Sliding Mode Control Part 5 pot

Sensorless First- and Second-Order Sliding-Mode Control of a Wind Turbine-Driven Doubly-Fed Induction Generator 129 (a) Active power (kW) 200 −200 −400 Ps Ps ref −600 −800 10 12 14 16 Reactive power (kVAr) (b) 200 −200 −400 Qs Qs ref −600 −800 10 Time (s) 12 14 16 Fig 11 Active and reactive powers of the DFIG commanded by the 2-SMC controller (a) Rotor voltage (kV) 0.4 0.2 −0.2 −0.4 vr α vr β 10 12 14 16 (b) Rotor currents (A) 400 200 ir a ir b ir c −200 −400 10 Time (s) 12 14 16 Fig 12 Rotor voltage components fed into the SVM algorithm, and resulting three-phase rotor current by the 1-SMC correspond to the gating signals of the RSC IGBTs, no additional modulation techniques —such as pulse-width modulation (PWM) or SVM— are required In contrast, the 130 Sliding Mode Control control signals produced by the proposed 2-SMC correspond to continuous voltage direct and quadrature components to be applied to the rotor by means of the RSC, which implies the use of intermediate SVM modulation Furthermore, providing an additional procedure for bumpless transition between the algorithms devoted to synchronization and power control is indispensable for the case of the 2-SMC, but it is not required for the 1-SMC scheme Regarding parameter tuning, only the c constants included in the four switching functions considered need to be tuned for the case of the 1-SMC It therefore turns out that satisfactory parameter adjustment is easily achieved by mere trial and error However, in addition to those c constants, the λ and w gains present in the STAs must also be tuned for the 2-SMC variant Even though, as stated in (Bartolini et al., 1999), it is actually the most common practice, trial and error tuning is not particularly effective in this latter case, as it may become highly time-consuming Therefore, it is believed that there exists a strong need for development of alternative methods for STA-based 2-SMC tuning Concerning the switching frequency of the RSC IGBTs, it is fixed at kHz in the case of the 2-SMC On the contrary, it turns out to be variable, within the range from to 20 kHz, for the 1-SMC algorithm This feature complicates the design of both the back-to-back converter feeding the DFIG rotor and the grid-side AC filter, since broadband harmonics may be injected into the grid As a result of the 25-μs sample time selected for the 1-SMC scheme, which leads to the aforementioned maximum switching frequency of 20 kHz, chatter observable in stator-side active and reactive powers is somewhat lower than ±3% of the DFIG 660-kW rated power Even a lower level of chatter arises from application of the SVM-based 2-SMC algorithm put forward Furthermore, that chatter, or at least great part of it, is caused by the SVM, not by the 2-SMC algorithm itself Apart from the superior optimum power curve tracking achieved with both alternative SMC designs, the dynamic performance resulting from realization of the proposed 1-SMC scheme is noticeably better than that to which application of its 2-SMC counterpart leads In effect, focusing on the state in which the DFIG stator is disconnected from the grid, HIL emulation results demonstrate that synchronization is reached faster by employing the 1-SMC algorithm On the other hand, the power exchange between the DFIG and the grid taking place at the initial instants after connection is significantly lower when adopting the 1-SMC algorithm put forward, hence evidencing that its dynamic performance is also better for the stage during which power control is dealt with The excellent dynamic performance reachable by means of its application supports the 1-SMC approach as a potential candidate for DFIG control under grid faults, where rapidity of response becomes crucial The main conclusions drawn from the comparison conducted in this section are summarized in Table 1-SMC ALGORITHM A LGORITHM COMPLEXITY Relatively simple Not required PWM/SVM B UMPLESS PROCEDURE Not required Straightforward PARAMETER TUNING S WITCHING FREQUENCY Variable from to 20 kHz ±3% of the rated power C HATTER LEVEL D YNAMIC PERFORMANCE Excellent 2-SMC ALGORITHM More complex Required Required Complex Fixed at kHz Lower Very good Table Comparison between the two SMC algorithms put forward Sensorless First- and Second-Order Sliding-Mode Control of a Wind Turbine-Driven Doubly-Fed Induction Generator 131 References Abo-Khalil, A G., Lee, D.