Position control of PMSM in sliding mode

198 Advances in Electrical and Electronic Engineering POSITION CONTROL OF PMSM IN SLIDING MODE P. Briš, J. Vittek, P. Makyš Department of Power Electrical Systems, Faculty of Electrical Engineering, University of Žilina Veký diel, 010 26 Žilina, tel.: +421 41 513 2260, mail: [email protected] Summary In the paper control of linear permanent magnet synchronous motor (PMSM) based on the principles of sliding mode control (SMC) with respect of vector control principles is carried out. The presented simulations comprise position quantization due to assumed experimental verification on the bench which consists of linear PMSM and incremental position sensor. Simulation results compare two methods for obtaining of position derivatives needed for SMC algorithm. The first method exploits a filtering observer and second one uses numerical derivations and first order filters. 1. INTRODUCTION This paper presents SMC of a linear PMSM. Primary part of this linear PMSM is moving one and incremental position sensor yields quantized signal of position x m . Control of PMSM is based on principles of vector control, i.e. designed control technique divides PMSM into two channels. In d-channel the current component, i d , is controlled at the zero value to control machine magnetic flux. In the q-channel position, x m , is controlled on a demanded value with the specified dynamics to control machine force. Both channels for control exploit principles of SMC, where only the plant rank, r, has to be known. As a result the control value derivatives up to the r-1 rank must create the feedback. Status of the system is then kept exactly in the r-1dimensional switching boundary within the rdimensional sub-state space. Feedback system dynamics is determined only by boundary alone and doesn’t depend on the system parameters and external disturbances and so a desirable robustness is ensured. For more detailed description of control theory see [1], [2] and [3]. 2. THEORY A. Design of Sliding Mode Control Algorithm Differential equations of linear PMSM extracted from [4] are as follows: ( )      +−−= ddPMmqsq q q iLv r p iRu Ldt di ψ 1 , (1)      +−= qqmdsd d d iLv r p iRu Ldt di 1, (2) [ ] qdqdqPMe iiLLi r p F)( 2 3−+= ψ , (3) ( ) Le m FF m dt dv −= 1, (4) m m v dt dx = , (5) where: πτ p pr= , (6) A simple form of SMC is as follows: ( ) θ Ssignuu qdemq max__ = , (7) Demanded voltage, uq_dem, is control variable, switching between maximal value uq_max and -uq_max which are imposed by the hardware. Rank r=3 results from PMSM equations for qchannel. Then two derivatives of position, xm, are needed for the feedback. Corresponding switching function, S  , is defined as: mmmdemm xwxwxxS &&& 21_ −−−= θ , (8) The switching function switches whenever S  changes its sign. The equation of the switching boundary can be derived if S  is set to zero. The control system is designed by the method of pole assignment, where the system has n poles of closed loop, i.e. n = r-1. The poles have equal real parts placed at, ω 0= − 1/Tc, which ensure the fastest settling time of the step response without overshoot. Ideal transfer function with respect of Dodds formula for specified settling time, Ts, is as: ( ) ( ) ( ) 1 9 4 81 4 1 15,1 1 1 2 2 2 _ ++ =             + + = = ss n n s demm m T s T s n T s sx sx , (9) The modification and the transformation into the time domain yields: 0 81 4 9 4 2 _ =−−− m s m s mdemm x T x T xx &&& , (10) Equating right hand sides of (10) and (8) yields control algorithm feedback gains: 81 4 , 9 4 2 21 ss T w T w== , (11a,b) Position control of PMSM in sliding mode 199 Disadvantage of the control based on SMC principles is control chattering. This control chattering can be alleviated by inserting a pure integrator at the plant input. But this smoothing integrator together with the plant creates a ‘new’ plant to be of 1 rank greater than the original