Modelling, Simulation and Speed Control of the Two-Mass Drive System
Shikha Sharma, Amit Gangopadhyay
ABSTRACT - A modern two mass system is made out of an engine which is associated with the load machine through a shaft. Once in a while the shaft may be hardened, yet in applications, for example, rolling mill drives, automated arms, conveyor belts and so on. These elastic joints are susceptible to torsional vibrations that breaking point the execution of modern drives. Because of this the engine velocity may be diverse from the load speed. To tackle this issue speed controllers are utilized which will change the load speed and the engine speed with that of the reference speed. Fuzzy logic controllers was utilized before. The limitation of these controllers is the lack of analytical tuning. Lately, neural systems have been requisitioned the control of two mass systems. Here, the synthesis of the control structure with proportional–
integral controller with proportional-derivative controller can be used to stabilize the two mass drive systems. Finally evaluation of PD controller is done and error estimation of ModelSim PD controller is done for hardware implementation on FPGA. The results of error and speed obtained under different parameters like 𝝉1, 𝝉2, ωr etc.
KEYWORDS- Neural Networks (NNs), PD Controller, Drive System.
I. INTRODUCTION
In the late years necessities for the ideal activity in enduring and element conditions of modern electrical drives are getting to be more stringent. The point of such systems is to minimize the length of time of the transient process, the perfect following of the given direction of the pace (or position), power to parameter change of the controlled system [1]. The necessities specified above lead the architects to grow new strategies and control calculations of the drives. Also, they oblige precise demonstrating and working conditions to acquire high accuracy of control, which is regularly associated with disposal of the streamlining presumptions.
Manuscript
Shikha Sharma, M. Tech student, Department of Electronics and Communication Engg. , Mangalayatan University, Aligarh, U.P, India
Amit Gangopadhyay, Head, Department of Electronics and Communication Engg., Mangalayatan University, Aligarh, U.P, India
A drive system is composed of a motor connected to a load machine through a shaft. In many industrial drives, like rolling mill drives or robot arms, the mechanical part of the system containing a long shaft between the motor and the load machine must be taken into account. Especially in the drive systems with high performance speed and torque regulation, the motor speed is different from the load speed during transients [2].The speed difference results in the coupling shaft stresses, which influence this mechanical coupling in a negative way. Additionally, speed oscillations decrease the quality of the product and can influence the stability of the control system. So the suitable control structure ensuring the vibration dumping must be used, which require additional feedbacks from such state variables as torsional torque, load speed and/or disturbance torque [2].
In numerous applications joined with electrical drives, algorithmic techniques are requisitioned the non- quantifiable state variables estimation, for instance, the Kalman filters [3, 4] and the Luenberger observers [5].
However, the algorithmic estimators require the scientific model and parameter learning of the system, which could change during the system operation—so to acquire the great estimation quality the parameters of the state estimators must be tuned on-line (by on-line plant parameters' recognizable proof alternately estimation). Alternative ways of solving this problem are estimators based on neural networks (NNs). Such estimators do not need a mathematical model and parameters of the system, only the training data are required for the estimator design.
Moreover, the generalization ability causes that neural estimators are less sensitive to parameters or measurement signals uncertainties [5].
In this paper, the NN-based speed estimation for the uncommon sort of the drive system with flexible association between the determined motor and burden is analyzed. In such a sort of drive, the flexibility of the interfacing shaft between the drive engine and loading machine can amazingly break down precision of the rate or position control, and in exceptional cases can provoke the loss of stability of the drive system. Thus, the nature of the mechanical technique can be inside and out lessened. Along these lines, the suitable control structures ensuring the vibration dumping must be used, which oblige additional inputs from such state variables as torsional torque, weight speed, and/or irritation torque [2]. Nonetheless, the direct inputs from those state variables are all the time unimaginable, on the grounds that estimations of those signs are troublesome, savvy, and lessen the framework dependability[7]. In any case, these estimators require the scientific model and parameter learning of the framework,
which could change amid the framework operation—so to get the great estimation quality. In alternate words—on account of these state estimators, the ideal learning of the system parameters is needed, as they are generally delicate to parameter changes [6].
However, to remove complex mathematical modelling to reduce cost of drive system and to enhance the system stability proportional-derivative (PD) controller in cascade with proportional-integral controller is used with optimized torque control.
The main objective of this thesis is to implement PD controller for the Speed Estimation of the Two-Mass Drive System that reduces the difference in speed of two motor i.e drive side and load side as compare to NN estimator speed.
