Model Based Predictive Speed Control of a Drive System with

14th International Power Electronics and Motion Control Conference, EPE-PEMC 2010

Model Based Predictive Speed Control of a Drive System with Torsional Loads – A Practical Approach Nils Hoffmann, Sönke Thomsen, Friedrich W. Fuchs Christian-Albrechts-University of Kiel, Institute for Power Electronics and Electrical Drives, Kiel, Germany, e-mail: [email protected] Abstract––This paper presents the selection and the implementation of a model based predictive speed control the Generalized Predictive Control - for a drive system with torsional loads. In drive applications where long or thin drive shafts are used torsional oscillations can occur due to the finite torsional stiffness of the shafts material. These unwanted oscillations can stress both mechanical and electrical drive components. Further, when shaft torsion is neglected during the control synthesis, the achievable bandwidth of the speed control is dramatically reduced. To avoid these problems a model based predictive speed control is designed that takes the torsional shaft behavior into account. To ensure a practical and realizable speed control, an aposteriori actuating value limitation in addition to the classical GPC approach is presented. The predictive speed control is verified in simulations and measurements. Additionally to that a comparative study to a PI based speed control is carried out.

combination of the model based predictive control approach, an accurate underlying system model and the utilization of the widespread FOC structure leads to a practical control concept which reduces the influences of the non ideal transmission behavior of the drive system.

Keywords—AC machine; Adjustable speed drive; Asynchronous motor; Control of Drive; Test bench; Torsional resonance.

I.

Fig. 1. .Schematic drawing of a drive system with torsional load and the resultant torsional vibration model

INTRODUCTION

Adjustable speed drives (ASDs) composed of induction machines (IM) and frequency converters are widespread in industrial applications [1]. With the development in power electronic components and micro processors it is possible to realize competitive, energy-saving and dynamic drive solutions. In general the driving torque is transferred to the load side with mechanical transmission elements. These elements can have a non ideal transmission behavior, e.g. a not negligible inertia, finite torsional stiffness, friction or backlash. The non ideal transmission of the driving torque to the load side can lead to serious problems in the mechanical drive components, e.g. torque impulses or mechanical vibrations, especially during torsional oscillation excitations. These aforementioned problems can appear in electric vehicles [2], rolling mill [3] or windmill [4] applications. Field-oriented control (FOC) is the state of the art in modern control of ASDs [5]. FOC consists of cascaded control loops with inner current control loops and with outer speed (or torque) and flux control loops respectively [6]. With a proper speed control synthesis it is possible to reduce the problems introduced by the non ideal transmission behavior of the mechanical drive components, e.g. presented in [7]-[9]. Thus speed control design has to be done carefully. An essential requirement for a speed control synthesis is an accurate modeling of the non ideal mechanical drive components. Here, a Model Based Predictive Control method (MBPC) [10] is selected. The

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MBPC in addition with a FOC structure has already been used in [11] for controlling electrical drives where a mechanical model without torsion is used for control synthesis. Further in [12] a predictive speed control for torsional vibration suppression using Dynamic Matrix Control (DMC) method [13] in addition to a PI-controller (Proportional-Integral-DMC) is presented. In that publication the mathematical basics of PI-DMC is carried out as well as promising simulation results. In [14] and [15] another possibility for controlling a drive system with mechanical elasticity of the driveshaft with finite stiffness is presented. There, a Model Predictive Control (MPC) algorithm is developed and validated in simulations and experiments. The shown MPC algorithm includes a constrained optimization problem. To reduce the online computation effort for solving that constrained optimization problem the predictive control is formulated as an explicit MPC algorithm with the use of multiparametric programming to solve the optimization problem. The aforementioned predictive control strategies for controlling a drive system with mechanical elasticity have two fold challenges. Either no actuating value limitations are considered in the predictive control synthesis [12] or the use of additional toolboxes to solve a complex constrained optimization problem and therefore reducing the online computation effort is required [14], [15]. This paper aims to overcome to these two challenges.

