84 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003 Brief Papers_______________________________________________________________________________ Robust Speed and Torque Estimation in Electrical Drives by Second-Order Sliding Modes Giorgio Bartolini, Alfonso Damiano, Gianluca Gatto, Ignazio Marongiu, Alessandro Pisano, and Elio Usai Abstract—This paper presents the synthesis and practical implementation of a robust digital differentiator that provides the first and second derivative of a sampled smooth signal. The robustness of the proposed digital device, based on second-order slidingmodes (2-SMs), is analyzed with respect to measurement errors. Experimental results on an induction motor (IM) drive show that fast and accurate estimates of speed and torque can be obtained in several operating conditions by double differentiation of the encoder position measurement. Index Terms—Electrical drives, induction motor (IM), nonlinear estimation, numerical differentiators, second-order sliding mode (2-SM). I. INTRODUCTION IGH-PERFORMANCE control of induction motor (IM) drives requires the knowledge of rotor flux, electromagnetic torque, and shaft speed [23]. Because of the difficulty to directly measure these quantities the use of nonlinear observers [16], [18], especially those based on high-gain, adaptive, or sliding-mode control [22], [20], [19], [9] appear to be a convenient choice when uncertainties and disturbances must be faced, and is attracting a number of research efforts [15], [12]. In this paper, we propose the substitution of the speed and torque measurement (or estimation) devices by a real-time numerical differentiator that processes the shaft position signal. DC or brushless tachometers, resolvers, and encoders are the usual devices for measuring shaft position/speed of motor drives. In particular, digital incremental encoders are widely used due to their low cost and to the digital nature of the output signal. However, in the low-speed region they suffer from some drawbacks that may cause a speed ripple with unpredictable frequency and magnitude. The simplest approach to speed estimation using sampled position measurements provided by an encoder is the classical backward-difference method (BDM), where the number of pulses generated by the encoder during each sampling interval is divided by the sampling period. Near the zero speed a little H Manuscript received February 2, 1999; revised November 28, 2000. Manuscript received in final form September 3, 2002. Recommended by Associate Editor S. Bortoff. The authors are with the Department of Electrical and Electronic Engineering, University of Cagliari, 09123 Cagliari, Italy (e-mail: giob@diee.unica.it; alfio@diee.unica.it; gatto@diee.unica.it; marongiu@diee.unica.it; pisano@diee.unica.it; eusai@diee.unica.it). Digital Object Identifier 10.1109/TCST.2002.806441 increment of pulses is detected at each sampling instant, and BDM may experience unacceptable errors that can largely overcome 100%, the higher the sampling rate the larger the error. Increasing the encoder resolution improves the accuracy but does not remove completely the low-speed error, and suitable techniques have been proposed to face the difficulties associated to the low-speed measurement [17], [11]. In [11], an adaptive speed/torque observer for ac drives was proposed which on-line identifies the mechanical inertia. Near the zero velocity, the observed torque gives the speed estimation scheme the lacking information. The idea of using observers as “aiding sources” when a direct speed measurement is not effective (e.g., at low speed) is certainly interesting, but at the same time it motivates a research activity aimed to improve the basic differentiation algorithms, and this is the scope of this paper. Moreover, as many motor drives are controlled by means of a DSP, the discrete-time implementation of real-time differentiators must be considered. In [6] it has been shown that the bilinear discrete implementation of the continuous-time high-gain observer encompasses other classes of numerical differentiators: the classical BDM, the inverse-integrator model [1], and that based on spline interpolation [7]. It was claimed in [6] that the spline interpolation method should be preferred in the presence of significant noise. Sliding-mode differentiators (SMDs) [24], [14], [2] offer an interesting tradeoff between accuracy and noise-immunity. A systematic comparison among existing numerical differentiators is out from the scope of this paper. We only analyze the possibility of using an SMD based on the suboptimal second-order sliding mode (2-SM) control algorithm [5], [2] to estimate the shaft speed and acceleration on the basis of the sampled encoder measurements. The proposed device requires a very small computational effort, and its dynamic features can be defined by means of few constant tuning parameters. In