Appropriate Tuning and Robust Design of a Generalized Predictive ...

Appropriate Tuning and Robust Design of a Generalized Predictive Speed Controller for Drive Systems with Resonant Loads Nils Hoffmann

Sönke Thomsen

Friedrich W. Fuchs

Student Member, IEEE [email protected]

Student Member, IEEE [email protected]

Senior Member, IEEE [email protected]

Christian-Albrechts-University of Kiel, Institute for Power Electronics and Electrical Drives Kaiserstraße 2, D-24143 Kiel, Germany Abstract – This paper presents an analysis of the appropriate tuning and robust control design for a model based predictive speed controller – the Generalized Predictive Controller (GPC) – used in a drive system with a resonant load. The resonance results from a finite stiffness of a long drive shaft. The achievable control quality of a model based predictive controller dependents on the appropriate choice of the control design parameters and the accuracy of the applied system model. If the predictive controller is not tuned adequately or unmodeled dynamics occur in the control system, the speed control performance must be reduced dramatically to avoid instability. To analyze these issues an overview about the influence of the GPC design parameters to the control performance is carried out. Moreover a possibility to tune the predictive control to be robust against model mismatches and control plant parameter uncertainties is presented in this paper. The theoretical approaches are verified in experiments with a 5.5 kW drive system. Finally, a study of the impact of backlash to the proposed speed control is presented.

algorithms has opened up many application areas in control of power electronics and electrical drives [3]. In general the achievable control performance of a model based predictive control algorithm depends on two aspects: First, the accuracy of the applied system model for the predictive control synthesis [4] and second, an appropriate tuning of the underlying control tuning parameters [5]. If the applied plant model for the predictive controller synthesis does not contain all relevant system dynamics or the predictive controller parameters are not chosen adequately the achievable control performance can be significantly decreased.

Index Terms—AC machine; Adjustable speed drive; Asynchronous motor; Control of Drive; Robust control; Robustness; Test bench. Fig. 1. Two-inertia torsional oscillation model of a drive-system with resonant loads

NOMENCLATURE M N J Ψ cT dT X* XS XN XDrive XShaft XLoad xˆ(t + k | t)

Torque Rotational Speed Inertia Magnetic Flux Torsional Stiffness Internal Damping Reference Value of Control-signal X Value of X referred to the Stator-side Nominal/Rated value of X Value of X referred to the Drive-side Value of X referred to the Shaft Value of X referred to the Load-side Predicted (at time t) value of x looking k sampling steps into the future

I.

INTRODUCTION

The basic principle of model based predictive control (MBPC) was known since the late seventies and was based on industrial research and development [1] [2]. Since that time further development of model based predictive control

978-1-4244-5287-3/10/$26.00 ©2010 IEEE

In Fig. 1 the two-inertia torsional oscillation model [6] of a drive-system with resonant load is presented. Usually the driving torque is transferred to the load side with a drive shaft that has non-ideal transmission behavior. This non-ideal driving torque transmission is mainly a consequence of a not negligible inertia, friction in the slide bearings and a finite material stiffness of the drive shaft. Based on the application this non ideal torque transmission can lead to serious problems in the drive system such as torque impulses or mechanical vibrations caused by torsional shaft oscillations. Applications where these aforementioned problems could be a matter of particular interest are electric vehicle [7], rolling mill [8] or windmill applications [9]. A combination of both the model based predictive control approach and the underlying two-inertia oscillation model can lead to a proper speed control design that takes the torsional dynamics of the drive mechanics into account. In [10]-[12] it is analyzed that such predictive speed control approaches result in high dynamic speed control performances with reduced stress of the mechanical and