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The proposed MBPC speed control consists of the well known Generalized Predictive Control (GPC) method which takes the torsional behavior of the driveshaft with finite stiffness into account, cf. Fig. 1. Moreover only one speed control constraint is considered, the actuating value limitation. The proposed solution for introducing this constraint aposteriori to the control loops is simple but effective. The predictive speed control is used in a well known FOC scheme whereas the conventional speed controller is replaced with a predictive speed controller. These simplifications lead to a computational effective and practical implementation of a model based predictive speed control for a drive system with torsional loads. The proposed control concept is verified in simulations as well as in experiments. To emphasize the improvement of using the predictive speed control a comparative study to a PIbased speed control is presented. Both the PI-based speed control and the GPC speed control methods use only the drive-side speed measurement to control the drive speed. This work is structured as follows. In chapter II the drive system is described and the model of the mechanical system is introduced. In chapter III the choice of the MBPC method GPC is briefly discussed and the basic mathematical equations of this predictive control method are shown. Chapter IV summarizes the chosen FOC structure. Moreover the detailed implementation of the predictive speed control is discussed as well as of the PIbased speed control. The test bench, simulation and experimental results are presented in chapter V. The paper will be closed by a conclusion presented in chapter VI. II. SYSTEM DESCRIPTION AND MECHANICAL SYSTEM MODEL The investigated drive system is shown in Fig. 2. A three-phase PWM rectifier is connected to the mains through an L-filter feeding the DC-link. A PWM inverter is used to create variable stator-voltages used to drive the induction motor. Further the mechanical drive system consists of clutch (with backlash), a long driveshaft, a flywheel and a load (in case of laboratory tests a load motor). As denoted in Fig. 1 and Fig. 2 respectively the mechanical system consists of more than one transmission element with finite stiffness. Hence it may not be evident to describe the mechanical system dynamics with a twoinertia torsional oscillation model as presented in Fig. 3. The development of Fig. 3 starts with the investigation of the mechanical drive system shown in Fig. 1 (top). This system can be modeled with a discrete five-inertia torsional oscillation model [16], cf. Fig. 1 (bottom). This discrete five-inertia system has five degrees of freedom. To reduce the systems state variables and therefore to simplify the control synthesis these degrees of freedom are reduced to the smallest possible amount. In [16] an overview of methods for reducing the degrees of freedom for discrete torsional oscillation system is presented. This overview leads to use of the reduction method of Rivin/Di [17] for

the proposed drive system. This method is chosen because of its simplicity and clarity but other methods could also be possible. PWM inverter

PWM rectifier

drive motor stator currents

L-filter

M

Udc

gear box flywheel

speed signal

Load torsional driveshaft

gate signals

Control Fig. 2. Topology of investigated drive system

1 J*Load

1 J*ASM

Δα

Fig. 3. Block diagram of driveshaft including mechanical elasticity

The reduction method of Rivin/Di leads to a two-inertia torsional oscillation system for the investigated drive system, whereas the reduced system parameters are summarized in Table III (bottom). The transfer function of the resultant two-inertia torsion oscillation system is presented in (1). Gmech ( s) =

N ASM 1 = * * M ASM s J ASM + J Load

(

Gd ( z ) =

J* d T* + s 2 Load * cT cT* (1) * * * d J J Load 1 + s T* + s 2 * ASM * * cT cT ( J ASM + J Load ) 1+ s

)

⎫⎪ z − 1 ⎧⎪ −1 ⎧ G ( s) ⎫ ⋅ Z ⎨L ⎨ ⎬ ⎬ z ⎪⎩ ⎩ s ⎭ t = k ⋅Ts ⎪⎭

(2)

In predictive control theory mostly discrete formulated transfer functions are used [10]. Therefore the discrete-time equivalent to the continuous-time transfer function (1) taking the sample and hold effect into account can be discretized with the use of (2), whereas Gd denotes the discrete transfer function, Z the Z-transformation, L-1 the inverse Laplace-transformation and Ts the applied sample time for discretization. III. CHOICE OF THE MBPC METHOD AND GPC CONTROL LAW The underlying concept of MBPC (also called longrange predictive control) was known since the late seventies. Many papers and books about the basic theory as well as specific applications have been published about MBPC. Furthermore, many different methods have been developed, enhanced or mixed in the past forty years. This leads to a wide range of predictive control algorithms that could be possible to use for the introduced problem.