particular, it appears well suited to be used at low speed, since the associated error is mainly located at the high frequencies. This paper is organized as follows: In Section II, the 2-SM approach to real-time differentiation is detailed. In Section III, the proposed differentiator scheme is completed in order to provide also an estimate of the second derivative. In Section IV, the overall scheme is summarized and some implementation issues are discussed. Finally, some experimental results are given in Section V. 1063-6536/03$17.00 © 2003 IEEE IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003 85 provided that the control amplitude satisfies the relationship (7) are suitable positive constants. where , , , and Proof: See the Appendix, within which the procedure to is described. compute the constants , , , and If the sampled input sequence is not affected by measurement errors, Proposition 1 ensures that after a finite transient Fig. 1. Differentiator scheme. (8) II. APPROXIMATE DIGITAL DIFFERENTIATION VIA 2-SMs The 2-SM methodology can be applied to design a digital device that gives an estimate of the first derivative of a sampled signal having bounded second derivative [2]. The structure of the proposed differentiator consists of a double integrator driven by a digital variable-structure controller (VSC) (see Fig. 1). The design objective is to constrain the outputs of the integraand , to track the input signal and its derivators, is the piecewise-contive, respectively. The driving signal denote the stant output of the digital VSC. Let sampled value of a generic signal at the sampling instant , , being the sampling period. From the scheme in Fig. 1, it results (1) , and put . The control Define and , with task is fulfilled by steering to zero unmeasurable. The associated dynamics is represented by (2) Provided that a positive constant is known such that (3) a number of 2-SM control algorithms [5] can be applied to stabilize system (2). The theoretical properties, and the actual performance, are comparable from almost all points of view. The “suboptimal” 2-SM control algorithm will be considered in the present treatment. The following proposition is proved. Proposition 1: Given (2)–(3), then the application of the digital control signal (4) where is evaluated as follows: gives an estimate of the derivative of the input signal that is affected by an high-frequency chattering error component. Note that due to the piece-wise constant form of the differthe overall scheme can be implemented entiator control without any discretization error by means of the following simple digital algorithm: (9) The effect of unknown-but-bounded (UBB) measurement noise with maximum amplitude affecting the input sequence deserves to be investigated. We shall demonstrate that the proposed differentiator is regular with respect to the presence of noise, i.e., the differentiation error is bounded and tends to zero if both and tend to zero. If the digital peak-detector (5) receives as input a noisy sequence it may detect fictitious singular points with a potentially destabilizing effect. In order to avoid this drawback, one can as follows: prefilter the sequence set set if otherwise. (10) The nonlinear filter (10) performs a piecewise constant approx. If the peak-detector (5) is imation of the input sequence applied to the filtered sequence , the estimated singular will differ from the actual ones by an error that depoints pends not only on the sampling period [as in (21)] but also on the noise magnitude . More specifically, if the presence of an UBB measurement noise is taken into account, and the prefilter (10) is used, then (21) turns out to be modified as follows: (11) if If the proof of Proposition 1 is rewritten using (11) in place of (21), it yields a boundary layer of size otherwise (5) guarantees the finite-time attainment of the following conditions: (6) (12) which means that the “noise-free” differentiation accuracy (8) turns out to be worsened, due to the measurement noise, in accordance with (13) 86 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003 where and are proper positive constants. The evaluation and follows the same procedure as in the proof of of Proposition 1 but is slightly more involved since system (26) is affected by , and the solution to the numerical optimization problem for evaluating and requires the actual noise magnitude to be prespecified. Note that in the actual case coin, where is the number cides with the encoder resolution of ppr (pulses per revolution) of the considered device. These robustness issues, which are a common feature of sliding differentiators based on 2-SMs, have been thoroughly analyzed in [14] making reference to a family of differentiators based on a different 2-SM controller, the so-called “supertwisting” algorithm. III. ESTIMATING THE SECOND DERIVATIVE Fig. 2. It is well known that during a sliding