PI Control, PI-Based State Space Control, and ... - Semantic Scholar

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 8, AUGUST 2011

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PI Control, PI-Based State Space Control, and Model-Based Predictive Control for Drive Systems With Elastically Coupled Loads—A Comparative Study Sönke Thomsen, Student Member, IEEE, Nils Hoffmann, Student Member, IEEE, and Friedrich Wilhelm Fuchs, Senior Member, IEEE

Abstract—Three different control methods for the speed control of drive systems with elastically coupled loads are presented and compared. In drive applications where the load is connected to the driving motor with a drive shaft that has a finite stiffness, unwanted mechanical dynamics can occur. These unwanted dynamics can stress both the mechanical and electrical drive components. Furthermore, the shaft torsion, if neglected in the control synthesis, can dramatically reduce the achievable control performance. To overcome these challenges, the design, analysis, and comparative study of three speed control methods for a drive system with resonant loads are carried out. The considered control methods are the following: a conventional proportional-integral (PI) control, a PI-based state space control, and a model-based predictive control. To ensure a suitable basis for their comparison, the three different speed control methods are designed with equal bandwidths and are verified with the same test setup. Furthermore, all speed control methods presented use only the drive-side speed measurement to control the drive speed. Index Terms—AC machine, adjustable speed drive, asynchronous motor, control of drive, predictive control, state feedback, test bench, vibrations.

I. I NTRODUCTION

T

HE realization of competitive, energy-saving, and dynamic drive solutions is a matter of common interest. The use of adjustable speed drives composed of induction machines and frequency converters in addition to the further development of machines, power electronic components, and microprocessors fulfill these high requirements of modern drives. A typical adjustable speed drive topology which is widespread in industrial applications [1] consists of an inverter-fed ac motor and a load. The load is connected to the motor via

Manuscript received August 13, 2010; accepted October 7, 2010. Date of publication October 28, 2010; date of current version July 13, 2011. This work was supported in part by the German Research Foundation DFG (Deutsche Forschungsgemeinschaft) and in part by CEwind e.G. Center of Excellence in Wind Energy of Universities in Schleswig-Holstein by the European Union (EFRE) and the state of Schleswig-Holstein, Germany. The authors are with the Institute for Power Electronics and Electrical Drives, Christian-Albrechts-University of Kiel, 24143 Kiel, Germany (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2010.2089950

the transmission elements which have a nonideal transmission behavior such as finite torsional stiffness. This finite stiffness can lead to unwanted torsional oscillations and can stress both the mechanical and electrical components of the drive system. These problems occur, for example, in electric vehicles [2], rolling mill [3], or windmill [4] applications. For high dynamic speed control of drive systems with elastic coupling, the nonideal behavior has to be considered in the control synthesis. Control structures with proportional-integral (PI) controllers are usually used in the industry for the speed control of drive systems with elastically coupled loads [5]. Various design methods for tuning the PI control to reduce the mechanical oscillations have been reported in the literature, cf., [6] or [7]. However, the PI-based speed control (PISC ) method without additional feedback of the mechanical system states provides only a constricted pole placement of the closed-loop speed control, and it is not able to effectively damp the torsional oscillations [8]. Occasionally, PI-derivative (PID) controllers are used in industrial applications for the reduction of the mechanical vibrations [9], [10]. However, the PID control method is very sensitive to the measurement noise [6]. One possibility for reducing the mechanical oscillations is the feedback of additional system states. These additional system states have to be measured or estimated by observers. The feedback of the shaft torque has been analyzed in [3] and [10]–[12]. A systematic analysis of speed control with different additional feedback is presented in [13]. Nevertheless, the poles of the closed-loop speed control and the resulting system dynamics cannot be set freely. A promising approach for the suppression of the mechanical vibrations is the feedback of all system states (state space (SS) control [14]). The SS control approach has been extensively studied in the academic research literature [15], and it is also attracting a growing interest for industrial applications [5]. The SS control method allows a free pole placement of the closed-loop control and, thus, a theoretically free choice of the system dynamics. The disadvantage of the SS control is the high implementation effort during the design of the control parameters and, moreover, the measurement of all system states or their reconstruction from measurable signals by observers. The PI-based SS speed control (PI-SSSC ) for drive systems