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Based on the literature [18]-[20] the MBPC algorithm of Generalized Predictive Control (GPC) [21] [22] is chosen. The basic formulation and a more detailed explanation of the most important selection criteria for this prediction control approach are summarized in the following paragraphs. T ABLE I SUMMARIZATION OF B ASIC MBPC M ETHODS Abbreviation Meaning of Abbreviation DMC Dynamic Matrix Control MAC Model Algorithmic Control EHAC Extended Horizon Adaptive Control EPSAC Extended Predictive Self-Adaptive Control GPC Generalized Predictive Control Multipredictor Receding Horizon Adaptive MURHAC Control PFC Predictive Functional Control

process output sequence (here the measured drive-speed NASM); e(t) describes the process distortion; z-1 the backshift operator and d the dead time of the system. The GPC prediction equation is shown in (5) whereas the polynomials Ek and Fk are derived from a Diophantine equation cf. [21]. e(t ) (3) A( z −1 ) y (t ) = z −d B ( z −1 )u (t − 1) + T ( z −1 ) Δ

Δ = 1− z

(4)

−1

yˆ (t + k t ) = Fk ( z −1 ) y (t ) + Ek ( z −1 ) B( z −1 )Δu (t + k t ) J ( N1 , N 2 , N u ) =

N2

∑ δ [ yˆ (t + j t ) − w(t + j ) ]

2

j = N1 Nu

+ ∑ λ [Δu (t + j − 1) ]

(5)

(6)

2

j =1

Table I summarizes some of the most historically important MBPC methods. A comparison between these methods was made to choose the best MBPC approach to control a drive system with torsional loads. Two main selection criteria are considered while comparing these control approaches. First, the two-inertia oscillation dynamics (especially the integrative behavior cf. (1)) of the mechanical part of the control plant and second, the assessed computation effort of the basic control algorithm has to be taken into account while selecting the model based control method. As highlighted in (1) the mechanical part of the control plant consists of an integrative term. This leads to the exclusion of the DMC, MAC and PFC methods respectively because only open-loop stable plants can be controlled with these approaches. Due to the assessed computation effort of the basic control algorithm the MURHAC method can be excluded. The remaining MBPC methods are the EHAC, EPSAC and the GPC. As stated in [19] the GPC approach can be seen as an extension of both the EHAC and EPSAC method. Thus for the sake of generality the GPC method is chosen for controlling the considered drive system. At this point it should be emphasized that only the basic publications of the presented MBPC were taken into account for clarity of the underlying principles and definitions. Of course other modified methods based on these basic MBPC algorithms could be possible to control the considered drive system. In the following paragraphs the basic idea of the GPC approach will be summarized. A GPC algorithm is composed of three parts [20]: A prediction equation, a cost function and a resultant control law. In (3) the considered 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 respectively. The polynomial T can be treated as a design polynomial to filter higher-frequency disturbances caused by model mismatch [23]. Furthermore, u(t) describes the control sequence (here the q-component of the stator reference current iS,q*); y(t) describes the

To derive a control law from the prediction equation (5) for the GPC algorithm a cost function has to be defined cf. (6). From (6) it can be seen that the quadratic cost function takes the predicted control error (ŷ-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. T ABLE II SUMMARIZATION OF GPC DESIGN PARAMETER Notation Parameter N1 Minimum costing horizon N2 Maximum costing horizon Np Prediction horizon Nu Control horizon Command-weighting sequence δ Control-weighting sequence λ T(z-1) Filter polynomial

Δ u (t )

y (t )

Fig. 4. Classical GPC control law [20]

Using the prediction equation (5) and the defined cost function (6) a GPC control law can be derived. In [20] a complete mathematical description of the GPC algorithm is given, therefore only the basic block diagram of the GPC control law is presented in Fig. 4 whereas K is derived by the control law with assuming a constant future reference trajectory. IV. CONTROL STRUCTURE As stated in the introduction of this paper, the predictive speed controller is implemented with a state of the art FOC structure [6]. This section summarizes the overall control structure of the drive system as well as the specific speed control plant. Furthermore, the design of the GPC speed control is summarized and a PI-based speed control is

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introduced. The PI speed-control is established to compare the resultant speed control of both the PI and the GPC speed-controller dynamics in chapter V. Both proposed speed-control methods are using only the drive-side speed measurement to control the drive systems speed.

The inner current control plant consists of a dead time introduced by the PWM-updating and sampling routine (7) and the first order time-delay dynamics of the induction machine. To simplify the control synthesis of the proposed GPC speed control the closed loop current control dynamics GCC are approximated as a dead time element, cf. (7). This simplification is legitimate since the bandwidth of the current control will be higher than the bandwidth of the speed control. As it can be seen in (3) this simplification can be easily included to the CARIMA process model with the choice of d equal to the dead time of the PWMupdating and sampling routine.

