Parameter Estimation and Model Based Control Design of ...

Parameter Estimation and Model Based Control Design of Drive Train Systems

MATS TALLFORS

Licentiate Thesis Stockholm, Sweden 2005

TRITA-S3-REG-0502 ISSN 1404-2150 ISBN 91-7283-969-4

KTH Signaler, Sensorer och System SE-100 44 Stockholm

SWEDEN

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan framlägges till offentlig granskning för avläggande av teknologie licentiatexamen fredagen den 25 februari 2005 i sal Q2, Osquldas väg 10, Kungliga Tekniska Högskolan, Stockholm. © Mats Tallfors, 2005 Tryck: Universitetsservice US AB

Abstract The main control task in many speed-controlled drives is to eliminate or reduce the load speed error caused by the load torque disturbance and reduce oscillations as quickly as possible. This thesis addresses different aspects of identification and control of such resonant elastic systems. In most industrial applications it is not practical to measure the load speed. Instead, we advocate model based control design that optimizes load speed while using motor speed as the feedback signal. For this to be possible one needs a mechanical model of the system and we suggest finding the mechanical parameters by estimation from experimental data. Hence a method has been developed which finds the mechanical parameters, including backlash, through a series of three dedicated experiments. At first this procedure is developed for the situation of one manipulated input, the motor torque, and one measured output, the motor speed. For drive systems with a very large motor in comparison to the load, it becomes very difficult to estimate all mechanical parameters from motor speed measurements only. An alternative estimation method has been developed for this purpose, using an additional sensor for the shaft torque. One more rather specific control problem is treated in the thesis, namely for drive systems with tandem coupled motors, where control structures have been developed with and without an extra sensor for shaft torque. Keywords: Control, Drive train, Estimation, Identification, Mechanical, Modelling, Shaft torque, Tandem, Two-mass system

Acknowledgements

First of all I want to thank ABB for the financial support of my research. I also want to show my gratitude to the two persons who have helped me and supported me from the very start of this work and all the way to this thesis: I want to thank Mattias Nordin for all his optimism and fantastic ability to explain complicated theories with simple examples. Also, big thanks to Alf Isaksson, who have been a great inspiration and support with his wide experience of control related problems. Furthermore, I want to thank the persons who have shown up during the way: A thanks goes to Håkan Hjalmarsson for his help to finish this thesis. Also, thank you goes to Magnus Hahlin and Per-Olov Gelin for creating an opportunity for me to complete this work. I have met a great deal of skilled persons on ABB during the years. I want to thank them for sharing their experience and interest in their work. I also want to thank all the nice people on the control department on S3 for all their kindness and helpfulness during my short visits. Finally I want to thank Bertil Hök, who once planted the idea that I should apply for a PhD program Västerås, Januari 2005 Mats Tallfors

Contents 1 Introduction 1 1.1 Outline and contributions 3 1.2 Relation to previous work 4 2 Control of elastic resonant systems 7 2.1 Models ..8 2.2 Performance 10 3 Selection of Optimization Criterion 13 3.1 Robustness 15 3.2 Models 16 3.3 Selection of Optimization Criterion 20 3.4 Conclusions 24 4 Multivariable control on tandem coupled motors 25 4.1 Models 28 4.2 Control Configurations 30 4.3 Optimization Criterion 31 4.4 Results 32 4.5 Conclusions 40 5 Identification of mechanical parameters 43 5.1 Background 43 5.2 Two-mass linear model 44 5.3 Black-box model 45 5.4 Identification of static gain 48 5.5 Grey-box identification 49 5.6 Backlash estimation 49 5.7 Experiment design 50 5.8 Summary and software tool 56 5.9 Real data example 56 5.10 Conclusions 60

Contents

6 Identification with shaft torque measurements 63 6.1 Static test 65 6.2 Dynamic test without gap openings 66 6.3 Dynamic test with gap openings 70 6.4 Implementation 71 6.5 Results 72 6.6 Conclusions 88 7 Conclusions 89 7.1 Future work 90 Bibliography 93

Chapter 1 Introduction This thesis addresses different aspects of identification and control of resonant elastic systems. These systems can be found in numerous speed-controlled applications in industry, such as rolling mills, cement mills, paper machines, large fans and industrial robots. The main control task is to eliminate or reduce the load speed error caused by the load torque disturbance and reduce oscillations as quickly as possible. This is in order to improve the quality of the produced product (e.g. steel thickness in a rolling mill). Reduced oscillations will dramatically decrease the wear and thus prolong the life length of the equipment significantly. The fact that there is often an anti-resonance close to the resonance, and that significant backlash often is exhibited, makes this a challenging control problem. Over the years many different control approaches have been applied to this problem. However, for industrial practice PI control remains the dominant method. For the design of the PI parameters a number of tuning methods are available. Increasingly model-based algorithms for tuning are preferred. Many of them correspond to optimizing some performance criterion. Therefore as a first topic in this thesis we have chosen to evaluate and compare different optimization criteria for the tuning of PI parameters.

2 1 Introduction

The comparison has been made with respect to both performance as well as stability robustness. Often only one measured quantity, the motor speed, is used for feedback control. What we want to control, however, is the load speed. In most industrial applications it is not practical to measure the load speed. Instead in this thesis we advocate model based control design that optimizes load speed while using motor speed as the feedback signal. For this to be possible a mechanical model of the system is required. Traditional black-box input-output models will not be sufficient. It has been assumed in this thesis that the majority of drive systems can be sufficiently well represented by a two-mass mechanical model. The traditional way of obtaining the mechanical data has been to find them via data sheets. This is often a cumbersome and time-consuming exercise, and, especially for the backlash, will never give an up to date picture of the real state of the drive. Instead we here suggest finding the mechanical parameters by estimation from experimental data. A method has been developed which finds the mechanical parameters, including backlash, through a series of three dedicated experiments. At first this procedure is developed for the situation of one manipulated input, the motor torque, and one measured output, the motor speed. For drive systems with a very large motor in comparison to the load, it becomes very difficult to estimate all mechanical parameters from motor speed measurements only. The system looks almost like a one-mass system, with very little oscillations apparent in the motor speed. To obtain good results some additional measurement signal is necessary. Of course measurement of load speed would be very useful, but, is as already remarked above, typically not realistic. Therefore tests have been made on a test-rig using one manipulated input, the motor torque, and two measured outputs, the motor speed and the shaft torque. An identification procedure has then been developed for this particular combination of measured signals. One more specific control problem is treated in the thesis, namely drive systems with tandem coupled motors. As a final investigation we also study how big an improvement would result if the motors use separate

1 Introduction 3

controllers with feedback from the same motor speed sensor and also with an extra feedback signal, the shaft torque.

1.1 Outline and contributions In relation to the literature this thesis has the following main contributions:

A An evaluation and comparison of classical optimization criteria for this particular application.

B Two new control concepts for tandem coupled motors. C New three stage experimental design with time domain estimation

of mechanical parameters. D Parameter estimation using shaft torque measurements.

Contribution A is presented in Chapter 2 and Chapter 3. This contribution has also been published as M. Tallfors, M.C. Nordin, and A.J. Isaksson (2003). Selection of Optimization Criterion for Speed Controlled Resonant Elastic Systems, The Fifth International Conference on Power Electronics and Drive Systems, Singapore. Contribution B is presented in Chapter 4, and has also been published as M. Tallfors, M.C. Nordin, A.J. Isaksson (2004). Multivariable Control on Tandem Coupled Motors in Rolling Mills. 11th IFAC Symposium on Automation in Mining, Mineral and Metal processing, Nancy, France. Contribution C is presented in Chapter 5. A shorter version of this contribution was also published in A.J. Isaksson , R. Lindkvist, X. Zhang, M.C. Nordin and M. Tallfors (2003). Identification of Mechanical Parameters in Drive Train Systems, 13th IFAC Symposium on System Identification, Rotterdam, The Netherlands.

4 1 Introduction

Contribution D corresponds to the most recent research and is presented in Chapter 6. This contribution has not yet been published at a conference or in a journal

1.2 Relation to previous work Control design based on optimisation of performance indices has a long history, see for example (Newton et al. 1957), but is also presented in more recent textbooks, such as (Dorf and Bishop 1995). There are also many publications making more general comparisons of the performance of different tuning methods for PI and PID controllers. See, for example, (Isaksson and Graebe 1999) or (Kristiansson 2003). Such comparisons are typically broader than the evaluation made here and aimed at finding the best method for a wide range of transfer functions. To our knowledge this is the first comparison of PI controller designs, focusing specifically on selection of optimization criteria for drive train systems. The control of tandem coupled motors with only speed measurements is an example of a 2x1 control system. Such systems can be found in other applications. For example, Allison and Isaksson (1997) studies so called mid-ranging control, where typically a set point of 50 percent is used for one input and where the other input deals with the transient. What is special in this application is the particular requirement for the steady state motor torques to be equal. When feedback is made from shaft torque as well, the system has two inputs and two outputs, and for such multivariable control problems there are many standard methods, see the books (Skogestad and Postlethwaite, 1996) or (Goodwin et al. 2001) for example. However, we have not seen any treatment of the specific problem of tandem coupled motors in the literature. Moreover, the particular contribution here is to use a control configuration with very few design parameters. To directly estimate parameters in a model derived from first principles is an established field within system identification, called either semi-

1 Introduction 5

physical modelling or grey-box identification. See for example the books by Bohlin (1991) or Ljung (1999). To identify the mechanical parameters for drive systems has been studied before using frequency domain techniques in, for example, (Nordin and Bodin 1995) and (Hovland et al. 1999). This appears to be the first application of time-domain techniques and the use of dedicated experiments is new for drive systems (but has certainly been used for other problems). Finally, identification using more than one output signal is of course not new. However, as far as we have been able to search the literature, to use the combination of motor speed and shaft torque for drive system identification is novel.

Chapter 2 Control of elastic resonant systems As mentioned in the previous Chapter, speed controlled resonant elastic systems can be found in numerous applications in industry, such as rolling mills, cement mills and paper machines. In this Chapter speed control under load torque disturbance is treated. Especially rolling mills may serve as an example of an application where load torque disturbance rejection may be of primary concern. Servo performance is usually tuned with a pre-filter in a two-degree of freedom controller and is not treated here. In the literature many different approaches have been applied to the control of drive train systems. The most common method used is still some version of PI control, see e.g. (Brandenburg and Schäfer 1989), (de Santis 1994), (Ji and Sul 1995), (Joos and Sicard 1992), (Koyoma and Yano 1991), and (Pollman et al. 1991). Albeit PI control has its limitations, see e.g. (Baril and Galic 1994), (Hori et al. 1994), and (Koyama and Yano 1991), we have here chosen to focus our comparison on this controller type.

