) 4C C According to this expression, the gain choice assumes the knowledge in advance o the Lipschitz constant o the (n+) order derivative o the useul signal. But, i the signal is noisy, the Lipschitz constant is a priori unknown and the choice becomes even more diicult. Thereore, the main drawback o such algorithm is the setting o its gains to keep good perormances even when the requency o the input signals o dierentiator changes or i the input spectrum has rich requencies. It is not always easy to tune the parameters values or a given bandwidth o the input signal. From (6), it can be seen that the terms z i introduce the integral components and act as estimators o the input signal derivative. Theoretically, the ideal sliding mode must ensure the irst terms o equations (6) to zero in inite time. However, the ideal sliding mode never be realized in the practice owing to the dierent origin o inaccuracy, such as a measurement errors. Moreover, the presence o sign(.) unction in these terms leads to high requency oscillations. Indeed, this chattering eect can deteriorate the precision o the estimated signal. Thereore, it is not easy to adjust the gains to reach a good compromise between accuracy and robustness to noise ratio. Then, to avoid this problem, a new dierentiator is proposed called Dynamic Gains Super Twisting dierentiator. C. Dynamic Gains Super Twisting dierentiator In [4], the Dynamic Gains Super Twisting (DGST) dierentiator is proposed to acilitate the adjustment o dierentiating gains while providing a good compromise between accuracy and noises robustness. This new scheme is based on adding a linear unction to the second term o the algorithm. This algorithm keeps the same notations as the classic ST. The proposed dierentiator is given by the ollowing system, where (t) is an input signal o the algorithm: z v v 2 ˆ z sign( z ) k( z ) z z v v ˆ sign( z ) (8) i

3 Where k is the convergence positive gain o the The parameter o each algorithm is taken as: the gain o the dierentiator Kalman ilter K. The dynamic gains ˆ ˆ,.95 The Super-Twisting are deined by: dierentiator parameters are: 4, 7. The value o ˆ 2 sign ˆ ; and ˆ the Dynamic Gains Super Twisting gain is k ; t 6..5 input ˆ ˆ ; and ˆ ; t. (9).5 For the proo convergence o the algorithm, see [5]. III. ACADEMIC EXAMPLE: MECHANICAL SYSTEM "MASS, SPRING, DAMPER" To study the eectiveness o the presented dierentiators, a mechanical system is used; see (Fig.) to compute speed o the mass m. Fig.. Mass- Spring- Damper System The mathematical model o the system is as ollows: my( t) y( t) ky( t) F( t) () Where: k : spring stiness (N / m); : spring damping (Ns / m); m : mass o the object (Kg); Ft (): external orce(n). Then the state representation is written as ollows: x( t) Ax( t) Bu( t) y( t) Cx( t) With: A k, B, C m m m () IV. SIMULATION RESULTS AND ANALYSIS Let consider the noisy input signal dierentiator shown in (Fig.2). For the simulation tests, the SIMULINK is used with 3 sampling period T equal to s. The parameters o the e system are chosen as: m kg k N m N s m,., Fig. 2. Input signal : position y(t) o the mass The comparison criteria adopted is the absolute value o magnitude error and the delay (pick to pick) between the signals. For the estimation o the irst derivative in presence o Gaussian white central noise, the simulations show that: The (ig.3.a) shows the Super-Twisting dierentiator presents a maximum error times more than that given by the Dynamic Gains Super Twisting dierentiator this big dienrence is because the presence o a linear part k ( z ) in the anallytic expression o Dynamic Gains Super Twisting dierentiator; the adding o this continuous term enssures smoothing o noise at the output thanks to the low value o convergence gains. The maximum error o the Dynamic Gains Super Twisting dierentiator is more important than that given by kalman ilter; in contrast i we add a iltering stage on the algorithm o Gains Super Twisting dierentiator the accuracy will be improved. For the phase shit, the Kalman ilter has the lowest phase schit. Finally, the Kalman ilter presents the best result because it puts in their optimal conditions that is to say with Gaussian white central noise. Or in the opposite case (with other type o noise) the Dynamic Gains Super Twisting dierentiators may be presents more eicient results. Table.I summarizes the results obtained during the simulations. TABLE I ESTIMATION ERRORS AND PHASE SHIFT FOR THE ESTIMATE OF THE FIRST DERIVATIVE Algorithm KALMAN ST DGST e max Phase shit ( )