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with elastic coupling has already been analyzed (e.g., in [8] and [16]–[20]). Observer techniques for the estimation of the additional system states, e.g., Luenberger observer, extended Luenberger observer, disturbance observer, or Kalman filter, have been studied in [21]–[24]. Other possible control methods for damping the torsional oscillations which do not necessarily need additional feedback (besides the measured drive-side speed) are the model-based or nonlinear control methods such as the sliding mode control (SMC), H∞ control, flatness-based control (FBC), or modelbased predictive control (MBPC). SMC is a discrete variable structure control method. The objective of the SMC is to drive the plant’s state trajectory onto a prespecified surface in the SS and to maintain the plant’s state trajectory on this surface for all subsequent time [25]. The SMC for elastically coupled drive systems has been analyzed in [26]–[30]. H∞ control enables the involvement of both performance and robustness specifications with the control synthesis [25]. The control requirements are defined using the ∞-norm of the closed loop. The challenge is to find a controller that minimizes this norm. H∞ control of the two-mass systems is presented in [31]. FBC uses the structural system property of a class of nonlinear systems. The concept of FBC allows designing the control as a combination of a nominal feedforward control and a feedback stabilizing control. The feedforward control provides a nominal input trajectory which forces the system to the desired output trajectory in the nominal case [32]. The FBC for the resonant drive systems has been analyzed in [33]. PISC , PI-SSSC , and MBPCSC are selected for this comparative study due to the fact that PI control is commonly used in the industry, i.e., the PI-SS promises great benefits in the achievable dynamic, and the MBPC can improve the dynamic with the use of a plant model without additional feedback and observers. Lately, the MBPC approach has been studied intensively for power electronic applications [34] and especially for drive systems with elastic coupled loads [35]. In [36], the MBPC approach was proven to be suitable for drive applications in terms of the achievable control performance, robustness against model mismatches, and unexpected dynamics, as well as in terms of its online computation burden. The MBPC speed control has already been used in [37] for a stiff drive system and in [35] and [38]–[40] for an elastic coupled drive system. The proposed control approaches are based on a deeper theoretical basis, starting with the well-known PISC across the PI-SSSC up to the MBPCSC . Except for the PISC , the scientific analysis and research of the considered speed control methods for a drive system with elastic coupling is not yet concluded, and it is still of high practical and scientific interest. Different publications and approaches such as those presented in the previous paragraphs can be found for every proposed speed control method, but until now (as far as the authors know), no comparative study between these control approaches has been presented in the literature. This paper presents a survey and a

Fig. 1.

Topology of the drive system.

comparative study to facilitate the choice of control approach for controlling a drive system with elastic coupled loads. Based on the given mechanical system, available sensor system, and achievable computation power for the control implementation, this paper determines which of the considered control approaches is the optimal choice in terms of the dynamic behavior for its drive system speed control. Furthermore, the theoretical and practical analysis will highlight that the choice of the optimal speed control approach is dependent on the complexity of the applied control structure and the desired control performance. The considered control methods will be compared in terms of the achievable dynamic performance indices and the stress on the mechanical system during the reference and load steps. Furthermore, the control approaches are compared in terms of their stability properties and online computation effort. The complexity of the control design process and the possibilities of influencing the dynamic speed control performance will be separately studied for each speed control approach. To ensure a practical and suitable basis for the proposed comparative study, all speed control methods are tuned to have the same bandwidth. Furthermore, only the drive-side speed is measured. This paper is structured as follows. In Section II, the mechanical system description and the torsional load model will be introduced. The speed control approaches that are to be considered are summarized in Section III. Furthermore, that section will present a comparison of the control approaches based on the proposed theoretical analysis. Section IV introduces the test bench, the measurement equipment, the measurement results obtained, and a comparison of the measurement results. A summary of the achieved results based on the theoretical and practical analyses will be presented in Section V. The conclusion is given in Section VI.

II. S YSTEM D ESCRIPTION AND T ORSIONAL L OAD M ODEL A typical drive system that contains a gear, a torsional shaft (elastic coupling), and an additional inertia is shown in Fig. 1. The driving motor is fed by a PWM inverter to create the variable stator voltage and frequency. The motor is connected to a load via a gearbox, a long torsional drive shaft, and an additional inertia. Thus, the mechanical part of these drive systems consists of several inertias and transmission elements with finite stiffness.