-C & Lee, S.-H (2006) Grid connection of doubly fed induction generators in wind energy conversion system, Proceedings of the CES/IEEE 5th International Power Electronics and Motion Control Conference (IPEMC 2006), Shanghai, China, vol 3, pp 1–5 Arnaltes, S & Rodríguez, J L (2002) Grid synchronisation of doubly fed induction generators using direct torque control, Proceedings of the IEEE 28th Annual Conference of the Industrial Electronics Society (IECON 2002), Seville, Spain, pp 3338–3343 Åström, K J & Hägglund, T (1995) PID Controllers: Theory, Design and Tuning, Instrument Society America, USA Bartolini, G., Ferrara, A., Levant, A & Usai, E (1999) On second order sliding mode ă ă ă controllers, in K Young & U Ozguner (eds), Variable Structure Systems, Sliding Mode and Nonlinear Control, Springer Verlag, London, UK, pp 329–350 Beltran, B., Ahmed-Ali, T & Benbouzid, M E H (2008) Sliding mode power control of variable-speed wind energy conversion systems, IEEE Transactions on Energy Conversion 23(2): 551–558 Beltran, B., Ahmed-Ali, T & Benbouzid, M E H (2009) High-order sliding-mode control of variable-speed wind turbines, IEEE Transactions on Industrial Electronics 56(9): 3314–3321 Beltran, B., Benbouzid, M E H & Ahmed-Ali, T (2009) High-order sliding mode control of a DFIG-based wind turbine for power maximization and grid fault tolerance, Proceedings of the IEEE International Electric Machines and Drives Conference (IEMDC 2009), Miami, USA, pp 183–189 Ben Elghali, S E., Benbouzid, M E H., Ahmed-Ali, T., Charpentier, J F & Mekri, F (2008) High-order sliding mode control of DFIG-based marine current turbine, Proceedings of the IEEE 34th Annual Conference of the Industrial Electronics Society (IECON 2008), Orlando, USA, pp 1228–1233 Blaabjerg, F., Teodorescu, R., Liserre, M & Timbus, A V (2006) Overview of control and grid synchronization for distributed power generation systems, IEEE Transactions on Industrial Electronics 53(5): 1398–1409 Ekanayake, J B., Holdsworth, L., Wu, X & Jenkins, N (2003) Dynamic modeling of doubly fed induction generator wind turbines, IEEE Transactions on Power Systems 18(2): 803–809 Kuo, B C (1992) Digital Control Systems, Oxford University Press, New York, USA Levant, A (1993) Sliding order and sliding accuracy in sliding mode control, International Journal of Control 58(6): 1247–1263 Ogata, K (2001) Modern Control Engineering, Prentice Hall, Englewood Cliffs (New Jersey), USA Peña, R., Cárdenas, R., Proboste, J., Asher, G & Clare, J (2008) Sensorless control of doubly-fed induction generators using a rotor-current-based MRAS observer, IEEE Transactions on Industrial Electronics 55(1): 330–339 Peña, R., Clare, J C & Asher, G M (1996) Doubly fed induction generator using back-to-back PWM converters and its application to variable-speed wind-energy generation, IEE Proceedings - Electric Power Applications 143(3): 231–241 Peresada, S., Tilli, A & Tonielli, A (2004) Power control of a doubly fed induction machine via output feedback, Control Engineering Practice 12(1): 41–57 132 Sliding Mode Control Rashed, M., Dunnigan, M W., MacConell, P F A., Stronach, A F & Williams, B W (2005) Sensorless second-order sliding-mode speed control of a voltage-fed induction-motor drive using nonlinear state feedback, IEE Proceedings-Electric Power Applications 152(5): 1127–1136 Susperregui, A., Tapia, G., Zubia, I & Ostolaza, J X (2010) Sliding-mode control of doubly-fed generator for optimum power curve tracking, IET Electronics Letters 46(2): 126–127 Tapia, A., Tapia, G., Ostolaza, J X & Sáenz, J R (2003) Modeling and control of a wind turbine driven doubly fed induction generator, IEEE Transactions on Energy Conversion 12(2): 194–204 Tapia, G., Santamaría, G., Telleria, M & Susperregui, A (2009) Methodology for smooth connection of doubly fed induction generators to the grid, IEEE Transactions on Energy Conversion 24(4): 959–971 Tapia, G., Tapia, A & Ostolaza, J X (2006) Two alternative modeling approaches for the evaluation of wind farm active and reactive power performances, IEEE Transactions