8 2 Control of elastic resonant systems

2.1 Models There are several benefits if a model of the closed loop system is available with sufficient accuracy. Firstly it is easy to select tuning parameters that make measured motor speed look good at speed steps and load torque disturbances, secondly it is possible to see how load speed and shaft torque behave during working conditions. Also from a model it is possible to calculate robustness and sensitivity to different disturbances like misalignments and motor torque ripple. Figure 2.1 presents a general control configuration for a speed-controlled drive train, where

Tl load torque [Nm] Tm motor torque [Nm] ωm motor speed [rad/s] ωl load speed [rad/s] P process with drive train and converter K controller

P

K

ωlΤl

ωmΤm

Figure 2.1:General control configuration of speed controlled drive train.

Mechanical two-mass model Mechanical systems are here represented by two-mass systems consisting of motor and load inertia, a connecting shaft, a speed sensor on the motor, a torque actuator (here an electric drive system, called converter) and a speed controller, see Figure 2.2. This is a usual model for two-mass systems, see (Nordin 2000) for example.

2 Control of elastic resonant systems 9

K(s) C(s)-

+

MotorSpeed

ωωωωm

MotorTorque

Tm

SpeedReference

ωωωωref

Load TorqueDisturbance

Tl

MotorJm cm

LoadJl cl

Shaftks cs

LoadSpeed

ωωωωl

Controller Converter

Mechanical systemG(s)

Figure 2.2: Speed controlled motor connected to load with an elastic shaft. Load speed is usually not measured on industrial systems.

The model of the mechanical system, G is defined by the following dynamic equations

(2.1)

where

mJ the motor moment of inertia [kgm2]

mc the viscous motor friction [Nm/(rad/s)]

lJ the load moment of inertia [kgm2]

lc the viscous load friction [Nm/(rad/s)]

sk the shaft elasticity [Nm/rad]

sc the shaft damping [Nm/(rad/s)] The following transfer functions form the input-output relations,

(2.2)

=

=

l

m

lllm

mlmm

l

m

l

m

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T

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sGsG

T

TsG

)()(

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TkcccJ

ωωθθωωω

θωωω

−=

−++−=+−++−=

&

&

&

)(

)(


Converter The converter is modeled as

(2.3)

representing the characteristic time delay, rise time and overshoot of an electric drive system. Notice that it is non-minimum phase. Controller PI controllers are very common for speed control applications. The PI controller here is parameterized by the transfer function

(2.4)

2.2 Performance The quality of load torque disturbance rejection of the closed loop system can be measured in several ways. Three performance factors are used in this thesis: overshoot, settling time and impact drop (ID), see Figure 2.3. The impact drop of motor speed is a common performance criterion of systems that relates to, but is not exactly the same as the integrated absolute error. Small ID is important especially when several drives are connected through the processed material e.g. the rolled steel. There an ID means that the tension and mass flow will change. Impact drop is here defined as

∫=t

tdtteID

0

)(max ,

where e(t) represents motor speed error em(t)= ωref(t)- ωm(t) or load speed error el(t)= ωref(t)-ωl(t) for a load torque step. The unit of ID is here

)2())2(()( 2222 ωζωωωτ +++−= ssssC d

K(s)=Kp(1+1/(sTi)).

2 Control of elastic resonant systems 11

expressed in rad/Nm. ID is the maximum angular error compared to an ideal angle given from reference speed when a torque step of one Nm is applied to the load. ID is sometimes expressed in terms of speed error in percentage of nominal speed integrated over time when a step of nominal torque is applied to load. Nominal speed and nominal torque is in that case collected from the ratings of the converter. Another difficulty is that usually the design of a system is judged from motor speed measurements of the real process that is affected by the load torque disturbance. Figure 2.4 illustrates that ID on motor speed and ID on load speed may differ for cases with light load and low Ti/Kp. The x-axis in the contour plot indicates the Ti value of the controller and the y-axis indicates the Kp value. In the contour plots, Each iso-line represents a constant impact drop (solid lines) or a constant maximum sensitivity (dashed lines, values in dB to the right). Both the axis and the iso-lines are logarithmically scaled. Impact drop of motor speed decreases almost uniformly with Ti/Kp until the system eventually becomes unstable in the left figure. The right figure shows that impact drop of load speed on the other hand first decreases rapidly like the motor speed when Ti/Kp is decreased, and then levels out around 0.00004 rad/Nm until the system eventually becomes unstable. Actually, the improvement in load speed impact drop levels out at some level when Ti/Kp is decreased, while motor speed impact drop does not. The model in the example above is tuned according to “Light Load”, IAE criterion in Table 3.2 and the drive train is parameterized according to “Light Load” in Table 3.1 using the mechanical model in (2.1). The converter (2.3) has been parameterized according to Section 3.2. However, ID alone does not say anything about damping of the closed loop system poles. The overshoot and settling time indicates how oscillatory the system is. Oscillating behavior is one of the most important aspects that the operators and process engineers find to be bad. One reason is that oscillations may open backlashes and increase the wear in the systems. A “good” control design therefore needs to take this aspect into consideration.


Figure 2.3: Performance factors. impact drop, overshoot and settling time.

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Figure 2.4: Isoline plot of impact drop of motor and load speed as function of Ti and Kp (solid lines) for light load case. Maximum sensitivity = 12 and 6 dB are also indicated by dashed lines.

Impact Drop on Motor Speed Impact Drop on Load Speed

Chapter 3 Selection of Optimization Criterion The purpose of this Chapter is to compare classical minimization criteria with respect to load disturbance rejection of speed controlled resonant elastic systems. The purpose of the comparison is to investigate what “good control” of these systems is, as opposed to finding the minimum of an arbitrarily selected optimization criterion. In many cases the “optimum” solution, i.e. the solution with minimized optimization criterion is judged to be “bad” by the naked eye of an experienced process engineer. They usually use rules of thumb or their professional skill when manually tuning the system, based on measurements of motor speed. Here we try to find criteria that also give good control with respect to such performance measures. In the examples in Figure 3.1 we see a very fast response, but with a tendency to oscillate to the left. In the same figure we see a slower, but smother response to the right. Imagine a case there the faster response is fulfilling the contract of a maximum impact drop of 3×10-5 rad/Nm with a broad margin, but the slower example is barely making it. Here however, the experienced process engineer would probably prefer the smoother one. The model in the example above is tuned according to “Heavy Load”, IAE criterion in Table 3.2 and the drive train is parameterized according to

14 3 Selection of optimization criterion

“Heavy Load” in Table 3.1 using the mechanical model in (2.1). The converter (2.3) has been parameterized according to Section 3.2. Another interesting factor is that in most industrial applications only the motor speed is measured, but the relevant process signal is in fact the load speed. Therefore the design that gives the best directly measurable performance is not necessarily the best design for the process.

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Figure 3.1: Two examples of load torque disturbance rejection. Left: Fast but oscillating response. Right: Slow, but smooth response.

In this Chapter we have chosen to study criteria with respect to “good control”. Three selected criteria have been calculated for load speed error, el(t)=ωref(t)-ωl(t), where ωref(t) is the speed reference and ωl(t) is the load speed. The criteria are (e(t)= el(t)):

• Integrated error

∫∞

=0

)( dtteIE ,

• Integrated squared error

3 Selection of optimization criterion 15

∫∞

=0

2 )( dtteISE

• Integrated absolute error

(3.1)

3.1 Robustness Since the models are based on simplifications and assumptions, robustness considerations need to be introduced. For example, shaft stiffness may vary with load, due to non-linear behavior of material, backlash in couplings or gearboxes etc, see (Nordin 2000), (Chestnut and Mayer 1955) and (Cosgriff 1958). Some of the selected optimization criteria give robustness for some cases. IE always give unstable solutions without constraints. ISE and IAE give sufficient robustness by themselves only for some cases with light load. There are several methods to achieve robustness. One possibility is to include motor torque reference in the criterion. Other common methods are based on limiting the closed loop response for different kinds of disturbances. In, for example (Åström and Hägglund 1995), it is suggested to use the maximum sensitivity

(3.2)

The sensitivity function is defined as

S(s)=1/(1+L(s)) where L(s) is the loop transfer function

L(s)=Gmm(s)C(s)K(s). Gmm(s) is the transfer function of the mechanical system from motor torque to motor speed according to (2.2), C(s) is the converter according to (2.3)

∫∞

=0

)( dtteIAE

Ms=max|S(iω)|


and K(s) is the controller according to (2.4). In the sequel we have used the constraint Ms≤2, i.e. an upper limit of 6 dB is assumed to give acceptable robustness.

3.2 Models The same models as in Section 2.1 have been used. Converter The converter model (2.3) has been parameterized with the following values:

• ω=433 [rad/s] • ζ=0.69 • τd=0.002 [s]

representing the characteristic time delay, rise time and overshoot of a DC-drive system. Notice that it is non-minimum phase. The values vary with the implementation of the DC-drive control. Mechanical two-mass model Three cases of two mass systems according to the model (2.1), have been studied:

• GL, light load with Jl=Jm/4, • GM, medium load with Jl=Jm and • GH, heavy load with Jl=4Jm.

Mechanical data for the three cases are presented in Table 3.1. The resonance frequency of all systems is 18.8Hz. Characteristic for systems with light load and low damping is that it is difficult to create well-damped closed loop poles. Also, when robustness aspects are considered, it does not pay to use a high order controller. With heavy load


on the other hand, a high order controller may increase the performance considerably, and still take robustness into account. The Bode plot in Figure 3.2 shows the frequency response from motor torque to motor speed for the three cases.

Description Symbol [unit]

Light Load GL

Medium Load GM

Heavy Load GH

Motor inertia Jm [kgm2] 20 20 20 Load inertia Jl [kgm2] 5 20 80 Motor dampning cm [Nm/(rad/s)] 0.006 0.006 0.006 Load damping cl [Nm/(rad/s)] 0.0015 0.006 0.0240 Shaft stiffness ks [Nm/rad] 56000 140000 224000 Shaft damping cs [Nm/(rad/s)] 56 140 224 Resonance freq. [Hz] 18.8 18.8 18.8 Relative damping [%] 5.92 5.92 5.92

Table 3.1: Mechanical data for the three two mass systems


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agni

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]

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102

103

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45

90

Pha

se [d

eg]

Frequency [rad/sec]

Light LoadMedium LoadHeavy Load

Figure 3.2: Bode plot from Motor Torque to Motor Speed of the three mechanical systems according to Table 3.1.