1) First we modelled and implement the two mass drive system with controller in Matlab R2013 Simulink software and get the different waveforms that estimate the low error or good syzchronisation between two systems.
2) Then we implement the two mass drive system with NN estimator in Simulink software and error is estimated which is quite larger than controller model.
3) Finally we modelled the PD controller with ModelSim plus SE5.6 software to provide the virtual hardware environment.
II. MATHEMATICAL MODEL OF TWO-
MASS SYSTEM
In this paper, the generally utilized scientific model of the drive system with the flexible coupling is considered.
Ordinarily, such commute is examined as a system made out of two masses joined by a flexible shaft, where the first mass speaks to the moment of inertia of the drive and the second mass refers the moment of inertia of the heap side. The framework is depicted by the accompanying state mathematical statement (in every unit system), where nonlinear phenomena, similar to backlash or friction are dismissed [2]
𝜏1𝑑𝜔1 𝑡
𝑑𝑡 = 𝑇𝑒 𝑡 − 𝑇𝑠 𝑡 (1) 𝜏2
𝑑𝜔2 𝑡
𝑑𝑡 = 𝑇𝑠 𝑡 − 𝑇𝐿 𝑡 (2) 𝜏𝑐
𝑑𝑇𝑠 𝑡
𝑑𝑡 = 𝜔1 𝑡 − 𝜔2 𝑡 (3) where, ω1 and ω2 are motor and load speeds , Te , TS , TL are electromagnetic, shaft, and load torques [in per unit system] 𝜏1, 𝜏2 are the mechanical time constants of the motor and load machine, and 𝜏c is the stiffness time constant.
Fig 1: The block diagram of the drive system with NN estimator The electrical drive system is normally controlled in the established cascade structure, which comprises of two control loops: the internal control loop encases the torque controller, the power converter, and the electromagnetic part of the engine. The PI torque controller is normally balanced by well known modulus basis, to give sufficiently quick torque regulation also, regularly is approximated by a first- order filter with little time constant [8, 9]. The external control loop incorporates the mechanical piece of the drive, the speed sensor, and the PI speed controller (Fig. 1), ordinarily balanced by the symmetry standard or pole placement technique. This established structure functions admirably just for some inertia ratio (T1/T2) of the two-mass system [10]. The load machine speed must be assessed by the neural speed estimator in light of estimations of the engine electromagnetic torque (current) and driven engine speed. After a suitable improvement of the inside electromagnetic torque control circle, a quick reaction of the engine torque can be obtained [11]. In this way, this optimized part of the control structure, with the neglecting time constant, can be represented by the transfer function
G(s) = 1
Parameters of the outer velocity control loop are planned utilizing the pole placement method. In this way, the PI speed controller parameters and gain of additional feedback from the load speed are as follows:
𝐾𝑃= 4𝜖𝑟𝜔𝑟𝜏1
1+𝑘1 (4)
𝐾𝐼= 𝜏1
(1+𝑘1)2𝜏2𝜏𝑐 (5) 𝑘1= 𝜖𝑟24𝜏1−𝜏2
𝜏1+𝜏2 (6)
where ωr, Ɛr are assumed resonance frequency and dumping factor of the closed-loop system. The equations describing the several gains in the control structure are dependent on the chosen values of the damping coefficient Ɛr and the resonance pulsation ω1 of the system. Therefore, there is a possibility for setting of the dynamic performance of the electric drive.
In addition to this we derived the resonant frequency from the control structure of two mass drive system model to get the approximate speed which helps in error estimation of two mass drive system. The equation is
ωr < τ1+τ2
2τ1τ2τc (7)
With the help of equation (7) we easily calculate the resonant frequency for the two mass drive system at different time constants. In this paper, the two mass drive system tested with two cases one with single time constant and twice bigger mechanical time constant (inertia) of the load side, to check its robustness to parameter changes of the drive system.
Case 1: When τ1= τ2= 203ms and τc= 2.6ms, then according to equation (7)
ωr < 203 +203
2×203 ×203 ×2.6
ωr < 406 ×10−3
214286 .8 ×10−9
ωr < 0.6371
0.01463
ωr < 43.54 s−1
From above equation we find that in this case ωr should be less than ―43.54s−1‖ for good synchronization speed in two mass drive systems. So we are using ―40s−1” resonant frequency in this paper.
