Design and Implementation of a Robust Controller for a Synchronous Reluctance Drive

Transcription

1 I. INTRODUCTION Design and Implementation of a Robust Controller for a Synchronous Reluctance Drive MING-TSAN LIN Tung Nan Institute of Technology Taiwan TIAN-HUA LIU, Senior Member, IEEE National Taiwan University of Science and Technology A robust controller design for a synchronous reluctance drive system is presented. Based on a simplified model of the system, a robust position controller has been derived. A digital signal processor (DSP), TMS--C, is used to implement the control algorithm. Furthermore, all the current, velocity, and position control loops are executed by the DSP. The system, as a result, is very flexible. Although the hardware circuit of the system is very simple, the synchronous reluctance drive system can accurately control a one-axis table. In addition, the system also has good transient response, load disturbance response, and tracking ability. Several experimental results validate the theoretical analysis. Manuscript received August 7, ; revised April 17, 1; released for publication July, 1. IEEE Log No. T-AES/7/4/1998. Refereeing of this contribution was handled by W. M. Polivka. This work was supported by the National Science Council, Republic of China, under Grant NSC 87-1-E and NSC 89-1-E Authors addresses: M.-T. Lin, Department of Electrical Engineering, Tung Nan Institute of Technology, Taiwan; T.-H. Liu, Department of Electrical Engineering, National Taiwan University of Science and Technology, 4 Keelung Rd., Section 4, Taipei, Taiwan 16, R.O.C /1/$17. c 1 IEEE The synchronous reluctance motor (SRM) has been recognized to have many advantages. For example, its structure is rugged and simple. In addition, its rotor does not have any winding or magnetic material. The SRM, therefore, is easy to manufacture. Prior to ten years ago, the SRM was regarded as inferior to other types of ac machines due to its lower average torque and larger torque pulsation. Recently, however, researchers have proposed many methods to improve the characteristics of the motor as well as the drive system [1, ]. The SRM has been shown to be suitable for ac drive systems for several reasons. First, it is not necessary to compute the slip of the SRM as it is with the induction motor. As a result, there is no parameter sensitivity problem. Next, it does not require any permanent magnetic material as the permanent magnet synchronous motor does. Many researchers have applied vector control to the SRMs. For example, Xiang, et al. discussed an incremental torque control method. The idea is good but requires complicated computations []. Vagati, et al. presented a high performance control of an SRM. The control method chooses a d-axis flux as a constant, and adjusts a q-axis current to control the torque of the motor [4]. Kang, et al. proposed a direct torque control of an SRM based on the stator vector [5]. These studies, however, focused on vector control but not controller design. To obtain a high performance drive system, the controller design is very important, however, there are only a few papers which have studied this topic. For example, Liu, et al. proposed a fuzzy sliding-mode controller design for an SRM drive [6]. With this system, however, it is necessary to estimate the acceleration of the motor. Moreover, only a speed-loop controller is designed to achieve both a fast transient response and a good load disturbance response. The fuzzy sliding-mode control algorithm, therefore, is very complicated and is difficult to implement. Lin, et al. proposed a forward-loop H 1 controller and a load compensator to improve the dynamic performance of an SRM drive. The forward-loop H 1 controller was designed by using the frequency domain analysis technique. In addition, this system required two separated controllers [7]. A state-space H 1 controller to improve the transient response and load disturbance response of a position drive system is proposed here. Only one controller is required. Moreover, the H 1 controller is designed by using the state-space analysis technique. The design procedures for the control algorithm are complicated, however, the implementation of the control law is very simple. In addition, the closed-loop SRM can drive a one-axis precision table with a satisfactory performance. For example, it has a fast response and good load disturbance rejection capability. The steady-state position error is 1 bit. In addition, the 144 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 7, NO. 4 OCTOBER 1