4 out Fig.3.a. Super Twisting dierentiator Analytic derivation Super-Twisting Dierentiator.25 Super-Twisting error x 4 Fig. 4.a. Super Twisting error out Analytic derivation Dynamic Gains Super-Twisting Dierentiators.2 Dynamic Gains Super-Twisting error time 4 Fig. 3.b. Dynamic Gains Super Twisting dierentiator Fig.4.b. Dynamic Gains Super Twisting error Analytic derivation Kalman ilter Kalman error.2. out time Fig.3.c. Kalman dierentiator Fig. 3. Dierentiator outputs Fig.4.c. Kalman error Fig.4. Error curve

5 In table II, the dierent simulation results provided in this paper or the dierent algorithms dierentiation are given. A comparative summary was compiled to highlight the advantages and disadvantages o each method TABLE II COMPARISON OF DIFFERENT SCHEMAS OF DIFFERENTIATION Algorithm precision phase shit robustness ease o implement ation calculati on time ST DGST KALMAN V. CONCLUSION In this paper we proved that the adjustment o the Kalman ilter is not easy. Indeed, the optimality o its accuracy is ensured that despite very strong assumption. In practice, various perturbations signals are not necessary to be white. The Dynamic Gains Super Twisting dierentiator may give better results than the kalman ilter in the case adds a ilter to it. It is possible to use cascade algorithms in order to compute acceleration o the mass. In th uture works, it is an important to study these dierentiators or dierent kinds o noises and also input signals. [] J. A. Moreno, M. Osorio, A Lyapunov approach to second-order sliding mode controllers and observers, Proceedings o the 47th IEEE conerence on Decision and Control, Cancun, Mexico, December 9-, 28. [2] M. Taleb, A. Levant and F. Plestan, Electropneumatic actuator control: solutions based on adaptive twisting algorithm and experimentation, Control Engineering practice, vol. 2, 22, pp [3] F. Plestan, Y. Shtessel, V. Bregeault and A. Poznyak, Sliding mode control with gain adaptation - application to an electropneumatic actuator, Control Engineering Practice, vol. 2, June 22. [4] L. Sidhom, X. Brun, M. Smaoui, E. Bideaux and D. Thomasset, Dynamic Gains Dierentiator or Hydraulic System Control, Journal o Dynamic Systems, Measurement, and Control Vol. 37, Issue 4, 25. [5] A.Levant, Robust Exact Dierentiation via Sliding Mode Technique, Elsevier Science Ldt. All rights reserved Prented in Graet Britain, vol 4, No.3, pp , 998. [6] Levant, A. "Robust exact dierentiation via sliding mode technique". Automatica, 998, vol. 34, n_ 3, pp REFERENCES [] Luma K. Vasiljevic and Hassan K. Khalil, Dierentiation with High- Gain Observers the Presence o Measurement Noise», Proceedings o the 45th IEEE Conerence on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 3-5, 26. [2] Miki Livne and Arie Levant, Homogeneous Discrete Dierentiation o Functions with Unbounded Higher Derivatives, 52nd IEEE Conerence on Decision and Control December -3, 23. Florence, Italy,23 [3] Vinay Chawda, Ozkan Celik and Marcia K. O Malley, Application o Levant s Dierentiator or Velocity Estimation and Increased Z-Width in Haptic Interaces, Department o Mechanical Engineering and Materials Science Rice University, Houston, TX 775 USA,2.Salim IBRIR, Logica-France, New Dierentiator or Control and observation Application, Proceeding o the American Control Conerence Arlington,VA June 25,27,2. [4] Francisco de Asís GARCIA COLLADO, Brigitte D ANDRÉA- NOVEL2, Michel FLIESS3, Hugues MOUNIER4 «Frequency analysis o algebraic derivators», 29 [5] Johann Reger, Hebertt Sira Ramirez and Michel Fliess, On nonasymptotic observation o nonlinear systems,25 [6] R. E. Kalman, "A New Approach to Linear Filtering and Prediction Problems", Transactions o the American Society o Mechanical Engeneers Journal o Basic Engeneering, Vol. 82, pp , 96 [7] A. N. Ramjattan, P. A. Cross, "A Kalman ilter model or an integrated land vehicle navigation system", Journal o Navigation, 995 [8] A. Levant, "Sliding Order and Sliding Accuracy in Sliding Mode Control", International Journal o Control, Vol. 58, No. 6, pp , 993. [9] S. V. Emelyanov, S. K. Korovin, AA. Levantovsky, "Higherorder Sliding Modes in the Binary Control Systems", Soviet Physics, Vol. 3, No. 4, pp , 986. [] A. Davila, J.A. Moreno and L. Fridman, Variable gains Super- Twisting algorithm : a Lypunov based design, American Control Conerence, Baltimore, MD, USA, 3 June- 2 July, 2, pp

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