one. Therefore, it is necessary to repeat already introduced approach for r=4, from (7) to (11). In addition of this, if signum function is replaced by a proportional high gain, K SM , then the control algorithm is as follows: dtx T x T x T xxKu m s m s m s mdemmSMdemq         −−−−= &&&&&& 216122 32 __ , (12) This means 1 rank higher derivation inputs to the control algorithm against the original plant. By integration of SMC algorithm (12) the highest derivative can be cancelled and the final position SMC algorithm is as follows: ( )      −−−−=  m s m s m s mdemmSMdemq x T x T x T dtxxKu &&& 216122 32 __ , (13) This way, using the rearrangement of control system a smoothing integrator doesn’t increase the number of derivatives in the feedback. The position control channel block diagram is shown in Fig. 1, which corresponds to (13) with mentioned rearrangement: u q_dem x m d x m /dt x m _ dem 2 s T 12 2 s T x m d 2 x m /dt 2 s 1 K SM s s x m Control Channel r = 3 216 3 s T Fig. 1. SMC control loop for position x m channel Similar way the SMC algorithm is derived for current component, i d control channel, which is as follows: ( )      −−=  d si ddemdiSMdemd i T dtiiKu 3 ___ , (14) The current component, i d control channel block diagram corresponding with (14) is shown in Fig.2: si T 3 1 u d_dem i d i d i d_dem i d Control Channel r = 1 K SM_i s 1 Fig. 2. SMC loop for current i d channel B. Position derivatives needed for SMC algorithm One way to gain needed position derivatives, which are velocity, v m , and acceleration, a m , for SMC feedback position loop is using of observer. Detailed description of observer is in [3]. Therefore next section presents only short extract. The observer is capable to produce filtered estimate of position, m x ˆ , estimate of velocity, m v ˆ , and acceleration, m a ˆ , including estimate the external load force, L F ˆ . State equations of the observer are as follows: ( ) xF L ek dt Fd += −0 ˆ (15) ( ) xvLqPM m ekFi r p mdt vd +      −+= ˆ 2 31 ˆ ψ (16) xxm m ekv dt xd += ˆ ˆ (17) The external force is assumed to be constant over a time interval that is short if compared with observer time constant, T so . This assumption forms (15). The equations (16) and (17) are based on the PMSM model, (3), (4), (5). As current component, i d , is controlled at the zero value then equation for electromagnetic force, F e , can be reduced. Observer state equations include correction loops based on position error, mqmx xxe ˆ _ −= , which is defined as difference between the measured position and its estimate. Block diagram of the described observer is shown in Fig. 3. s 1 s 1 k F k v k x qm x _ m x ˆ m v ˆ s 1 m 1 e x m x ˆ m v ˆ m a ˆ L F ˆ − qPM i r p m ψ 2 31 Fig. 3. Block diagram of observer The correction loop gains k x , k v and k F are designed by pole-placement. Equating the desired characteristic polynomial ( LHS of (18)) and the characteristic polynomial of observer of Fig. 3 ( RHS) of (18)): m k kskss T s T s T sF vx sososo +++=+++ 23 32 23 21610818 (18) Desired characteristic polynomial was derived again with the aid of settling time formula (9) to yield a correction loop settling time, T so . 200 Advances in Electrical and Electronic Engineering Then correction loop gains are as follows: so x T k18 = , 2 108 so v T k= , 3 216 so F T m k = . (19) Thank to inverse estimated load force, - L F ˆ , produced in the observer the characteristic polynomial (RHS of (18)) includes only plus signs and all the correction loop gains have positive values. Block diagram of parallel SMC structure of SMPM with PWM modulation and with exploitation of filtering observer is shown in Fig. 4. u q_dem u a,b,c dqabc dem u a,b,c u d_dem i a,b,c i d_dem PWM x m_dem i d FILTERING OBSERVER m a ˆ m x ˆ m v ˆ qm x _ i q abcdq SMC of i d SMC of x m F L F e Driven Mech. Load AC motor Fig. 4. Parallel SMC structure with filtering observer The input to the filtering observer is