The sensitivity to model variations in the closed loop system depends on which parameter that is varied. Because we have chosen to select Ms as robustness criterion, it is possible to observe Ms in Nichols plots in order to see how it varies with variations in model parameters. Ms corresponds to the iso-lines in the Nichols plot. The iso-lines in the figures are marked with –6, -3 ,1 0, 1, 3, 6, 12 and 20 dB. The open loop gain, L(s) and the phase are presented as a line, beginning with low frequencies in the top of the figure and high frequencies in the bottom. The larger level in the iso-lines that is touched by the open loop line L(s), the larger Ms. A variation of ± 20% in shaft stiffness and load inertia for PI tuned closed loops (controller parameters according to IAE criterion in Table 3.2) are presented in the Nichols chart in Figure 3.3. In this case, Ms is not affected much by the applied variations (see how L(s) touches the Ms=6 dB curve, which is the higest value of Ms in this plot, independently of the applied


variations) for any loop transfer function. These variations illustrate what happens if load inertia is varied. This is usual in e.g. rolling mills, when work roll changes are preformed and when shaft stiffness varies with working load (i.e. the shaft torque operating point). Other parameter variations have more impact on the maximum sensitivity. An increase in τd from 0.002 s to 0.004 s corresponds to moving the RHP zero from 1000 rad/s to 500 rad/s. This variation of τd corresponds to uncertainty in delay in the converter. Jm is decreased to 80% of its original inertia. Shaft stiffness and damping are here modified in order to keep resonance frequency and damping constant. The variations are presented in a Nichols chart in Figure 3.4. Ms is very sensitive to these variations. The variation in motor inertia illustrates that the system is more sensitive to error in motor inertia than error in load inertia. Also, any variations of static gain (not presented in the figure) in the converter directly causes variations of robustness corresponding to moving the Nichols chart up or down. One interesting conclusion is that usually the performance of the converter is very important. See also (Baril and Galic 1994).

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Figure 3.4: Nichols chart of loop transfer functions with added model variations. τd is increased from 0.002 s to 0.004 s and Jm is decreased to 80% with corresponding shaft modification in order to keep resonance frequency and damping constant.

3.3 Selection of Optimization Criterion The results from optimization of the criteria are presented in Figure 3.5-Figure 3.7. The upper plots in the figures show isobars for the criteria and the lower plots show load torque step response. The performance factors are presented for each criterion in the boxes between isobars and load torque step responses. The results from optimization of the criteria for the light load case are presented in Figure 3.5. The impact drop does not vary significantly. ISE has the lowest overshoot with IAE on second place. The IAE has by far the shortest settling time. The author’s opinion is that the ISE and IAE are good designs. Here the robustness constraint is not active for ISE and IAE.


The results from optimization of the criteria for the medium load case are presented in Figure 3.6. IE has a slightly smaller impact drop than the two others, but has very big overshoot and settling time. Here both IAE and ISE work well. The robustness constraint is active for all criteria. The results from optimization of the criteria for the heavy load case are presented in Figure 3.7. The impact drop of the IAE is larger than the two others. On the other hand the overshoot and settling time is much shorter, making it the best overall design in the author’s opinion. The robustness constraint is active for all criteria.

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Ti [s]

Kp

IE

2e-0

05

4e-0

05

8e-0

05

0.0001 6dB

12dB

UnstableArea

← Min: 2.54e-005

0 0.05 0.1 0.15 0.2-2

-1.5

-1

-0.5

0

0.5

1x 10

-3

Time [s]

Spe

ed [r

ad/s

]

IE

Impact Drop = 3.81e-005 rad/Nm Overshoot = 41.6 % Settling Time = 0.142 s


0.01 0.02 0.05 0.1 0.2

1500

2000

3000

4000

6000

Ti [s]

Kp

ISE8e

-008

8e-008

6dB

12dB

UnstableArea

← Min: 5.63e-008

0 0.05 0.1 0.15 0.2-2

-1.5

-1

-0.5

0

0.5

1x 10

-3

Time [s]

Spe

ed [r

ad/s

]

ISE



0.01 0.02 0.05 0.1 0.2

1500

2000

3000

4000

6000

Ti [s]

Kp

IAE

8e-005

0.00

016

0.00

016

6dB

12dB

UnstableArea

← Min: 5.15e-005

0 0.05 0.1 0.15 0.2-2

-1.5

-1

-0.5

0

0.5

1x 10

-3

Time [s]

Spe

ed [r

ad/s

]

IAE



Figure 3.5: Optimization results on light load.


0.01 0.02 0.05 0.1 0.2

1500

2000

3000

4000

6000

Ti [s]

Kp

IE

1e-0

05

2e-0

05

4e-0

058e

-005

0.00016

6dB

12dB

UnstableArea

← Min: 1.53e-005

0 0.05 0.1 0.15 0.2-8

-6

-4

-2

0

2

4x 10

-4

Time [s]

Spe

ed [r

ad/s

]

IE



0.01 0.02 0.05 0.1 0.2

1500

2000

3000

4000

6000

Ti [s]

Kp

ISE

1e-0

08

2e-0

0 8

2e-008

4e-008

8e-008

6dB

12dB

UnstableArea

← Min: 1.12e-008

0 0.05 0.1 0.15 0.2-8

-6

-4

-2

0

2

4x 10

-4

Time [s]

Spe

ed [r

ad/s

]ISE



0.01 0.02 0.05 0.1 0.2

1500

2000

3000

4000

6000

Ti [s]

Kp

IAE

4e-0

05 4e-0

05

8e-005

8e-0

05

0.00016 6dB

12dB

UnstableArea

← Min: 2.46e-005

0 0.05 0.1 0.15 0.2-8

-6

-4

-2

0

2

4x 10

-4

Time [s]

Spe

ed [r

ad/s

]

IAE



Figure 3.6: Optimization results on medium load.

0.01 0.02 0.05 0.1 0.2

1500

2000

3000

4000

6000

Ti [s]

Kp

IE

1e-0

05

2e-0

05

4e-0

058e

-005

0.00016 6dB

12dB

UnstableArea

← Min: 1.53e-005

0 0.1 0.2 0.3 0.4-4

-3

-2

-1

0

1

2x 10

-4

Time [s]

Spe

ed [r

ad/s

]

IE



0.01 0.02 0.05 0.1 0.2

1500

2000

3000

4000

6000

Ti [s]

Kp

ISE

5e-009

1e-008

2e-008

4e-008

8e-008

6dB

12dB

UnstableArea

← Min: 7.37e-009

0 0.1 0.2 0.3 0.4-4

-3

-2

-1

0

1

2x 10

-4

Time [s]

Spe

ed [r

ad/s

]

ISE

Impact Drop = 2.55e-005 rad/Nm Overshoot = 35 % Settling Time = 0.354 s


0.01 0.02 0.05 0.1 0.2

1500

2000

3000

4000

6000

Ti [s]

Kp

IAE

2e-005

4e-0

05

8e-005

8e-0

05

0.0001 6dB

12dB

UnstableArea

← Min: 3.64e-005

0 0.1 0.2 0.3 0.4-4

-3

-2

-1

0

1

2x 10

-4

Time [s]

Spe

ed [r

ad/s

]

IAE



Figure 3.7: Optimization results on heavy load.


Case Crit-erion

Kp Ti [s] Impact Drop [rad/Nm]

Over Shoot [%]

Settling-time [s]

IE 1936 0.01464 3.811×10-5 41.63 0.142 ISE 1634 0.02442 3.875×10-5 20.70 0.126

Light Load

IAE 1809 0.02139 3.847×10-5 28.39 0.064 IE 1948 0.01586 2.303×10-5 50.14 0.276 ISE 2641 0.04097 2.360×10-5 6.444 0.096

Medium Load

IAE 2669 0.04478 2.391×10-5 2.718 0.068 IE 2270 0.02458 2.299×10-5 50.33 0.526 ISE 2466 0.03554 2.548×10-5 34.96 0.354

Heavy Load

IAE 2606 0.05590 3.028×10-5 16.93 0.282

Table 3.2: Controller values from the minimum criterions at the three cases according to Table 3.1.

Some of the criteria are more difficult to calculate than others. IAE have to be calculated numerically from time domain functions. ISE problems can be formulated as least squares problems with analytical solutions (Newton et al. 1957) and IE is simply Ti/Kp (Åström, and Hägglund 1995). The constraint in maximum sensitivity also complicates the search for the minimum solution. One candidate approach in order to maintain robustness, but without any explicit robustness constraint is to include a time weighting in the criterion, e.g.

∫∞

=0

2 )( dttetSEIT kk

However, Figure 3.8 shows an example where k=8 for the heavy load case. IT8SE on load speed does not give sufficient robustness. Also Ti turns to be quite large, around 0.075 s, compared to the Ti presented for ISE and IAE in Table 3.2.


0.01 0.02 0.05 0.1 0.2

1500

2000

3000

6000

Ti [s]

Kp

IT8SE

1e-015

1e-015

1e-0

151e-015

1e-0 12

1e-012

1e-0

1 21e

-012

1e-009

1e-0

091e

-009

1e-006

6dB

12dB

UnstableArea

← Min: 1.1e-017

Figure 3.8: Contour plot for IT8SE criterion of load speed error as a function of Kp and Ti for heavy load (Jl = 4Jm). The optimal solution is marked with a + sign. The interpretation of the contour plot is presented in the remark to Figure 2.4.

3.4 Conclusions Generally, it is difficult to find one “perfect” criterion that handles all cases in the best way. A major conclusion is that if one criterion should be chosen, IAE calculated for load speed is the criterion that is most general among the tested criteria. These tunings also look best to the naked eye. Also, if the load is known to have low inertia compared to the motor, then it is not necessary to use a constrained search. If the load is known to be heavy compared to the motor inertia, however, a robustness constraint is needed. Another finding is that, when manually tuning a controller with light load it might be tempting to decrease Ti/Kp more than necessary. The actual load speed impact drop improvement actually levels out at some value of Ti/Kp. Other alternative criteria candidates have been evaluated, and rejected. For example, a time weighted integrated squared error (ITkSE) criterion may help to get rid of the robustness constraint, but unfortunately sometimes still leads to insufficient robustness and bad performance of the optimized systems.

Chapter 4 Multivariable control on tandem coupled motors Today most rolling mills are using DC-motors. Since the motor lifetime is longer than the converter or electronics lifetime, the converter is commonly replaced or revamped and the motor is kept. Therefore it is interesting to study how improved control, applied to existing motors and mechanics, can improve performance. Many of these DC-motors are tandem coupled and suffer from insufficient dynamic performance, due to increased production and quality requirements. This is especially true in hot rolling mill applications such as Wire Rod Mill block drives, shear drives, Hot Strip Mill main stands and various types of reversing mills, where the load torque goes from zero to maximum torque in tens of milliseconds. Today the industry standard is a PI controlled single-input single-output (SISO) system with one speed sensor and the two motors in a master-slave configuration as shown in Figure 4.1. In this Chapter two distinctly new control configurations are proposed for tandem coupled DC motors. The results are compared to standard SISO configuration results.