2 Fig. 1. Block diagram of drive system. position control system has good tracking ability. It can track a time-varying command well. To the best of the authors knowledge, this is a new control algorithm to drive an SRM. This is the first paper to propose applying the state-space H 1 controller to an SRM drive system. The experimental results show that the proposed drive system is successful and performs well in industrial position control. II. MATHEMATICAL MODEL A. System Description The block diagram of the SRM drive system is shown in Fig. 1. The system includes four major parts: the motor, the inverter, the sensors, and the digital signal processor (DSP). The DSP executes the control algorithms, the coordinate transformation, and outputs the three-phase switching patterns to the inverter. The inverter is a voltage-source current-regulated inverter. The inverter forces the three-phase real currents to follow their current commands. Then, the inverter drives the SRM. The encoder, which is mounted on the shaft of the motor, detects the motor position signal. Finally, the DSP reads the feedback position signal, and a closed-loop system is thus obtained. B. Motor and Load Dynamics The mathematical model of an SRM without damping winding can be described, in the synchronous d e -q e reference frame, by the following nonlinear differential equations: pi ds = µ v ds r s i ds + P! r L qs i qs µ pi qs = v qs r s i qs P! r L ds i ds =L ds (1) =L qs () where p is the differential operator d=dt, i ds and i qs are the d e -q e axis stator currents, v ds and v qs are the d e -q e axis stator voltages, r s is the stator resistance, P is the number of poles,! r is the mechanical speed of the rotor shaft, and L ds and L qs are the d e -q e axis stator inductances. The electromagnetic torque of the motor is T e = P (L ds L qs )i ds i qs () Fig.. Block diagram of controls system. where T e is the torque. The dynamic equations of the speed and position of the SRM are and p! r = 1 J (T e T l B! r ) (4) pµ r =! r (5) where J is the inertia constant of the motor and load, T l is the external load torque, B is the viscous frictional coefficient of the motor and load, and µ r is the shaft position of the motor. The main purpose of this research is to develop a high performance controller for an SRM drive. As a result, the position-loop controller design is focused on here. In order to reduce the required computing time of the DSP, a simple three-phase independent current-regulated control method is used here. The current control is executed by the DSP with a 5 ¹s sampling interval. Then, the three-phase currents i as, i bs, i cs can follow the three-phase current commands i as, i bs, i cs. After the coordinate transformation from the a-b-c stationery frame to the d e -q e synchronous frame, we can conclude that the d e -q e axis currents i ds and i qs follow the d e -q e axis current commands i ds and i qs very well. The torque command (ideal torque), therefore, can be expressed as T e = P (L ds L qs )i ds i qs = K t i qs (6) where T e is the torque command, i ds is the d-axis current command, Kt is the torque constant, and i qs is the q-axis current command. III. CONTROL ALGORITHM Fig. shows the block diagram of the SRM control system. The uncontrolled plant is a second-order system with a transfer function K t =((Js+ B)s). The system is a simplified system that is based on the current-regulated field-oriented control drive system. The purpose of this work is to design a state-space H 1 controller. The W 1, W,and LIN & LIU: DESIGN AND IMPLEMENTATION OF A ROBUST CONTROLLER 145

