Nonlinear control of a synchronous reluctance drive
system with reduced switching frequency
T.-H. Liu, M.-T. Lin and Y.-C. Yang
Abstract: A novel PWM switching method and control algorithm for synchronous reluctance drive systems is proposed. First, in different switching modes, the current slope of the stator current can be systematically derived. The current slope is related to the dc-bus voltage, operating conditions, and parameters of the motor. Then, by computing the derivation of the current slope command and the real current slope, an optimum switching state can be determined and used to trigger the inverter. After that, the nonlinear controllers for an adjustable speed drive system and a position control system are proposed. Using the proposed method, the synchronous reluctance drive system performs very well. It has smaller current harmonics, lower switching frequency, and less switching loss when compared to the hysteretic or bang-bang current control. In addition, a fast transient response, good load disturbance rejection ability, and good tracking performance of speed control and position control can be achieved. No extra hardware is required. Several experimental results validate the theoretical analysis.
1 Introduction
The synchronous reluctance motor (SynRM) has been recognised to have many advantages due to its simple and rugged structure. In addition, the SynRM has no winding or magnetic material on its rotor. The SynRM is shown to be very suitable for ac drive systems due to many factors. First, the field-oriented control of the SynRM does not require computing slip frequency as required for the induction motor. As a result, there is no parameter sensi-tivity problem[1]. Next, the SynRM does not require any permanent magnetic material as the permanent-magnet synchronous motor does[2]. Moreover, the SynRM is more suitable for a sensorless drive due to the obvious difference in d-axis inductance and q-axis inductance. As a result, it is possible to use the inductances to estimate the shaft position of the SynRM[3–6].
To achieve a high performance drive system, the motor design, PWM switching strategy, and controller design are all important. Several researchers have studied and proposed many different techniques in these fields. In PWM techniques, the most popular method is the hysteretic or bang-bang current regulated control. This method is very simple; however, it provides a highly variable PWM switching frequency. In addition, its current ripple and harmonic content are large[7]. To solve this problem, the space vector modulation is proposed. The performance of the space vector modulation is excellent; nevertheless, this method requires a lot of complicated computations. As a
result, a digital signal processor for space vector modulation is required [8–9]. Research on PWM strategies in induc-tion motors or permanent magnet synchronous motors is popular; however, research on PWM strategies for synchro-nous reluctance motors is not common. This has motivated us to study the PWM strategy for SynRM drives. On the other hand, in the fields of motor design, motor drive, and controller design, several techniques have been proposed for SynRM drives. For example, Platt designed a new reluctance motor with strong rotor anisotropy[10]. Vagati et al. proposed a flux-observer-based control scheme to achieve a high performance SynRM drive system [11]. Sul et al. developed a high dynamic torque control for a SynRM [12]. Liu et al. implemented an HN
to improve the dynamic response of a position control system for a SynRM [13]. These papers [11–13], however, only focus on linear controller design. To improve the performance of the SynRM drive system, nonlinear controllers are more effective and have been proposed. For example, Shyu et al. implemented a combination of the classical state feedback and the variable structure control [14]. The idea is good: however, a chattering problem appears in steady-state. Sul et al. proposed a nonlinear input-output linearisation technique for a SynRM. The results achieved optimal efficiency. However, the computation of the control algorithm is very complicated[15].
In this paper, a novel PWM scheme for the SynRM is proposed to improve the torque response and reduce the switching frequency. The method is based on the measure-ment of the current slope. Although the computation is simple, the performance of the proposed method is satisfactory. Experimental results show that this method has a lower switching frequency, lower harmonics, and better current tracking ability. In addition, in order to improve the transient response, load disturbance rejection and tracking ability, a nonlinear controller is proposed here. To the best of the authors’ knowledge, this is the first time that the novel switching method and the nonlinear controller for a SynRM drive system have been proposed. The details follow.
