SLIDING MODE CONTROL OF LINEAR SYNCHRONOUS MOTOR SERVODRIVE

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1 Journal of Engineering Sciences, Assiut Universit, Vol. 34, No. 4, , Jul 2006 SLIDING MODE CONROL OF LINEAR SYNCHRONOUS MOOR SERVODRIVE Electrical Engineering Deartment, Aswan Facult of Engineering, South Valle Universit, Aswan, Egt (Received Aril 2, Acceted Ma 3, 2006) ABSRAC his aer rooses a method to design sliding mode control (SMC) for the osition tracking control of a ermanent magnet linear snchronous motor (PMLSM). he controller is designed to achieve accurate erformance in the resence of unknown disturbance and arameter uncertainties. he comonents of the control law, the equivalent comonent and the robust comonent, are designed such that the nominal sstem exhibits desirable dnamics and the reaching condition is guaranteed. Switching surface is designed based on ole lacement and generalized matrix inverse. Simulation results show that the suggested rocedure rovides high erformance dnamic characteristics and robust with regard to external disturbance and arameter variations. KEYWORDS: Sliding mode control, Pole assignment, Descritor sstems, Control of electric drives, Observers,. INRODUCION he control of motors used in high erformance servo drives is one of the most fundamental roblems that considered in the field of automatic control. he controller design for a lant must satisf that the oututs are tracking an inut reference value and must be indeendent of the arameter uncertaint and disturbances as much as ossible. Variable structure control with sliding mode has been known for its robustness roerties such as insensitivit to arameter variations, external disturbance rejection and satisfactor dnamic resonses [see, e.g. -3]. hese robustness roerties make sliding mode control a good candidate for industrial alications. Sliding mode control generall involves two main stes: firstl, the selection of a sliding surface which induces assigned stable reduced order dnamics, and secondl, the snthesis of a control law to force the closed loo sstem trajectories onto and subsequentl remain on the sliding surface [4-6]. Several researches deal with the design and the alications of SMC on the control of motors [e.g.,3,7,8]. In [] a roosed sliding mode control method based motion control drive design in accordance with a traezoidal velocit rofile for ermanent magnet snchronous motor is given. A robust controller is designed in [7] b using roortional integral (PI) control algorithm 255

2 256 and a sliding mode control. In [8] a robust controller is designed b emloing variable structure and linear quadratic method for PMSM. Also, robust control method for PMLSM based on recurrent fuzz neural network using sliding mode control is given in [3]. In this aer a method is roosed to design the control law and the switching herlane of the sliding mode control. he robustness against matched disturbance and the sstem stabilit are discussed. Simulation results of the PMLSM servodrive are given and its show good erformance for the tracking in resence of inut disturbance and arameter uncertainties. For the sake of comarison uroses, simulations are carried out for the same reference trajectories with a classical roortional-integral (PI) controllers. It should be noted that, the PI-te controller is still the most oularl used for motor drives, eseciall among ermanent magnet snchronous motor drives; this is due to its relativel simle imlementation. However, PI-te control methods are not enough to accommodate the variations of external disturbance and arameter uncertainties during oeration. his is obvious from the given simulated results. 2. SYSEM DESCRIPION AND PROBLEM FORMULAION Consider a dnamical sstem described b where x ( = Ax( + B( u( + f ( x, ) ( = Cx( (b) x n ( R is the state vector, outut vector and f u m ( R is the control inut, (a) ( R is the m ( t, x) R reresents the uncertainties and /or/ disturbances. Note that f ( t, x) is assumed unknown but bounded, i.e. f ( t, x) f ( where f o ( reresents known uer bound and. denotes the Euclidean norm. o A n n R, n m n B R, andc R, are real constant matrices of the nominal sstem. Define the integral vector of tracking error as [8] where e t t ζ ( = e ( dt = ( d ( τ )) dτ (2) 0 0 ( R is the tracking error, R d ( is the desired reference inut and ζ ( R. Equivalentl, (2) can take the following differential form ζ = d ( = d Cx( (3) he newl defined augmented sstem can be rewritten in the following form where z ( = Mz( + H ( u( + f ( x, ) + N d (4)

