Polynomial Interpolation in 5-AXIS Milling

LNU-FUZZY NETWORK AS A MATHEMATICAL ADAPTIVE MODEL OF A HYDRAULIC SYSTEM MARTIN VESELY FMV-Design, Czech Technical University in Prague Faculty of Mechanical Engineering Department of Instrumentation and Control Engineering DOI : 10.17973/MMSJ.2018_11_201856 e-mail : [email protected] Model adaptive controllers such as Model Predictive Control or Model Reference Adaptive Control need a precise mathematical model of the controlled system adaptable in real-time. Systems consisting of a hydraulic 4way proportional valve and a linear motor have non-linear behaviour such as hysteresis of and valve, death zone of a valve spool, time delay of a data transfer and control unit, dependence on coils temperature and oil temperature and nonlinear flow characteristics. This paper introduces modified Neuro-Fuzzy network as a mathematical adaptive model of a hydraulic system with above mentioned properties. The paper presents the basic architecture of Neuro-Fuzzy network which consists of artificial neural units a fuzzy layer and introduces modifications focused on identification. The basic real-time learning method such as Normalized Gradient Descent is introduced specially for the designed Neuro-Fuzzy Network. Identification and real time learning abilities of the model were tested on the hydraulic stand. KEYWORDS neuro-fuzzy network, 4-way proportional valve, adaptive model 1 Nonlinear adaptive controllers need a precise mathematical model of the controlled system adaptable in real-time. This article presents the basic architecture of Neuro-Fuzzy network which consists of artificial neural units [YANG 2016],[GUPTA 2004] and fuzzy layer, and introduces modifications focused on identification of hydraulic systems. The basic real-time learning method such as Normalized Gradient Descent [BUKOVSKY 2016] is introduced specially for the designed Neuro-Fuzzy Network. Identification and real-time learning abilities of the model were tested on the hydraulic stand. 2 PROPERTIES OF 4-WAY PROPORTIONAL VALVE One of the most important property of the valve is their flow characteristic, dependence of flow through the valve on the spool position or current to the coil. We expect that spool is much faster than the hydraulic motor so it is possible to neglect dynamic of the spool. The flow through the valve isn't directly proportional to the current in the coil for following reasons: • Position of the spool depends on force-current characteristics of the coil and the temperature of the coil. • The channel opening is not directly proportional to the position of the spool • The flow depends on Bernoulli's equation • The Figure 1 presents measured flow-characteristics of the valve. There is obvious offset depending on minimum opening of the channels, hysteresis depending on the spool and the magnet friction and nonlinear characteristics. INTRODUCTION Systems consisting of a hydraulic 4-way proportional valve and a linear hydraulic motor are usually controlled by PID or their modifications. Designed controller has to fulfil requirements for the control criteria and also has to be designed with robust control for changing behaviour of the controlled system. Using classical control strategies can result in following: • Each controller for each machine has to be set separately. • The controller of machine has to be reconfigured during the machine lifetime. • There are requirements for linearity of the systems components. All above mentioned points increase the price of the machine or the price for their service. That’s the motivation for designing nonlinear adaptive controllers for hydraulic systems. Nonlinear adaptive controllers such as Model Predictive Control (MPC) [YANG 2016] or Model Reference Adaptive Control (MRAC) [HE 2012], [YANG 2011] are able to change their parameters during process according to changing systems behaviour. They are also able to partly calculate with nonlinear systems. Using nonlinear adaptive controllers is not meant to promise high precision of controlling but it promises low price of machines. Figure 1. Current-Flow Characteristic of a Valve Hydraforce SP08-47CSpool-Type, 4-Way, 3-Position, Closed Center 3 LNU-FUZZY MODEL The main advantage of Artificial Neural Network is their universality of use and their ability to learn. But they are not as suitable for using in real-time