26 4 Multivariable control on tandem coupled motors

The proposed controller structures are based on standard P/PI-controllers. These controllers are industrial standard and well known for process engineers who may need to tune the controllers by hand. Extra emphasis is placed on using configurations that can be widely used in the industry in the future. Therefore, the performance of advanced control strategies, like H∞ or LQG, is not investigated. Note that no effort is made to optimize speed step performance. The proposed control configurations are not optimized in order to handle load sharing between the motors at speed reference steps. There are several possible solutions for that case, using two degree of freedom, but they are not investigated here. We will study:

1. Multi-input single-output (MISO): Existing sensor configurations, but one separate controller for each motor, instead of the hard master-slave connection as in Figure 4.2.

2. Multi-input multi-output (MIMO): An additional shaft torque sensor is used in conjunction with one separate controller for each motor. These sensors are becoming cheaper and cheaper and are increasingly used also for supervision systems. Measured shaft torque is used for feedback control as in Figure 4.3.

The proposed control configurations are not optimized in order to handle load sharing between the motors at speed reference steps. There are several possible solutions for that case, but they are not investigated here.

4 Multivariable control on tandem coupled motors 27

K1(s) C(s)-

+

Motor 1Speed

ωωωωm

Motor 1Torque

Tm1

SpeedReference

ωωωωref


Tl

Motor 1Jm1 cm1

LoadJl cl

LoadShaft

ks2 cs2

LoadSpeed

ωωωωlPI

ControllerConverter


C(s)

Converter

Motor 2Jm2 cm2

MotorShaft

ks1 cs1

Motor 2Torque

Tm2

Figure 4.1: Standard master-slave connection (SISO).

K1(s) C(s)-+

Motor 1Speed

ωωωωm

Motor 1Torque

Tm1

SpeedReference

ωωωωref


Tl

Motor 1Jm1 cm1

LoadJl cl

LoadShaft

ks2 cs2

LoadSpeed

ωωωωlPI



C(s)

Converter

Motor 2Jm2 cm2

MotorShaftks1 cs1

Motor 2Torque

Tm2

K2(s)

PIController

Figure 4.2 Separate motor controllers (MISO).

K1(s) C(s)-

+

Motor 1Speed

ωωωωm

Motor 1Torque

Tm1

SpeedReference

ωωωωref


Tl

Motor 1Jm1 cm1

LoadJl cl

LoadShaftks2 cs2

LoadSpeed

ωωωωlPI



K3 C(s)

PController

Converter

LoadShaft

TorqueTls

Motor 2Jm2 cm2

MotorShaft

ks1 cs1

Motor 2Torque

Tm1

Figure 4.3: Extra torque sensor (MIMO).


4.1 Models Mechanical model Multi-mass models have been studied extensively. See for example: (Butler et al. 1992), (Honjyo and Watanabe 1975), (Gudehus 1983), (Nordin 2000), (Stamerjohanns and Rohde, 1983), (Tessendorf. 1992), (Weisshaar et al. 1998), (Wright 1976) and (Hori et al. 1994). The mechanical system in Figure 4.1 - Figure 4.3 is defined by

(4.1)

where

ωm1 motor 1, speed [rad/s] ωm2 motor 2 speed [rad/s] ωl load speed [rad/s] Tm1 applied motor 1 torque [Nm] Tm2 applied motor 2 torque [Nm] Tl applied load torque [Nm]

G can be calculated from the dynamic equations

(4.2)

=

l

m

m

l

m

m

TT

TsG

2

1

2

1

)(ω

ωω

lms

mms

lsslslmsll

mssss

lsmsmmmsmm

mssmsmsmmm

TkcccJ

Tkk

cccccJ

TkcccJ

ωωθωωθ

θωωωθθ

ωωωωθωωω

−=

−=

−++−=+−+

+++−+=+−++−=

22

211

2222

22211

221211122

11121111

)(

)(

)(

&

&

&

&

&


where

Jm1 the motor 1 moment of inertia [kgm2] Jm2 the motor 2 moment of inertia [kgm2] Jl the load moment of inertia [kgm2] cm1 the viscous motor 1 friction [Nm/(rad/s)] cm2 the viscous motor 2 friction [Nm/(rad/s)] cl the viscous load friction [Nm/(rad/s)] ks1 the shaft motor 1 – motor 2 elasticity [Nm/rad] ks2 the shaft motor 2 – load elasticity [Nm/rad] cs1 the shaft motor 1 – motor 2 damping [Nm/(rad/s)] cs2 the shaft motor 1 – motor 2 damping [Nm/(rad/s)]

Three different mechanical cases have been selected, representing a wide range of different rolling mill applications. The main factors affecting the results are the relations between inertias and shaft stiffness. The following three cases where the improvement is significant have been tested:

• G1: Motors and load have same inertia and viscous friction. Both shafts have same stiffness and damping. See Figure 4.4

• G2: Load inertia and viscous friction is four times larger than the inertia and viscous friction on each motor. Both shafts have same stiffness and damping. See Figure 4.5

• G3: Load inertia and viscous friction is four times larger than the inertia and viscous friction on each motor. The shaft between motor 2 and the load has double stiffness and damping compared to the shaft between the motors. See Figure 4.6

Motor 1 Load

LoadShaft

Motor 2

MotorShaft

Figure 4.4: Sketch of G1.


Motor 1 Load

LoadShaft

Motor 2

MotorShaft


Motor 1 Load

LoadShaft

Motor 2

MotorShaft


Cases with very high shaft stiffness between motors, weak shaft to load or small load inertia are not investigated. The complete mechanical data for the three cases are presented in Table 4.1. Electrical model The main dynamics of the electrical system comes from the converter dynamics. Here, the converter is modeled as in (2.3), with the parameter values according to the converter in Section 3.2. The nominal motor power rating in this investigation is 6.5 MW and the nominal speed is 630 rpm: The nominal torque is accordingly 100 kNm.

4.2 Control Configurations For all evaluated control configurations, the speed controller K1(s) is a PI-controller with the transfer function K1(s)=Kp1(1+1/(sTi1)) using a speed sensor on motor 1 as input and the converter to motor 1 as output. The different control configurations are as follows:

• SISO: K1(s) is used on both converters/motors, see Figure 4.1.


• MISO: In the MISO configuration shown in Figure 4.2, K1(s) is used for converter/motor 1. K2(s) is also a PI-controller, K2(s)=Kp2(1+1/(sTi2)) with speed sensor signal from motor 1 as input and converter/motor 2 as output. K2(s) is restricted in such way that Kp2=KKp1 and Ti2=KTi1, where K is a tuning parameter, used in order to achieve equal load on both motors at steady-state.

• MIMO: The MIMO case is shown in Figure 4.3. K1(s) is here complemented with a gain K3 using feedback from torque sensor between motor 2 and load. K3 is set to 0.5 in order to distribute load torque approximately equal between both motors at steady state for a load disturbance.

Symbol [unit] G1 G2 G3 Jm1 [kgm2] 1×103 1×103 1×103 Jm2 [kgm2] 1×103 1×103 1×103 Jl [kgm2] 1×103 4×103 4×103 cm1 [Nm/(rad/s)] 0.3 0.3 0.3 cm2 [Nm/(rad/s)] 0.3 0.3 0.3 cl [Nm/(rad/s)] 0.3 1.2 1.2 ks1 [Nm/rad] 8×106 8×106 8×106 ks2 [Nm/rad] 8×106 8×106 1.6×107 cs1 [Nm/(rad/s)] 8×103 8×103 8×103 cs2 [Nm/(rad/s)] 8×103 8×103 1.6×104 Res. freq. 1 [Hz] 14.2 10.6 12.8 Res. freq. 2 [Hz] 24.7 23.4 27.3 Rel. damp. 1 [%] 4.47 3.34 4.03 Rel. damp. 2 [%] 7.75 7.34 8.59

Table 4.1: Mechanical data for the three drive trains

4.3 Optimization Criterion To evaluate the performance of various controllers is not trivial. The prevailing industry criterion, also known in various norms and standards,


is the impact drop ID, see Section 2.2. Impact drop is in this Chapter presented in percent of nominal speed, multiplied by time, which explains the unit %s. The quality of load torque disturbance rejection of the closed loop system can be measured in several ways. The three performance factors, overshoot, settling time and impact drop, used in this comparison are also presented in Section 2.2. The free parameters in the selected control configurations have been optimized using integrated absolute error (IAE) criterion according to (3.1) together with maximum sensitivity constraint defined in (3.2). This method has been found to give good designs for speed control of elastic systems presented in Chapter 3. The optimizations also facilitate a more fair comparison of the different cases than manual tuning would give.

4.4 Results In this Section the three control configurations are compared for each of the three mechanical cases. G1: Medium load, medium load shaft stiffness Table 4.2, together with Figure 4.7– Figure 4.9, summarize the results for the three control configurations for G1. The most important conclusions are: MISO:

• ID decreased from 0.128 %s to 0.0842 %s. • Overshoot decreased from 16.2 % to 3.74 %. • Settling time decreased from 0.239 s to 0.138 s. • Overshoot Tm1 increased from 52.6% to 109%

MIMO: • ID decreased from 0.128 %s to 0.067 %s. • Overshoot decreased from 16.2 % to 1.98 %. • Settling time decreased from 0.239 s to 0.077 s.


• Overshoot Tm1 increased from 52.6% to 85.6%. • No remaining oscillations.

SISO MISO MIMO Kp1 27 62.9 78.5 Ti1 [s] 0.0444 0.0692 0.0598 K - 0.636 - Kp2 - 40 - Ti2 [s] - 0.0441 - K3 - - 0.5 IAE [%s] 0.153 0.0894 0.0684 ID [%s] 0.128 0.0842 0.067 Overshoot ωl [%] 16.2 3.74 1.98 Settling time [s] 0.239 0.138 0.077 Overshoot Tm1 [%] 52.6 109 85.6 Overshoot Tm2 [%] 52.6 50.6 34.9

Table 4.2: Performance for G1

0 0.1 0.2 0.3 0.4 0.5-20

-15

-10

-5

0

5x 10

-3

Time [s]

Spe

ed [

p.u.

]

Motor 1 SpeedMotor 2 SpeedLoad Speed

0 0.1 0.2 0.3 0.4 0.50

0.5

1

Tor

que

[p.u

.]

Time [s]

Motor 1 TorqueMotor 2 Torque

Figure 4.7: Load torque step for: SISO configuration on G1.


0 0.1 0.2 0.3 0.4 0.5-20

-15

-10

-5

0

5x 10

-3

Time [s]

Spe

ed [

p.u.

]


0 0.1 0.2 0.3 0.4 0.50

0.5

1

Tor

que

[p.u

.]

Time [s]


Figure 4.8: Load torque step for MISO configuration on G1.

0 0.1 0.2 0.3 0.4 0.5-20

-15

-10

-5

0

5x 10

-3

Time [s]

Spe

ed [

p.u.