T.-H. Liu and Y.-C. Yang are with Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan 106, Republic of China
M.-T. Lin is with Department of Electrical Engineering, Tung-Nan Institute of Technology, Taipei, Taiwan 222, Republic of China
E-mail: [email protected]
rIEE, 2006
IEE Proceedings online no. 20050124 doi:10.1049/ip-epa:20050124
2 Switching strategy
2.1
Dynamic model of a SynRM
The mathematical model of a SynRM in d-q synchronous frame can be expressed as
did dt ¼ ðoeLqiq rsidþ vdÞ=Ld ð1Þ and diq dt ¼ ðoeLdid rsiqþ vqÞ=Lq ð2Þ where d
dtis the differential operator, idis the d-axis current,
iqis the q-axis current, oeis the electrical speed, Lqis the q-axis inductance, Ld is the d-axis inductance, rs is the stator resistance, vdis the d-axis voltage and vqis the q-axis voltage. The electro-magnetic torque expressed in the d-q synchronous frame is Te ¼ 3 2 P0 2 ðLd LqÞidiq ð3Þ
where Teis the electro-magnetic torque of the motor, P0is the number of poles of the motor, idis the d-axis equivalent current and iq is the q-axis equivalent current. The rotor speed and position of the motor can be expressed as
d dtor ¼ 1 JðTe Tl BorÞ ð4Þ and d dtyr ¼ or ð5Þ
where J is the inertia constant of the motor and load, Tl is the external load torque, B is the viscous frictional coefficient of the motor and load, yris the mechanical rotor position and oris the mechanical rotor speed. The electrical rotor speed and position are
oe ¼ P0 2 or ð6Þ and ye ¼ P0 2 yr ð7Þ
where oeis the electrical rotor speed and yeis the electrical rotor position of the motor.
2.2
Proposed switching strategy
In this paper, a new switching strategy for the SynRM is proposed. The details are discussed here. If the inverter is switched in mode A+, which is shown in Fig. 1, the switching state is (1, 0, 0).
From Fig. 1, we can easily obtain van ¼ 2 3Vdc ð8aÞ vbn ¼ 1 3Vdc ð8bÞ and vcn ¼ 1 3Vdc ð8cÞ
The relationship of the voltages between the a-b-c axis to d-q axis is vd vq ¼ sin ye sinðye 2p 3Þ sinðyeþ 2p 3Þ
cos ye cosðye2p3Þ cosðyeþ2p3Þ
" # van vbn vcn 2 6 4 3 7 5 ð9Þ
By substituting (8a), (8b) and (8c) into (9), one can obtain vd ¼ 2 3Vdcsin ye ð10aÞ and vq ¼ 2 3Vdccos ye ð10bÞ
After that, by substituting (10a) and (10b) into (1) and (2), we can derive that the dynamic equations of the d-q axis currents can be expressed as:
did dt mode Aþ ¼ oeLqiq rsid Ld þ2 3 Vdcsin ye Ld ð11Þ and diq dt mode Aþ ¼ oeLdid rsiq Lq þ2 3 Vdcsin ye Lq ð12Þ When the inverter is switched in a zero-voltage mode ( free-wheeling mode), the switching state is (1, 1, 1) or (0, 0, 0). Then, the motor is three-phase shorted and is disconnected with the input voltage. The input voltage, therefore, is equal to zero. By substituting Vdc¼ 0 into (11) and (12), one can obtain did dt mode 0¼ oeLqiq rsid Ld ð13Þ diq dt mode 0 ¼ oeLdid rsiq Lq ð14Þ Combining (11)–(14), one can easily obtain
did dt mode Aþ ¼ did dt mode 0þ 2 3 Vdcsin ye Ld ð15Þ diq dt mode Aþ ¼ diq dt mode 0 þ2 3 Vdccos ye Lq ð16Þ By using the same method, one can derive that the switching modes B+ and C+ have the following dynamic equations: did dt mode Bþ ¼ did dt mode 0 þ2 3 Vdcsinðye2p3Þ Ld ð17Þ fb fc fd V3 V2 V1 V4 V5 V6 fq fa ∆idLd ∆iqLq e 0°
Fig. 1 The circuits of different modes
a Mode A+
b Mode B+
diq dt mode Bþ ¼ diq dt mode 0 þ2 3 Vdccosðye2p3Þ Lq ð18Þ and did dt mode Cþ ¼ did dt mode 0 þ2 3 Vdcsinðye2p3Þ Ld ð19Þ diq dt mode Cþ ¼ diq dt mode 0 þ2 3 Vdccosðye2p3Þ Lq ð20Þ
To realise the proposed method, first we measure the current slope in the free-wheeling switching state. Then, we compute the desired d-axis current slope between the current command i
d and the real current id, and the desired q-axis current slope between the current command i
q and
the real current iq, which are expressed as: did dt ¼ i d id T ð21Þ and diq qt ¼ i q iq T ð22Þ where (did/dt) *
is the desired current slope, and T is the switching interval of the inverter. After that, we can compute the current deviation between the desired current slope and the current slope at mode 0:
Did ¼ did dt did dt mode 0 T ð23Þ and Diq ¼ diq dt diq dt mode 0 T ð24Þ
The required voltage vector angle is g ¼ tan1 LdDid LqDiq
þ ye ð25Þ