3 W are weighting functions. The x 1, x, x,andx 4 are state variables. The z 1, z,andz are weighted state variables. The r is the reference input command, the e is the position error, the d is the equivalent external load disturbance, the u is the control input, the y is the output of the system, and the K 1 is the feedback gains of the H 1 state-feedback controller. The H 1 state-feedback controller is designed as follows. First, with x 1 =! r, x = µ r,andx with x 4 as the internal variables, we can obtain the following dynamic equations: and _x 1 B _x J _X = _x 5 = a s 54 _x 4 1 a T K t K t J + r J + u d x 1 x x x = AX + B 1 w + B u (7) 6 Z = 4 z 1 z z b s = (b T a T ) 6 7 r d u 5u d x 1 x x x = C 1 X + D 11 w + D 1 u (8) e =[ 1 ] 6 4 x 1 x x x 4 r 7 +[1 ] 5 d = C X + D 1 w (9) B J A = a s 5 1 a T K t J B 1 = (1) (11) K t J B = (1) b s 6 7 C 1 = 4 5 (1) (b T a T ) 6 7 D 1 = 4d u 5 (14) r w = d (15) C =[ 1 ] (16) D 1 = [1 ] (17) where A, B 1, B, C 1, C, D 1,andD 1 are relative matrices or vectors of the dynamic equations, a s, b s, a T, b T, d u,and are the parameters of the weighted functions, X is the vector of the state variables, and w is the reference command and disturbance vector. A. Controller Design First, we can define the performance index as [8] [1] ³ = Z Tf (Z T Z w T w)dt < (18a) where ³ is the performance index, T f is the terminal time, Z is the vector of weighted state variables including z 1, z, z,and is a selected constant which is related to the norm of the system. The purpose of the proposed control algorithm is to find a control-input u to keep the ³ negative. Then, the closed-loop system can be a passive, stable system. From (18a), it is easy to understand that the weighted states are bounded if the input is bounded and the norm of the closed-loop system is limited. The calculus of variations is applied here to obtain an H 1 robust controller. Here, the drive system has a fixed starting time but an unspecified termination time. Moreover, the dynamic equation of the plant is shown as (7). The controller is designed as follows. First, from (7) and (18a), we can combine the terminal equation and equality constraints via Lagrange multipliers to obtain the cost function: ³ = T f (AX T + B 1 w + B u) + Z Tf f(z T Z w T w)+ T[(AX + B 1 w + B u) _ X]gdt < where f and are Lagrange multipliers for the terminal condition and the equality constraint (18b) 146 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 7, NO. 4 OCTOBER 1

4 separately, and T is the transposition for a matrix or vector. Then, from (8) and (18b), we can define the Hamiltonian function as H =(C 1 X + D 1 u) T (C 1 X + D 1 u) w T w + T(AX + B 1 w + B u) (19) where H is the Hamiltonian function. The design of the H 1 controller considers the suitable w and u to achieve min u max w ³<. The u is called a minimizing player, which functions to minimize the performance index ³. On the other hand, the w is called a maximizing player, and functions to maximize the performance index ³. As a result, this control algorithm is based on the two-player game theory [11, 1]. From (18b), we can derive the first variation of the performance index and express it as the following equation [1]: ±³(X,u,w, )= Z Tf ±X + _ Hj t=tf ±t + ±X T T f (A T f (T f )) () where ± is the operator of the first variation, and X Tf is the state at final time. Moreover, the bounded conditions at the initial time and final time include and X() = X (1) (T f )=A T f () _Xj t=tf =(AX + B 1 w + B u)j t=tf = () H(X,u,w, )j t=tf =: (4) The optimal condition occurs when the first variation of the performance is zero. As a result, by letting ±³ = and substituting (1) (4) into (), we can easily derive the three necessary equations determining the optimal control and state + _ =: (7) In order to solve the optimal condition, first, substituting (19) into (5), one can obtain the optimal control input u and express it as u = [(D T 1 D 1 ) 1 D T 1 C 1 X + 1 (DT 1 D 1 ) 1 B T ]: (8) Second, substituting (19) into (6), one can derive the maximum value of w and express it as w = 1 BT 1 (9) and then substituting (8) and (9) into (19), we can derive the optimal value of the Hamiltonian matrix as H(X,u,w, )=X T [C T 1 C 1 CT 1 D 1 (DT 1 D 1 ) 1 D T 1 C 1 ]X T 4 B 1 BT B (DT 1 D 1 ) 1 B T + T[A 1 B (DT 1 D 1 ) 1 D T 1 C 1 ]X X T [ 1 CT 1 D 1 (DT 1 D 1 ) 1 B T ] : () Third, substituting (19) into (7), one can easily obtain = [C1 T C 1 CT 1 D 1 (DT 1 D 1 ) 1 D1 T C 1 ]X [A B (D T 1 D 1 ) 1 D T 1 C 1 ]T : (1) Combining (8), (9), and (7), one can obtain _X =(A B (D1 T D 1 ) 1 D1 T C 1 )X + 1 B 1 BT 1 1 B (DT 1 D 1 ) 1 B T =(A B (D1 T D 1 ) 1 D1 T C 1 )X + 1 µ 1 B 1 BT 1 B (DT 1 D 1 ) 1 B T : () Finally, letting is equal to Px and assuming P is a constant matrix, one can derive _ =P X _ () where P is a square matrix. Combining (1), (), and (), we can derive [14] P(A B (D T 1 D 1 ) 1 D T 1 C 1 )+(A B (DT 1 D 1 ) 1 D T 1 C 1 )T P µ 1 + P B 1 BT 1 B (DT 1 D 1 ) 1 B T P +(C T 1 C 1 CT 1 D 1 (DT 1 D 1 ) 1 D T 1 C 1 )=: (4) The system is a time-invariant system. In addition, in this work, we select T f as the final time at which the SRM drive system reaches steady-state condition. The matrix P can be expressed as the following Riccati [15] A B P =Ric(M)=Ric4 (D1 T D 1 ) 1 D1 T C 1 1 B 1 BT 1 B (DT 1 D 1 ) 1 B T 5 (5) C1 T(I DT 1 (DT 1 D 1 ) 1 D1 T )C 1 (A B (D1 T D 1 ) 1 D1 T C 1 )T LIN & LIU: DESIGN AND IMPLEMENTATION OF A ROBUST CONTROLLER 147