THOMSEN et al.: PI CONTROL, PI-BASED SS CONTROL, AND MBPC FOR DRIVE SYSTEMS

Fig. 2.

Block diagram of the two-inertia oscillation model. TABLE I PARAMETERS OF THE M ECHANICAL PART

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Fig. 3. Block diagram of the proposed rotor-flux-oriented drive control structure.

The transfer function of the reduced mechanical two-inertia model is summarized in G(s) =

1 + s dcTT + s2 JcTL ΩM (s) 1 = · . JM JL MM (s) s(JM + JL ) 1 + s dcT + s2 c (J T T M +JL ) (2)

The ratio R between the load- and drive-side inertias is R = JL /JM = 3.81. III. P ROPOSED S PEED C ONTROL A PPROACHES

For an appropriate control synthesis, a model of the system has to be determined. The consideration of all inertias and transmission elements of the mechanical system leads to a multimass model and, due to the high number of system states, to a complex control synthesis. However, this multimass drive system has one dominant mechanical resonant frequency at fres = 41.3 Hz, and the next much more damped resonant frequency is at approximately 400 Hz. In this case, the model of the mechanical system can be reduced to a simple two-mass model without an appreciable influence on the resultant control performance. Hence, the multimass model of the mechanical part is reduced to a two-mass model with the method of Rivin and Di [41]. This reduction leads to a two-inertia model with a drive-side inertia JM , a load-side inertia JL , and an elasticity coefficient cT that represents the torsional stiffness of the drive shaft. The block diagram of the mechanical two-inertia model is shown in Fig. 2. The elastic shaft is presented with torsional elasticity cT and internal damping dT , whereas for the proposed approach, the damping coefficient is estimated. MM , MS , and ML are the electromagnetic torque of the motor, the shaft torque, and the load torque, respectively, which can be considered as an external disturbance. ΩM and ΩL represent the motor and load speeds, respectively. Nonlinearities such as backlash and friction are neglected in this mechanical system model. The parameters of the mechanical part and the parameters of the reduced two-inertia model are specified in Table I. The theoretical mechanical resonance frequency of the reduced model can be expressed as in (1). Note that the theoretical resonance frequency of the reduced two-inertia model corresponds with the measured resonance frequency of the multimass system of fres = 41.3 Hz (JM + JL ) · cT 1 fres = = 40.1 Hz. (1) 2π JM JL

The different speed control methods are implemented in a conventional field-oriented control (FOC) structure. The block diagram of the rotor-flux field-oriented drive control (R-FOC) structure is shown in Fig. 3. A. Conventional PI Speed Control The PI controller is widespread for speed control in the industry because of its simple implementation and tuning of the control parameters [42]. The design of the control parameters is typically done using standard optimization methods, e.g., the symmetric optimum criterion [42]. This method is based on the idea of finding a controller that makes the frequency response of the closed-loop control as close to one as possible for low frequencies [25]. However, the default setting of the symmetrical optimum is not appropriate for drive systems with elastic couplings [43]. An appropriate method for the PI speed control of the resonant drive systems with ratio R ≥ 2 is tuning the poles of the closed-control loop to an identical damping coefficient [6]. Based on the PI controller structure presented in (3), the controller parameters (4) and (5) are derived to achieve identical damping coefficients D = D1 = D2 ∗ (s) MM kI = kP + ∗ ΩM (s) − ΩM (s) s kP = 2JM D(ωz1 + ωz2 ) JM JL 2 2 ωz1 · ωz2 . kI = cT

GP I (s) =

(3) (4) (5)

The eigenfrequencies ωz1 and ωz2 presented in (6) are derived from the system parameters and the selected damping coefficient D JL JL 2+4∓ 2 − 4D JM JM − 4D ωz1/z2 = 2 JcTL ωz1 = 61.22 rad/s

ωz2 = 215.77 rad/s.

(6)

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first-order time-delay with a time constant TC , the characteristic polynomial of the closed-loop Q(s) is given by (8), where the internal damping dT is neglected [20] Q(s) = s5 +s4 · Fig. 4. Block diagram of the conventional PI speed control.