Finally, the voltage vector of the inverter can be selected according to the vector angle g. In Fig. 1, for example, the vector V1is selected when g is between 301 and 301. In the real world, if the required voltage amplitude is small, we choose a free-wheeling voltage vector to reduce the current harmonics.
3 Nonlinear controller design
Many papers have proposed different control methods to improve the dynamic performance of synchronous reluc-tance drive systems. For example, Thanaa et al. proposed a sliding mode controller for position control of a vector-controlled synchronous reluctance drive system. The results are satisfactory; however, the control method requires knowing the parameter variation bounds of the motor. In addition, a boundary layer is required to alleviate the chattering problem [16]. Lee et al. used the feedback
linearisation technique to obtain an efficiency-optimised direct torque control. The idea is of great interest; however, the controller is very complicated and requires a lot of computations[15]. Liu et al. proposed an HN
controller to improve the transient response and load disturbance response of a position control system for a synchronous reluctance motor. The experimental results are satisfactory; however, the controller is complicated [17]. Chiang et al. proposed an integral variable structure controller with grey prediction for a synchronous reluctance drive system. The method can significantly reduce chattering and steady-state error; however, the control method requires a lot of computing time using a digital signal processor [18]. Recently, Shyu et al. proposed a state feedback controller combined with a variable structure controller for position control of a synchronous reluctance motor. The idea is new and interesting; however, a chattering problem exists due to the high frequency switching of the variable structure controller[19].
In order to reduce the complexity of the controllers mentioned above [15–19], this paper proposes a simple nonlinear controller for a SynRM drive system. Only the error signal is used here. By combining the integral and proportional items of the error signal and its nonlinear operation, a control input can be obtained. A systematic stability analysis of the control system is discussed. The proposed method can be applied for a speed control system and a position control system. To the authors’ best knowledge, this is the first time that the proposed method has been applied to a SynRM drive system. The details are shown as follows.
3.1
Speed-loop controller
The block diagram of the closed-loop speed control system is shown in Fig. 2a. To obtain a nonlinear controller, the control input of the drive system is selected as
u ¼ Ge
@V @eþ Gi
Z @V
@edt ð26Þ
where u is the control input, Geis the proportional gain, Gi is the integral gain, V is the Lyapunov function, e is the error between the command and output, d is the differential operator and t is time. In this paper, the Lyapunov function is selected as[20]: VðeÞ ¼ 3 4Kp0e 4 3þ1 2Kp1e 2þ3 4Ki0e 4 3þ1 2Ki1e 2 ð27Þ
The general results in stability theory were developed by Lyapunov. As we know, Lyapunov formulated a basic concept of stability and derived sufficient stability condi-tions for dynamic systems, described by ordinary differ-ential equations. In recent years, there has been considerable research in developing innovative non-quadratic forms of Lyapunov candidate. The application of efficient and flexible software, such as MATLAB, has changed the aim of the Lyapunov theory. Owing to the improvement of the computing ability for software, the value of Lyapunov’s theory is shifted from the descriptive theory to the design of stabilising controllers. The Lyapunov concept is a viable tool in solution of stabilisation and optimisation problems, encountered in the wide areas of control [20]. The basic motivation of this paper is to apply the Lyapunov concept to design nonlinear controllers for a synchronous reluctance drive system. As a result, the fractional powers of the error signal are selected. It is demonstrated that the application of the complicated Lyapunov candidates is an important step. This allows one to design a high performance nonlinear controller. In this paper, the Lyapunov concept is used to
design both a speed controller and a position controller for the synchronous drive systems.