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electrical drive system components in comparison to conventional mechanical resonance suppression approaches, e.g. [6], [13] or [14]. To achieve these control performance requirements without decreasing the whole speed control dynamics a well tuned and robust model based predictive speed control must be designed. Since the late seventies many different new, modified or combined model based predictive control algorithms have been developed by both the academic and industrial research communities. The wide range of these predictive control approaches and their main distinctive features is the objective of many survey studies, e.g. [15]-[17]. Among other model based predictive control algorithms the Generalized Predictive Control (GPC) seems to be a very promising and well explored approach due to the underlying system model and the possibility of an analytical calculation of the control law [18] [19]. Furthermore in [20] it is proven that the online calculation time of a GPC speed controller is short enough for use in drive applications. In a further study of model predictive control for electrical drives [21] it is emphasized that the GPC approach is very attractive because it is not necessary to solve very complex linear programming (LP) or quadratic programming (QP) problems online. One main drawback (which is also highlighted in [21]) of the original GPC approach are the not considered system constraints during the analytical (offline) solution of the optimization problem which results in the final applied control law. Once the system constraints (e.g. an actuating value limitation) are neglected during the control law calculation the achievable predictive control performance is not optimal for every system state. Therefore in [11] as well as in [21] model predictive control (MPC) algorithms are presented where a complex piecewise affine optimization problem is solved online taking several system constraints into account. This formulation of the model based predictive control as an explicit MPC approach is done to reduce the online computation effort of solving the resultant constrained optimization problem.

This paper aims to produce a new solution to these aforementioned problems. Based on the model based predictive control approach of GPC an aposteriori implementation of an actuating value limitation is added to the proposed speed control loop to realize a practical control concept. Based on this theoretical background the influence of the GPC control design parameters will be studied in theory and in experiments. Moreover a possibility to transform the time-based GPC approach into a frequencydomain based transfer function expression to carry out robustness and stability analysis is presented in this paper. Based on a sensitivity function analysis in the frequencydomain, in addition to a plant parameter uncertainty analysis, unstructured robustness of the proposed GPC speed control will be proven. Beyond that the structured robustness of the predictive speed control is studied in relation to an unmodeled backlash of a clutch in the mechanical drive system. This paper is structured as follows: In chapter II the underlying torsional load model and the main concept of the proposed GPC approach are summarized. In chapter III and IV the appropriate tuning and the robust design respectively are discussed in detail. Experimental results are given in chapter V to verify the theoretical analysis, ending with a conclusion in chapter VI. II. TORSIONAL LOAD MODEL, CONTROL STRUCTURE & THE PREDICTIVE SPEED CONTROL APPROACH The underlying concept of the chosen Generalized Predictive Control (GPC) was first published by Clarke et al. [22] [23]. The GPC approach is composed of three main parts: A prediction equation which is derived from an assumed process model, a cost function to penalize future output errors and control actions respectively. Finally a control law based on the prediction equation and cost function is obtained using the idea of predicted free and forced system responses [4]. These three main parts of the GPC algorithm will be briefly introduced in the following paragraphs. Further the implementation of the actuating value limitation will be introduced and discussed.

1 J Load

1 JDrive Δα

(a)

(b)

Fig. 2. Block diagram of (a) the driveshaft with mechanical elasticity and (b) the proposed rotor-flux oriented dive control structure

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The underlying CARIMA (Integrated Controlled AutoRegressive Moving Average) process model of the GPC approach is presented in (1) whereas A describes the denominator polynomial of the open loop plant transfer function and B describes the numerator polynomial respectively. As it will be highlighted in chapter IV the polynomial T can be treated as a design polynomial to filter higher-frequency disturbances caused by model mismatch. Furthermore u(t) describes the control sequence; y(t) describes the process output sequence; e(t) describes the process distortion; z-1 the backshift operator and d the dead time of the system. e(t ) (1) A( z −1 ) y (t ) = z −d B ( z −1 )u (t − 1) + T ( z −1 ) Δ (2) Δ = 1 − z −1 Fig. 2 (a) presents the block diagram of the mechanical drive system including a drive shaft with finite stiffness and internal damping. To obtain this model the shafts inertia is neglected as well as a possible non-linear system behavior due to friction in the slide bearings or a backlash in the drive clutches. The associated linear transfer function of the mechanical drive system is summarized in (3). As it can be concluded from (1) the GPC approach requires the formulation of the systems transfer function in Z-domain. The discrete transfer function Gd(z-1) relating to the continuous transfer function G(s) is obtained by using the transformation law presented in (4). In (4) Z denotes the Z-transformation, L-1 the inverse Laplace-transformation and Ts the applied sample time for discretization. N 1 G(s) = Drive = M Drive s J Drive + J Load