process including in simple control units because they can be too large and their learning can be too computationally demanding. Fuzzy sets represent intuitive approach for when you one has mainly only theoretical or verbal knowledge. For example in our case: "The slope of the Current-Flow characteristic depends on actual current and its derivation." According that it is possible to build a space of fuzzy values. By combining the simplest neural model Linear Neural Units (LNUs) and output fuzzy layer a mathematical model was built with following advantages: MM SCIENCE JOURNAL I 2018 I NOVEMBER 2573 Even though the model is nonlinear it is linear according to LNUs weights, so it is possible to teach LNUs with simple optimization algorithms that search only for local minimums such as Normalized Gradient Descent (NGD). The size of the model is smaller than neural networks with hidden layers Building the model can be done intuitively according to technical knowledge of the system and it is also possible to use machine learning. Linear Neural Unit (LNU) is the simplest HONU model [BUKOVSKY 2007]. The formula without activation function is following: Figure3. LNU-Fuzzy Model Fuzzy membership yi ( k )   wi ( k )  xi ( k ) (1) y i is the neural output, w is a vector of neural weights. x is a input vector into LNU and for systems with 1 Where degree of freedom can be follows: y position, nu i is calculated as samples of control input and n the  2 (3) number of is fuzzy sets and S i is the center position of the fuzzy set  i . The final output of the model is n is the vector of recent u( k )  u( k )  u( k 1) , Where  2n2  u( k  )  Si calculated as a center of maxim as follows: T   y( k  ny ) ,K , y( k 1) , u( k   nu 1) , K , u( k  )  Where to a given fuzzy set i  e j 1 x( k )   x1( k ) ,K , xn ( k )  i follows: q ny T (2) y( k )  y samples of piston u is the vector of recent  is the input delay of the system. The reason why u is used here instead of u is in hysteresis. Hysteresis moves the absolute value of the input signal, but u is independent of that. The slope of the Current-Flow characteristic depends on actual input, hence the fuzzy variable was chosen as the delayed input u( k  ) . The space of the fuzzy variable is shown in Chyba! Nenalezen zdroj odkazů. where each of fuzzy set i i(k ) i 1 i(k ) (4) n  i 1 i(k ) LNU-Fuzzy model given be (1),(2) and (4) can be rewritten to the matrix form as follows: y( k )  α( k )  W( k )  x( k ) (5) Where the neural weights w ji of the matrix W have to learn in real-time. For that Normalized Gradient Descent algorithm was chosen given by (6). belongs to W( k )  W( k 1)  norm ( k 1)  one of the LNU. The final built Neuro-Fuzzy model is in Chyba! Nenalezen zdroj odkazů.. Q( k 1)  Where Q( k 1) w( k 1) 1 2 1 eref ( k 1)   yref ( k 1)  y( k 1)  2 2 (6) (7) Q is the cost function where eref is the reference error yref and the output from between real output of the system the model y . Putting (5) and (7) into (6) we get the following formula for weights adaption. W( k )  W( k 1)  norm ( k 1)  eref ( k 1)  αTk 1  xTk 1 (8) According [3] and (7) is the normalized learning rate follows for this case: norm( k 1)  Figure 2. Fuzzy Sets Where    α ( k 1)  α T ( k 1)  xT( k 1)  x( k 1) (9)  is the learning rate and  is dumping small value. 4 IDENTIFICATION OF THE HYDRAULIC STAND The experimental adaptive identification was tested on hydraulic stand shown in Figure 4 with hydraulic schema Figure 5. The lower hydraulic motor controlled by 4-way proportional valve in Figure 1 with the input signal u. The output signal was MM SCIENCE JOURNAL I 2018 I NOVEMBER 2574 the position of the lower piston y. The upper hydraulic motor was used as a constant load controlled by using pressure reducing valve. This arrangement of two hydraulic motors against each other simulating a hydraulic press. Important settings of the hydraulic stand and LNU-Fuzzy model are in Table 1.   