Case 2: When τ1= 203 , τ2= 406ms and τc = 2.6ms, then according to equation (7)
ωr < 203 +406
2×203 ×406 ×2.6
ωr < 609 ×10−3
428573 .6 ×10−9
ωr < 0.780384
0.020702
ωr < 37.69 s−1
From above equation we find that in this case ωr should be less than ―37.69s−1‖ for good synchronization speed in two mass drive systems. So we are using ―30s−1 ”resonant frequency in this paper.
III. PROPOSED WORK
As we know that there are many advantageous features of NNs (like the approximation, generalization of data, parallel processing) that cause they are a useful computational tool in practice. Inspite of this Neural Network estimation method have some problems like it required the mathematical model and parameter knowledge of the system. They usually sensitive to parameter changes that causes stability problem sometimes. Moreover their
designing methodology and practical realization is very complex and costly.
To overcome these problems Proportional Derivative (PD) controller is used in cascade with Proportional Integral controller (PI) with optimized torque control loop as shown in Fig 2.
The aim of using P-D controller is to increase the stability of the system by improving control since it has an ability to predict the future error of the system response. In order to avoid effects of the sudden change in the value of the error signal, the derivative is taken from the output response of the system variable instead of the error signal.
Therefore, D mode is designed to be proportional to the change of the output variable to prevent the sudden changes occurring in the control output resulting from sudden changes in the error signal. In addition D directly amplifies process noise therefore D-only control is not used.
Fig 2: The block diagram of the drive system with PD controller
IV. PROCEDURE
In this paper we are doing implementation of two mass system on MATLAB function called SIMULINK. Simulink, developed by MathWorks, is a data flow graphical programming language tool for modeling, simulating and analyzing multidomain dynamic systems. Its primary interface is a graphical block diagramming tool and a customizable set of block libraries.
There are three major steps in this paper:
1. Implementation of two mass system with NN estimator in Simulink software according to block diagram shown in fig 1.With the help of differents blocks present in simulink library, implement two mass system model in addition to this neural network feed forward network is added with the help of Matlab program using functions gensim and trainlm etc. Then run the
simulink model and get the different waveform results as shown in results section.
2. Second step is to implement the two mass system with PD controller as shown in fig 2 in Simulink software with different blocks in simulink library and after running this model we get the different systems waveform with error estimation and also calculate the error value through matlab programming. Figures of waveform is shown in results.
3. Lastly, we model this PD controller Simulink model with ModelSim SE5.6 version software which provides the hardware environment of two mass system. ModelSim blocks are generated at the input of the system with compile HDL design and Launch HDL simulator. With the help of cosimWizard and vsim function we Starts and configures the ModelSim simulator (vsim) for use with the MATLAB and Simulink features of HDL Verifier.
After running simulink model of PD controller modelsim two mass system the different waveforms will apper with sum of error value which show the virtual hardware implementation of two mass system these waveforms are shown in results.
V. RESULTS AND DISCUSSION
The NN estimators are tested in the control structure presented in Fig.1. The NN estimator was tested for nominal (𝜏2 = 𝜏2N) and twice bigger (𝜏2 = 2 𝜏2N) mechanical time constant(inertia) of the load side, to check its robustness to parameter changes of the drive system. It should be specified, that the NN was not trained for this greater time constant.
The drive system is carrying out the cyclic opposite with different reference speed values. In the wake of setting the velocity on the required level, the step change of the load torque is applied as shown in below figures the estimation errors of the load speed and shaft torque are very small.
Here we define two different cases with different time constants value and according to this different resonant frequency as we already discussed in section II.
All three implementation results are given below with both cases.
A. NN Estimator Simulation Results
Case 1 :When 𝜏1 = 203ms,𝜏2= 203ms and ωr = 40s-1
Fig 3: Estimation error of load speed and shaft torque for 𝜏1 = 203ms, 𝜏2= 203ms is 9.58
Case 2: When 𝜏1 = 203ms,𝜏2= 406ms and ωr = 30s-1
Fig 4: Estimation error of load speed and shaft torque for 𝜏1 = 203ms, 𝜏2= 406ms is 100.7080
Estimation error, for the test with nominal parameters of the two-mass drive system is equal 9.58 and increase of the time constant to 𝜏2 = 2 𝜏2N leads to increase of this error value i.e 100.7080
B. PD Controller Simulation Results
The PD Controller with PI Controller are tested in the control structure presented in Fig.1 in Matlab/ Simulink with different time constant as we done with NN estimator.