(

Gd ( z −1 ) =

dT 2 J Load +s cT cT J Drive J Load d 1+ s T + s 2 cT cT ( J Drive + J Load ) 1+ s

)

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

Gd ,PWM ( z −1 ) =

(

)

U s ( d , q ) ( z −1 ) U S* ( d ,q ) ( z −1 )

= z −k ≈ Gd ,CC ( z −1 )

(3)

(4)

processor (5). To simplify the GPC speed control synthesis the closed loop current control dynamics GCC are approximated as a dead time element, cf. (5). This simplification is based on the assumption that the bandwidth of the current control is tuned to be much higher than the bandwidth of the speed control. Therefore the inner current control and the outer speed control can be treated as decoupled loops. In the underlying CARIMA model (1) this simplification can be easily included with the choice of d equal to the dead time of the PWM-updating and sampling routine. yˆ (t + k t ) = Fk ( z −1 ) y f (t ) + E k ( z −1 ) B ( z −1 ) Δ u f (t + k − 1)

(6)

1 1 (7) y (t ) and Δu f (t ) = Δ u (t ) T ( z −1 ) T ( z −1 ) Taking these simplifications into account the GPC prediction equation can be derived from the CARIMA process model and the discretized two-inertia oscillation model. The GPC prediction equation of the systems output behavior y (here: the motors rotational speed NDrive) is shown in (6) whereas the polynomials Ek and Fk are derived from a Diophantine equation cf. [22]. Moreover the control actuating value u (here: the q-component of the stator reference current is,q*) and the actual measured system output y are weighted with the T design-polynomial, cf. (7) and (8) respectively. 1 Δ u f (t ) = iS ,q * (t ) − iS ,q * (t − 1) T ( z −1 ) (8) 1 y f (t ) = N ( t ) Drive T ( z −1 ) y f (t ) =

[

]

To obtain a control law based on the presented prediction equation (6) a cost function is defined (9). From (9) it can be seen that the quadratic cost function takes the predicted control error into account as well as the control signal change to reach the commanded value. Further the future reference trajectory w can be established freely. J ( N1 , N 2 , N u ) =

N2

∑δ ( j) [ yˆ (t + j t ) − w(t + j ) ]

j = N1

(5)

Nu

+ ∑ λ ( j )[Δu (t + j − 1) ]

2

(9)

2

j =1

To create a model based predictive speed control that can be used in a wide range of drive applications that already exist in the field the well known rotor-flux field-oriented control (R-FOC) structure [24] is chosen for controlling the motor speed NDrive. The proposed R-FOC control structure is presented in Fig. 2 (b). R-FOC consists of cascaded control loops with inner current control and outer speed and flux control respectively. In this paper the synthesis of the PIbased flux and current control is not discussed, further explanation can be found in literature, e.g. in [24]. The inner current control contains a dead time introduced by the PWM-updating and sampling routine of the DSP

u(t ) = u(t − 1) + K T ( w − f )

(10)

A complete mathematical description of obtaining the GPC control law from the presented prediction equation (6) and the cost function (9) using the principle of the free and forced system response can be found in literature, e.g. in [18]. In (10) the resultant control law of the GPC approach is summarized. It is worth mentioning that the actual actuating value u(t) is depended on the former actuating value u(t-1) and a vector K derived as the solution of the optimization problem as well as on the future reference trajectory w (here: assumed to be constant) and the free system response f.

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T ABLE I SUMMARIZATION OF GPC TUNING PARAMETERS Notation Parameter N1 Minimum cost horizon N2 Maximum cost horizon Np Prediction horizon Nu Control horizon Command-weighting sequence δ Control-weighting sequence λ T(z-1) Filter polynomial

Fig. 3. GPC speed control with proposed actuating value limitation network

As stated in the introduction of this paper the next step of the GPC speed control synthesis is an aposteriori implementation of the actuating value limitation. This aposteriori implementation is mainly to reduce the online computation effort needed for the proposed GPC speed control algorithm. If this actuating value limitation would be considered during the formulation of the optimization problem (9) to calculate the optimal future system behavior (i.e. the optimal sequence of future actuating values) a constrained optimization problem would result. Such a constrained optimization problem is complex in its definition and the required online computation effort would increase significantly if no additional measures (e.g. shown in [11] or [21]) are taken into account. (11)