5 where Ric expresses the Riccati equations and M is an extended matrix. Substituting =PX into (8), we can obtain the optimal control u in another form [15] u = [(D1 T D 1 ) 1 D1 T C 1 +(DT 1 D 1 ) 1 B T P]X = k 1 X (6) where u is the optimal control input, and k 1 is the H 1 feedback gain. The k 1 is a vector which includes four gains k 11, k 1, k 1,andk 14 to feedback x 1, x, x,andx 4.Thex 1, x, x,andx 4 are the state variables of the proposed system. In Fig., the state variable x is the shaft position of the motor and it can be obtained from the encoder. The state variable x 1 is the velocity of the motor. As a result, it can be obtained by computing x = t. By using bilinear transformation, the state variable x is obtained from the position error signal e, and the state variable x 4 is obtained from the position of the motor. According to Fig., the transfer matrix between the output x and the input r, and the output x and the input d can be derived as T(s)=[ x r = x d ] =( C (si (A + B K 1 )) 1 B 1 ) Kt J K 1 (s + a T ) s 4 + s + s + 1 s + Kt J (s + a s )(s + a T ) s 4 + s + s + 1 s + (7) = B J + a s + a T K t J K 11 (8) = B J (a s + a T )+a s a T K t J ((a s + a T )K 11 + K 1 ) (9) 1 = B J a s a T + K t J (K 1 K 14 (a s + a T )K 1 a s a T K 11 ) (4) = K t J (a T K 1 a s K 14 a s a T K 1 ): (41) Similarly, the sensitivity matrix is derived as Fig.. Proposed closed-loop drive system. of the sensitivity function determines the steady-state error of the closed-loop system. In addition, the maximum gain of the sensitivity function determines the overshoot of the closed-loop system. B. Selection of the Weighting Functions For a control system, if we are given the time domain specification, then, we can determine the corresponding requirements of the specification in frequency domain. In this work, a standard second-order system is chosen as the approximate corresponding frequency domain, and can be described as T(s)= s +»! n +! n (4) where» is the damping ratio of the closed-loop system, and! n is the natural frequency. It is well known that the rise time, settling time, and percent overshoot are related to the damping ratio and natural frequency. As a result, by suitably selecting the damping ratio and the natural frequency of the transfer function, a satisfactory performance can be obtained. Moreover, the sensitivity function S(s) can be described as S(s)=1 T(s)! n = s +»! n s s +»! n s +! n (44) h e S(s)= r e i d =(I + C (si (A + B K 1 )) 1 B 1 ) " = 1 K t J K 1 (s + a T ) s 4 + s + s + 1 s + 1+ Kt # J (s + a s )(s + a T ) : (4) s 4 + s + s + 1 s + The transfer matrix describes the input-output relationship of a closed-loop system. For example, the bandwidth, the resonant frequency, and the resonant peak determine the performance of the closed-loop system. On the other hand, the sensitivity matrix describes the relationship between the error signal and the input of a closed-loop system. The dc gain The next step is to select the weighting functions. The weighting functions are related to the transfer functions R(s) andt(s), and the sensitivity function S(s). In addition, selection of the weighting functions depends on the designer s experience. This selection, however, is not obtained by trial and error. Several guidelines can be used here [15]. 148 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 7, NO. 4 OCTOBER 1