Case 1 :When 𝜏1 = 203ms,𝜏2= 203ms and ωr = 40s-1
Fig 5: Estimation error of load speed and shaft torque of PD Controller for 𝜏1 = 203ms, 𝜏2= 203ms is 0.03393
Case 2 : When 𝜏1 = 203ms,𝜏2= 406ms and ωr = 30s-1
Fig 6: Estimation error of load speed and shaft torque of PD Controller for 𝜏1 = 203ms, 𝜏2= 406msis 0.26039
As we clearly see from these error estimation results that estimation error of load speed and shaft torque in PD controller is much less than in NN estimator results.
Now we are using modelsim for virtual hardware implementation for getting the error estimated output of load speed and shaft speed with PD Controller.
C. Modelsim Results
Now we are doing final simulation of PD controller with Modelsim that creates virtual hardware environment for error estimation of load speed and shaft torque of two mass drive system.
In this paper we are using ModelSim SE5.6 version offers high-performance and advanced debugging capabilities, while ModelSim PE is the entry-level simulator for hobbyists and students. ModelSim SE is used in large multi-million gate designs, and is supported on Microsoft Windows and Linux, in 32-bit and 64-bit architectures Simulation is done for both values of time constant as we done before. Here we show modelsim environment of version 6.5se of PD controller.
After running simulink model of PD controller modelsim two mass system the different waveforms will apper with sum of error value which show the virtual hardware implementation of two mass system.After running the model the Modelsim window parameters are shown as in figure 7.
Fig 7: Modelsim environment version 6.5se after running two mass model
Case 1 : When 𝜏1 = 203ms,𝜏2= 203ms and ωr = 40s-1
Fig 8: Modelsim Response for error estimation of load speed for τ1= 203ms, τ2= 203ms and error is 0.36286
Case 2 : When 𝜏1 = 203ms,𝜏2= 406ms and ωr = 30s-1
Fig 9: Modelsim Response for error estimation of load speed for τ1= 203ms, τ2= 406ms and error is 1.04071
Here, yellow color shows speed response of motor 1 and pink color waveform shows speed response of motor 2 and sky blue color shows the difference in their speed which is very small.
In both cases estimated error of two mass drive system is small but it is higher than the real hardware setup of two mass drive system DC motor.
D. COMPARISION OF ERROR
Table 7.1: Comparision of errors when 𝝉1 = 203ms, 𝝉2 = 203ms and 𝝎𝒓 = 𝟒𝟎 𝒔−𝟏
S No. NN
Estimator error
PD Controller
error
Modelsim Error
1 9.58 0.03393 0.36286
This table is for case 1 in which parameters are τ1 = 203ms, τ2 = 203ms and ωr = 40 s−1
Here, we find that the error of nn estimator is higher than PD controller error and modelsim error so we find that addition of PD controller by replacing NN estimator is a good choice.
Table 7.2: Comparision of errors when 𝝉1 = 203ms, 𝝉2 = 406ms and 𝝎𝒓 = 𝟑𝟎 𝒔−𝟏
S No. NN
Estimator error
PD Controller
error
ModelSim Error
1 100.7080 0.26039 1.04071
This table is for case 2 in which parameter are τ1 = 203ms, τ2 = 203ms and ωr = 40 s−1
This also shows that PD controller two mass system results are much better than NN estimator error results.
In comparision to both tables we saw that case 1 having 𝜏1 = 203ms, 𝜏2 = 203ms so we take 𝜔𝑟 = 30 𝑠−1 and case 2 having 𝜏1 = 203ms, 𝜏2 = 406ms and 𝜔𝑟 = 30 𝑠−1according to equation (7) we saw that as the time constant doubles the error estimation also slightly increases but in case of NN estimator it increases more than PD controller which indicates that two mass drive system with PD controller is more robust to changes of the load side inertia than two mass drive system with NN estimator.
VI. CONCLUSION
Application of neural estimators in the drive system with elastic coupling enables very good estimation quality There are many advantageous features of NNs (like the approximation, generalization of data, parallel processing) that cause they are a useful computational tool in practice.
NN estimator generates low estimate error inspite of this method have some problems like it required the mathematical model and parameter knowledge of the system. They usually sensitive to parameter changes that causes stability problem sometimes. Moreover their designing methodology and practical realization is very complex and costly. So, to overcome these issues or to increase the stability and reduce the error od two mass drive system PD controller is used .The two mass drive system has been modelled in Simulink where the motor, load and shaft are modelled as integral elements. Hence, we get small estimated error in case of PD controller than NN estimator.
In addition to this we are doing simulation of PD controller with Modelsim SE6.5 for virtual hardware environment of two mass drive system dc motor and we estimated small error without implementation of such a real hardware setup which is very complex and costly.
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