]


0 0.1 0.2 0.3 0.4 0.50

0.5

1

Tor

que

[p.u

.]

Time [s]


Figure 4.9: Load torque step for MIMO configuration on G1.


G2: Heavy load, medium load shaft stiffness Table 4.3 together with Figure 4.10– Figure 4.12 summarizes the results of the three control configurations for G2. The most important conclusions are: MISO:

• ID decreased from 0.194 %s to 0.0991 %s. • Overshoot decreased from 30 % to 8.76%. • Settling time decreased from 0.623 s to 0.308 s. • Overshoot Tm1 increased from 47.8% to 71.1%.

MIMO: • ID decreased from 0.194 %s to 0.0982 %s. • Overshoot decreased from 30 % to 1.98%. • Settling time decreased from 0.623 s to 0.16 s. • Overshoot Tm1 increased from 47.8% to 53.6%.


SISO MISO MIMO Kp1 25.2 90.9 76.5 Ti1 [s] 0.0608 0.128 0.106 K - 0.767 - Kp2 - 69.7 - Ti2 [s] - 0.0979 - K3 - - 0.5 IAE [%s] 0.276 0.109 0.1 ID [%s] 0.194 0.0991 0.0982 Overshoot ωl [%] 30 8.67 1.98 Settling time [s] 0.623 0.308 0.16 Overshoot Tm1 [%] 47.8 71.1 53.6 Overshoot Tm2 [%] 47.8 42.6 34.9


0 0.1 0.2 0.3 0.4 0.5-20

-15

-10

-5

0

5x 10

-3

Time [s]

Spe

ed [

p.u.

]


0 0.1 0.2 0.3 0.4 0.50

0.5

1

Tor

que

[p.u

.]

Time [s]


Figure 4.10: Load torque step for SISO configuration on G2.


0 0.1 0.2 0.3 0.4 0.5-20

-15

-10

-5

0

5x 10

-3

Time [s]

Spe

ed [

p.u.

]


0 0.1 0.2 0.3 0.4 0.50

0.5

1

Tor

que

[p.u

.]

Time [s]



0 0.1 0.2 0.3 0.4 0.5-20

-15

-10

-5

0

5x 10

-3

Time [s]

Spe

ed [

p.u.

]


0 0.1 0.2 0.3 0.4 0.50

0.5

1

Tor

que

[p.u

.]

Time [s]




G3: Heavy load, high load shaft stiffness Table 4.4 together with Figure 4.13 – Figure 4.15 summarizes the results of the three control configurations for G3. The Most important conclusions are: MISO:

• ID decreased from 0.14 %s to 0.0801 %s. • Overshoot decreased from 31.4 % to 5.29%. • Settling time decreased from 0.534 s to 0.176 s. • Overshoot Tm1 increased from 48.5% to 76.4%.

MIMO: • ID decreased from 0.14 %s to 0.0757 %s. • Overshoot decreased from 31.4 % to 1.52%. • Settling time decreased from 0.534 s to 0.154 s. • Overshoot Tm1 increased from 48.5% to 48.8%.

SISO MISO MIMO Kp1 28.7 96.7 82.4 Ti1 [s] 0.0503 0.118 0.0926 K - 0.573 - Kp2 - 55.4 - Ti2 [s] - 0.0674 - K3 - - 0.5 IAE [%s] 0.205 0.0842 0.0767 ID [%s] 0.14 0.0801 0.0757 Overshoot ωl [%] 31.4 5.29 1.52 Settling time [s] 0.534 0.176 0.154 Overshoot Tm1 [%] 48.5 76.4 48.8 Overshoot Tm2 [%] 48.5 28.2 29.2



0 0.1 0.2 0.3 0.4 0.5-20

-15

-10

-5

0

5x 10

-3

Time [s]

Spe

ed [

p.u.

]


0 0.1 0.2 0.3 0.4 0.50

0.5

1

Tor

que

[p.u

.]

Time [s]


Figure 4.13: Load torque step for SISO configuration on G3.

0 0.1 0.2 0.3 0.4 0.5-20

-15

-10

-5

0

5x 10

-3

Time [s]

Spe

ed [

p.u.

]


0 0.1 0.2 0.3 0.4 0.50

0.5

1

Tor

que

[p.u

.]

Time [s]




0 0.1 0.2 0.3 0.4 0.5-20

-15

-10

-5

0

5x 10

-3

Time [s]

Spe

ed [

p.u.

]


0 0.1 0.2 0.3 0.4 0.50

0.5

1

Tor

que

[p.u

.]

Time [s]



4.5 Conclusions The major conclusion is that there is a great advantage in using improved control configurations when revamping tandem coupled DC-motors. When using the existing sensor configuration the dynamic performance is improved significantly, with up to at least a factor of two. Adding the extra torque sensor can reduce shaft torque oscillations to almost zero. Below, the results are summarized per control configuration:

• MISO: The impact drop is reduced with a factor around 2, overshoot and settling time is reduced considerably compared to the SISO configuration. No extra sensor is required. The price is a larger overshoot in torque in motor 1. Also, a third tuning parameter is required together with the standard PI controller parameters Kp1 and Ti1.


• MIMO: The performance factors are only a little improved compared to the MISO configuration, but the system will become more well damped from oscillations and the big overshoot in motor 1 is somewhat reduced. No extra controller parameter needs to be tuned except for the standard PI controller parameters Kp1 and Ti1. The improvements should be weighted against the fact that an extra sensor is required.

The results are valid for a wide range of mechanical configurations. For extreme cases, e.g. with a very high shaft stiffness between the two motors, the system essentially behaves as a normal two-mass system, and the improvements are small for those cases.

Chapter 5 Identification of mechanical parameters The benefits using models during tuning of the closed loop are discussed in Chapter 2 and Chapter 3. Today the modeling is dependent on the collection of mechanical data of motor, as well as load, from data sheets provided by the suppliers. This is often a cumbersome and time-consuming effort. Moreover, the gearboxes normally have significant gear play, which introduces backlash into the drive system. This backlash increases with wear and usually cannot be accurately determined without performing experiments. Hence, the gathering of mechanical data (including gap size) has been identified as a bottleneck in the tuning procedure. Therefore it has been suggested to estimate the mechanical parameters based on a fit to experimental data instead.

5.1 Background Many industrial drives consist of motor, flexible shafts, gearboxes and load, which form a multi-mass system. Backlash is introduced due to existence of elements like gearboxes and flexible couplings. In this case, the system model becomes non-linear.

44 5 Identification of mechanical parameters

The working hypothesis of this thesis is that in many cases the model can be simplified to a two-mass system where the first mass represents the motor, the second mass represents the load, and the shaft is considered as mass and inertial free. Such drive train system serves as a model for rolling mill, paper machine, large fan, etc. The main task is then to estimate the unknown parameters of the two-mass model by making a numerical fit to experiment data.

5.2 Two-mass linear model In drive train systems, backlash introduces a non-linearity which makes the mathematical model much more difficult to estimate. The following linear model of an uncertain two-mass system without backlash is equal to the model presented in (2.1), but here with separated shaft torque, Ts [Nm], and idle torque, T0 [Nm]:

(5.1)

with transmitted shaft torque according to

(5.2)

where the system input is mT , the motor torque, and the measured output is

mω , the motor speed. The other two signals in the model are

lmd θθθ −= , the angle difference between motor and load , and lω , the load speed. The parameters to estimate are:

0T the idle torque [Nm]


lmd

sllll

msmmmm

TcJ

TTTcJ

ωωθωωωω

−=

+−=−+−−=

&

&

& 0

)( lmsdss ckT ωωθ −+=

5 Identification of mechanical parameters 45





sc the shaft damping [Nm/(rad/s)] The suggested approach is to fit parameters directly in the model (5.1)-(5.2) by solving the system of differential equations and forming a non-linear least-squares criterion by comparing the computed motor speed to the measured one. There are, however, at least two issues that complicate matters. Firstly, we need to somehow obtain initial estimates of the seven parameters listed above. Secondly, when there is significant backlash in the system we need to introduce a model for this as well and estimate a gap parameter. How to deal with these problems is described in the subsequent Sections.

5.3 Black-box model Since the gap free two-mass model (5.1)-(5.2) in the previous Section is linear, a natural idea is to study its transfer function. The uncompensated transfer function of the open loop, )(sPum , from mT to

mω is given by:

(5.3)

with

A natural approach to finding the initial parameter estimates would be by estimating a general third-order discrete-time transfer function and convert it to a general continuous-time one in the form

)(

)()(

2

sd

ksccsJsP ssll

um

+++=

slmslsmlmsml

slmsmllm

kccscccccckJJ

sccJccJsJJsd

)())((

))()(()( 23

+++++++++++=


(5.4)

Comparison of equations (5.3) and (5.4) then gives the following system of equations:

(5.5)

(5.6)

(5.7)

(5.8)

(5.9)

(5.10)

The idea is then to use estimates of the a- and b-parameters to solve this system of equations for Jm, Jl, cm, cl, ks and cs. The system of equations can indeed be solved in, for example, Maple using so-called Groebner bases. However, it turns out that for typical values of the mechanical parameters the solution is very sensitive. In particular the inertia dampings cl and cm are extremely sensitive to small variations in the estimated discrete-time parameters. The reason is that they are small in comparison to the other parameters, and basically only influence the static gain

lm cca

b

+= 1

0

0

012

23

012

2)(asasas

bsbsbsG

+++++

=

mJb

12 =

lm

sl

JJ

ccb

+=1

lm

s

JJ

kb =0

l

sl

m

sm

J

cc

J

cca

++

+=2

lm

lssmlmlms

JJ

ccccccJJka

))((1

++++=

lm

lms

JJ

ccka

)(0

+=


The static gain is, however, very difficult to estimate from dynamic experiments, since it is so small that for short data records the system acts almost like an integrating system. Unfortunately the bad accuracy of the inertia dampings also influences the estimate of the other mechanical parameters to such an extent that it was found that a method based on solving the complete system of equations is not feasible. Instead an alternative approach was taken, neglecting cl and cm altogether for the initialization. Then the system (5.3) has a pure integration, i.e. it has the form

(5.11)

where

(5.12)

(5.13)

(5.14)

(5.15)

(5.16)

Notice that in equations (5.12) to (5.16) there are now five equations but only four unknown parameters. Several alternatives are therefore possible (including solving the over determined system of equations in a least-

mJb

12 =

lm

s

JJ

cb =1

lm

s

JJ

kb =0

lm

lms

JJ

JJca

)(2

+=

lm

lms

JJ

JJka

)(1

+=

)()(

122

012

2

asass

bsbsbsG

++++

=


squares sense). We have tried several combinations and in the end settled for using equations (5.12), (5.14), (5.15) and (5.16). Hence the equation (5.13) is excluded in the solution. It now remains to find initial estimates for the inertia dampings cl and cm , since they were excluded in the black-box identification. Notice also that since mean values are typically subtracted from the signals before black-box identification no estimate of the idle torque 0T is obtained either. To solve these two remaining problems we suggest introduction of yet another dedicated experiment described in the next Section.