6 Fig. 4. Weighting functions. The selection of the weighting functions W 1 (s), W (s), and W (s) must satisfy the following three equations: ¾(s(j!)) jw 1 1 (j!)j (45) ¾(R(j!)) jw 1 (j!)j (46) ¾(T(j!)) jw 1 (j!)j (47) where ¾ is the upper boundary of the sensitivity function or transfer functions, and R(s) isthetransfer function between the reference input and control u. However, from (4) (44), it is easy to understand that the sensitivity function S(s) is a high pass function and the transfer function T(s) is a low pass function. Then, in order to satisfy the inequality equations (45) and (47), the W 1 (s) should be a low pass function; on the other hand, the W (s) should be a high pass function. A very important point in choosing the W 1 (s) 1 and W (s) 1 is that the db crossover frequency of W 1 (s) 1 must be sufficiently below the db crossover frequency of W 1 (s). Otherwise, the performance requirements of (45) and (47) cannot be achieved. The W 1 (s) is suitably chosen to avoid the control input of the drive system becoming over-saturated. From (45) (47), we can obtain the following equations W 1 (s)s(s) W (s)r(s) 1 (48) W (s)t(s) where k k is the H 1 norm. The proportional-integral (PI) controller with the external load estimator has been proposed for a speed drive system. This system, however, still requires tuning of the bandwidth of the external load estimator. In addition, this system is suitable for a speed control system but not a position control system. The major reason is that a low pass filter is required to remove the high frequency noise of the external load estimator. As a result, a speed loop has a time lag. This time lag negatively affects the performance of the position response [16]. We, therefore, propose the state space H 1 approach in this work. C. Determination of the According to the Bounded Real Lemma, the system can be stable if and only if the following two equations are satisfied [17]: Re(Eig M)) 6= (49) and P> (5) where Re(EigM) is the real part eigenvalues of its relative matrix M. ThematrixM isshownin(5). Equation (49) implies that no eigenvalues of matrix M exist in the imaginary axis. In addition, (5) describes that the matrix P must be a positive definitive matrix. By substituting (1) (17) into (49), we can obtain the following equations: B=J (51) a s (5) a T (5) a s + a T (b s = ) +(B=J) > (54) a s (a T +(B=J) ) (b s B=( J)) + a T ((B=J) (b s = ) ) ( Kt =( J)) +( Kt =(d u J)) > (55) (1= 4d u )( a ( a T s b s )d u (B=J) +( a s +( )b s + b T ) ( d u )(K t =J) > (56) LIN & LIU: DESIGN AND IMPLEMENTATION OF A ROBUST CONTROLLER 149

7 Fig. 5. Position control (5 mm) with moving table. (a) Position. (b) Speed. (c) Current. (1= 4d u )( a T b s + ( a s b s )b T ) ( d u )(K t =J) > : (57) From (51) (57), we can find the minimum value of the parameter. Then, we substitute the into (5) to obtain the matrix P. IfP is not a positive definite matrix,weneedtoincrease and then substitute it into (5) to obtain a new P. The recursive procedure is continuous until the P is a positive definite matrix. As a result, here we use an effective way to obtain the. IV. EXPERIMENTAL RESULTS A. Implementation The block diagram of the implemented drive system is shown in Fig. 1. The system consists of 15 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 7, NO. 4 OCTOBER 1