+ s2 · + s·

1−k4 JL (kP −k1 +cT TC )+JM cT TC +s3 · TC JM JL TC

(JM +JL ) cT +JL kI −JL cT k2 −(JM +JL ) cT k4 JM JL TC

cT kP −cT k1 −cT k3 cT kI + . JM JL TC JM JL TC

(8)

Furthermore, a desired polynomial P (s) with five zeros, which determines the poles pz0 to pz4 of the closed-loop control, is introduced in

Fig. 5. Block diagram of the PI-based SS speed control.

A good tradeoff between the reference reaction and the disturbance rejection can be obtained with a damping coefficient D = 0.707, which leads to the control parameters shown in kP = 13.24 Nms

kI = 446.49 Nm.

(7)

The resulting bandwidth of the PI control is 12.6 Hz. The bandwidth of the speed control is a quantity for the speed performance of the control system. For a good comparability of the considered speed control methods, all controllers are tuned to have the same bandwidth of 12.6 Hz. In [44], a comparative study of the PI control, the PI-based SS control, and the MBPC is carried out with a closed-loop bandwidth of 8.7 Hz. The main disadvantage of the conventional PI control method is the limited pole placement possibilities and, thus, a limited speed control dynamics. Fig. 4 shows the block diagram of the conventional PI speed control. An antiwindup limitation network is implemented, but it is only suggested as a saturation block. B. PI-Based SS Speed Control The PI-based SS control method includes the feedback of all mechanical system states [20]. Fig. 5 shows the block diagram of the PI-based SS speed control, with an additional observer for the shaft torque and the load speed. Thus, only the motor-side speed has to be measured for speed control. The advantage of SS control is a theoretically free pole placement of the closed-loop speed control if there is no limitation of the actuating value. Therefore, a high dynamic and a high damping of the torsional oscillations can be achieved. The difficulty of SS control is the determination of the control parameters. The design of the control parameters is done using pole placement of the closed-loop poles. Therefore, the characteristic polynomial Q(s) of the closed-loop system is needed. Considering the mechanical part as a two-inertia model (2) and with the delay of the inner current control loop approximated as a

P (s) = (s − pz0 ) (s − pz1 ) (s − pz2 ) (s − pz3 ) (s − pz4 ) . (9) The poles of the closed-loop control are characterized by the damping coefficients D1 and D2 and by the eigenfrequencies ωz0 , ωz1 , and ωz2 pz0 = ωz0

pz1/z2 = ωz1 −D1 ± j 1 − D12 pz3/z4 = ωz2 −D2 ± j

1−

D22

.

(10)

Comparing the coefficients of the characteristic polynomial Q(s) with the coefficients of the desired polynomial P (s) leads to a system of equations. The control parameters can be obtained by solving these equations. Due to the length of the expressions, the equations of the control parameters are not explicitly shown. The control parameters kP , kI , and k1 . . . k4 are dependent on the system parameters JM , JL , cT , and TC and on the desired poles of the closed-loop control pz0 to pz4 . Thus, the dynamics of the speed control can be adjusted with the damping coefficients D1 and D2 and by the eigenfrequencies ωz0 , ωz1 , and ωz2 . An appropriate approach is to place all poles, except the dominant pole pair, far to the left side in the complex s-plane. In that case, the dynamic is primarily influenced by the dominant pole pair pz1/z2 . However, the speed control becomes very sensitive to the measurement noise if the poles are placed too far to the left in the s-plane. Consequently, a tradeoff has to be found. Satisfactory results and a bandwidth of 8.7 Hz have been achieved with the parameters presented in ωz0 = 1000 rad/s ωz1 = 95 rad/s ωz2 = 250 rad/s

D1 = 0.4 D2 = 0.707.

(11)

These desired poles of the closed-loop control lead to the control parameters of the PI-SSSC presented in kP = 0.75 Nms kI = 225.69 Nm

k1 = −4.22 Nms k2 = 0.16

k3 = −0.78 Nms

k4 = 0.46.