[

]

* * * * * ⎧iS*,q (t) &ΔiSq ,if iS*,q(t)∈ −iSq ,Lim(t) = iS,q (t) −iS,q (t −1) ,max,iSq,max ⎪* * * * * * iSq ,if iS*,q(t) >iSq ,Lim(t) = ⎨iSq,max &ΔiSq,Lim(t) = iSq,max−iS,q (t −1) ,max * * * * * ⎪-i* Δ = − − − < − & i ( t ) i i ( t 1 ) , if i ( t ) i Sq Sq Lim Sq S q S q Sq , max , , max , , ,max ⎩

parameters to tune the speed control performance. These tuning parameters are summarized in Table I. In the following paragraphs the influence to the speed control performance and the choice of these parameters will be discussed in detail. A. Minimum and Maximum Cost Horizons The choice of the minimum (N1) and maximum (N2) cost horizons can be explained intuitively. The cost horizons define the observation window where the future (predicted) control deviation is penalized by the GPC cost function weighting factor δ cf. (9). Therefore the minimum value of N1 is defined by the dead time d of the considered control plant. It is unnecessary to weight control deviations during the dead time because a possible change of the actuating value cannot affect the output of the control system. The number of dead time sampling points d for the proposed control plant is constant. Thus for the sake of simplicity the minimum cost horizon is chosen to be equal to this dead time d cf. (12). (12) N =d 1

Bearing in mind the classical GPC approach [22] the solution of the optimization problem is obtained offline. Therefore the calculation effort during the online calculation is significantly reduced and mainly determined by the vector multiplication presented in (10). In [25] one possibility to include such an actuating constraint during the implementation of the predictive controller is presented. The main idea is not only to limit the actual actuating value, but also the change of the actuating value which is needed to calculate the free system response has to be limited cf. (11). With this simple modification of the classical GPC approach the proposed speed control will be able to deal with actuating value constraints without a significant reduction of the control performance. As stated in [25] this solution is in fact an optimal solution in the context of the predictive control cost function definition with actuating value limitation. The modified GPC speed control with the proposed limitation network is illustrated in Fig. 3.

The maximum cost horizon N2 has to be less to or equal to the defined prediction horizon Np. In [18] it is recommended to select to maximum cost horizon equal to the prediction horizon (13). The following paragraphs will demonstrate that this choice of N2 does not reduce the achievable speed control performance. (13) N =N 2

B. Control and Prediction Horizon The control horizon Nu was already defined implicitly during the control law derivation. To derive the optimal actuating value u(t) via the control law (10) only the previous actuating value u(t-1) is taken into account. Using this calculation rule the control horizon was implicitly set equal to one (14). Further simulations and experiments have shown that using a control horizon greater than one increases the implementation and calculation effort significantly while the speed control performance is only increased marginal. (14) N =1 u

III. APPROPRIATE TUNING OF THE GPC SPEED CONTROL

TDrive =

The GPC approach offers based on the underlying process model (1) and the defined cost function (9), seven design

p

J Drive N N ,Drive M N ,Drive

, TLoad =

J Load N N , Drive M N , Drive

, TC =

M N , Drive cT N N , Drive

(15)

The most important factors for determining the prediction horizon Np are the applied sampling time Ts for the speed

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control, the available processing power and the time constants of the considered control plant. In (15) the characteristic time constants of the considered torsional mechanical drive system are presented. To guarantee a proper speed control performance the prediction horizon Np should be chosen relative to the slowest time constant of the mechanical system [22]. Taking this requirement into account the minimal required amount of sampling points Np,min (sampling time Ts) that has to be considered in the prediction can be calculated relating to (16). In (16) it is assumed that the load side time constant TLoad is the slowest mechanical time constant. max(TDrive , TLoad , TC ) TLoad (16) N p ,min ≈ = TS TS N p,max = f ( tcalc )

(17)

T ABLE II E XAMPLE OF M INIMUM AND M AXIMUM PREDICTION HORIZONS Minimum Prediction Maximum Prediction Sampling Horizon Horizon Frequency fs [kHz] Np,min. /tcalc [μs] Np,max /tcalc [μs] 2 150 / 29 1000 / 113 3 200 / 35 600 / 73

In [5] it is emphasized that the achievable control performance is increased for an increased prediction horizon (Np≥Np,min). More precisely, with an increasing prediction horizon the settling time tst of the speed control increases slightly while the percentage overshoot OSmax decreases. Therefore the maximal prediction horizon Np,max is limited only to the available online processing power. In Table II an example of the (rounded) minimal and maximal predication horizons as well as the overall calculation time tcalc to derive on control step for the proposed test drive system (detailed explanation of test bench cf. [12]) is summarized.