General Block Diagram Fig. 5 presents the general block diagram of the proposed drive control where the torsional load mechanics and the position of the speed controller in the proposed structure are highlighted. The driving motor dynamics as well as the rotor-flux dynamics are modeled as a first order time delay [24]. The q-component of the rotor-flux is equal to zero due to the orientation of the control to the rotor-flux d-component. An inverse rotor-flux oriented machine model is used to calculate the transformation angle ϑk and the estimated rotor flux d-component ΨR,dcalc whereby the stator-currents iS,abc and the drive-speed NASM are measured. In this paper the PI-controller synthesis of the flux and current controllers are not discussed, further explanation can be found in literature, e.g. in [6],[24].

Gd ( z −1 ) =

U s ( d ,q ) ( s ) U

GPWM ,d ( z ) =

* S ( d ,q )

U s ( d ,q ) ( z ) U S* ( d ,q ) ( z )

⎛ 1 ⎞⎟ K p , fc ⎜1 + ⎜ sT ⎟ i , fc ⎠ ⎝

ΨR* ,d

(s )

i

* Sd ,Lim

* i Sq , Lim

N *ASM

=

1 ≈ Gd ,CC ( z ) zk

⎛ 1 ⎞⎟ K p ,cc ⎜⎜1 + ⎟ ⎝ sTi ,cc ⎠

⎛ 1 ⎞⎟ K p ,cc ⎜⎜1 + ⎟ ⎝ sTi ,cc ⎠

(

T ( z −1 ) = 1 − 0.97 z −1

(8)

* uSd ,Lim

uS* ,α

* uSq ,Lim

(10)

)

2

Once the discrete transfer function of the speed control plant is derived the GPC control law can be obtained by solving the unconstrained optimization problem defined in the cost function offline. More detailed explanation how the control law is derived from the known control transfer function and the defined T filter polynomial can be found in [20].

(7)

= e − s k ⋅Ts ≈ GCC ( s )

(9)

Taking these simplifications into account and using the discretized model of the two-inertia oscillation system the discrete control transfer function (9) of the mechanical system can be derived. Here, the coefficients b0 and b1 are equal to zero due to an assumed dead time delay of two sample periods for the proposed drive system. In [20] the role of the T polynomial is described. It is stated that no unified guidelines exist in literature to select this filter polynomial. In [23] the robustness effects of the filter polynomial to the GPC control synthesis is analyzed. This analysis leads to the selection of the T filter polynomial according to (10).

Model Based Predictive Speed Control In Fig. 5 the cascaded FOC drive control structure is presented. The inner current controllers and the outer flux and speed controllers are shown in this block diagram. The purpose of this paper is to design a GPC speed controller that takes the torsional behavior of the long drive shaft into account. The following paragraphs summarize the MBPC speed controller synthesis. GPWM ( s ) =

Ω M ( z −1 ) b0 + b1 z −1 + b2 z −2 + b3 z −3 + b4 z −4 = M * ( z −1 ) a0 + a1 z −1 + a 2 z −2 + a3 z −3

αβ

uS*,β

e−skTs

uS ,β

VΨR 1 + sT Ψ R

iS , d

uS ,α VASM 1 + sTASM

iS , q

3 pLh ΨR 2 LR

M∗

ΨR,d

N ASM

ϑk

ΨR,calc d

ϑk

1

z

Fig. 5. Block diagram of proposed rotor-flux field-oriented drive control (dynamics denoted) with proposed GPC speed-controller emphasized

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As highlighted in Fig. 5 an actuating value limitation has to be taken into account to design a realizable GPC speed control for the drive system with torsional loads. There are two possible ways to implement the actuating value limitation to a MBPC algorithm. Either the limitation is considered in the formulation of the optimization problem as an additional constraint or the limitation is considered aposteriori during the implementation of the proposed GPC speed controller. If the actuating value limitation is considered in the formulation of the optimization problem to derive the GPC control law a constrained optimization problem has to be solved. This constrained optimization problem is very complex in its definition and therefore the implementation and the online computation effort to derive the optimal control law for the GPC speed control increases significantly. To reduce the implementation and the online computation effort but still be able to deal with actuating value limitations this constraint is implemented aposteriori to the speed control system in this paper. In [25] the theoretical background of implementing an actuating value limitation aposteriori to the MBPC system is presented. As it can be seen from Fig. 4 the calculation of the free system response is dependent on the change of the actual actuating value Δu. Therefore the proposed solution is to not only limit the actuating value but also limit the change of the actuating value needed to calculate the free system response. For the considered drive system the speed control actuating value is the q-component isq* of the reference stator-current. Therefore the aposteriori implementation of the actuating value limitation can be formulated cf. (11). The resultant structure of the GPC speed control is emphasized in Fig. 5. (11)