(12)


A disturbance observer [45] is implemented for the reconstruction of the shaft torque and load speed. The disturbance observer consists of a well-known Luenberger observer which contains the mechanical two-mass model (2), whereas the internal damping dT is neglected, and the inner current control loop is approximated as a first-order time delay. The observer is extended to a disturbance model, whereas the disturbance is assumed to be constant. The disturbance observer is tuned using pole placement. All of the poles of the observer are placed twice as far to the left as those of the closed-loop speed control.

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TABLE II S UMMARY OF THE GPC D ESIGN PARAMETERS

J(N1 , N2 , Nu ) =

N2

δ [ˆ y (t + j |t ) − w (t + j)]2

j=N1

C. Generalized Predictive Speed Control The underlying concept of MBPC (also known as long-range predictive control) has been known since the late seventies. Many publications and books about its basic theory as well as specific applications have been published by now. Furthermore, many different methods have been developed, enhanced, or mixed in the past 40 years. This leads to a wide range of predictive control algorithms that could be used for the problem being addressed in this paper. Based on the literature [46]–[48], the MBPC algorithm of generalized predictive control (GPC) [49], [50] is chosen to control the drive-side speed of the drive system with elastic coupled load considered here. The GPC algorithm is composed of three parts [48]: a prediction equation, a cost function, and a resultant control law. In (13), the plant model of the chosen GPC algorithm is summarized as an integrated controlled autoregressive moving average (CARIMA) model. A describes the denominator polynomial of the open-loop plant transfer function, and B describes the numerator polynomial. The polynomial T can be treated as a design polynomial to filter the higher frequency disturbances caused by the model mismatch [51]. Furthermore, u(t) describes the control sequence (here, the q-component of the stator reference current i∗S,q ), y(t) describes the process output sequence (here, the measured drive speed NM ), e(t) describes the process distortion, z −1 describes the backshift operator, and d describes the dead time of the system. The GPC prediction equation is shown in (15) where the polynomials Ek and Fk are obtained from a Diophantine equation, cf., [49] A(z −1 )y(t) = z −d B(z −1 )u(t − 1) + T (z −1 ) Δ = 1 − z −1

e(t) Δ

(13) (14)

yˆ (t+k |t ) = Fk (z −1 )y(t)+Ek (z −1 )B(z −1 )Δu (t+k |t )

+

Nu

λ [Δu (t + j − 1)]2 .

To derive a control law from the prediction equation (15) for the GPC algorithm, a cost function has to be defined, cf., (16). From (16), it can be seen that the quadratic cost function takes the predicted control error (ˆ y − w) into account, as well as the control signal change Δu, to reach the commanded value. In Table II, the associated GPC design parameters are summarized. As highlighted in Fig. 6, an actuating value limitation has to be taken into account to design a realizable GPC speed control for the drive system with torsional loads. To reduce the implementation and the online computation effort, this constraint is implemented a posteriori to the speed control system in this paper. In [52], the theoretical background in implementing an actuating value limitation a posteriori to the MBPC system is presented. The calculation of the free system response is dependent on the change of the actual actuating value Δu. Therefore, the proposed solution is not only to limit the actuating value but also to limit the change of the actuating value needed to calculate the free system response. For the drive system under consideration here, the speed control actuating value is the q-component i∗sq of the reference stator current. Therefore, the a posteriori implementation of the actuating value limitation can be formulated as in (17), shown at the bottom of the page. The resultant structure of the GPC speed control is shown in Fig. 6. To summarize the MBPCSC control design and to ensure a suitable bandwidth for the proposed comparative study, the GPC design parameters are chosen as in N1 = d = 2 λ(j) ≡ λ = 10

N2 = Np = 600

δ(j) ≡ δ = 3.5

Nu = 1 2 T (z −1 ) = 1 − 0.97z −1 .