By substituting (27) into (26), one can obtain u ¼ iq ¼ GeðKp0þ Ki0Þe 1 3þ GeðKp 1þ Ki1Þe þ GiðKp0þ Ki0Þ Z e13dtþ GiðKp1þ Ki1Þ Z e dt ð28Þ In order to simply the expression of (28), we define
Kpn ¼ GeðKp0þ Ki0Þ ð29Þ
Kpe ¼ GeðKp1þ Ki1Þ ð30Þ
Kin ¼ GiðKp0þ Ki0Þ ð31Þ
Kie ¼ GiðKp1þ Ki1Þ ð32Þ
Then, the control input of the drive system can be expressed as u ¼ Kpme 1 3þ Kpeeþ Kin Z e13dtþ Kie Z e dt ð33Þ
We can observe that the first and third terms are increased as compared with the traditional PI controller. In fact, the first and third terms are used to provide an extra effort to improve the dynamics of the system including transient response and load disturbance response.
3.2
Position-loop controller
The block diagram of the closed-loop position control system is shown in Fig. 2b. From Fig. 2b, we can obtain
iq ¼ Kpðor KvorÞ ð34Þ
By substituting (34) into (3) and (4), we can derive or ¼ Kt0 J Kpðo r KvorÞ TL J ð35Þ Js + B 1 r ∆r Te TL r ∗r r r s 1 b + equation (40) Kp Kv
SynRM simplified model
− − + − + ∗r (Ld − Lq) P0 2 2 3 Js + B 1 (Ld − Lq) P0 2 2 3 ∗r r ∆r i∗ Te TL r a equation (26)
SynRM simplified model
+ − + − q id∗ iq∗ id∗
Fig. 2 The block diagram of the proposed system a Speed control system
b Position control system
0 0.03 0.06 0.09 0.12 0.15 −200 −150 −100 −50 0 50 100 150 200 s vab ,V a 0 0.03 0.06 0.09 0.12 0.15 −200 −150 −100 −50 0 50 100 150 200 s vab ,V b
Fig. 3 The voltage vabat a steady-state a Proposed
Define the position error as
ey ¼ yr yr ð36Þ
By taking the derivative of (36), we can obtain e y ¼ y r y r ð37Þ
To obtain a nonlinear position controller, the control input of the position control system is selected as
u ¼ Gep @Vðey;orÞ @ey þ Gip Z @Vðe y;orÞ @ey dt þ Gxp @Vðey;orÞ @or ð38Þ
where Gep is the proportional gain of the position-loop controller. Gip is the integral gain of the position-loop controllers, and Gxpis the feedback gain of velocity.