C. Command and Control-Weighting Sequence The weighting sequences δ(j) and λ(j) are used to penalize a predicted future control deviation as well as a possible future change of the actuating value in the cost function (9). Moreover it is possible to vary those sequences over the prediction horizon. In the following steps it is assumed that these weighting sequences are chosen to be constant (18).

λ ( j) ≡ λ > 0 and δ ( j ) ≡ δ > 0

(18)

The weighting factor δ penalizes future control derivation. Thus for an increasing value of δ (assuming λ to be constant) the actuating energy increases when trying to reduce the control derivation. This leads to an increasing maximum overshoot as well as a decreasing settling time tst of the speed control. The weighting factor λ penalizes changes of the actuating value. Therefore an increasing value of λ (assuming δ to be constant) leads to a decreased maximum overshoot and an increased settling time of the speed control.

IV. ROBUST DESIGN BASED ON FREQUENCY DOMAIN ANALYSIS In general the achievable quality of a predictive control depends on the quality of the applied system model. If the system is not modeled accurate enough the control performance will decrease drastically. This can lead in the worst cases to unstable control operation points when structured plant uncertainties like not-considered nonlinearities or unmodeled dynamics accrue in the controlled system. Additionally, unstructured uncertainties have to be taken into account during the control synthesis when physical parameters of the system vary during the operation or they are not exactly determinable. Therefore it is desirable to design the proposed GPC speed control which is robust against these aforementioned uncertainties of the control plant. The first step for the robust speed control design is to transfer the time-based GPC equations into the frequency domain. In [4] the chosen approach to express the closed-loop relationships based on a transformation of the GPC control law to a classical pole placement structure is presented. In Fig. 4 (a) the resultant pole-placement structure is illustrated whereby the polynomials S(z-1) and R(z-1) are obtained by the transformation law (cf. [4]), the T(z-1) filter-polynomial remains unaffected. Basically, as stated in [27], the prediction error caused by model mismatches relating high frequencies can be influenced by the filter-polynomial T(z-1), the lower frequency disturbances are removed by the integrative term Δ in the underlying CARIMA process model, cf. (1). The high frequency disturbances are of practical interest because they contain unmodeled system dynamics (e.g. PWM-saturation effects, non-linear sampling effects or measurement noise). With a proper selection of T(z-1) it is possible to reject these high frequency disturbances. Moreover the filter-polynomial T(z-1) can be treated as a design parameter to influence the robust stability of the predictive control [4]. The next step of the robust predictive speed control design is a proper selection of the filter-polynomial. Although some approaches to select the filter-polynomial exist in literature, no unified guideline has been completely established until now [4]. Therefore for the proposed approach the selection of the filter polynomial is divided in two basic steps. Based on [27] the basic form of the polynomial will be chosen. In the next step the influence of this selected filter-polynomial form to the critical speed controls sensitivity functions for different choices of the filter parameters will be studied. This leads to a final selection of T(z-1). To verify this filter-polynomial selection the influence of uncertain load inertia JLoad to the pole-zero locations of the closed-loop system will be carried out. Further, to prove the controls structured robustness, an additional backlash which was not considered in the control synthesis will be introduced to the mechanical system plant.