on Energy Conversion 21(4): 901–920 Utkin, V., Guldner, J & Shi, J (1999) Siliding Mode Control in Electromechanical Systems, Taylor & Francis, London, UK Utkin, V I (1993) Sliding mode control design principles and applications to electric drives, IEEE Transactions on Industrial Electronics 40(1): 23–36 Vas, P (1998) Sensorless Vector and Direct Torque Control of AC Machines, Oxford University Press, New York, USA Xu, L & Cartwright, P (2006) Direct active and reactive power control of DFIG for wind energy generation, IEEE Transactions on Energy Conversion 21(3): 750–758 Yan, W., Hu, J., Utkin, V & Xu, L (2008) Sliding mode pulsewidth modulation, IEEE Transactions on Power Electronics 23(2): 619–626 Yan, Z., Jin, C & Utkin, V I (2000) Sensorless sliding-mode control of induction motors, IEEE Transactions on Industrial Electronics 47(6): 1286–1297 Zhi, D & Xu, L (2007) Direct power control of DFIG with constant switching frequency and improved transient performance, IEEE Transactions on Energy Conversion 22(1): 110–118 Part Sliding Mode Control of Electric Drives Sliding Mode Control Design for Induction Motors: An Input-Output Approach Universidad John Cortés-Romero1 , Alberto Luviano-Juárez2 and Hebertt Sira-Ramírez3 Nacional de Colombia Facultad de Ingeniería, Departamento de Ingeniería Eléctrica y Electrónica Carrera 30 No 45-03 Bogotá 1,2,3 Cinvestav IPN, Av IPN No 2508, Departamento de Ingeniería Eléctrica, Sección de Mecatrónica Colombia 2,3 México Introduction Three-phase induction motors have been widely used in a variety of industrial applications Induction motors have been able to incrementally improve energy efficiency to satisfy the requirements of reliability and efficiency, Melfi et al (2009) There are well known advantages of using induction motors over permanent magnet DC motors for position control tasks; thus, efforts aimed at improving or simplifying feedback controller design are well justified There exists a variety of control strategies that depend on difficult to measure motor parameters while their closed loop behavior is found to be sensitive to their variations Even adaptive schemes tend to be sensitive to speed-estimation errors, yielding to a poor performance in the flux and torque estimation, especially during low-speed operation, Harnefors & Hinkkanen (2008) Generally speaking, the designed feedback control strategies have to exhibit a certain robustness level in order to guarantee an acceptable performance It is possible to (on-line or off-line) obtain estimates of the motor parameters, Hasan & Husain (2009); Toliyat et al (2003), but some of them can be subject to variation when the system is undergoing actual operation Frequent misbehavior is due to external and internal disturbances, such as generated heat, that significantly affect some of the system parameter values An alternative to overcome this situation is to use robust feedback control techniques which take into account these variations as unknown disturbance inputs that need to be rejected In this context, sliding mode techniques are a good alternative due to their disturbance rejection capability (see for instance, Utkin et al (1999)) In this chapter, we consider a two stage control scheme, the first one is devoted to the control of the rotor shaft position This analog control is performed by means of the stator current inputs, in a configuration of an observer based control The mathematical model of the rotor dynamics is a simplified model including additive, completely unknown, lumping nonlinearities and external disturbances whose effect is to be determined in an on-line fashion by means of linear observers The gathered knowledge will be used in the appropriate canceling of the assumed perturbations themselves while reducing the underlying control problem to a simple linear feedback control task The control scheme thus requires a rather reduced set of parameters to be implemented 136 Sliding Mode Control The observation scheme for the modeled perturbation is based on an extension of the Generalized Proportional Integral (GPI) controller, Fliess, Marquez, Delaleau & Sira-Ramírez (2002) to their dual counterpart: the GPI observer which corresponds to a class of extended Luenberger-like