(15)

i∗Sq,bg

⎧ ∗ ⎨ iSq = i∗Sq,max ⎩ ∗ −iSq,max

(16)

j=1

& Δi∗Sq,bg (t) = i∗Sq (t) − i∗Sq (t − 1), & Δi∗Sq,bg (t) = i∗Sq,max − i∗Sq (t − 1), & Δi∗Sq,bg (t) = −i∗Sq,max − i∗Sq (t − 1),

(18)

if i∗Sq (t) ∈ −i∗Sq,max , i∗Sq,max if i∗Sq (t) > i∗Sq,max if i∗Sq (t) < −i∗Sq,max

(17)

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Fig. 6. GPC speed controller with an actuating value limitation network. TABLE III S TABILITY AND ROBUSTNESS PARAMETERS OF THE S PEED C ONTROL M ETHODS FOR THE N OMINAL AND U NCERTAIN S YSTEMS

Fig. 8. Fig. 7. Bode diagram of the open-loop transfer functions (simulations). (Black) Standard PI speed control. (Red) PI-based SS speed control. (Green) GPC speed control.

Mechanical part of the test bench in the laboratory. TABLE IV S YSTEM PARAMETERS

Further information and additional analysis on the proposed MBPCSC is presented in [53] and [54]. D. Comparison of the Speed Control Methods Concerning Theoretical Analysis This section gives a brief comparison of the speed control methods presented in this paper, concerning the possibilities of the controller designs, the complexity of the implementation and control parameter tuning, and their stability properties. Concerning the possibilities of the controller design, the PI control method has two control parameters (kI and kP ), and for feedback, it only has that of the motor speed. Consequently, the choice of the closed-loop poles and the achievable dynamics are restricted. The PI-based SS control method includes feedback of all mechanical system states and leads to a theoretically free placement of the closed-loop poles and an enhanced dynamic. The structure of the GPCSC is similar to the structure of the PISC , and it includes only the feedback of the motor speed. Thus, the pole placement of the closed-loop poles is also restricted. However, in addition, the GPCSC contains a predictive calculation of the actuating signal based on the system model. Thus, an improvement of the dynamic can be achieved.

Due to the fact that the PI controller consists only of two control parameters, this method is the easiest to implement and to design. Comparing the PI-based SS control method and the generalized predictive controller in terms of their complexity of implementation does not lead to a clear decision. The PI-SSSC control contains six control parameters and the disturbance observer. The GPCSC has six design parameters and a filter polynomial. The design of the PI-SSSC is primarily done with two parameters: the damping coefficient D1 and the eigenfrequency ωz1 . The design of the GPCSC is basically done with three parameters: the prediction horizon Np , the command-weighting sequence δ, and the control-weighting sequence λ. Nevertheless, all control parameters of PI-SSSC and GPCSC have to be adequately chosen. Depending on the specific knowledge one might have, it might be easier for some users to implement an SS controller and for other users to implement a predictive controller. However, the complexity of


Fig. 9.

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Measured step response (40–43 rad/s). (Black) Standard PI speed control. (Red) PI-based SS speed control. (Green) GPC speed control.

Fig. 10. Measured step response (40–60 rad/s). (Black) Standard PI speed control. (Red) PI-based SS speed control. (Green) GPC speed control.

Fig. 11. Measured disturbance rejection (load step from 0 to 10 Nm). (Black) Standard PI speed control. (Red) PI-based SS speed control. (Green) GPC speed control.

the implementation and tuning of the PI-SSSC and the GPCSC is much higher than that of a conventional PISC , as shown in Sections III-A–C. Further information about the design of PI-SSSC and GPCSC can be found in [20], [53], and [54]. The stability analysis of these control methods is done by analyzing the Bode diagram of the open-loop control systems. Fig. 7 shows the open-loop frequency responses of the proposed speed control methods. They are determined using a simplified model containing the approximated inner current control loop and the mechanical two-mass model (2). This figure presents the magnitude and phase plots of the control methods with the nominal systems model. The gain gm and phase margins ϕm are determined for these control methods and are summarized in Table III. It can be seen that all of these control methods have stable performances. The highest gain and phase margins are achieved with PI-SSSC , followed by PISC . The least stability properties are obtained with GPCSC . The robustness concerning parameter uncertainties is analyzed by varying the system parameters of the mechanical system JM , JL , and cT , whereas the controllers and observer

are tuned for the nominal system. The ranges of the parameter variations are given in JM = JM ± 0.3 · JM JL = JL ± 0.5 · JL cT = cT ± 0.5 · cT .