3 SLIDING MODE CONROL OF LINEAR SYNCHRONOUS. 257 x z = R ζ ) B H = R 0 A 0, M = C R 0 0, N = I R ) m ) ) ) herefore, the roblem is to design a switched robust control that drive the lant state to the switching surface and maintain it on the surface uon intercetion. A Launov aroach is used to characterize this task. Also, switching surface that gives desired sstem dnamics on sliding will be designed. Moreover, simulation results will be given to clarif the effectiveness and robustness of the suggested method. 3. SLIDING MODE CONROL DESIGN Define the sliding function, σ (z), as σ ( z ) = Sz( (5) m ) where S R matrix and will be determined such that the sliding mode dnamics in the sliding surface S = { z R ) σ = Sz( = 0} (6) have (n+-m) desired eigenvalues. he choice of S is often referred to as herlane design. his will be discussed latter. Now, the controller is designed to drive the sstem trajectories into sliding motion and once the sstem is in sliding motion, the controller should be kee the sstem in sliding motion desite the resence of uncertainties/ and/ or disturbances. he reachabilit condition is chosen to make the sliding manifold attractive to the remaining state sace. his reachabilit condition makes sure that the selected Launov function [4] satisf that V = σ σ / 2 (7) σ σ < 0 (8) So, define the control law as u = u E + u R (9) where u E is the equivalent control that ma be obtained from a conventional method of the linear sstem theor alied to the nominal sstem and u R is the robust control which is switching in nature. he control law will be develoed to guarantee the reaching condition (8). Using (4) and (6), u can be obtained as follows ) = E σ ( z S z = SMz + SHu E + SN d (0) herefore, ue = ( SH ) SMz ( SH ) SN d (),

4 258 Also, u R is assumed as follows u R = ( µ + ρβ )( SH ) sgn( σ ) (2) in which µ > 0, β f, ρ SH, and sgn(σ ) is the sign function of σ that defined as +, sgn( σ ) =, Consequentl, the control law will be if if σ > 0 σ < 0 u = ( SH ) SMz ( SH ) SNd ( µ + ρβ )( SH ) sgn( σ ) (3) Lemma: he control vector reresented b (3) is satisfing the reaching condition given in (8). Proof: Using (4), (7) and (3), ields σ = z ( Mz + H ( u + f ) Nd ) [ M z H ( SH ) SMz H ( SH ) SNd σ σ S = σ S + = σ S H ( µ + ρβ )( SH ) sgn( σ ) + Hf + [ ( µ + ρβ )sgn( ) + ( SH ) f ] = σ σ = ( µ + ρβ ) σ sgn( σ ) + σ ( SH ) f = ( µ + ρβ ) σ + σ ( SH ) f ( < 0 µ + ρβ ) σ + σ SH f ] N d whereas σ σ Now, the sliding matrix S is chosen b assigning the sstem dnamic on sliding motion. So, in surface reresented b (6), the sstem dnamics roerties are determined b suitable choice of (n+-m) eigenvalues, sa, λ, λ, 2..., λn+ m. he rest of m eigenvalues are assigned to be zeros [9]. However, in [0], these m eigenvalues are laced at some stable value, λ, and called the sliding margin. So, substituting from () into (4), then the sstem dnamics on sliding is described b z = ( M HK) z (4) where K = ( SH ) SM (5)

5 SLIDING MODE CONROL OF LINEAR SYNCHRONOUS. 259 is the state feedback gain matrix. Clearl, the matrix K can be found using ole lacement method b choice the desired sstem eigenvalues. herefore, if the eigenvalues λ, λ,..., 2 λn + m, λ,..., λ are suitabl chosen, then the matrix S will be determined such that S( M HK) = λ S (6) Letting M ) C = M HK, and Y = ( λ I n+ M C, then (6) reduces to YS = 0 (7) Using the generalized matrix inverse [], the general solution of (7) for S is S g = ( I + Y Y ) W (8) n g where Y is the generalized inverse of Y and W is an (n+) m arbitrar matrix that must be chosen to get matrix S such that (SH) is invertable matrix. 4. PMLSM SERVODRIVE MODEL AND SIMULAION o demonstrate the effectiveness and robustness of the roosed aroach, simulation is done on the PMLSM servodrive. he model of the PMLSM servodrive is shown in Fig. [3,2], i q k F + _ w Ms + D v s Fig. : PMLSM servodrive model where k F is the thrust coefficient, i q is the command thrust current, M is the total mass of the moving element sstem, D is the viscous friction and iron loss coefficient, w is the external disturbance, v is the linear velocit and is the outut osition. Consider the assumtions as in [3, 2], the nominal values of the arameters are k F = 20 N / A, M = 0.254Ns / V and D = N / V Letting x = and x2 = v, then the state sace reresentation will take the following form x x 0 x 0 D k F = + ( u + f 0 x 2 2 M M where f = w/ k. Choosing the desired eigenvalues as 30, 35, 0, and letting F ) (9)

6 260 W = [ 8 5 0] then the switching matrix will be S = [ ] Also, taking µ = 0. 2, and using Matlab ackage, the simulated results of the outut ( x ), control (u), robust control and sgn(σ ) (sign of sigma) for a rectangular osition command inut (command = 4 mm) are given according to the following cases: (i) Nominal values of arameters without disturbances is resented in Fig. 2. (ii) Uncertaint in the mass, M = 2*nominal, and disturbance w = 20 is alied in the shown intervals of time is shown in Fig. 3. (iii) Uncertaint in the mass, M=3*nominal, and disturbance as in case (ii) is shown in Fig. 4. Obviousl, from the shown figures, the outut osition is tracking the command inut in desite of the existence of disturbance and arameter uncertainties. his clarifies the robustness roert of the suggested sliding mode control design aroach, in comarison with the conventional PI controller, K = and K = 0. 9, which is shown in Figs i Fig. 2: simulation resonse of nominal sstem.