observers, Luviano-Juárez et al (2010) Such observers were introduced in, Sira-Ramirez, Feliu-Batlle, Beltran-Carbajal & Blanco-Ortega (2008) in the context of Sigma-Delta modulation observer tasks for the detection of obstacles in flexible robotics Under reasonable assumptions, the observation technique consists in viewing the measured output of the plant as generated by an equivalent perturbed pure integration dynamics with an additive perturbation input lumping, in a single function, all the nonlinearities of the output dynamics The linear GPI observer, is set to approximately estimate the states of the pure integration system as well as the evolution of the, state dependent, perturbation input This observer allows one to approximately estimate, on the basis of the measured output, the states of the nonlinear system, as well as to closely estimate the unknown perturbation input The proposed observation scheme allows one to solve, rather accurately, the disturbance estimation problem Here, these observers are used in connection with a robust controller design application within the context of high gain observation This approach is prone to overshot effects and may be deemed sensitive to saturation input constraints, specially when used in a high gain oriented design scheme via the choice of large eigenvalues Such a limitation is, in general, an important weakness in many practical situations However, since our control scheme is based on a linear observer design that can undergo temporary saturations and smooth “clutchings" into the feedback loop, its effectiveness can be enhanced without affecting the controller structure and the overall performance We show that the observer-based control, overcomes these adverse situations while enhancing the performance of the classical GPI based control scheme The linear part of the controller design is based on the Generalized Proportional Integral output feedback controller scheme established in terms of Module Theory In the second design stage, the designed current signals of the first stage are deemed as reference trajectories, and a discontinuous feedback control law for the input voltages is sought which tracks the reference trajectories Since the electrical subsystem is faster than the mechanical, we propose a sliding mode control approach based on a class of filtered sliding surfaces which consist in regarding the traditional surface with the addition of a low pass filter, without affecting the relative degree condition of the sliding surface The “chattering effect" related to the sliding mode application is eased by means of a first order low-pass filter as proposed in, Utkin et al (1999) GPI control has been established as an efficient linear control technique (See Fliess et al., Fliess, Marquez, Delaleau & Sira-Ramírez (2002)); it has been shown, in, Sira-Ramírez & Silva-Ortigoza (2006), to be intimately related to classical compensator networks design The main limitation of this approach lies in the assumption that the available output signal coincides with the system’s flat output (See Fliess et al.Fliess et al (1995), and also Sira-Ramírez and Agrawal, Sira-Ramírez & Agrawal (2004)) and, hence, the underlying system is, both, controllable and, also, observable from this special output Nevertheless, this limitation is lifted for the case of the induction motor system The controller design is carried out with the philosophy of the classical field oriented controller scheme and implemented through a flux simulator, or reconstructor (see Chiasson, Chiasson (2005)) The methodology is tested and illustrated in an actual laboratory implementation of the induction motor plant in a position trajectory tracking task The rest of the chapter is presented as follows: Section describes each of the methodologies to use along the chapter such as the sliding mode control method, the Generalized Proportional Integral control and the disturbance observer The modeling of the motor and the problem formulation are given in Section 3, and the proposed methodologies are joined to solve the Sliding Mode Control Design for Induction Motors: An Input-Output Approach 137 problem in Section The results of the approach are obtained in an experimental