In this paper, the Lyapunov function of the position control system is selected as[20]:
Vðey;orÞ ¼ 3 4Kp0e 4 3 yþ 1 2Kp1e 2 y þ3 4Ki0e 4 3 yþ 1 2Ki1e 2 yþ 1 2Kx0o 2 r ð39Þ
Substituting (39) into (38), we can obtain u ¼ GepðKp0þ Ki0Þe 1 3 yþ GepðKp1þ Ki1Þey þ GipðKp0þ Ki0Þ Z e13 ydtþ GipðKp1þ Ki1Þ Z eydtþ GxpKx0or ð40Þ
In order to simplify the expression of (40), we define Kpnr ¼ GepðKp0þ Ki0Þ ð41Þ Kper ¼ GepðKp1þ Ki1Þ ð42Þ Kinr ¼ GipðKp0þ Ki0Þ ð43Þ Kier ¼ GipðKp1þ Ki1Þ ð44Þ and Kxp r ¼ GxpKx0 ð45Þ
By substituting (41)–(45) into (40), we can derive the control input as u ¼ or ¼ Kpnre 1 3 yþ Kperey þ Kinr Z e 1 3 ydtþ Kier Z eydtþ Kxp ror ð46Þ 0 0.03 0.06 0.09 0.12 0.15 −2.5 −2.0 −1.5 −1.0 −0.5 0 0.5 1.0 1.5 2.0 2.5 s ia , A − 2.5 −2.0 −1.5 −1.0 −0.5 0.5 1.0 1.5 2.0 2.5 ia , A a 0 0.03 0.06 0.09 0.12 0.15 s b 0
Fig. 4 The current iaat a steady-state a Proposed b Bang-bang 2 3 4 5 6 7 8 9 10 11 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10 000 bang-bang control proposed method current, A
switching frequency, times
/s
Fig. 5 The switching frequency to different loads
0 0.01 0.02 0.03 0.04 0.05 0.06 0 1 2 3 4 5 6 7 8 s bang-bang current control torque, N-m proposed
4 Experimental results
The proposed system uses a 32-bit digital signal processor, TMS 320C30, to determine the PWM switching states and execute the control algorithms. The dc bus voltage of the inverter is 150 V. The sampling interval of the current-loop is 100 ms, and the sampling intervals of the speed- and position-loop are 1 ms, respectively. The motor is a 3-phase, 4 pole, rated speed 1800 r/min, 0.75 hp. It was manufactured by the Reliance Electric Company. The parameters of the motor are: rs¼ 2 O, Ld¼ 0.148 H, Lq¼ 0.0672 H, J ¼ 0.00239 kgm2, B¼ 0.012 N.m/s, Po¼ 4 poles.
Several experimental results are shown here. Figures 3a and 3b show the measured line-to-line voltage of the motor. Figure 3a is the result of the proposed PWM control. Figure 3b is the waveform of the bang-bang PWM control. As can be observed, the proposed PWM method has a lower switching frequency than the traditional bang-bang control. Figures 4a and 4b show the measured a-phase currents of the motor. Figure 4a is the current waveform of the proposed control method and Fig. 4b is the waveform of the bang-bang control. The proposed
method has lower harmonics than the bang-bang
control. Figure 5 compares the switching times/s of the proposed method and bang-bang control. When the motor is operated at a heavy load, both of them have very similar switching times; however, when the motor is operated at a light load, the bang-bang control has twice the number of switching times as compared to the proposed method. 0 1 2 3 4 5 6 7 8 0 200 400 600 800 1000 1200 1400 1600 1800 2000 s PI controller nonlinear controller a 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0 100 200 300 400 500 600 nonlinear controller PI controller s 0 1 2 3 4 5 6 0 10 20 30 40 50 60 70 b nonlinear controller PI controller s c r , rpm r , rpm r, rpm
Fig. 7 The measured transient speed responses a 1500 r/min b 500 r/min c 50 r/min 0 1 2 3 4 5 6 7 8 9 0 100 200 300 400 500 600 nonlinear controller PI controller s r, rpm
Fig. 8 The measured load disturbance responses
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 r , rpm r , rpm −600 −400 −200 0 200 400 600 −600 −400 −200 0 200 400 600 ∗ r r s a s b r r∗
Fig. 9 The measured sinusoidal speed responses a Nonlinear
Figure 6 shows the tracking performance of the torque. Again, the proposed method performs better than the bang-bang control although the proposed method has a lower switching frequency than the traditional bang-bang control.