7 SLIDING MODE CONROL OF LINEAR SYNCHRONOUS. 26 Fig. 3: simulation resonse with disturbance w=20 added from t=3 to t=7, and M=2nominal value. Fig. 4: simulation resonse with disturbance w=20 added from t=3 to t=7, and M=3 nominal value

8 262 d + K K + i s iq k F + _ w Ms + D s Fig. 5: PMLSM servodrive model with PI-te controller. ime (sec) Fig. 6: simulation resonse of nominal sstem with PI controller. ime (sec) Fig. 7: simulation resonse of sstem with PI controller, w=20 added from t=3 to t=8, and M=2 nominal value.

9 SLIDING MODE CONROL OF LINEAR SYNCHRONOUS. 263 ime (sec) Fig. 8: simulation resonse of sstem with PI controller, w=20 added from t=3 to t=8, and M=3 nominal value. 5. CONCLUSIONS he design of a robust controller for a lant to solve the servomechanism roblem has been considered. he roosed SMC control law can force the sstem trajectories onto the sliding motion and subsequentl maintain the trajectories on this motion. Also, this control law can reject the arameter uncertainties and the external disturbance so that the outut fulfills satisfactor erformance. he design technique is simle and straightforward. Simulation results show the high outut erformance eseciall in resence of external disturbance and arameter uncertainties. Comarison with conventional PI-te controllers is included. REFERENCES [] C.-K. Lai and K.-K. Shu, A novel motor drive design for incremental motion sstem via sliding mode control method, IEEE ransactions on Industrial Electronics, Vol. 52, No. 2, , [2] K.D. Young, V.I. Utkin, and U. Ozguner, A control engineer s guide to sliding mode control, IEEE ransactions on Control Sstem echnolog, Vol. 7, No. 3, , 999. [3] F.-J. Lin, C.-H. Lin and P.-K. Huang, Recurrent fuzz neural network controller design using sliding mode control for linear snchronous motor drive, IEE Proc. Control heor and Alications, Vol. 5, No. 4, , 2004.

10 264 [4] V.I. Utkin, Sliding modes and their alication in Variable Structure, Mir Publishers Moscow, 978. [5] K. Erbatur, M. O. Kanak and A. Sabanovic, A stud on robustness roert of sliding mode controllers: A novel design and exerimental investigations, IEEE ransactions on Industrial Electronics, Vol. 46, No.5, [6] J.Y. Hung, W.B. Gao, and J.C. Hung, Variable structure control: A surve, IEEE ransactions on Industrial Electronics, Vol. 40, No. 2,. 2-22, 993. [7] J.-L. Chang, Design of a robust controller using onl outut feedback to solve the servomechanism roblem, IEE Proc. Control heor and Alications, Vol. 50, No., , [8] K.-K. Shu, C.-K. Lai, Y-W sai and D.-I Yang, A newl robust controller design for the osition control of ermanent magnet snchronous motor, IEEE ransactions on Industrial Electronics, Vol. 49,No. 3, , [9] J. Ackerman and V. Utkin, Sliding mode control design based on Ackerman s formula, IEEE ransactions on Automatic Control, Vol. 43, No. 2, , 998. [0] Q.P. Ha, H. rinh, H.. Nguen and H.D. uan, Dnamic outut feedback sliding mode control using ole lacement and linear functional observers, IEEE ransactions on Industrial Electronics, Vol. 50, No. 5, , [] C.R. Rao and S.K. Mitra, Generalized Inverses of Matrices and its Alications, New York, Wile, 97. [2] F.-J. Lin, C.-H. Lin and C.-M. Hong, Robust control of linear snchronous motor servodrive using disturbance observer and recurrent neural network comensator, IEE Proc. of Electrical Power Alications, Vol. 47, No. 4, , اا كاااا ھها ا ط ا ا %$"ا#" كا اا/(ا.- ذوا*(ط)ا '&. و% % ات & %0 أدا هو%$" د 0 و 7 دد>; : / 9 م وأ= و 7 د %*ات &ا/ات ا (<م. و ) law ( control وان C/ ا اء ) control ( equivalent وا نا اى ) control ( robust & -9 ا (<م ا 9 دي ) sstem ( nominal ا كا 6 (/ ا-ب وأ= %0 I طا H ) condition ( reaching.و% % P-N اق & N 6. امط /ا#"اM وروا 9 سا 9 م J ت. و/ C ا (Q3 وا N /ت ا#ا % ال RA %$ Cأنا - ا %= Cأدا 3 A وة ل و 7 دد>; : / 9 م وو 7 د %* ا$را/ات. و&رھه ا (Q3 /" Q3 اN 6. امات /C عا( N U وا/; و 7 6 أنھها - أ=; > Hو 7 دد>; : / 9 موو 7 د %*ا$را/ات.