5.4 Identification of static gain As pointed out in the previous Section we need another experiment to estimate the sum of viscous motor friction and load friction, as well as the offset created by the idle torque. The model (5.1)-(5.2) implies the following static relationship

0)( TccT mlmm ++= ω where mω is static motor speed (rad/s) and mT is static motor torque (Nm). Hence, the aim of the static experiment is to find the slope lm ccc += and offset 0T of a straight line. If a number of observations can be made at different operating points, then c and 0T can be estimated by forming an over determined system of equations, which is easily solved in a least-squares sense. Since only the total damping can be estimated, some further assumption has to be made before initializing the individual inertia dampings. We have chosen as a first guess to distribute the damping relative to the size of the corresponding inertia, i.e.

lm

mm JJ

cJc

+= ;

lm

ll JJ

cJc

+= .


5.5 Grey-box identification Once initial estimates of the mechanical parameters have been obtained using the procedures described in Sections 5.3 and 5.4, fine tuning is made by fitting the mechanical parameters directly, using a least-squares criterion

∑ −=k

mm kkV 2)),(ˆ)(( θωω

where )(kmω denotes the (discrete-time) measurement of motor speed and

),(ˆ θω km denotes the motor speed simulated through the model (1)-(3) at the same time instants, using the current mechanical parameter values here denoted by the vectorθ . In a pre-study the general purpose Matlab based program MoCaVa -- (Bohlin and Isaksson, 2003) -- was used to verify the feasibility of the approach. For the final tool a modified version of the Matlab toolbox Diffpar -- (Edsberg and Wedin, 1993, 1995), (Edsberg and Wikström, 1995) -- has instead been used to develop a graphical user interface dedicated to the drive train problem.

5.6 Backlash estimation Once the linear mechanical parameters have been obtained, another experiment is made that is guaranteed to open up the gap. The model (5.1) is modified by replacing (5.2) with the following backlash model:

(5.17)

)(/ lmssd kcz ωωθ −+=

ααδα

αδ

−<++−+=

−≥=++=

zifzzkT

zifzzkT

ss

ss

))2(2(2

1

)(2

1

2

2


This corresponds to a dead-zone model; corrected for the effect of damping, see Nordin (2000). Furthermore a smoothness parameterδ has been introduced to give a continuous derivative. Unfortunately this instead causes a discontinuity of sT at α−=z . However, if a very small value of δ is used, this does not seem to cause any numerical problems. In the current implementation the linear parameters are fixed and only the gap parameterα is allowed to vary. Since obtaining an initial estimate for α is very difficult, the program instead initializes the search at several user chosen starting values. Once again prototyping was made using MoCaVa, while the end product is built on Diffpar.

5.7 Experiment design To summarize the procedure that has been developed over the last Sections, what we suggest is a sequence of three dedicated experiments:

1. Static experiment. First, collect data that represent different operating points, in order to estimate inertia dampings and idle torque.

2. Dynamic linear experiment. Perform a tailored experiment that guarantees that no gap openings are encountered. Hence linear transfer function estimation should be applicable. Then use black-box identification to find initial mechanical parameters. Fine tune the mechanical parameters on the same data using the mechanical model.

3. Dynamic nonlinear experiment. Perform another experiment that with certainty contains gap openings, which enables estimation of the gap size.

Below we will elaborate a bit more on each one of these experiments. The static experiment


The purpose of the static experiment is to find a straight line to represent the static gain of the transfer function. Hence we need to visit enough operating points to make a fit of this line possible. The experiment may be performed as one sequence of set point steps or as different experiments with different fixed set points in each experiment. Therefore an experiment is conducted with the speed controller in automatic, where a sequence of setpoint steps are made spanning the operating range. The torque and speed signals are then run through a method described in Cao and Rhinehart (1995) that automatically finds data points where the system is in steady-state. The data in all sufficiently long steady-state intervals are then averaged to form one observation of the straight line The dynamic experiment without gap openings So far we have assumed that an experiment can be performed which does not open the gap. A couple of questions naturally arise:

• Is this possible? • If so how much excitation will it provide?

First of all one can conclude that an experiment like this has to be performed in open loop, i.e. we want to design the shape of the motor torque directly. As soon as feedback is allowed we cannot guarantee that gap openings are avoided. A natural candidate would be to apply a step in motor torque. The idea here is that if first a steady state constant speed is maintained, then the gap is already closed. If then a motor torque step is applied to increase the speed hopefully the gap does not open. For a two-mass model with some simplifying assumptions this can be shown mathematically to hold. Based on the results in (Jewik et al. 1969) about torque amplification factor (TAF), if friction and damping is zero, i.e. cm=cl=Tm0=Tl0=cs=0


Ts(t)= 2Jl/(Jm+Jl)(1-cos(ωt))Tm This implies Ts ≥0, which is exactly the condition for the gap to remain closed. Unfortunately it is not true for general two-mass systems. It is not difficult to construct an example which when simulated experiences gap opening. Table 5.1 and Figure 5.1 show the results of such a simulation. Notice, however, that is a rather extreme case, since there is no damping in the shaft and no friction in the load, but a very large viscous friction in the motor. Figure 5.2 show measurements where the backlash is opened in a motor torque step, in this case, due to overshoot in the converter. For systems with higher number of inertias than two, there is no guarantee that the backlash is closed at motor torque step even for systems where the friction and damping is zero. Jewik et al. (1969) also show that for example for three-mass systems:

( ) ( ) ( ) m

lm

ls Ttt

JJJ

JtT

−

−−

+++

= 221

22

22

121

22

21

2

coscos1 ωωω

ωωωω

ω

where J2 is the intermediate inertia, Ts is the shaft closest to the load, ω1 is the first (lowest) resonance frequency and ω2 is the second resonance frequency. Thus to find a point where Ts(t)<0, set cos(ω1 t))= -1, cos(ω2 t))=1, which leads to

( ) m

lm

ls T

JJJ

JtT

21

22

21

2

2

ωωω−++

−=

In practice, at least for rolling mills, there is a usual mechanical design criterion to build drive trains that fulfils ω2 ≥ 2ω1 and therefore limits the minimum value of Ts(t) above. Also, usually the friction on load side give a steady-state shaft torque, and the damping in the shaft is usually large enough in order to give some margin between the shaft torque peaks in the oscillation during motor torque step and the zero-torque level in the shaft shown in Figure 5.3.


A step is of course not a very rich signal. However, applying superposition using the same reasoning as above if one step does not open the gap, then a sequence of positive steps still does not open the gap. A practical limitation, though, is that sooner or later the speed will reach a level where the experiment has to be interrupted. Remark: In the next Chapter use of a torque sensor is proposed. It is then possible to decide directly form the measurements if the shaft torque passes zero-level, and thus if the backlash remained closed during the entire experiment. Jm Jl cm cl ks cs Tm0 Tl0 1 1 10 0 10000 0 0 0

Table 5.1: Mechanical parameters.

Figure 5.1: Simulation of motor torque step with shaft torque passing zero-torque, which results in backlash opening.


Figure 5.2: Measurement of motor torque step with shaft torque touching zero-torque, which results in backlash opening.


Figure 5.3: Simulation of motor torque step with shaft torque passing steady-state shaft torque, but not zero-torque level, which do not result in any backlash opening.

The dynamic experiment with gap openings To perform an experiment that opens the gap is much easier. What we recommend, and have typically used, is an open-loop experiment with alternating negative and positive steps in motor torque. However, a closed-loop experiment is equally applicable. In this thesis no effort has been made to study optimal design of this experiment, which is potential topic for further investigation. It is obvious, however, that it is possible obtain very good excitation since, contrary to the linear experiment, there is really no limitation on the duration of the experiment.


5.8 Summary and software tool At this point a summary of the developed approach may be appropriate. The results in Sections 5.3-5.7 describe a procedure based on three dedicated experiments using four identification methods that should be run sequentially:

• A steady-state experiment with controller in automatic, yielding estimates of lm cc + and 0T .

• A dynamic backlash free experiment followed by • initial estimation by black-box identification. • fine tuning using Diffpar and the mechanical model (5.1)-(5.2),

yielding estimates of Jm, Jl, cm, cl,, ks and cs. • A dynamic experiment with guaranteed backlash to estimate the

gap parameter α , keeping all other parameters fixed. A user interface has been created that reflects this four step procedure, where the second and third step share the same data. In each step the user can choose whether to use the results from the previous step, or to manually insert values. Thus making it possible to, for example, identify mechanical parameters from dynamic data even if a steady-state experiment is not available. The main window of this identification tool is shown in Figure 5.8.

5.9 Real data example Real data has been collected on a test rig at ABB Automation Technologies, as well as from both rolling mills and paper mills. The results so far have been very encouraging. We will here show the results from one set of data from the test rig, consisting of one AC motor on the drive side, and a DC motor as load. The system should be possible to model very well using a two-mass model. From data sheets the motor inertia is given as Jm = 0.40, the load inertia Jl = 5.6 and shaft stiffness ks = 3300 (with negligible shaft inertia). The test rig also has a rubber flexible coupling, which has a similar effect as a gear play.


First the steady-state experiment is shown in Figure 5.4, where in the lower plot the automatically found steady-state intervals are marked with a fat line. Obviously there is very little viscous damping (i.e. the upper line has almost zero slope), while there is significant static friction resulting in a non-zero idle torque. These facts are also reflected in the obtained parameter estimates 00184.0=c and =0T 15.14.

Figure 5.4: Steady state identification for test rig data. Lower plot: Scaled motor speed and corresponding scaled motor torque. Upper plot: Mean speed and torque for each found steady-state interval (“+” sign) and least square solution (solid line).

In Figure 5.5 and Figure 5.6 the data and model fit for the backlash free experiment are shown. Figure 5.5 shows the fit after initial black-box identification and Figure 5.6 the fine tuned fit after linear grey-box estimation.


Figure 5.5: Data and model fit of black-box identification for test rig data (data solid, simulation dashed).

Figure 5.6: Data and model fit for fine tuning on backlash free test rig data (data solid, simulation dashed).


The estimation results for the mechanical parameters are summarized in Table 5.2. Notice that even though from visual inspection the fit looks very good already after the initial black-box estimation, the fine tuning in fact reduces the loss function by almost 25 %. According to the “true” values given above both the load inertia and shaft stiffness estimates are quite acceptable, whereas the motor inertia is estimated about 50 % too large. In this case, however, it is probably fair to claim that the estimated parameters better represent the real system than the ones obtained from data sheet.