Figures 7a, 7b, and 7c show the measured transient speed responses of different speed commands. According to the measured results, the proposed nonlinear controller per-forms better than the PI controller at different operating speeds. Figure 8 shows the load disturbance responses when a 2 N-m load is added to the motor at the speed of 500 r/min. Again, the nonlinear controller has a lower speed dip than the PI controller. Figures 9a and 9b show the measured sinusoidal speed responses. As can be observed the nonlinear controller tracks the sinusoidal command well; however, the PI controller produces a lagging res-ponse. Figures 10a and 10b show the measured triangular speed responses. The nonlinear controller performs better again. Figures 11a, 11b, and 11c show the measured responses of position, velocity, and q-axis current of the nonlinear controller as the shaft of the motor is controlled from 180 degrees to180 degrees. Figure 12 shows the load disturbance position response when a 2 N-m load is added. Figures 13a and 13b show the measured sinusoidal position responses of the two controllers. The nonlinear controller performs better. Figures 14a and 14b show the measured triangular position responses. A similar conclusion can be obtained. 0 1 2 3 4 5 6 7 8 −600 −400 −200 0 200 400 600 s a 0 1 2 3 4 5 6 7 8 −600 −400 −200 0 200 400 600 s b r , rpm r , rpm r r r∗ r∗
Fig. 10 The measured triangular speed responses a Nonlinear b PI 0 0.1 0.2 0.3 0.4 0.5 0.6 −200 −150 −100 −50 0 50 100 150 200 s a 0 0.1 0.2 0.3 0.4 0.5 0.6 −800 −700 −600 −500 −400 −300 −200 −100 0 100 200 s r , rpm b 0 0.1 0.2 0.3 0.4 0.5 0.6 −10 −8 −6 −4 −2 0 2 4 6 8 10 s iq , A c r , deg
Fig. 11 The measured transient position responses a Position b Velocity c q-axis current 0 1 2 3 4 5 6 0 50 100 150 200 250 s r , deg
Fig. 12 The measured load disturbance position response at 2 N-m load
5 Conclusions
A new PWM strategy and a novel control algorithm have been proposed. The PWM strategy and the control algorithm are implemented by a 32-bit digital signal processor. As a result, the hardware is very simple and a fully digital control system is achieved. In addition, the proposed method can be applied to a speed control system and a position control system. Experimental results show that the proposed system has satisfactory performance and can be applied to a synchronous reluctance drive system.
6 Acknowledgment
The authors would like to acknowledge the financial support of the National Science Council in Taiwan, Republic of China, through grant NSC 93- 2213- E- 011-012.
7 References
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20 Lyshevski, S.E.: ‘Control systems theory with engineering applications’ (Birkhauser, Boston, 2001) 0 1 2 3 4 5 6 7 8 −200 −150 −100 −50 0 50 100 150 200 r 0 1 2 3 4 5 6 7 8 −100 −200 −150 −50 0 50 100 150 200 r , deg r r , deg s a s b r∗ r∗
Fig. 13 The measured sinusoidal position responses a Nonlinear b PI r , deg 200 150 100 50 0 −50 − 100 − 150 − 200 200 150 100 50 0 −50 −100 −150 −200 θr , deg 0 1 2 3 4 5 6 7 8 s a 0 1 2 3 4 5 6 7 8 s b r r r∗ r∗