position, x m_q , which is quantized, as the output of the incremental position sensor. The quantization step is determined by the resolution of the used sensor. Another way to gain the desired position feedback derivatives is numerical derivation. Before derivation the quantized position signal is filtered to gain smooth function of, x m_f . Then the desired velocity is computed via numerical derivation and again filtered for elimination of noise. Filtered velocity, v m_f , is again numerically derivates to gain desired acceleration, a m_f , for feedback. All the filters exploited are of the first order. Block diagram of parallel SMC structure with exploitation of numerical derivation to gain demanded feedbacks is shown in Fig. 5. u q_dem u a,b,c dqabc dem u a,b,c u d_dem i a,b,c i d_dem PWM i d abcdq SMC of i d 1 1 1 +sT f 1 1 2 +sT f s x m_dem qm x _ F L F e fm a _ fm v _ s SMC of x m fm x _ Driven Mech. Load AC motor Fig. 5. Parallel SMC structure based on numerical derivation 3. EXPERIMENTAL RESULTS The parameters of used linerar PMSM are: winding resistance, R s =0.44  , winding inductances, L d =157  H, L q =141.3  H, permanent magnet flux,  PM =0.066 Wb, pole pair number, p=3, pole pitch,  p =25mm, and weight of primary part, m=1.483kg. All state variables introduced in theoretical section have zero initial states. The simulations presented further have following parameters: prescribed settling time of position, T s =15 ms, position SMC algorithm gain, K SM =7.10 6 , prescribed settling time of d-current, T si =5 ms, d-current SMC algorithm gain, K SM_i =50. Computational step of control algorithm is 0,128 ms and SMPM model computational step is 10 -6 s. Demanded position step, x m_dem =5 mm, overall simulation time is 0,08 s and at the t=40 ms load force, F L , is changed from zero value to 80 N to achieve approximately the nominal current 7A. Simulation results for ideal SMC algorithm, where SMC algorithm and PMSM model without PWM modulation, without abc-dq transformations and position quantization, are shown in Fig. 6. Position derivatives as inputs of SMC algorithm are obtained directly from PMSM model. It can be seen from Fig. 6, that the demanded d,q-voltages and corresponding currents are smooth thanks to smoothing effect of the added integrator. The function of position x m has also smooth behavior and as it’s evident from Fig. 6a the difference between real and ideal response x id is very small therefore the demanded dynamics of position was also achieved. a) position response b) velocity as f(t) c) acceleration as f(t) d) demanded d-voltag e) demanded q-voltage f) d,q-currents Fig. 6. Simulation results for ideal SMC In next simulations PWM modulation, abc-dq transformations and position quantization is performed and DC-bus voltage U dc =24 V and quantization is taken into account with computational step 10  m. Simulation results with filtering observer are shown in Fig. 7. Settling time of observer is T so =5ms. Simulation results with numerical derivation for reconstruction of position derivatives 0 0.02 0.04 0.06 0.08 -0.2 0 0.2 0.4 0.6 v m (t) 0 0.02 0.04 0.06 0.08 -10 0 10 20 30 t [s] i d , i q [A] i d (t) i q (t) 0 0.02 0.04 0.06 0.08 -0.04 -0.02 0 0.02 t [s] u d [V] u d (t) 0 0.02 0.04 0.06 0.08 -100 0 100 200 t [s] a m [ms - 2 ] a m (t) 0 0.02 0.04 0.06 0.08 -2 0 2 4 6 x 10 -3 x m , x id [m ] t [s] 2( x id - x m ) x m (t) x id (t) 0 0.02 0.04 0.06 0.08 -5 0 5 10 15 t [s] u q [V] u q (t) Position control of PMSM in sliding mode 201 are shown in Fig. 8, where time constants of the first order filters are T f1 = 0.001/3 and T f2 = 0.002/3. a) position response b) position detail c) velocity as f(t) d) acceleration as f(t) e) demanded d,q-voltage f) d,q-currents Fig. 7. Simulation results for position SMC with exploitation of filtering observer a) position response b) position detail c) velocity as f(t) d) acceleration as f(t) e) demanded d,q-voltage