(19)

The gain and phase margins have been determined for every parameter variation. The minimum values of the amplitude and phase margin of the uncertain system are presented in Table III. It can be seen that all of the considered control methods yield stable control loops for the uncertain system. Again, PI-SSSC achieves the highest, and PISC achieves the second highest gain and phase margins. GPCSC also yields a stable control but with less robustness than PISC and PI-SSSC . IV. M EASUREMENTS Measurements were taken for each control method under the same conditions, such as the actuating value limitation and

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TABLE V M EASURED C ONTROL P ERFORMANCES FOR THE PI C ONTROL , GPC, AND PI-BASED S TATE S PACE S PEED C ONTROL T UNED W ITH THE S AME BANDWIDTH OF 12.6 Hz

the sampling frequency. For a good comparability of the speed control methods under consideration, all controllers were tuned to the same bandwidth of 12.6 Hz. Of course, another basis for the comparative study could have been possible. Consequently, other results concerning the large-signal behavior, the smallsignal behavior, and the disturbance rejection might be possible. Furthermore, only the drive-side speed measurement was used for all control methods to control the drive speed. The load-side speed NL and the shaft torque MS are measured to analyze the proposed speed control methods. A. Test Bench Description To verify the theoretical approach, the measurement results are presented. The mechanical part of the test bench is shown in Fig. 8. The setup consists of a 5.5-kW converter fed induction motor and a converter fed servo induction machine, connected by a long drive shaft. The servo induction machine is used to induce a variable disturbance torque. The parameters of the drive and load machine are presented in Table IV. A flywheel is used on the load side to increase the ratio R between the load- and drive-side inertias to R = JL /JM = 3.81. The shaft torque is measured with a torque sensor to determine the shaft stress. Incremental encoders are used for speed measurement with 10 000 pulses/revolutions on the motor and load sides. The different speed control methods are implemented, superimposed to a conventional field-oriented current control (FOC) structure that is oriented to the rotor flux. The control algorithms are implemented on a dSPACE DS1103 PPC controller

board. The sampling frequency is set to 3 kHz for each control method. B. Comparison of the Measurement Results The measurement results of the different speed control methods are shown in Figs. 9–11. Fig. 9 shows the results of a reference step from 40 to 43 rad/s (small-signal behavior—actuating value not in limitation), the results of a reference step from 40 to 60 rad/s (large-signal behavior—actuating value in limitation) are shown in Fig. 10, and Fig. 11 shows the results of a disturbance rejection during a load step from 0 to 10 Nm. The achieved control performances are characterized by the percentage overshoot P O of the drive and load speeds, rise tr and settling times ts of the drive speed, and the stress of the drive shaft, i.e., the maximum occurring shaft torque MS and the maximum occurring rate of change of the shaft torque dMS /dt. These values are summarized in Table V for each control method. Looking at the small-signal behavior, it can be concluded that the PISC leads to a higher P O and a longer settling time than PI-SSSC and GPCSC . Furthermore, the torsional oscillations are obtained with PISC . This leads to a significantly higher shaft torque and a much higher dMS /dt. The overshoot, settling time, and stress of the drive shaft are considerably reduced with the proposed PI-SSSC and GPCSC . GPCSC achieves the shortest settling time, but it tends to achieve a higher maximum shaft torque than PI-SSSC . PISC leads to the worst control performance even for the large-signal behavior. PI-SSSC leads to the lowest overshoot


TABLE VI C ALCULATION T IME OF THE T OTAL FOC FOR THE P ROPOSED S PEED C ONTROL A PPROACHES

TABLE VII C OMPARISON OF THE P ROPOSED S PEED C ONTROL A PPROACHES

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VI. C ONCLUSION A detailed comparison of three different control methods for the speed control of drive systems with elastic coupling with the conventional PI control, the PI-based SS control, and the MBPC (GPC) has been presented. The control methods considered in this paper were designed and compared in terms of their dynamic behavior, the stress of the drive shaft, and their stability as well as their robustness properties. Furthermore, the online calculation time, the possibilities of the controller design, and the complexity of the implementation and tuning of each control method were compared. For comparability, all control methods were tuned to the same bandwidth of 12.6 Hz, and the measurements were taken for each control method under the same boundary conditions. It can be concluded that the PI control method is well known and is very easy to design and to implement. However, only a poor speed control performance can be achieved as a consequence of its constricted pole placement. The control performance can be considerably increased by the use of the PI-SSSC due to its free pole placement. However, an observer for the estimation of the nonmeasured states is required. Similar good control performances can be achieved with the modelbased control method (GPC). Thereby, additional observers are not required. However, the processing power may be a problem. The implementation effort and the complexity relating to the control synthesis of the PI-SSSC and GPCSC are much higher than that of the conventional PISC . Finally, the user has to decide depending on the required control performance and the available control hardware, which, of all of the control methods presented in this paper, is the most convenient one. R EFERENCES