8 TABLE I Parameters of Motor r s L ds.18 H L qs.57 H J.94 N m/s (with one-axis moving table) J L.51 N m/s (with one-axis table and external inertia) Kt :4848 N m/a B.1 N m/s four major parts: a DSP system, an inverter, an SRM, and some sensors. A TMS--C DSP system is used here. The DSP system executes the position control algorithm, the d-q axis to a-b-c axis coordinate transformation, and the current-regulated pulsewidth modulation (PWM) switching algorithm. Finally, the DSP determines the firing signals and sends them to the inverter. A closed-loop system is thus achieved. The three-phase current deviations of the motor are small because the DSP can complete execution of the current-regulated algorithm in only 5 ¹s. In addition, the DSP completes execution of the position-loop control in 1 ms. Most of the jobs are implemented by software, the hardware circuit, therefore, is very simple. The inverter is realized by using six insulated gate bipolar transistors (IGBTs). The IGBTs have the following specifications: 5 A continuous rating current, 1 A maximum peak current, and 1 V rating voltage. The motor is a -phase, 4 pole, V,.75 HP SRM, with 18 r/min rated speed, manufactured by the Reliance Electric Company. The parameters of the motor are shown in Table I. The d-axis current command is A. The sensor of the system includes two parts: the Hall-effect current sensors and the encoder. The Hall-effect sensors are type LP-1P, made by LEM Company. The bandwidth of the Hall-effect sensors is about 1 khz. Only two current sensors are required because the drive system is a three-phase balanced system. The DSP reads the output signals of the Hall-effect sensors via analog-to-digital (A/D) converters. The A/D converters are 1 bit, with a ¹s conversiontime. An absolute encoder is mounted on the shaft of the motor. The encoder originally outputs position pulses with one index pulse for every mechanical revolution. However, it can output 8 pulses with an index pulse by using a multiplier circuit. A one-axis precision table is used to evaluate the performance of the SRM drive. In addition, the inertia of the system is varied by coupling different external mechanisms. B. Experimental Results Some experimental results are shown here. In this paper, by selecting the transfer function as 194=(s +5:4s + 194), we can obtain the Fig. 6. Position response at Nt m load. weighting as W 1 (s)=b s =(s + a s )=1=(s +:1), W (s)=d u =:5, and W (s)= (s + b T )=(s + a T )= 9(s + 1)=(s + 1). Then, we can obtain =:98 and the feedback gain as K 1 =[ :15 :668 1:69 :975]. Based on this controller, several measured results are shown here. Fig. 4 shows the relationships between the S(s), W 1 1 (s), T(s), and W 1 (s). The amplitude of the S(s) isbelow W 1 1 (s). Similarly, the amplitude of the T(s) isbelow W 1 (s). Fig. 5 shows the position control response for the one-axis table with short distance moving. In the real system, the measured responses are slower than the ideal responses because the real system has a 15 A current limit. Fig. 5(a) is the position response. The proposed H 1 controller has a faster response than the PI controller does. Fig. 5(b) is the relative velocity response. Fig. 5(c) is its relative q-axis current. Fig. 6 shows the position response of load disturbance rejection. An external N.m load is added here. Again, the proposed H 1 controller performs better. Fig. 7(a), (b), (c) show the position response, velocity response, and q-axis current response of position control for the one-axis table linked with an external inertia. Although the total inertia is increased more than twice, the changes in the responses are not obvious. The reason is that the proposed system is very robust. Fig. 8 shows the line voltage of the motor. It is a PWM waveform. Fig. 9 shows the responses of the table with long distance moving. Fig. 9(a) is the position response. The position is linearly increased because the motor is operated at a maximum velocity. Fig. 9(b) is the relative velocity response. The velocity of the motor is near 11 r/min. Fig. 9(c) is the q-axis current. The peak current is about 15 A, which occurs when the motor accelerates and decelerates. Fig. 1 shows the tracking responses. Fig. 1(a) is the position response. Fig. 1(b) is its relative velocity response. Fig. 1(c) is its q-axis current. LIN & LIU: DESIGN AND IMPLEMENTATION OF A ROBUST CONTROLLER 151

9 Fig. 7. Position control (5 mm) with moving table and external inertia. (a) Position. (b) Speed. (c) Current. In the proposed system, there are several parameters, for example, a T, b T,, d u, a s,andb s. However, the most important parameters, which obviously affect the performance of the drive system, are b s and a s. In order to analyze the system easily, certain parameters are selected as constant values. The fixed parameters are shown as follows: a T = 1, b T = 1, = 9, and d u =:5. Fig. 11 shows the 15 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 7, NO. 4 OCTOBER 1