f) d,q-currents Fig. 8. Simulation results for position SMC with numerical derivation for position derivatives Fig. 7b shows quantization of PMSM model position as well as observed position, m x ˆ . Similar way in Fig. 8b are for short time interval shown quantized position, x m_q and filtered position, x m_f . Although behaviors of velocity and acceleration in simulation with numerical derivation have a certain delays versus PMSM model, ( see Fig. 8c,d ) the position response shows only a small transient in position decrease due to loading by the same step of external force, F L it’s compared with the simulation with observer ( see Fig. 7b and Fig. 8b ). This position decrease in simulation with numerical derivation, Fig. 8a, is comparable with the decrease at ideal SMC shown in Fig. 6a. At simulation with observer the position transient is bigger due to certain error in observer estimated values, especially in velocity, ( see Fig. 7c ), during L F ˆ transient. But interval with constant F L hasn’t any visible delay between model and its estimated behavior ( see Fig. 7c,d ). It is valid for both simulations of Fig. 7 and Fig. 8 that chattering of u q isn’t caused by SMC but this chattering is a result of feedback derivatives. Also u d is chattering, which is caused by chattering of i d due to PWM modulation. 4. CONCLUSION Based on presented simulation results for linear PMSM position control it can be concluded that chattering of u d isn’t significant and although i d as an input of SMC algorithm would be filtered the chattering of i d will be the same as a result of PWM. A more significant is chattering of u q resulting in chattering of position in steady-state. This can be eliminated by better filtering of position derivatives. But in simulation with the numerical derivation it’s at the expense of control stability. For both simulations is also valid that transients in position error due to external force loading are more significant. As conclusion is valid the smoother are derivatives as inputs of SMC algorithm the smoother is also i q current and the only chattering presented is due to PWM modulation. Acknowledgement The authors wish to thank Slovak Grant Agency VEGA for funding the project No. 4087/07 ‘Servosystems with Rotational and Linear Motors without Position Sensor’. REFERENCES [1] Utkin V.. I.: Sliding Mode Control Design Principles and Applications to Electric Drives, IEEE Trans. on Industrial Electronics, vol.40, pp.2336, 1993. [2] Kardoš, J.: Teória systémov s premenlivou štruktúrou a  asovo suboptimálne riadenie polohy. HMH ATP Journal 2007. [3] Vittek, J., Briš, P., Štulrajter, M., Makyš, P.: Preliminary Verification of the Chattering Free Slinding Mode Control of the Drive Position employing PMSM, OPTIM 2008, Brasov, Romania, May 2008. [4] Žalman, M., Jovankovi  , J.: Nové trendy v riadení lineárnych pohonov, AT&P journal 2/2006, pp.6770. 0 0.02 0.04 0.06 0.08 -10 0 10 20 30 t [s] i d , i q [A] i q (t) i d (t) 0 0.02 0.04 0.06 0.08 -10 0 10 20 30 t [s] u d , u q [V] u q (t) u d (t) 0 0.02 0.04 0.06 0.08 -100 0 100 200 300 t [s] a m , a m_f [ms - 2 ] a m _f (t) a m (t) 0 0.02 0.04 0.06 0.08 -0.2 0 0.2 0.4 0.6 t [s] v m , v m_f [ms - 1 ] v m_f (t) v m (t) 0 0.02 0.04 0.06 0.08 -2 0 2 4 6 x 10 -3 x m , x id [m ] t [s] x id (t) x m (t) 2( x id - x m ) 0.05 0.055 0.06 4.9 4.95 5 5.05 x 10 -3 x m , x m_q , x m_f [m ] x m (t) x m _q (t) x m_f (t) t [s] 0 0.02 0.04 0.06 0.08 -10 0 10 20 30 i d , i q [A] i q (t) i d (t) t [s] 0 0.02 0.04 0.06 0.08 -10 0 10 20 u d , u q [V] t [s] u q (t) u d (t) 0 0.02 0.04 0.06 0.08 -100 0 100 200 t [s] a m , a m ^ [ms - 2 ] a m ^(t) a m (t) 0 0.02 0.04 0.06 0.08 -0.2 0 0.2 0.4 0.6 t [s] v m , v m ^ [ms - 1 ] v m ^(t) v m (t) 0 0.02 0.04 0.06 0.08 -2 0 2 4 6 x 10 -3 x m , x id [m ] x id (t) x m (t) 2( x id - x m ) t [s] 0.04 0.045 0.05 0.055 0.06 4.9 4.95 5 5.05 x 10 -3 t [s] x m , x m_q , x m ^ [m ] x m (t) x m_q (t) x m ^(t)