and to the shortest settling time. In terms of the stress of the drive shaft, PI-SSSC tends to achieve the highest stress. PISC and GPCSC achieve a comparable low stress of the drive shaft. Analyzing the disturbance rejection, it is analyzed that PI-SSSC leads to a faster rejection of the disturbances than PISC or GPCSC . Again, PISC yields the longest settling time (Table VI). V. S UMMARY OF THE C OMPARISON The results obtained based on the theoretical and practical analyses are summarized in Table VII. The measurement results confirmed that the conventional PISC leads to a low control performance. Significantly better results concerning the dynamic behavior were achieved with the GPCSC and PI-SSSC . Both PISC and PI-SSSC have high stability and robustness properties. The proposed GPCSC is stable and robust with respect to the parameter uncertainties presented, but it yields less robustness than PISC and PI-SSSC . The limitation of the GPCSC is the required online calculation time. Depending on the prediction horizon, the GPCSC needs much more processing power than PISC or PI-SSSC .

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Sönke Thomsen (S’07) was born in Kiel, Germany, in 1980. He received the Dipl.Ing. degree from the Christian-Albrechts-University of Kiel, Kiel, in 2007. Since 2007, he has been a Graduate Research Assistant with the Institute for Power Electronics and Electrical Drives, Christian-Albrechts-University of Kiel. His primary research interests include the control of electrical drive systems, power converters, and identification methods. Mr. Thomsen is a Student Member of the IEEE Industrial Electronics Society, IEEE Power Electronics Society, and Verband der Elektrotechnik Elektronik Informationstechnik.


Nils Hoffmann (S’09) was born in Halle, Germany, in 1983. He received the Dipl.Ing. degree from the Christian-Albrechts-University of Kiel, Kiel, Germany, in 2009. In September 2009, he started to work as a Graduate Research Assistant with the Institute for Power Electronics and Electrical Drives, ChristianAlbrechts-University of Kiel. His main research interests include the control of grid-connected PWM-converters and drives as well as renewable energies and power quality in distributed power generation networks. Mr. Hoffmann is a Student Member of the IEEE Power Electronics Society and the IEEE Industrial Electronics Society.

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Friedrich Wilhelm Fuchs (M’96–SM’01) received the Dipl.Ing. and Ph.D. degrees from the RheinischWestfälische Technische Hochschule University of Technology, Aachen, Germany, in 1975 and 1982, respectively From 1975 to 1982, he was a Scientific Assistant in research for automotive ac drives with the Rheinisch-Westfälische Technische Hochschule University of Technology. From 1982 to 1996, he was engaged in the research on the development of power electronics and electrical drives: first, in a medium-sized company and later as a Managing Director with the Converter and Drives Division (presently Converteam), AEG, Berlin. Since 1996, he has been with the Christian-Albrechts-University of Kiel, Kiel, Germany, as a Full Professor and as the Head of the Institute for Power Electronics and Electrical Drives. He is the author or coauthor of more than 150 papers. His research interests include power semiconductor applications, converter topologies, and variable speed drives as well as their control. Other research interests include renewable energy conversion, especially wind and solar energies, on the nonlinear control of drives as well as on the diagnosis of drives and fault tolerant drives. Many of his research projects were carried out with industrial partners. His institute is a member of CEwind e.G., which is the registered corporative competence center of research in wind energy in universities and the competence center for power electronics in Schleswig-Holstein (KLSH), where he is the Chairman of the Board. Dr. Fuchs is the Convener in standardization for Dialog Kompetenz Engagement and International Electrotechnical Commission and a member of Verband der Elektrotechnik Elektronik Informationstechnik and EPE.