10 the system of different b s.forasmallb s, the inertia varies within a wide range. On the other hand, for alargeb s, the allowed inertia variation is obvious reduced. V. CONCLUSIONS Fig. 8. PWM voltage of SRM. typical frequency response of the sensitivity function. In this figure, the parameter a s is.1, b s is 1, and is.995. It includes a dc gain and a maximum gain which is the peak gain of the frequency response. Fig. 1 shows the variations of the dc gain of the sensitivity function. By increasing b s and decreasing a s, the dc gain of the sensitivity function decreases. As a result, the steady-state error of the system can be reduced. Fig. 1 shows the variations of the gains of the sensitivity function while b s is adjusted. The parameter a s is set as.1. From this figure, when the parameter b s increases, the dc gain reduces. Therefore, the steady-state error of the drive system is reduced as well. On the other hand, when the parameter b s is increased, the maximum gain of the drive system is increased. The overshoot of the system, therefore, is increased too. Fig. 14 shows the gains of e(s)=d(s),whicharethedcgainand the maximum gain of the position error and the external load. The parameter a s is set as.1. When the parameter b s is increased, the dc gain and the maximum gain between the position error and the external load are reduced. As a result, the load disturbance rejection capability of the drive system is obviously improved. Fig. 15 shows the relationship between the H 1 norm and the parameters b s and b T. The parameter a s is.1 and the parameter is 9. If the b s increases, the parameter is increased as well. However, by suitably adjusting the parameter b T, the parameter can be reduced. The parameter a T was selected as b T here. The H 1 norm is an important factor for the drive system. Generally speaking, the parameter is selected below 1. to obtain satisfactory performance of the drive system. Fig. 16 shows the variations of the and the performance when J and B are changed. The inertia changes from.1 time to 1 times. In addition, the viscous coefficient also varies from.1 time to 1 times. Fig. 17 shows the root loci of This paper proposes an H 1 controller design for a fully digital synchronous reluctance drive. The parameters of the controller can be obtained after suitably selecting the weighting parameters of the drive system. Selection of the weighting parameters requires some guidelines and experience. The basic guidelines for selecting the weightings are discussed. The controller is easily implemented by using a DSP. The experimental results show that the whole drive system has good transient response, load disturbance response, and tracking ability. In addition, the proposed drive system can control a one-axis table with satisfactory performance. This paper proposes a new direction in the design and implementation of a new position control algorithm for the synchronous reluctance drive. REFERENCES [1] Xu, L., Xu, X., Lipo, T. A., and Novotny, D. W. (1991) Vector control of a synchronous reluctance motor including saturation and iron loss. IEEE Transactions on Industry Applications, 7, 5 (Sept./Oct. 1991), [] Lagerquist, R., Boldea, I., and Miller, T. J. E. (1994) Sensorless control of the synchronous reluctance motor. IEEE Transactions on Industry Applications,, (May/June 1994), [] Xiang, Y. Q., and Nasar, S. A. (1997) A fully digital control strategy for synchronous reluctance motor servo drives. IEEE Transactions on Industry Applications,, (May/June 1997), [4] Vagati, A., Pastorelli, M., and Franceschini, G. (1997) High-performance control of synchronous reluctance motors. IEEE Transactions on Industry. Applications,, 4 (July/Aug. 1997), [5] Kang, S. J., and. Sul, S. K. (1998) Highly dynamic torque control of synchronous reluctance motor. IEEE Transactions on Power Electronics, 1, 4(July 1998), [6] Liu, T. H., and Lin, M. T. (1996) A fuzzy sliding-mode controller design for a synchronous reluctance motor drive. IEEE Transactions on Aerospace and Electronic Systems,, (July 1996), [7] Lin, M. T., and Liu, T. H. (1998) Design and implementation for a digital synchronous reluctance drive. IEEE Transactions on Aerospace and Electronic Systems, 4, 4 (Oct. 1998), [8] Zhou, K., Doyle, J. C., and Glover, K. (1996) Robust and Optimal Control. Englewood Cliffs, NJ: Prentice-Hall, LIN & LIU: DESIGN AND IMPLEMENTATION OF A ROBUST CONTROLLER 15