framework, as depicted in Section Finally some concluding remarks are given Some preliminary aspects 2.1 Sliding mode control using a proportional integral surface: Introductory example Consider the following first order system: ˙ y = u + ξ (t) (1) where y is the output of the system, ξ (t) can be interpreted as a disturbance input (which may be state dependent) and u ∈ {−W, W } is a switched class input We propose here to take as a sliding surface coordinate function the following expression in Laplace domain s: s+z e s ∗ e = y−y σ=− (2) with z > The switched control is defined as u = Wsign(σ), W>0 (3) We propose the following Lyapunov candidate function: V= σ (4) ˙ ˙ whose time derivative is V = σσ From (2) ˙ ˙ σ = − e − ze (5) We have ˙ ˙ σσ = − σe − zeσ ˙ ˙ = − σy + σy∗ − zeσ ˙ = −W | σ| − σξ (t) + σy∗ − zeσ ˙ since the term − σξ (t) + σy∗ − zeσ does not depend on the input, by setting W in such a way ˙ that we can ensure that V < 0, the sliding condition for σ is achieved The classical interpretation of the output feedback controller suggests, immediately, the following discontinuous feedback control scheme: ξ (t) y∗ (t) u − e n( s ) σ Wsign(σ) Plant d(s) + y(t ) Fig GPI control scheme 138 Sliding Mode Control where n (s) = s + z regulates the dynamic behavior of the tracking error and d(s) = s acts as a “filter" of the sliding surface The equivalent control is obtained from the invariance conditions: ˙ σ=σ=0 i.e, ˙ u eq = y∗ − ze (6) in other words, the proposed sliding surface has, in the equivalent control sense, the same behavior of the traditional proportional sliding surface of the form σ1 = ze However, the closed loop behavior of the system with the smooth sliding surface, presents some advantages as shown in, Slotine & Li (1991) Since this class of controls induce a “chattering effect", to reduce this phenomenon, we insert in the control law output a first order low-pass filter, which, in some cases, needs and auxiliary control loop (as shown in the integral sliding mode control design, Utkin et al (1999)) In our case, the architecture of the control system based on two control loops and disturbance observers will act as the auxiliary control input 2.2 Generalized Proportional Integral Control GPI control, or Control based on Integral Reconstructors, Fliess & Sira-Ramírez (2004), is a recent development in the literature on automatic control Its main line of development rests within the finite dimensional linear systems case, with some extensions to linear delayed differential systems and to nonlinear systems (see Fliess et al., Fliess, Marquez, Delaleau & Sira-Ramírez (2002), Fliess et al., Fliess, Marquez & Mounier (2002) and Hernández and Sira-Ramírez, Hernández & Sira-Ramírez (2003)) The main idea of this control approach is the use of structural reconstruction of the state vector This means that states of the system are obtained modulo the effect of unknown initial conditions as well as constant, ramp, parabolic, or, in general, polynomial, additive external perturbation inputs The reconstructed states are computed solely on the basis of inputs and outputs These state reconstructions may be used in a linear state feedback controller design, provided the feedback controller is complemented with a sufficient number of iterated output, or input, integral error compensation which structurally match the effects of the neglected perturbation inputs and initial states To clarify the idea behind GPI control, consider the following elementary example, ă y = u+ξ y (0) = y0 ˙ ˙ y (0) = y0 (7) with ξ being an unknown constant disturbance input The control problem consists in obtaining an output feedback control law, u, that forces y to track a desired reference trajectory, given by y∗ (t), in spite of the presence of the unknown disturbance signal and the unknown ˙ value of y(0) Let ey y − y∗ (t) be the reference trajectory tracking error and let u ∗ be a feed-forward input ă ă nominally given by y (t) = u ∗ (t) The input error is defined as eu u − u ∗ (t) = u − y∗ (t) Integrating equation (7) we have, 149 Sliding Mode Control Design for Induction Motors: An Input-Output Approach i∗ S ω∗ ω ˆ ψR ˆ di ∗ s dt GPI Controller Outer Loop iS ˆ ξ1 ˆ ψR uS ˆ ξ GPI Observer