【正文】 s, it can provide a reference input which satisfies the dynamic and stable performance of the whole system. In [10], the modified continuous second order smooth trajectory was presented and the short time trajectory stable problem was tackled. Based on the continuous time results, Zanasi has proposed a discrete time nonlinear control law that guarantees the minimum time global stabilization of a chain of two discrete integrators with bounded input [11]. The smooth trajectory filter was applied into PMSM servo motion system and accurate simulation and experiment results were obtained in [12]. The designed modified second smooth filter structure in this paper is shown as Fig. 1. The two discrete integrators in Fig. 1 has different structure, this choice has been made in order to guarantee the same dynamic characteristic as the continuous STF. This controller structure is similar with which proposed in [11], the only difference between them is that they have diverse nonlinear feedback controller.Assume T is the sample period of controller, and let, then can be calculated as (1)So the dynamic state model of second order smooth trajectory filter C3 in Fig. 1 can be depicted as follow: (2)To do the vector transform for, and let (3) (4)Then (5)Where .Thus, equation (3), (4) and the follow equation constitute the controller C3 in Fig. 1. (6)Whereis the tracking error, is the velocity error, ,is the discretetime derivative of signal rn and σn =0 is the sliding mode surface.Fig. 1 Second discrete nonlinear smooth tracking filterAs is seen from the initial controller in [11] and the improved controller in this paper, both them have the same bound limitation of the derivatives of input , . However, the effect of the second derivative is not considered in the former under the precondition of the first derivative being piecewise constant, which limits the types of signal as those meeting the demand, such as square wave, slope wave, the sawtoothed wave and so on. On the contrast, the latter design proposed by this paper can be available to the broader range of signal than the former, only with the bound limitation of the derivatives of input as above, such as sine wave which can not fit for the former.III. MODEL OF BLDCMIn the ideal condition, the three phase voltage equations in a matrix form for the BLDCM are represented as (7)Where, ua, ub and uc are stator phase voltage。 r is the winding resistance。 ia, ib and ic are the line currents。 ea, eb and ec are the back emfs of the phases。 L is the selfinductance。 M is the mutual inductance。 and D is the differential operator.Further, the above equation can be simplified as (8)Where u is the terminal voltage。 i is the phase current。 r39。 is the equivalent phase winding resistance。 L39。 is the equivalent phase inductance。 ke is the back electromotive force constant。 and ω is the motor speed.And the mechanical equation of BLDCM is given asWhere Te is the electromagnetic torque, Tl is the load torque, J is the inertia of the motor, kt is the motor torque constant, P is the viscous coefficient and θ is the motor mechanical angular velocity.IV. BLDCM POSITION CONTROL BASED ON THE STFA. Design of the STF controllerFrom the above section, we can divide the nonlinear, strong coupled BLDCM control system into two parts as follows:I : , and II : .Let, thenEquation (12) is the typical cascade two order system posed by two integrators, so it can build a nonlinear state feedback controller based on the STF to obtain a desirable smooth reference signal x , and bounded first and second out derivatives , . Here the linear PID controller is still used for the current loop of BLDCM. From (12), the reference input of current loop i* can be achieved as (13).In order to ensure better dynamic performance of the control system, the calculation equation of i* is modified as followWhere , and Kp , Kv are constants. FF is the feedforward control part and the LR is the linear feedback control one. Then the discrete form of (14) can be written asB. Load torque observerConsidering the variable and unknown load torque and disturbance of moment of inertia, friction coefficient:Substitute (16) into (9), the following equation can be obtained.Where Td is the unknown load disturb torque.According to the estimate of Td caused by unknown changes of parameters and loads, this paper designs a reduced order disturbance observer by the assumption that the disturbance is unknown constant during each sufficiently fast sampling interval [13]. And it is expressed as follows:Where , , , L1 is the observer gain.So after introducing the load observer into the BLDCM control system, the reference input of current loop i* is rewritten as follow. (21)V. SIMULATION AND EXPERIMENTAL RESULTSTo verify the correctness of the above control scheme, the following simulation works are done in the MATLAB environment and the experimental is performed based on the TMS320LF2407A DSP under the same input signals. Both of them have the same motor parameters. The system simulation diagram is shown as Fig. 2. And the parameters of the motor are: rated Voltage U=20V, rated speed n=3000rpm, phase resistor R=, phase reluctance L=, backemf ke= ?s/rad, viscous coefficient B=104 kg?m?s/rad, torque coefficient kt=?m/A and inertia J=104 kg?m?s2/rad. The controller parameters are selected as Kp =500, Kv =, T=. And all the initial load is set to 0 and change to ?m at in all the following experiments.Fig. 3 shows the results of BLDCM position servo control system when the input reference is set to the step signal rn (n)=20sgn(nT) ，where is 300rad/s and is 15000rad/s2.Fig. 4 is the related results of BLDCM position servo control system when the input reference is set to the ramp signal rn(n)=200nT, where is 204rad/s and is 5106 rad/s2.Fig. 5 shows the results of BLDCM positi