11 Fig. 9. Position control with external inertia (4 mm). (a) Position. (b) Speed. (c) Current. 154 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 7, NO. 4 OCTOBER 1

12 Fig. 1. Position tracking responses. (a) Position. (b) Speed. (c) Current. LIN & LIU: DESIGN AND IMPLEMENTATION OF A ROBUST CONTROLLER 155

13 Fig. 14. Gains between position error and external load. Fig. 11. Sensitivity function of drive system. Fig. 1. DC gains between position error and position command. Fig. 1. Variations in gains between position error and position command. 156 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 7, NO. 4 OCTOBER 1

14 Fig. 15. Variation in. [9] Ball, J. A., Kachroo, P., and Krener, A. J. (1999) H 1 tracking control for a class of nonlinear systems. IEEE Transactions on Automatic Control, 44, 6(June 1999), [1] Lin, M. T., and Liu, T. H. (1999) Robust controller design for a synchronous reluctance drive. In Proceedings of IEEE Power Electronics Specialists Conference, Charleston, SC, June 7 July 1, 1999, [11] Basar, T., and Bernhard, P. (1995) H 1 -Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach. Boston: Birkhauser, [1] Rhee, I., and Speyer, J. L. (1991) A game theoretic approach to a finite-time disturbance attenuation problem. IEEE Transactions on Automatic Control, 6, 9(Sept. 1991), [1] Sage, A. P., and White, C. C. (1977) Optimum Systems Control. Englewood Cliffs, NJ: Prentice-Hall, Fig. 16. Variations in and overshoot when J and B are changed. (a). (b) Overshoot. Fig. 17. Root loci. LIN & LIU: DESIGN AND IMPLEMENTATION OF A ROBUST CONTROLLER 157

15 [14] Iwasaki, T., and Skelton, R. E. (1994) All controllers for the general H 1 control problem: LMI existence conditions and state space formulas. Automatica,, 8 (1994), [15] Doyle, J., and Zhou, K. (1999) Essentials of Robust Control. Englewood Cliffs, NJ: Prentice-Hall, [16] Umeno, T., and Hori, Y. (1991) Robust speed control of dc servomotors using modern two degrees-of-freedom controller design. IEEE Transactions on Industrial Electronics, 8, 5(Oct. 1991), [17] Zhou, K., Doyle, J., and Glover, K. (1996) Robust and Optimal Control. Englewood Cliffs, NJ: Prentice-Hall, Ming-Tsan Lin was born in Keeling, Taiwan, Republic of China, on February 7, He received the B.S., M.S., and Ph.D. degrees from National Taiwan University of Science and Technology, Taipei, Taiwan, Republic of China, in 199, 1994, and 1999, respectively, all in electrical engineering. Since 1999, he has been an Assistant Professor in Department of Electrical Engineering, Tung Nan Institute of Technology. His research interests include motor controls, application of control theory, and implementation of microprocessor-based systems. Tian-Hua Liu (S 85 M 89 SM 99) was born in Tao Yuan, Taiwan, Republic of China, on November 6, 195. He received the B. S., M. S., and Ph. D. degrees from National Taiwan University of Science and Technology, Taipei, Taiwan, in 198, 198, and 1989, respectively, all in electrical engineering. From August 1984 to July 1989, he was an instructor in the Department of Electrical Engineering, National Taiwan University of Science and Technology. He was a Visiting Scholar in the Wisconsin Electric Machines and Power Electronics Consortium (WEMPEC) at the University of Wisconsin, Madison, from September 199 to August 1991, and in the Center of Power Electronics Systems (CPES) at Virginia Tech from July 1999 to January. From August 1989 to January 1996, he was an Associate Professor in the Department of Electrical Engineering, National Taiwan University of Science and Technology, where since February 1996, he has been a professor. His research interests include motor controls, power electronics, and microprocessor-based control systems. 158 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 7, NO. 4 OCTOBER 1