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T ΔR

0,004 0.1 0.08

S T

0 0.06 -0,002

z∞,2

0.04 -0,004

(a)

Stability Bound

z∞ ,1

0,002

(b)

0.02 0.968

0.97

0.972

0.974

0 -0.02

z∞* , 2

-0.04 -0.06 -0.08

z ∞* ,1

Z-Plane

-0.1 0.94

0.96

0.98 Real Part

1

(a) (d) (c) Fig. 4. Sensitivity function analysis for differeent filter polynomials T(z-1): (a) Equivalent pole-placement structure and sensitivvity to: (b) actuating value noise, (c) multiplicative plant uncertainty and ((d) plant output noise

0.994

0.9852

0.99 0.9848 0.986 0.5

1

JLoad,real JLoad,nominal

1.5

0.5

(b)

1.5

(c)

f a uncertain load inertia Fig. 6. Robustness analysis for JLoad: (a) Pole-Zero Map for varied v load inertia, (b) and (c) change of absolute valu ue for critical pole-pairs

MShaft / Nm

Fig. 5. Mechanical part of the test-bench in the institutes laboratory with an adjustable backlash gap WL of the hiighlighted clutch

1

JLoad,real JLoad,nominal

Operation Point: UDC=565 V, ΨR=00.6 Vs, iS,max=8 A, fs=3 kHz, Np=600, λ=20, δ=0.4, T(z-1)=(1-0.97z-1)2, MLoad=0 Nm Fig. 7. Measured step response characteristics for proposed GPC speed control with additional backlash (backlash h gap WL=2.52°): (a) Rotational motor speeed NDrive, (b) speed control actuating value iS,q* and (c) shaft torque MShafft T ABLE III SYSTEM P ARAMETERS 44 V Value 44 Parameter Notation Drive motor: Squirrel-cage IM 43 43 5.5 kW Rated power PDrive 2 42 =0.4 =0.4 42 JDrive 0.03338 kgm Motor-side inertia (incl. clutch) =1 =1 41 41 Load motor: Servo IM =5 =5 40 40 6.44 kW Rated power PLoad 2 JLoad 0.12887 kgm Load-side inertia 39 39 -0.05 0 0.05 0.1 0.15 -0.05 0 0.05 0.1 0.15 (incl. torque senor t/s t/s and flywheel) Clutch: Operation Point: UDC=565 V, ΨR=0.6 Vs, iS,max=15 A, T((z-1)=(1-0.97z-1)2, MLoad=0 Nm 0 to 10.51° Backlash Gap WL (adjustable) Fig. 8. Measured step response of Fig. 9. Measured step response of Shaft: rotational motor speed for a different rotationaal motor speed for a different 17000 Nm/rad Stiffness cT weighting factor δ weighting factor d dT (estimated) 0.15 N Nms/rad Internal damping (fs=3 = kHz, λ=10, Np=600) (fs=3 kHz, λ=10, Np=200)

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A. Robustness Relating Unstructured Uncertainties In [27] it is emphasized that the form of T(z-1) should be chosen according to (19). The choice of τ depends on the required stability reserve for the critical high frequencies and the applied sampling time Ts. For the proposed control system τ is chosen to be equal to 0.97. To determined the degree n of T(z-1) a sensitivity function analysis is used. (19) T ( z −1 ) = (1 − τ z −1 ) n A complete description for a sensitivity function analysis for a model based predictive controller can be found in [5]. Based on the equivalent pole-placement structure (cf. Fig. 6 (a)) of the GPC in frequency-domain the sensitivity of the proposed GPC speed control relating multiplicative plant gain uncertainties SG, actuating value noise Sun and output measurement noise Syn can be derived, cf. (20)-(22). B( z −1 )S ( z −1 ) z −1 (20) S G ( z −1 ) = −1 R( z )ΔA( z −1 ) + B( z −1 )S ( z −1 ) z −1 T ( z −1 ) S ( z −1 ) A( z −1 ) R( z −1 )ΔA( z −1 ) + S ( z −1 ) B( z −1 ) z −1

(21)

T ( z −1 ) R( z −1 )ΔA( z −1 ) R( z −1 )ΔA( z −1 ) + S ( z −1 ) B( z −1 ) z −1

(22)