Fig. 14 The measured position responses a Nonlinear
8 Appendices
8.1
The selection of the parameters for the
proposed speed controller
The Lyapunov theorem is used to validate the stability of the drive system. We can take the derivation of the Lyapunov function and express it as:
VðeÞ ¼ ðKp0þ Ki0Þe
1
3eþ ðKp1þ Ki1Þee ð47Þ
The speed error is defined as
e ¼ or or ð48Þ
and the derivation of speed error is e ¼ or ¼ 1 JðTe TLÞ ¼ 1 JðK 0 tu TLÞ ð49Þ Kt0 ¼ 3 2 P0 2 ðLd LqÞid ð50Þ
where Kt0 is the torque constant. The viscous frictional coefficient B is ignored here because its value is small. By substituting (48)–(50) into (47), and doing some mathema-tical processes, we can derive
VðeÞ ¼ K 0 t J KpnðKp0þ Ki0Þe 2 3þ ½KpeðKp 0þ Ki0Þ n þ KpnðKp1þ Ki1Þe 4 3þ KpeðKp 1þ Ki1Þe2 þ KinðKp0þ Ki0Þe 1 3 Z e13dtþ KieðKp 0þ Ki0Þe 1 3 Z edtþ KinðKp1þ Ki1Þe Z e13dtþ KieðKp 1þ Ki1Þe Z edtoþ 1 JðKp0þ Ki0Þe 1 3þ1 JðKp1þ Ki1Þe TL ð51Þ By letting Kpno Kin Z e13dtþTL K0 t ð52Þ and Kpe4 Kie Z edt ð53Þ We can derive VðeÞoK 0 t J KinðKp0þ Ki0Þ e 2 3 Z e13dt þe 1 3 Z e13dt þKinðKp1þ Ki1Þ e 4 3 Z e13dt þ e Z e13dt þKieðKp0þ Ki0Þ e 4 3 Z edt þ e 1 3 Z edt þKieðKp1þ Ki1Þ e2 Z edt þ e Z edt þ ðKp0þ Ki0Þ þ e 2 3TL Kt0 e 1 3TL Kt0 þðKp1þ Ki1Þ e 4 3 TL Kt0 e TL Kt0 0 ð54Þ
When the drive system does not add an external load, the TL¼ 0. Then, (54) becomes VðeÞoK 0 t J KinðKp0þ Ki0Þ e 2 3 Z e13dt þe13 Z e13dt þKinðKp1þ Ki1Þ e 4 3 Z e13dt þ e Z e13dt þKieðKp0þ Ki0Þ e 4 3 Z edt þ e 1 3 Z edt þKieðKp1þ Ki1Þ e2 Z edt þ e Z edt o0 ð55Þ
8.2
The selection of the parameters for the
proposed position controller
The Lyapunov theorem is used to validate the stability of the position control system. We can take the derivation of the Lyapunov function and express it as
Vðey;orÞ ¼ ðKp0þ Ki0Þe 1 3 ye yþ ðKp1þ Ki1Þeye y þ Kx0oro r ð56Þ
By substituting (35) and (37) into (56), and doing some mathematical process, we can obtain
Vðey;orÞ ¼ ðKp0þ Ki0Þe 1 3 yor ðKp1þ Ki1Þeyor þ Kx0 KpKt0 J Kpnre 1 3 yorþ Kpereyor þ Kinror Z ey 1 3dtþ Kieror Z eydtþ Kxpro2r Kvo2r TLor J ð57Þ Next, by choosing Kx0 ¼ J KpGepKt0 ð58Þ Then, substituting (58) into (57), we can obtain
Vðey;orÞ ¼ 1 Gep Kinror Z ey 1 3dtþ Kieror Z eydtþ Kxpro2r Kvo2r TLor KpKt0 ð59Þ
In this paper, we select Kv4 Kinr Z e13 ydtþ Kier Z eydtþ Kxprþ TL KpKt0 ð60Þ
Substituting (60) into (59), we can obtain
V ðey;orÞ ¼ 1 Gep Kinr Z ey 1 3dt o2rþ Kinror Z ey 1 3dt Kier Z eydt o2rþ Kieror Z eydt TL KpK 0 t o2r TLor KpK 0 t 0 ð61Þ
From (59)–(61), we can obtain and define Z1ðtÞ ¼ 1 Gep Kinror Z ey 1 3dtþ Kieror Z eydtþ Kxpro2r Kvo2r TLor KpKt0 0 ð62Þ By integrating (62), we can derive
Z t
0
Z1ðtÞdt ¼ V ðeyðtÞ; orðtÞÞ V ðeyð0Þ; orð0ÞÞ ð63Þ
Because eyðtÞ and orðtÞ are bounded, from equation (63),
we can obtain lim t!1 Z t 0 Z1ðtÞdtðtÞdto1 ð64Þ Because Z1(t) and Z
1ðtÞ are bounded, and Z1(t) is a uniform and continuous function. From Barbalet’s lemma, we can obtain
lim
t!1z1ðtÞ ¼ 0 ð65Þ
Combining (61) and (65), we can conclude that the proposed position control system converges to a zero steady-state error. As a result, the position control system is an asymptotically stable system.