i

* Sq,bg

[

* * * * * * * ⎧iSq & Δ iSq , if iSq (t ) ∈ − iSq ,bg (t ) = iSq (t ) − iSq (t − 1) ,max , iSq, max ⎪* * * * * * = ⎨iSq,max & Δ iSq,bg (t ) = iSq,max − iSq (t − 1) , if iSq (t ) > iSq,max * * * * * ⎪- i * ⎩ Sq,max & Δ iSq,bg (t ) = −iSq,max − iSq (t −1) , if iSq (t ) < −iSq,max

control of a drive system with torsional loads based on restricted pole locations have been introduced. Further research based on this restricted pole locations is presented in [26]. In this work the PI speed control parameters are designed using a standard optimization method, the symmetric optimum criterion [27]. This method is based on the idea of determining the PI-controller parameters Kp and Ki in that way, that the frequency response from the setpoint to the plant output is as close to one as possible for low frequencies [28]. However, the default setting of symmetrical optimum is not appropriate for drive systems with elastic couplings [29]. The controller gain has to be reduced in order to avoid high stress of the mechanical parts. Therefore, the reset time Tr = KP/KI = 0.053 of the controller is kept constant whereas the proportional gain is reduced. Simulations and measurements have shown that satisfactory results can be achieved with the PI controller parameters presented in (12). GPI ( s) =

]

The proposed implementation of the actuating value limitation has two distinct advantages. First the implementation and the online computation effort is reduced to the same amount required when no constraint is considered in the GPC control law calculation. Second this solution remains the optimal solution in the context of the predictive control cost function formulation [25]. Taking the proposed aposteriori actuating value limitation network into account, an effective and practical implementation of the GPC speed control is achieved. PI-Based Speed Control As mentioned in the introduction of this paper, a PIspeed control will be used to compare with the resultant control dynamics of the GPC speed control. In [8] three possible tuning methods for a conventional PI-speed

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isq* (s) N

* ASM

(s) − N ASM ( s)

= Kp +

KI 138.1 = 7.36 + s s

T ABLE III SYSTEM P ARAMETERS AND USED N OTATION Parameter Notation Value Electronics Drive motor data: (used index: ASM) Type Squirrel-cage IM Rated power PASM 5.5 kW MASM Rated torque 36 Nm Rated speed 1455 RPM NASM fASM Rated frequency 50 Hz Load motor data: (used index: Load) Type Servo IM Rated power PLoad 6.4 kW MLoad Rated torque 24.6 Nm Rated speed 2490 RPM NLoad fLoad Rated frequency 85 Hz Mechanics Inertia drive motor 0.018 kgm2 Clutch data: (used index: Clutch) 0 to 10.51° Backlash WL (adjustable) JClutch 0.0158 kgm2 Inertia Drive shaft data: (used index: Shaft) 100 Nm Rated torque MShaft 0.0013 kgm2 Inertia JShaft Torsional stiffness cT 3094.2 Nm/rad Internal damping dT (estimated) 0.15 Nms/rad Inertia fly wheel JFW 0.11 kgm2 Inertia load motor JLoad 0.018 kgm2 Reduced System Parameters of Two-Inertia Oscillation Model Drive side data: 0.0338 kgm2 Inertia J*ASM Load side data: 0.1287 kgm2 Inertia J*Load Drive shaft data: 1700 Nm/rad Torsional stiffness c*T d*T (estimated) 0.15 Nms/rad Internal damping

(12)

65

65

PI-SC 60

50

/ rad/s

45

Load

60 55

GPC-SC

55 50 45 40

40

Simulation

Simulation 35 -0.05

0

0.05

0.1

0.15

0.2

0.25

35 -0.05

0

0.05

0.1

0.15

0.2

0.25

t/s (c)

t/s (a) 65

PI-SC 60 55

GPC-SC

50 45 40

Measurement 35 -0.05

0

0.05

0.1

0.15

0.2

0.25

t/s (b)

Fig. 6. Simulated (a,c,e) and meassured (b,d,f) step response (40 to 60 rad/s) for proposed PI- and GPC C-speed control (SC) Kp=7.36, KI = 138.1 (PI-SC C), Np=600, λ=10, δ=0.4 (GPC-SC), UDC=565 V, ΨR=0.6 Vs, imax=15 A, fs=3 kHz T ABLE IV M EASURED PERFORMANCE FOR PROPOSED PI- AND GPC-SPEED C ONTROL TUNED WITH S AME B ANDWIDTH A Drive-side Speed Current- and Torque Load Speed Performance Performance Speed Control max. max. max. PO# tr## ts### PO# (SC) dMShaft/dt MShaft isq,ref [%] [ms] [ms] [%] [Nm/ms] [A] [Nm] Small-signal behavior (Reference Sttep from 40 to 43 rad/s) UDC=565 V, ΨR=0.6 Vs, imax=15 A, A fs=3 kHz, cf. Fig. 6 PI-SC*