S un ( z −1 ) = − S yn ( z −1 ) =

The different gain responses of these aforementioned sensitivity functions for different choices of the degree n (n=0, 1 and 2) for the filter-polynomial T(z-1) is shown in Fig. 4 (b)-(d). From Fig. 4 (c) it can be concluded that the sensitivity against high frequency multiplicative uncertainties is significantly decreased for higher degrees n of T(z-1). Further the sensitivity against measurement noise of the systems output (here: the measured drive speed NDrive) Fig. 4 (d) and the sensitivity against actuating value noise (here the q-component of the stator reference current is,q*) Fig. 4 (b) is decreased. Therefore to ensure a high robustness and thus a low sensitivity relating to higher frequency parameter uncertainties the degree of the filter-polynomial T(z-1) is selected to be equal to two (n=2). To conclude the analysis of the robust design of the proposed GPC speed control relating unstructured plant uncertainties the selected filter-polynomial T(z-1)=(1-0.97z-1)2 is used to study the theoretical robustness of the control against an uncertain load inertia JLoad. Therefore the controller design is done with an assumed nominal load inertia JLoad,nominal. Afterwards the load inertia of a theoretical applied system plant JLoad,real is varied (in the range of 0.5≤JLoad,real/JLoad,nominal≤1.5) whereby the nominal controller design is kept unchanged. The theoretical achieved closedloop pole-zero map for the varied ratio of JLoad,real to JLoad,nominal (the arrowhead points to an increasing value) is presented in Fig. 6 (a). In Fig. 6 (b) and (c) respectively the absolute values of the critical pole-pairs are illustrated for the differed ratio of JLoad,real to JLoad,nominal. From these figures it can be concluded that the closed loop GPC speed control remains stable with a adequate distance to the stability

boundary (|z|=1) for the assumed variation range of the ratio JLoad,real to JLoad,nominal.

B. Robustness Relating Structured Uncertainties To analyze exemplarily the robustness of the proposed GPC speed control relative to structured uncertainties an additional clutch backlash (with an adjustable backlash gap WL) was added to the system cf. Fig. 5. This backlash was not considered during the control synthesis. In Fig. 7 the measured speed control performance with an applied backlash gap WL of 2.52° for a speed reference step from 40 to 50 rad/s and back to 40 rad/s is presented. The effect of the additional backlash in the clutch is shown when the drive system decelerates from 50 to 40 rad/s. The deceleration is caused by a negative drive torque MDrive* which results from the negative q-component of the stator reference current iS,q* commanded by the GPC speed control. Once the negative driving torque is applied to the mechanical system the clutch’s backlash opens and after the backlash gap is passed through the backlash is closed again. Depending on the considered operation point and the width of the backlash gap this process of closing and opening the backlash can be repeated several times. Further the achieved speed control performance is decreased significantly due to an additional oscillation in speed NDrive Fig. 7 (a) and a high shaft stress caused by high values of the shafts torque MShaft Fig. 7 (c). Nevertheless the GPC speed control remains stable and the commanded speed values are reached in the considered operation point even though the backlash was not considered in the GPC speed control synthesis. V. EXPERIMENTAL RESULTS Although one experimental result has already been presented during the robustness analysis relating structured plant uncertainties, in this chapter the test bench setup and more experimental results for the proposed GPC speed control will be shown. Fig. 5 presents a picture of the mechanical part of the laboratory test bench. The most important system parameters of the test bench are summarized in Table III. The mechanical resonance frequency of the laboratory system is approximately 40.3 Hz, where all side effects are taken into account. The control algorithms are implemented on a dSPACE DS 1103 board. To verify the proposed predictive speed control the load-side and the drive-speed signal and the shaft torque are measured. The influence of the command-weighting factor δ to the measured step response is presented in Fig. 8 and in Fig. 9 respectively for two different prediction horizons Np and a constant control-weighting factor λ equal to 10. From these figures it can be concluded that the theoretical interpretations of the influence of the command-weighting factor δ and the prediction horizon Np to the control performance are shown in practice. This is due to an increasing control overshoot and a decreasing control settling time for bigger values of δ in

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both presented measurements. Further for an increasing prediction horizon Np the achieved control quality increases which is due to a slightly deceased control overshoot and a smoother step response behavior in general for a higher value of the prediction horizon.