Black-box Grey-box Jm 0.601 0.587 Jl 5.554 5.179 cm 0.000180 0.000176 cl 0.0017 0.0017 ks 3445 3310 cs 3.324 3.115 Resonance freq. 12.68 12.60 Loss function 0.00856 0.00656

Table 5.2. Summary of estimation results for backlash free data.

Finally, an attempt was made to estimate also the gap parameter using the third set of experiment data (see Figure 5.7) yielding the result =α 4.3 degrees. Obviously the gap model is not able to describe the more damped behavior of the flexible coupling, but at least manages to quite well capture the correct resonance frequency.


Figure 5.7: Data with backlash and corresponding model fit after gap identification for test rig data (data solid, simulation dashed).

5.10 Conclusions In this Chapter, a method has been developed which finds the mechanical parameters, including backlash, through a series of three dedicated experiments. At first this procedure is developed for the situation of one manipulated input, the motor torque, and one measured output, the motor speed.


Figure 5.8: Main window of the Drive Train Identification Tool.

Chapter 6 Identification with shaft torque measurements On systems with small load inertia compared to motor inertia, motor speed does not contain sufficient information to estimate the mechanical parameters for a two mass model. The reason is that a complex zero pair will be close to the complex pole pair that dominates the resonance in the system. These zeros will therefore hide the most interesting information about the mechanical system. Viewed from motor speed the system only appears to be a one-mass system. This Chapter presents a method for identification of models using a shaft torque sensor together with a motor speed sensor. The benefit of this extra output from the system is that this transfer function does not usually contain any complex zeros and therefore, it is possible to obtain measurements where the dominating system resonance frequency will be possible to identify. An extra bonus is that it will be possible to separate static torque between motor inertia and load inertia. An alternative approach would be to measure load speed, but it is often more difficult to apply temporary speed sensors for the measurements, especially in rolling mills, where the environment around the load is quite rough. Figure 6.1 shows two examples with motor torque step on systems with load to motor inertia ratios around 1/8 and 8, It is obvious that motor speed

64 6 Identification with shaft torque measurements

measurements are not sufficient for identification the dominating resonance frequency when load inertia is low compared to motor inertia. In the figures also shaft torques are presented. Motor torque steps seem to excite the systems enough to reveal resonance frequency information in shaft torque for both examples.

1.4 1.5 1.6 1.710

12

14

16

18

Spe

ed [

rad/

s]

Time [s]

Motor Speed Act

1.4 1.5 1.6 1.70

50

100

150

200

Tor

que

[Nm

]

Time [s]

Shaft Torque ActMotor Torque Act

0.3 0.4 0.5 0.65

10

15

20

25

Spe

ed [

rad/

s]

Time [s]

Motor Speed Act

0.3 0.4 0.5 0.60

100

200

300T

orqu

e [N

m]

Time [s]

Shaft Torque ActMotor Torque Act

Figure 6.1: Left: Motor torque step with light load inertia. Right: Motor torque step with heavy load inertia.

Portable torque sensors are available for temporary measurement procedures. This means that it is possible to perform sufficient measurements on most of the existing speed controlled drive systems as well as during commissioning of new systems without introduction of any extra hardware cost. The following linear model of an uncertain two-mass system is similar to the one presented in (5.1)-(5.2), except that T0 is here called Tm0 and that Tl0 is also introduced in order to separate static friction between motor and load:

(6.1)

lmd

lsllll

mmsmmmm

TTcJ

TTTcJ

ωωθ

ωωωω

−=

−+−=−+−−=

&

&

&

0

0

6 Identification with shaft torque measurements 65

with transmitted shaft torque

(6.2)

where the system input is mT , the motor torque, and the measured output is

mω , the motor speed, and sT , the shaft torque. The other two signals in the model are lmd θθθ −= , the angle difference between motor and load, and

lω , the load speed. The parameters to estimate are:

0mT the idle torque on motor [Nm]

0lT the idle torque on load [Nm]






sc the shaft damping [Nm/(rad/s)] We will follow the procedure with three dedicated experiments proposed in Chapter 5. Below each of these steps is described in more detail.

6.1 Static test The first step is to identify cm, cl, Tm0 and Tl0 from constant speed measurements at different speed levels where ωm, Tm and Ts are measured. Equation (6.1) is here used in order to find simple equations to be solved using least squares method. Assuming stationary, 0=== dlm θωω &&& first

of all yields ωl=ωm, then

0

0

0

0

lsml

mmsmm

TTc

TTTc

−+−=−+−−=

ωω

)( lmsdss ckT ωωθ −+=


or

0

0

lmls

mmmsm

TcT

TcTT

+=+=−

ωω

where mω is static motor speed (rad/s), sT is static shaft torque e (Nm) and

mT is static motor torque (Nm). cm, cl, Tm0 and Tl0 can be determined in a least square sense if at least three sets of static measurements are collected.

6.2 Dynamic test without gap openings The test is done using a torque step from a none-zero speed where the backlash is assumed to be closed due to static and viscous friction in the load. More details about motor torque steps can be found in Section 5.7. Black-box identification with closed backlash To follow the idea from Chapter 5 we need the uncompensated transfer function of the open loop. The transfer function from mT to mω , )(sPum is already defined in (5.3), and the one from mT to sT is given by

(6.3)

with the same dominator as in (5.3), i.e.

)())((

))()(()( 23

lmsslsmlmlms

slmsmllm

ccksccccccJJk

sccJccJsJJsd

+++++++++++=

The method of finding the initial parameter estimates and estimate general third-order discrete-time transfer functions and convert them to general continuous-time ones is the same as in previous Chapter. The transfer function (5.11) could be used, but in this case, there is one transfer function for each output from the system, i.e. 6+6=12 estimates available to solve for the 6 mechanical parameters and two static torques. The two

)()(

)(2

sd

ckscckJscJsP lsslslsl

us

+++=


denominators would be equal in a perfect world, but in practice some of the estimated parameters are very sensitive to noise and should therefore not be used for estimation of mechanical parameters. The method presented here uses known information about the structure of the model and finds start values of the mechanical parameters step by step. Motor speed measurements A first order model is usually sufficient in order to find the total inertia of the system, if data is collected during a sufficiently long time interval compared to the complex zero, ωzero lJk /≈ . Bode plots of Pum for light

load and heavy load examples are shown in Figure 6.2. If cm=cl=0 and Jm>>Jl in (5.3) then the following model results

(6.4)

This may be identified using black-box identification

s

bsP =)(

but the estimate of Jtot will then be less accurate compared to when a first order model

as

bsP

+=)(

is identified using motor torque step input. Only Jtot=1/b=Jm+Jl is then estimated from P(s).

sJJsJJkscsJ

kscsJ

sJJksJJcsJJ

kscsJsP

lmlmssl

ssl

lmsmlslm

sslum

)(

1

))((

)()()(

2

2

23

2

+=

+++++

≈++++

++=


-80

-60

-40

-20

0

20

Mag

nitu

de (

dB)

10-2

100

102

-90

-45

0

45

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

-100

-50

0

50

Mag

nitu

de (

dB)

10-2

100

102

-90

-45

0

45

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec) Figure 6.2: Left: Bode plot of light load of Pum, right: Bode plot of heavy load of Pum.

Shaft torque measurements Assuming cm=cl=0 in (6.3), then

)/()()/()(

//

)()()(

2

2

lmlmslmlms

msms

lmslmslm

slslus

JJJJksJJJJcs

JksJc

JJksJJcsJJ

kJscJsP

+++++

=++++

+=

Thus use black-box identification using motor torque step input on a second order model

012

01)(asas

bsbsP

+++

=

Bode plots of Pus for light load and heavy load examples are shown in Figure 6.3.


-80

-60

-40

-20

0

20

Mag

nitu

de (

dB)

10-2

100

102

-180

-135

-90

-45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

-60

-40

-20

0

20

40

Mag

nitu

de (

dB)

10-2

100

102

-180

-135

-90

-45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec) Figure 6.3: Left: Bode plot of light load of Pus, right: Bode plot of heavy load of Pus.

The mechanical parameters may then be found from

b0=ks/Jm

a0=ks(Jm+Jl)/(Jm Jl)

a1=cs(Jm+Jl)/(Jm Jl)

or

Jl=Jtot b0/a0=1/b b0/a0

Jm=Jtot-Jl=1/b (1-b0/a0)

ks=b0 Jm=1/b (1-b0/a0) b0

cs=a1/a0 Accordingly, b1 is neglected.


Grey-box identification with closed backlash If the static experiment has generated sufficient accuracy of parameters cm, cl, Tm0 and Tl0, then only Jm, Jl, ks and cs need to be adjusted.

6.3 Dynamic test with gap openings Several contacts to both sides of the backlash are preferable. It is therefore important that the experiments are performed in such a way that there is enough excitation of the system and that both motor and load have about the same relation between inertia and decelerating force at the operating point. Motor torque limits may be adjusted in order to obtain sufficient behavior at backlash opening. Backlash play can be studied in the shaft torque measurements. If stage 1 and stage 2 of the identification procedure have generated sufficient accuracy of parameters cm, cl, Tm0, Tl0, Jm, Jl, ks and cs, then only α and possibly δ need to be adjusted. The backlash model is here modified a somewhat compared to the model (5.17) in Section 5.6. The difference here is that the discontinuity around θd=0 is removed and that the damping, cs is acting linearly with respect to the difference between motor speed and load speed, i.e. also when the backlash is opened.

α

ωωαδαδ

α

ωωδδ

−<

−+++−++++−=

−≥=

−+++++−=

zif

kczzaakT

zif

kczzaakT

lmssss

lmssss

)),(/2)2(2(21

)),(/2(2

1

22

22

dz θ=


The backlash model (5.17) has also been tested, but the result will then be less accurate on the test rig that has been used for tests. We do not have any good explanation to the result, but as the coupling halves slides to each other, there might still be some force transferred over the coupling also when the backlash is opened. The diameter of the coupling in the test rig is 170 mm and the shaft has a diameter of 22 mm. We believe that the damping in the shaft itself does not explain the total damping of the system.

6.4 Implementation The output error model in System Identification Toolbox in Matlab has been used for black-box identification. Grey-box optimization has been performed using Simulink and Optimization Toolbox in Matlab. The minimization criterion that have been used in the grey-box optimization is

( )∑ −+−= 22 )ˆ()ˆ( ssmm TTV µωω

where

mω Measured motor speed.

mω̂ Simulated motor speed.

sT Measured shaft torque

sT̂ Simulated shaft torque µ Weighting factor between motor speed error and shaft torque

error Some adjustments have been made manually on µ and the backlash angle α during the steps in the optimization in order to find local minimum points in the minimization criterion that corresponds to matching shaft torque peaks between measurements and simulation.