mode behavior the so-called “equivalent control” (which is the nonswitching control action that would maintain the system exactly on the actual sliding manifold) can be estimated by linear low-pass filtering of the discontinuous switching control [22]. The definition of the equivalent control has been recently generalized to the 2-SM setting [4], and it was also shown that a proper choice of the filter time-constant can minimize the difference between the equivalent control and its estimate. As far as the differentiator system (2) is concerned, the equivis the second derivative of , i.e., alent control (14) Under the further assumption that the equivalent control is globally Lipschitz, i.e., (15) it has been shown in [4] that after a finite transient, the signal defined by (16) is an approximation of , i.e., (17) and are proper constants, depending on and where , which can be computed according to the formulas given in [4, Th. 2]. is piecewise constant, filter (16) can be As the control exactly discretized as follows: (18) The effect of UBB measurement noise traduces in worsening the accuracy (17). In particular, if one rewrites the proof of Theorem 2 in [4] with (12) used instead of (6), it can be computed , which depends on and , such that a positive constant the accuracy of the second derivative estimate is upperbounded as follows: (19) Overall scheme for estimating speed and acceleration. IV. VARIABLE STRUCTURE SYSTEM (VSS)-BASED ROTOR SPEED AND ACCELERATION ESTIMATION The encoder signal consists of two square-waves whose frequency and reciprocal phase displacement indicates the speed magnitude and the direction of rotation, respectively. An highfrequency counter is properly updated according to the rising and/or falling edges of the encoder signals. Near zero speed the number of pulses detected during each sampling period is close to the unity, and even an error of 1 pulse can lead to unacceptable errors. In this condition, the frequency of the speed ripple is strictly linked to the ratio between the frequency of the square wave and the sampling frequency. By these considerations, it is clear that the effectiveness of the linear filtering of the estimated speed depends on the actual operating conditions, and that the low-speed ripple cannot be filtered out without unacceptable degradation of the measurement bandwidth. The proposed SMD attenuates the low-speed measurement drawback since the switching nature of the differentiator concauses the resulting error to be mainly located at high trol frequencies, and thus it is possible to filter it out efficiently. In Fig. 2, the overall structure of the proposed digital device is summarized in terms of discrete transfer functions. and represent the estimated velocity and acceleration, respectively, and the constant depends on the time constant of the digital low-pass filter [ , see (18)]. is expressed according to (4), and The switching control it commutes at high frequency between the constant, opposite, . values value directly affects the dynamic performances of The the differentiator, because it actually sets the range for the estimated acceleration (and, at the same time, the slew rate of the should estimated speed). To achieve the best performance be chosen taking into account the actual sampling period, the drive parameters and the characteristics of the hardware setup. will result in a larger bandwidth, but it will cause Increasing larger oscillations to affect the estimated speed and acceleration. is defined according to the required dynamic perforOnce mance, the measurement accuracy defines a lower threshold under which decreasing of the sampling time does not guarantee any improvement [see relationships (13) and (19)]. IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003 87 (a) 6 Fig. 3. 1500 r/min 2-Hz square-wave reference speed. Speed detected by BDM and by the VSS. V. EXPERIMENTAL RESULTS The proposed estimation algorithm has been implemented on a commercial DSP board (TMS320C31, 40 MHz clock frequency), and the shaft velocity and acceleration of a 15-Nm FO-controlled IM drive have been estimated in the open loop. The position signal was provided by a 5000-ppr incremental encoder. Different operating conditions have been analyzed. All the reported tests have been performed at the sampling frequency of value has been set at 12 000 rad s , and the 7.5 kHz. The acceleration filter time constant has been set at 4.53 ms. To further reduce the high-frequency residual chattering, the estimated speed has been filtered with time constant of 3.46 ms. As for this latter filter, it can be inserted without compromising the performance since the filtered signal is not fed back to the differenof the switching signal tiator. Remember that the amplitude must be sufficiently large to provide the desired bandwidth, and which propathe residual high-frequency components of gates through the