[6] [7] [8]

VI. CONCLUSION This paper summarizes the speed control synthesis for a model based predictive control approach – the Generalized Predictive Control (GPC) – used in a drive system with resonant load behavior. A simple approach for an aposteriori implementation of an actuating value limitation of the speed control is shown as well as an implementation of the predictive speed control in a rotor-flux oriented drive control structure. Further the GPC tuning parameters and their influence to the achievable control performance are presented. One approach to overcome the problems introduced by high frequency model mismatches to the GPC closed-loop speed control is presented. Based on a sensitivity function analysis the design filter-polynomial T(z-1) is chosen in a way which leads to a robust predictive speed control. The robust design is proven by a pole-zero map study with an uncertain load inertia. Furthermore measurements with an additional backlash in the mechanical system which was not considered in the control synthesis are presented to examine the system’s robustness against these structured model uncertainties. The presented theoretical and practical studies emphasize the ability of the GPC speed control to decrease the mechanical vibrations caused by torsional shaft oscillations without decreasing controller dynamics. Further, the GPC approach was shown to be very robust against structured and unstructured uncertainties. The addition of both the GPC speed control and the well known rotor-flux oriented drive control leads to a control concept that can be used for drive systems with torsional loads that already exist in the field.

[9]

[10] [11]

[12] [13] [14] [15] [16] [17] [18] [19] [20]

ACKNOWLEDGEMENT This work has been partly founded by the German Research Foundation (DFG). REFERENCES [1] [2] [3]

[4] [5]

Richalet, J., Rault, A., Testud, J.L. and Papon, J.: Algorithmic Control of Industrial Processes. In 4th IFAC Symposium on Identification and System Parameter Estimation, Tbilisi USSR, 1976. Cutler, C.R. and Ramaker, B.C.: Dynamic Matrix Control --- A Computer Control Algorithm. In Proc. of Automatic Control Conference, San Francisco, 1980. 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. 4312-4324, Dec. 2008. Camacho, E.F. and Bordons, C.: Model Predictive Control. 2.Edt. Berlin, Heidelberg, New York: Springer-Verlag, 2007, p. 13. Rossiter, J.A.: Model-Based Predictive Control – A Practical Approach. Boca Raton, London, New York, Washington D.C.: CRCPress, 2003, pp. 85-101.

[21] [22] [23] [24] [25] [26] [27]

771

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. 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 Powerbased Embedded Generator. In IEEE Trans. on Power Systems, vol. 18, no. 2, pp. 793-802, May 2003. 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. 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. Hoffmann, N., Thomsen, S. and Fuchs, F.W.: Model based Predictive Speed Control of a Drive System with Torsional Loads --- A practical Approach. In Proc. of EPE-PEMC 10, Ohrid, Macedonia, Sept. 2010. Ellis, G. and Lorenz, R.D.: Resonant Load Control Methods for Industrial Servo Drives. In Proc. of IAC 2000, vol. 3, pp. 1438-1445, Oct. 2000. Nordin, M. and Gutman, P.-O.: Controlling Mechanical Systems with Backlash --- A Survey. Automatica, vol. 38, no. 10, pp. 1633-1649, Oct. 2002. De Keyser, R.M.C., Van De Velde, Ph.G.A. and Dumortier, F.A.G.: A Comparative Study of Self-adaptive Long-range Predictive Control Methods. Automatica, vol. 24, no. 2, pp. 149 – 163, 1988. Garcia, C.E., Prett, D.M. and Morari, M.: Model Predictive Control: Theory and Practice – a Survey. 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. Bordons, C. and Camacho, E.F.: A Generalized Predictive Control for a Wide Class of Industrial Processes. In IEEE Trans. on Control System Tech., vol. 6, no. 3, pp. 372 – 387, May 1998. Clarke, D.W.: Application of Generalized Predictive Controlto Industrial Processes. In IEEE Control System Magazine, vol. 8, no.2, pp. 49 – 55, April 1988. Kennel, R. Linder, A. and Linke, M.: Generalized Predictive Control (GPC) – Ready for Use in Drive Applications?. In Proc. of Power Electronics Specialists Conf, PESC 2001, vol. 4, pp. 1839 – 1844, June, 2001. Linder, A. and Kennel, R.: Model Predictive Control for Electrical Drives. In Proc. of PESC 05, vol. 1, pp. 1793-1799, June 2005. 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. 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. Wang, L.: Model Predictive Control System Design and Implementation Using MATLAB®. Berlin, Heidelberg, New York,: Springer-Verlag, 2009, pp. 44-47. Levine, W.S.: The Control Handbook. 1.Edt. Boca Raton, London, New York, Washington D.C.: CRC-Press, 1996, p. 158. 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.