6.5 Results Real data has been collected on a test rig at ABB Automation Technologies. A picture of the rig is presented in Figure 6.4. We will here show the results from one set of data from the test rig, consisting of one AC motor and a DC motor connected with a weak shaft, three couplings; one with large backlash, and a torque sensor, see Figure 6.5. The system should be possible to model very well using a two-mass model. Identification is performed from both motor sides, using the technique presented earlier in this Chapter and it is therefore possible to compare the results.

Figure 6.4: Picture of the test rig.


AC- Motor(small inertia)

Torquesensor

Coupling

DC- Motor(large inertia)

Couplingwith

backlash

Coupling

Weakshaft

Figure 6.5: Test rig.

The size of the backlash is quite large. The reason is that the coupling originally was a rubber filled toothed coupling, or a flexible coupling. In order to be more similar to real life applications, i.e. to reduce the very big non- linearity in the stiffness, varying with applied torque, the rubber was taken away from the coupling. A large backlash makes it harder to perform measurements where backlash dynamics are sufficiently exited. A large number of contacts to both sides of the backlash are preferable. The tests are performed with an active PI-controller, but where the torque limits are adjusted in order to get contact bumps on both sides of the backlash. Steady-state experiments are presented in Figure 6.6 and Table 6.1. Tm0, Tl0, cm, and cl are presented in Table 6.2. There is a 6-7 Nm difference in friction in the working point in the steady-state tests depending on if they are performed from the AC or from the DC motor. This difference is reduced when the parameters are released in the grey-box identification with closed backlash on light load identification (Tm0 is increased from 24.2 Nm to 27.9 Nm in Table 6.2). Some of the idle torques Tm0, Tl0 in the Table are less than zero. This is not physically correct, but the model is built around a working point.


Figure 6.6: Steady-state experiments. Left: Light load. Right: Heavy load.

ωm Tm-Ts Ts Tm-Ts Ts 73.3 1.12 38.5 105 35.6 2.78 1.86 42.6 136 39.7 3.31 2.74 46.0 157 43.8 3.71

Table 6.1: Mean values from steady-state measurements.

Light load experiment from DC motor. The estimated mechanical parameters are presented in Table 6.2. The black-box identification gives a model with small deviation between measurement and simulation, see Figure 6.7. The main difference is that the model is accelerating slower than the test rig during a motor torque step. The amplitude of the oscillations in the shaft torque is slightly lower in the simulation than in the measurements. Grey-box fine-tuning of some of the mechanical parameters from the black-box result reduces the difference in acceleration between


measurements and simulation during motor torque step, see Figure 6.8. The amplitude on the oscillations in the shaft torque is still slightly lower in the simulation than in the measurements. Grey-box tuning with opened backlash indicates that the model behave similar to the test rig, see Figure 6.9. Experiments with more frequent backlash contact bounces would be preferred in order to get even better quality of the estimate. Black-

Box Light Load Closed Backlash

Grey-Box Light Load Closed Backlash

Grey-Box Light Load Opened Backlash

Black-Box Heavy Load Closed Backlash

Grey -Box Heavy Load Closed Backlash

Grey -Box Heavy Load Opened Backlash

Jm 4.94 4.78 4.78 2.14 0.710 0.566 Jl 0.655 0.646 0.646 4.17 4.87 5.01 ks 3.78×103 3.77×103 3.77×103 9.56×103 4.13×103 3.50×103 cm 0.154 0.107 0.107 0.0258 0.0258 0.0258 cl 0.0176 0.0307 0.0307 0.120 0.120 0.120 cs 0.083 0.083 0.863 2.49 2.01 0.470 Tm0 19.2 24.2 27.9 -0.799 -0.799 -0.669 Tl0 0.931 -1.02 -1.02 29.8 29.8 29.7 α 0 0 0.156 0 0 0.131 δ 1×10-9 1×10-9 8.32×10-7 1×10-9 1×10-9 6.34×10-5 Jtotal 5.59 5.42 5.42 6.32 5.58 5.58 fres 12.9 13.0 13.0 13.1 13.0 13.2 ζres 0.00106 0.00117 0.00958 0.0108 0.0200 0.00583

Table 6.2: Values of mechanical parameters from different steps of identification. The three rows in the bottom of the table shows total inertia, Jtotal and corresponding resonance frequency, fres and damping factor, ζres for a linear system.


Figure 6.7: Black-box identification, light load.


Figure 6.8: Grey-box identification, light load.


Figure 6.9: Grey-box identification, light load. Opened backlash.


Heavy load experiment from AC motor. Essentially this method with shaft torque sensor is tailored to be used for systems with light load. It was assumed that Jm>>Jl in order to perform the simplifications according to (6.4). Because this only affects the initial values for the grey-box it is still interesting to see how well the result will be. The estimated mechanical parameters for this case are also presented in Table 6.2. The black-box identification gives a model with correct resonance frequency, but too small amplitudes in the oscillations on both motor speed and shaft torque. Also, the acceleration is slower in the simulation than in the measurements, see Figure 6.10. This is not surprising because this method was not designed for heavy load cases. Grey-box fine-tuning of some of the mechanical parameters from the black-box result reduce most of the difference between measurements and simulation during motor torque step, see Figure 6.11. Note that only Jm, Jl, ks and cs are tuned compared to the black-box model. Grey-box tuning with opened backlash indicates that the model behave similar to the test rig, see Figure 6.12. Again more frequent backlash contact bounces in a shorter time period would be preferred in order to get even better quality of the backlash estimate.


Figure 6.10: Black-box identification, heavy load.


Figure 6.11: Grey-box identification, heavy load.


Figure 6.12: Grey-box identification, heavy load. Opened backlash.


Validation Normally the following methods are used for validation:

1. Estimate mechanical parameters from different sets of measurements and compare the parameter values.

2. Estimate mechanical parameters from one set of measurements and simulate the model based on a new set of measurements

For this particular test case, there are three further possible methods, Load speed is not included in the minimization criterion, but it is still available for observations.

3. Compare the results of estimation of parameters made from both

motors. 4. Estimate mechanical parameters from one motor and simulate the

model based on the opposite motor measurements, but where all parameters are switched between motor and load.

5. Compare how well the mechanical model emulates the load speed compared to measurements.

Comparison of parameters between motors (3) The estimated parameters in column 3 and column 6 in Table 6.2 are compared. Note that all mechanical parameters referred to, as motor in the light load case, should be compared with the parameters referred to as load in the heavy load case and vice verse. The difference between the parameters is quite small, except for the shaft damping, ks, which differs almost a factor two depending on if they are identified from AC motor or DC motor. Simulation of validation data on light load (2) The simulation of the model based, on mechanical data in column 6 in Table 6.2 is presented in Figure 6.13 on estimation data and in Figure 6.14 on validation data. The simulation data of shaft torque fits well to the


validation data until the three-second indication. Then the shaft torque peaks start to appear different in time between simulation and measurement. Comparison of measured and simulated load speed (5) If the simulation is close to the measurement, then the model does not only simulate motor speed and shaft torque well, but also simulates load speed well. This is one of the most important properties of the mechanical model, since as has been pointed out earlier, one important goal with the model is model based controller design with focus on load speed. Simulated load speed fits quite well to the estimation data until the three-second indication in Figure 6.13. Adjustment of backlash size In order to see how sensitive the model is to backlash-size the backlash parameter α is released. α then was increased from 0.156 to 0.170. After that, the model output is quite similar to measurements during the whole measurement. Figure 6.15.


Figure 6.13: Grey-box identification result on light load with opened backlash. Here is also load speed presented.


Figure 6.14: Grey-box validation result on light load with opened backlash. Here is also load speed presented.


Figure 6.15: Grey-box validation result on light load with opened backlash. Here is also load speed presented. Re-tuned backlash α improves the match between measurements and simulations.


6.6 Conclusions The estimation result is very good for the light load, i.e. from DC motor. Initial parameters, Jm and Jl are better estimated on heavy load, i.e. from AC motor side, using the method presented in Chapter 5. This was expected due to the fact that this method was not designed for heavy load. The final estimated mechanical parameters are yet close to the ones estimated from the DC motor side, except for the shaft damping, that differs a factor two.. When load speed is measured and compared to simulated load speed, it is shown that the model also simulates load speed well. This is one of the most important properties of the mechanical model because one important goal with the model is model based controller design with focus on load speed.

Chapter 7 Conclusions In this thesis we have studied different aspects of estimation of mechanical parameters and model based design of drive train systems that can be found in numerous of applications. In Chapter 3 it was found that it is difficult to find one “perfect” search criterion that finds good controller settings for all relations between load inertia and motor inertia in the drive train in the best way. A major conclusion is that if one criterion should be chosen, IAE calculated for load speed is the criterion that is most general among the tested criteria. These tunings also look best to the naked eye. Another conclusion is that robustness constraint is generally needed. In Chapter 4 it was found that there is a great advantage when using the proposed control configurations when revamping tandem coupled DC-motors. When the existing sensor configuration is used, the dynamic performance is improved significantly when the control of the motors are separated, with up to at least a factor of two with respect to impact drop. Adding the extra torque sensor in the feedback loop can reduce shaft torque oscillations to almost zero. The results are valid for a wide range of mechanical configurations. For extreme cases, e.g. with a very high shaft stiffness between the two motors, the system essentially behaves as a normal two-mass system, and the improvements are small for those cases.

90 7 Conclusions

Chapter 5 describes a way to identify mechanical parameters in drive train systems with help of measurements of motor torque and motor speed. For this purpose, three dedicated experiments is suggested. Chapter 6 modifies the method in Chapter 5 using measurements of shaft torque in order to also identify mechanical parameters for drive train systems with light load. The results of the method are evaluated on a test rig, both for light load case and heavy load case. The estimation result is very good for both cases, except that the shaft damping varies a factor two depending on how it was estimated. It is shown that load speed is modelled well. This is one of the most important properties of the mechanical model because one important goal is model based controller design with focus on load speed.

7.1 Future work Speed reference input needs to be treated in the multivariable control concept. Also, there are cases where three and even four motors are coupled in tandem. It would probably be an even larger benefit if the presented concept as developed further for these drives. It would be interesting to compare the result from the tandem drives with H∞ control in order to see how close the proposed configurations are to a optimal control of this application. The black-box estimation with shaft torque sensor should be modified in order to also handle heavy load cases. It has been found that identification may be disturbed by oscillations caused by motor speed dependent disturbances. It might therefore be interesting to include a speed varying notch filter in the model or directly on the measurements. Initial backlash estimation for the grey-box tuning procedure, based on energy balance has been sketched, but not fully evaluated. This type of method is useful in order to develop a tool for practical use.

7 Conclusions 91

The backlash and friction problem indicates that there is still more to do on the modeling of backlash. The numerics in the grey-box estimation need to be improved in order to be more robust.

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