0.01 Learning rate U p V2 p2 A U x P y in x T ny  20 u in x nu  30 Fuzzy sets n  20 V1 A B P T U Pressure 1 p1  50[bar ] Pressure 2 p2  20[bar ] Max. force V1 FV 1  1100[N] Max. force V2 FV 2  400[N] Max. speed v  150[mm  s p p1 1 ] Table 1. Hydraulic Stent and LNU-Fuzzy Model Settings Figure5. Simplified Hydraulic Scheme For identification 2 data sets were measured. Output data was compared with output from the model given as a prediction from the 0.2s data. The first data are from controlling without load and the LNU-Fuzzy model was pre-trained for it in Chyba! Nenalezen zdroj odkazů.. The second data was with the load and was tested if and how quickly is the model able to adapt to these changes. Chyba! Nenalezen zdroj odkazů. shows data with load and the model output trained on the data without load. The position of the contact of the pistons and higher inaccuracy of the model is visible. Figure4. Hydraulic Stand Figure6. Trained Model without a Load, - line reference data, -- line predicted output MM SCIENCE JOURNAL I 2018 I NOVEMBER 2575 • Model is not as big as universal neural networks with hidden layer or as a High Order Neural Units. Therefore it is more suitable for use in simple controllers. Disadvantages of this type of modelling must be mentioned, and are following: • The parameters of this type of the model don’t have any real meaning such as dumping, stiffness or mass. If the model doesn't work or stops working flawlessly it is hard to tell the reasons why or where the problem could be. Another problem can be brought by real-time learning. • For using in practise it should be ensured that the model can't be retrained to worse results. REFERENCES Figure7. Trained Model with a Load, - line reference data, -- line predicted output Figure 8 shows model adapting to new data with load. Comparing Figure 7 and Figure 8 shows the ability to adapt to new data or new behaviour with load. Figure 8. Adaptive Model with a Load, - line reference data, -- line predicted output 5 [Yang 2016] Yang, Y., Tan, S. C., Hui, A. S. Y. Adaptive reference model predictive control for power electronics. In: 2016 IEEE Applied Power Electronics Conference and Exposition (APEC): 2016 IEEE Applied Power Electronics Conference and Exposition (APEC). 2016, pp. 1169–1175. [He 2012] He, Qiang. The design of model reference adaptive control. In: IET Conference Proceedings; Stevenage, United Kingdom: The Institution of Engineering & Technology, 2012 [vid. 2017-04-11]. ISBN 978-1-84919-537-9. [Yang 2011] Yang, Y., Balakrishnan, S. N., Tang, L. and Landers, R. G. Electro-hydraulic piston control using neural MRAC based on a modified state observer. In: Proceedings of the 2011 American Control Conference: Proceedings of the 2011 American Control Conference. 2011, pp. 25–30. [Gupta 2004] Gupta, Madan, Liang Jin a Noriyasu Homma. Static and Dynamic Neural Networks: From Fundamentals to Advanced Theory. B.m.: John Wiley & Sons, 2004. ISBN 978-0 471-46092-3. [Bukovsky 2016] Bukovsky Ivo and Noriyasu Homma. An Approach to Stable Gradient Descent Adaptation of Higher Order Neural Units. IEEE Transactions on Neural Networks and Learning Systems. 2016 (DOI:10.1109/TNNLS.2016.2572310). [Bukovsky 2007] Bukovsky, Ivo, Zeng-Guang Hou, Jiri Bila a Madan M. Gupta. Foundation of Notation and Classification of Nonconventional Static and Dynamic Neural Units. B.m.: IEEE, 2007, s. 401–407 [vid. 2016-03-13]. ISBN 978-1-4244 1327-0. CONCLUSIONS This article presented one of the options for identification of a hydraulic system consisting of a 4-way electromagnetic proportional valve and a linear hydraulic motor. Using Linear Neural Units and fuzzy sets a nonlinear model was built. The main reasoning for that is using adaptive controllers such as MPC or MRAC. The article briefly introduces properties of a 4-way proportional valve which should be respected in the model. The article explained the basics of neural units and fuzzy sets and LNUFuzzy model was created. A specialized augmented Normalized Gradient Descent was used for LNU-Fuzzy model learning. In the last section the LNU-Fuzzy model was used for identification and their retraining after changing the load on the stand simulating a machine press. Using LNU-Fuzzy model as a model of hydraulic system brings following advantages: • Model is adaptable during process and during lifetime of the machine and it's not necessary to retune it. • Model is linear according to neural weights so it's able to learn by using simple local searching algorithms such as mentioned NGD. CONTACT: Ing. Martin Vesely FMV-DESIGN s.r.o. and Czech Technical University in Prague Faculty of Mechanical Engineering Department of Instrumentation and Control Engineering Technicka 1902/4, 166 07 Prague, Czech Republic MM SCIENCE JOURNAL I 2018 I NOVEMBER 2576