1.7

23.6

249.8

2.8

14.1

28.6

11.3

GPCSC**

0.7

27.5

248.5

2.2

10.3

21.3

2.6

T ABLE A V C ALCULATION T IM ME OF TOTAL FOC FOR DIFFERENT S PEED P C ONTROLLERS Calculation Prediction Samplinng Speed Control Time Horizon Frequency (SC) Method Np fs=1/Ts [kH kHz] tcalc [μs] PISC 3 13 200 29 2 400 34 600 72 GPCSC 200 35 3 600 73

Large-signal behavior (Reference Sttep from 40 to 60 rad/s) UDC=565 V, ΨR=0.6 Vs, imax=15 A, A fs=3 kHz, cf. Fig. 8 PI-SC*

3.6

94.5

225.6

4.7

14.3

42.5

12.5

GPCSC**

0.6

111. 5

137.8

1.3

14.3

44.5

7.5

# *

##

###

Percent Overshoot; Rise Time; Settling Time; T Definitions cf. [28] Kp=7.36, KI = 138.1, **Np=600, λ=10, δ=0.4

V. TEST BENCH DESCRIPTION, SIMULATION I AND MEASUREMENT RESSULTS Based on the presented plant modelinng and the control design the system has been simulated. The plant data were taken from the laboratory test setup. Thee simulation results are obtained with the MATLAB®/Sim mulink simulation environment. In the next paragraphss the test bench including the most important system parameters p will be introduced, and second, a comparative study between the PI-based and the GPC-based speed conttrol will be carried out with simulation and test data. Test Bench Description Fig. 7 presents a picture of the mechhanical part of the

Fig. 7. Mechannical part of test bench

laboratory test bench. The moost important electrical and mechanical system parameters are a summarized in Table III. Further the reduced system parameters relating the twoinertia system model are alsso presented in Table III (bottom). The mechanical resoonance frequency introduced by the long drive shaft of the laboratory system is approximately 41.3 Hz, wheere all side effects of the laboratory system are taken into account. The control algorithms are implemented onn a dSPACE DS 1103 board. To verify the proposed speed control methods, the loadside and the drive-side speeed signals are measured. Moreover the shaft torque MShhaft is measured to study the shafts stress during a speed refeerence step.

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20

15

/ rad/s ASM

isq,max 10

5

0

Simulation -5 -0.05

0

0.05

0.1

0.15

0.2

0.25

t/s (c) 35

Measurement

30

MShaft / Nm

25 20 15 10 5 0 -5 -0.05

0

0.05

0.1

0.15

0.2

0.25

t/s (f)

Fig. 8. Simulated (a,c,e) and measured (b,d,f) step response (40 to 43 rad/s) for proposed PI- and GPC-speed control (SC) Kp=7.36, KI = 138.1 (PI-SC), Np=600, λ=10, δ=0.4 (GPC-SC), UDC=565 V, ΨR=0.6 Vs, imax=15 A, fs=3 kHz

Comparative Study Both the GPC- and the PI-based speed controllers are tuned to have the same bandwidth of 8.7 Hz approximately. Further the presented results are divided into the smallsignal (actuating value not in limitation) and large-signal behavior (actuating value in limitation) to compare the resultant speed control performances. Fig. 6 presents the large-signal behavior for simulation and measurement of the drive system for a speed-reference step from 40 to 60 rad/s. In addition to that Fig. 8 presents the simulated and measured small-signal behavior for a speed-reference step from 40 to 43 rad/s. As it can be concluded from these figures the different speed control methods show very different control performances if the actuating value is less than the limitation. While the PIbased control acts abruptly to a control deviation, the GPC control acts much smoother to a control deviation. Therefore more torsional system oscillations can be seen if the PI-based speed control is used instead of the GPC speed-control. Further, the systems small-signal behavior changes if a GPC-speed-controller is used instead of a PIspeed controller. Table IV presents the evaluated control performances for the considered operating points of Fig. 6 and Fig. 8 respectively for each presented speed control method. Summarizing the presented simulation and measurement results as well as the evaluated control performance indices for the proposed PI-speed and GPC-speed control an improvement of the control performance can be seen for the use of the predictive speed control in comparison to the PI-based speed control. This is due to a reduced drive and load-side speed overshoot for every considered reference step. Further, the shaft stress relating to the maximum occurring shaft torque MShaft and the maximum occurring rate of change of the shaft torque dMShaft/dt can be reduced significantly for the use of GPC-speed control in