First Order Filter uS Induction Motor Second Stage ω iS Sliding mode Controller Inner Loop ω ˆ ξ1 iS Flux Reconstructor ω iS ˆ ψR First Stage Fig Control schematics Experimental results We illustrate the proposed control approach by some experiments on an actual induction motor test bed The experimental induction motor prototype includes the following parameters: J = 4.5 × 10−4 [Kg m2 ], n p = 1, M = 0.2768 [H], L R = 0.2919 [ H ], L S = 0.2919 [H], RS = 5.12 [ Ω], R R = 2.23 [ Ω] The flux absolute desired value was 0.5872 [Wb] The sliding mode surface parameter was z = 350 The output of the sliding control was filtered by means of a first order low pass filter of Bessel type with cut frequency of 750 rad/s The angle measurement was obtained using an incremental encoder with 10000 PPR The desired closed loop tracking error was set in terms of the characteristic polynomial Pθ (s) = (s2 + 2ζω n + ω n )(s + p), with ζ = 1, ω n = 330, p = 320, and the observer injection error characteristic polynomial was Pξ = (s2 + 2ζ ω n1 + ω n1 )4 , with ζ = 2, ω n1 = 27 ˆ The controller was devised in a MATLAB - xPC Target environment using a sampling period of [ms] The communication between the plant and the controller was performed by two data acquisition devises The analog data acquisition was performed by a National Instruments PCI-6025E data acquisition card, and the digital outputs as well as the encoder reading for the position sensor were performed in a National Instruments PCI-6602 data acquisition card The voltage and current signals are conditioned for adquisition system by means of low pass filters with cut frequency of [kHz] The interconnection of the modules can be appreciated in a block diagram form as depicted in figure The output reference trajectory to be tracked, was set to be a biased sinusoidal wave of the form: θ∗ = + sin(t − π/2) 0≤t 0, to both sides of (27) and formally dividing by (d/dt + c) transforms (27) to di1 = a + ωb + dt (28) Cascade Sliding Mode Control of a Field Oriented Induction Motors with Varying Parameters 161 where → exponentially and the functions a and b are linear combinations of the filtered stator current and stator voltage command: a = (c + α1 ) i11 + α2 i10 + α3 u10 + α4 u11 b = M ( β i11 + β i10 + β u10 ) where s i , i11 = i s+c s+c 1 s = u , u11 = u s+c s+c i10 = u10 (29) Here s denotes d/dt To obtain a reference model for the rotor resistance identification the part of the right-hand side of (28) containing R2 is separated: di1 = f + R2 f + ωM f + dt (30) where f = (c + ρ1 ) i11 + ρ2 u11 f = γ1 i11 + γ2 i10 + γ3 u10 (31) f = β i11 + β i10 + β u10 Coefficients in (31) are calculated according to the following formulae: ρ1 = − , σL1 n p R1 np β2 = , β3 = − , σL1 σL1 R1 , σL1 β1 = n p , γ1 = − L2 ρ2 = 1+ L2 m σL1 L2 , γ2 = ρ1 , L2 γ3 = ρ2 L2 Since ω is available for measurement the design of an R2 identifier is straightforward It is based on the MRAS identification approach (Sastry & Bodson (1989)) with(30) being a reference model Within this approach an identifier consists of a tuning model depending on an estimate of the unknown parameter and a mechanism to adjust the estimate This adjustment is performed to make the output of the tuning model asymptotically match the output of the reference model In our case the tuning model is given by ˆ d i1 ˆ ˆ = − L i1 − i1 + f + ωM f + R2 f dt (32) ˆ ˆ where L > is a constant and R2 is the estimate of R2 The dynamics of the error e = i1 − i1 is the following de ˆ = − Le + R2 − R2 f − dt (33) 162 Sliding Mode Control ˆ The adjustment equation for R2 is ˆ d R2 ˆ = − γ i1 − i1 dt ( T f2 , γ > (34)     ) ... regulation [A] i* (t) Sa iSa(t) ? ?5 [A] ? ?5 i* (t) Sb iSb(t) Time [s] Fig Current tracking  152 Sliding Mode Control u1 [V] 50 ? ?50 10 10 10 u2 [V] 50 ? ?50 u3 [V] 50 ? ?50 Time [s] Fig Stator voltages... New York, USA Levant, A (1993) Sliding order and sliding accuracy in sliding mode control, International Journal of Control 58 (6): 1247–1263 Ogata, K (2001) Modern Control Engineering, Prentice... and Chaos 20 (5) : 150 9U– 151 7  154 Sliding Mode Control Melfi, M., Evon, S & McElveen, R (2009) Induction versus permanent magnet motors, IEEE Industry Applications Magazine 15( 6): 28 – 35 Sira-Ramírez,