discrete integrator (see Fig. 2) can be attenuated by filtering. Direct comparison with the “unfiltered” curves clearly indicates the positive effect of post-filtering. The estimated velocity has been compared with that obtained byBDM.Twotestshavebeenperformed,imposingasquare-wave reference speed with amplitudes 1500 r/min and 4 r/min, respectively. It can be noted the perfect matching between VSS and BDM estimates when the speed is large (Fig. 3), while the behaviors are much different at low speed (Fig. 4). In the absence of external load torque, the estimated speed and acceleration can be used to estimate the electromagnetic , in accordance with torque (20) where is the shaft velocity estimate, is the estimated acceleration, is the total inertia, and is the damping factor. The electromagnetic torque observed by the sensorless (analog) FO IM drive has been compared with that obtained by (20) using the VSS estimates. Three experiments at high speed, (b) 6 Fig. 4. 4 r/min 2-Hz square-wave reference speed. Speed detected by (a) BDM and (b) VSS. with different wave-shape for the reference velocity, have been made (see Figs. 5–7). The torque observed by the VSS has smaller peaking errors, and the ripple is located at higher frequencies, as compared with those obtained by the servo drive. In the presented tests, the estimated signals are not used for feedback, and the proposed differentiator is actually in the open-loop. It must be underlined that the reduced computational burden allows the VSS algorithm to compute both speed and torque estimates in less than 4 s. VI. CONCLUSION This paper dealt with the robust speed and torque estimation in electrical drives by VSS approach. A digital double differentiator, based on 2-SMs, has been proposed. The encoder signal has been processed by the VSS algorithm in order to provide robust estimates of the shaft speed and acceleration, from which torque can be estimated in turns. Good performance in the low-speed region, robustness against measurement errors and small computational demand are the main features of the proposed method. Experimental tests have been conducted on an FO IM drive. 88 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003 (a) (a) (b) (b) 6 6 Fig. 5. 1500 r/min 2-Hz square-wave reference speed. Torque observed by (a) the servo drive and (b) VSS. Fig. 6. 1500 r/min 2-Hz sinusoidal reference speed. Torque observed by (a) the servo drive and (b) VSS. APPENDIX PROOF OF PROPOSITION 1 condition” then a first singular point is attained after a finite time. Our objective is to show that, from this point on, a sequence of singular values is generated, featuring the convergence toward a vicinity of zero, i.e., The convergence of the state trajectories to the origin of the plane is demonstrated by showing that the feedback law (4)–(5) causes subsequent crossings of the state trajectory with the abscissa axis (singular points of ), and that the control amplitude can be set in order to ensure that these subsequent crossings are nearer and nearer to the origin of the state plane. The actual system trajectory lies between limiting arcs deof the uncertain fined by the maximum constant bounds dynamics (2)–(3) (see Fig. 8). is an estimate of the last singular point of , , the last crossing point of the state trajectory with the abscissa axis. It is estimated by means of the peak-holder defined by (5), error such that which introduces a (22) (23) , , be the actual th singular value of . AsLet . Analogous consume, for the sake of simplicity, that siderations are also valid if . Taking into account the approximate detection of the singular points (21), and the switching delay due to sample-and-hold effect, it is possible to show that the subsequent singular point will be such that (21) It is not difficult to verify that, independently from the initial conditions, if the control amplitude satisfies the “dominance (24) IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003 89 (a) Fig. 8. Limit trajectories in the lower half-plane. , and it was then the first inequality of (6) holds with . shown also that the second of (6) is verified with In general, if the requirement of minimizing is dispensed the problem admits a solution prowith, for any vided that belongs to a suitable open set whose upper and lower bounds depend on (28) (b) 6 Fig. 7. 1500 r/min 2-Hz triangular reference speed. Torque observed by (a) the servo drive and (b) VSS. Let us introduce the normalized nonnegative variables defined as follows: (25) Considering (24), the contraction condition (22) is equivalent to the following system of inequalities: and are constants which do not depend on where the sampling period [3]. The precise expression of and as functions of the system parameters is quite complex, and it is not reported since it is not useful for practical design purposes. 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