comparison to the PI-speed control (assuming both controllers are tuned to the same bandwidth). The achieved improvement of the GPC-speed control performance does result in a higher computation effort, which can be concluded from Table V. This table summarizes the total computation time for the whole FOC algorithm including all transformations, controllers and PWM calculations for both the PI-speed controller and the GPC-speed controller. The computation time of the FOC with GPC-speed control tcalc is approximately two to six times higher than the calculation time of the FOC with the PI-speed controller. This is due to the additional calculation effort introduced to calculate the free system response which increases extensively for higher prediction horizons Np. VI. CONCLUSION This work presents a practical approach to use a model based predictive speed controller in the well known fieldoriented control structure for a drive system with torsional loads. The mechanical system is described in detail and one method to model this system with reduced degrees of freedom is discussed. Based on the mechanical system model and the assessed computation effort, the model based predictive control method of Generalized Predictive Control (GPC) is chosen. The basic principles and the underlying equations of GPC are introduced. Further the actuating value limitation is implemented aposteriori to the speed control loop which leads to a significant reduction of online computation effort. The theoretical design of the predictive speed control is verified in simulations and measurements. Moreover a comparison of the predictivespeed control to a PI-speed control is done to highlight the control performance improvement of the predictive control approach. The proposed model based predictive speed control shows good performance in simulation as well as in measurement. With the use of the predictive speed control

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the torsional oscillations in the drive system can be reduced without decreasing the speed control dynamics. In comparison to the PI-speed control, an improved reference reaction of the controlled drive system with torsional loads is achieved. Moreover the mechanical stress on the drive components, especially the stress on the long drive shaft, is reduced significantly by the predictive speed control in comparison to the PI-based speed control. Although the online computation effort of the proposed predictive speed control is still higher than the online computation effort of the PI-based speed control, the final computational burden is not excessive. Combining the well known field-oriented control with the predictive speed control, a simple and realizable control for drive systems with torsional loads is achieved. The conventional speed controller can be replaced by the predictive speed controller whereas only the drive-side speed has to be measured.

[17]

[18] [19] [20] [21] [22] [23] [24]

REFERENCES [1] [2] [3]

[4]

[5] [6] [7] [8] [9] [10]

[11] [12] [13] [14]

[15]

[16]

Blaabjerg, F. and Thoegersen, P.: Adjustable speed drives --- future challenges and applications. In Proc. of IPEMC 04, vol. 1, pp. 36--45, Aug. 2004. Götting, G. and De Doncker, R.W.: Active drive control of electric vehicles using a modal state observer. In Proc. of PESC 04, vol. 6, pp. 4585-4590, June 2004. Hori, Y., Sawada, H. and Chun, Y.: Slow Resonance Ratio Control for Vibration Suppression and Disturbance Rejection in Torsional System. In IEEE Trans. on Industrial Electronics, vol. 46, no. 1, pp. 162-168, Feb. 1999. Salman, S.K. and Teo, A.L.J.: Windmill Modeling Consideration and Factors Influencing the Stability of a Grid-connected Wind Power-based Embedded Generator. In IEEE Trans. on Power Systems, vol. 18, no. 2, pp. 793-802, May 2003. Leonhard, W.: Control of electrical drives. 3.Edt. Berlin, Heidelberg, New York: Springer-Verlag, 2001. Kazmierkowski, M.P., Krishnan, R. and Blaabjerg, F.: Control in Power Electronics --- Selected Problems. 1.Edt. San Diago, London: Academic Press, 2002, pp. 176-190. Ellis, G. and Lorenz, R.D.: Resonant Load Control Methods for Industrial Servo Drives. In Proc. of IAC 2000, vol. 3, pp. 14381445, Oct. 2000. Zhang, G. and Furusho, J.: Speed Control of Two-Inertia System by PI/PID Control. In IEEE Trans. on Industrial Electronics, vol. 47, no. 3, pp. 603-609, June 2000. Nordin, M. and Gutman, P.-O.: Controlling Mechanical Systems with Backlash --- A Survey. Automatica, vol. 38, no. 10, pp. 16331649, Oct. 2002. Cortes, P., Kazmierkowski, M.P., Kennel, R.M., Quevedo, D.E. and Rodriguez, J.: Predictive Control in Power Electronics and Drives. In IEEE Trans. on Industrial Electronics, vol. 55, no. 12, pp. 43124324, Dec. 2008. Linder, A. and Kennel, R.: Model Predictive Control for Electrical Drives. In Proc. of PESC 05, vol. 1, pp. 1793-1799, June 2005. Shi, P., Liu B. and Hou, D.: Torsional Vibration Suppression of Drive System Based on DMC Method. In Proc. of WCICA 08, vol. 1, pp. 4789-4792, June 2008. Cutler, C.R. and Ramaker, B.C.: Dynamic Matrix Control --- A Computer Control Algorithm. In Proc. of Automatic Control Conference, San Francisco, 1980. Cychowski, M., Delaney, K. and Szabat K.: Low-Cost HighPerformance Predictive Control of Drive Systems with Elastic Coupling. In Proc. of EPE-PEMC 08, vol. 1, pp. 2241-2247, Sept. 2008. Cychowski, M., Szabat, K. and Orlowska-Kowalska, T.: Constrained Model Predictive Control of the Drive System with Mechanical Elasticity. In IEEE Trans. on Industrial Applications, vol. 56, no. 6, pp. 1963-1973, June 2009. Dresig, H.: Schwingungen mechanischer Antriebssysteme --Modellbildung, Berechnung, Analyse, Synthese.(German) (Oscillations in mechanical drive systems --- modeling, calculation,

[25] [26] [27] [28] [29]

T5-156

analysis, synthesis) 2.Edt. Berlin, Heidelberg, New York: SpringerVerlag, 2006, pp. 119-124, pp. 153-172. Di, P.N: Beitrag zur Reduktion diskreter Schwingungsketten auf ein Minimalmodell. (German) (Contribution for reduction of discrete vibration systems to a minimal model) dissertation, Aachen, RWTH, Dr.-Ing. Diss., 1981. Garcia, C.E., Prett, D.M. and Morari, M.: Model Predictive Control: Theory and Practice --- a Survey. In Automatica, vol. 25, no. 3, pp. 335-348, 1989. Krämer, K. and Unbehauen, H.: Aspects in Selecting Predictive Adaptive Control Algorithms. In Proc. of American Control Conference, pp. 2396-2401, June 1992. Camacho, E.F. and Bordons, C.: Model Predictive Control. 2.Edt. Berlin, Heidelberg, New York: Springer-Verlag, 2007. Clarke, D.W., Mohtadi, C. and Tuffs, P.S.: Generalized Predictive Control- Part I. The Basic Algorithm. In Automatica, vol. 23, no.2, pp. 137-148, 1987. Clarke, D.W., Mohtadi, C. and Tuffs, P.S.: Generalized Predictive Control- Part II. Extensions and Interpretations. In Automatica, vol. 23, no.2, pp. 149-160, 1987. Robinson, B.D. and Clarke, D.W.: Robustness effects of a Prefilter in generalised predictive control. In IEE Proc. of Control Theory and Applications, vol. 138, no.1, pp. 2-8, Jan. 1991. Schröder, D.: Elektrische Antriebe – Regelung von Antriebssystemen. 2.Aufl. Berlin, Heidelberg, New York: SpringerVerlag, 2001. Wang, L.: Model Predictive Control System Design and Implementation Using MATLAB®. Berlin, Heidelberg, New York,: Springer-Verlag, 2009, pp. 45-46. Thomsen, S. and Fuchs F.W.: Speed Control of Torsional Drive Systems with Backlash. In Proc. of EPE 2009 Barcelona, ISBN: 9789075815009, Sept. 2009. Bolognani, S., Venturato, A. and Zigliotto, M.: Theoretical and Experimental Comparison of Speed Controllers for Elastic TwoMass-Systems. In Proc. of PESC 00, vol. 3, pp. 1087-1092, 2000. Levine, W.S.: The Control Handbook. 1.Edt. Boca Raton, London, New York, Washington D.C.: CRC-Press, 1996, p. 158. Wertz, H., Beineke, S., Frohleke, N., Bolognani, S., Unterkofler, K., Zigliotto, M. and Zordan, M.: Computer aided Commissioning of Speed and Position Control for Electrical Drives with Identification of Mechanical Load. In Proc. of Industry Applications Conference 99, vol. 4, pp. 2372-2379, 1999.