32.2 Neuron Cell A biological neuron is a complicated structure, which receives trains of pulses on hundreds of excitatory and inhibitory inputs. Those incoming pulses are summed with different weights (averaged) during the time period of latent summation . If the summed value is higher than a threshold, then the neuron itself is generating a pulse, which is sent to neighboring neurons. Because incoming pulses are summed with time, the neuron generates a pulse train with a higher frequency for higher positive excitation. In other words, if the value of the summed weighted inputs is higher, the neuron generates pulses more frequently. At the same time, each neuron is characterized by the nonexcitability for a certain time after the firing pulse. This so-called refractory period can be more accurately described as a phenomenon where after excitation the threshold value increases to a very high value and then decreases gradually with a certain time constant. The refractory period sets soft upper limits on the frequency of the output pulse train. In the biological neuron, information is sent in the form of frequency modulated pulse trains. This description of neuron action leads to a very complex neuron model, which is not practical. McCulloch and Pitts (1943) show that even with a very simple neuron model, it is possible to build logic and memory circuits. Furthermore, these simple neurons with thresholds are usually more powerful than typical logic gates used in computers. The McCulloch–Pitts neuron model assumes that incoming and outgoing signals may have only binary values 0 and 1. If incoming signals summed through positive or negative weights have a value larger than threshold, then the neuron output is set to 1. Otherwise, it is set to 0. (32.1) where T is the threshold and net value is the weighted sum of all incoming signals: (32.2) Examples of McCulloch–Pitts neurons realizing OR, AND, NOT, and MEMORY operations are shown in Fig. 32.1. Note that the structure of OR and AND gates can be identical. With the same structure, other logic functions can be realized, as Fig. 32.2 shows. The perceptron model has a similar structure. Its input signals, the weights, and the thresholds could have any positive or negative values. Usually, instead of using variable threshold, one additional constant input with a negative or positive weight can added to each neuron, as Fig. 32.3 shows. In this case, the FIGURE 32.1 OR, AND, NOT, and MEMORY operations using networks with McCulloch–Pitts neuron model. FIGURE 32.2 Other logic function realized with McCulloch–Pitts neuron model. T 1, if net T≥ 0, if net T< = net w i i=1 n ∑ x i = +1 +1 +1 A B C T = 0.5 A + B + C (a) +1 +1 +1 A B C T = 2.5 ABC (b) AND −1 A T = − 0.5 NOT A (c) NOT +1 +1 T = 0.5 WRITE 1 WRITE 0 −2 (d) MEMORY OR +1 +1 +1 A B C T = 1.5 AB + BC + CA (a) +1 +1 +2 A B C T = 1.5 AB + C (b) 0066_Frame_C32.fm Page 2 Wednesday, January 9, 2002 7:54 PM ©2002 CRC Press LLC 32.2 Neuron Cell A biological neuron is a complicated structure, which receives trains of pulses on hundreds of excitatory and inhibitory inputs. Those incoming pulses are summed with different weights (averaged) during the time period of latent summation . If the summed value is higher than a threshold, then the neuron itself is generating a pulse, which is sent to neighboring neurons. Because incoming pulses are summed with time, the neuron generates a pulse train with a higher frequency for higher positive excitation. In other words, if the value of the summed weighted inputs is higher, the neuron generates pulses more frequently. At the same time, each neuron is characterized by the nonexcitability for a certain time after the firing pulse. This so-called refractory period can be more accurately described as a phenomenon where after excitation the threshold value increases to a very high value and then decreases gradually with a certain time constant. The refractory period sets soft upper limits on the frequency of the output pulse train. In the biological neuron, information is sent in the form of frequency modulated pulse trains. This description of neuron action leads to a very complex neuron model, which is not practical. McCulloch and Pitts (1943) show that even with a very simple neuron model, it is possible to build logic and memory circuits. Furthermore, these simple neurons with thresholds are usually more powerful than typical logic gates used in computers. The McCulloch–Pitts neuron model assumes that incoming and outgoing signals may have only binary values 0 and 1. If incoming signals summed through positive or negative weights have a value larger than threshold, then the neuron output is set to 1. Otherwise, it is set to 0. (32.1) where T is the threshold and net value is the weighted sum of all incoming signals: (32.2) Examples of McCulloch–Pitts neurons realizing OR, AND, NOT, and MEMORY operations are shown in Fig. 32.1. Note that the structure of OR and AND gates can be identical. With the same structure, other logic functions can be realized, as Fig. 32.2 shows. The perceptron model has a similar structure. Its input signals, the weights, and the thresholds could have any positive or negative values. Usually, instead of using variable threshold, one additional constant input with a negative or positive weight can added to each neuron, as Fig. 32.3 shows. In this case, the FIGURE 32.1 OR, AND, NOT, and MEMORY operations using networks with McCulloch–Pitts neuron model. FIGURE 32.2 Other logic function realized with McCulloch–Pitts neuron model. T 1, if net T≥ 0, if net T< = net w i i=1 n ∑ x i = +1 +1 +1 A B C T = 0.5 A + B + C (a) +1 +1 +1 A B C T = 2.5 ABC (b) AND −1 A T = − 0.5 NOT A (c) NOT +1 +1 T = 0.5 WRITE 1 WRITE 0 −2 (d) MEMORY OR +1 +1 +1 A B C T = 1.5 AB + BC + CA (a) +1 +1 +2 A B C T = 1.5 AB + C (b) 0066_Frame_C32.fm Page 2 Wednesday, January 9, 2002 7:54 PM ©2002 CRC Press LLC 33 Advanced Control of an Electrohydraulic Axis 33.1 Introduction 33.2 Generalities Concerning ROBI_3, a Cartesian Robot with Three Electrohydraulic Axes 33.3 Mathematical Model and Simulation of Electrohydraulic Axes The Extended Mathematical Model • Nonlinear Mathematical Model of the Servovalve • Nonlinear Mathematical Model of Linear Hydraulic Motor 33.4 Conventional Controllers Used to Control the Electrohydraulic Axis PID, PI, PD with Filtering • Observer • Simulation Results of Electrohydraulic Axis with Conventional Controllers 33.5 Control of Electrohydraulic Axis with Fuzzy Controllers 33.6 Neural Techniques Used to Control the Electrohydraulic Axis Neural Control Techniques 33.7 Neuro-Fuzzy Techniques Used to Control the Electrohydraulic Axis C ontrol Structure 33.8 Software Considerations 33.9 Conclusions 33.1 Introduction Due to the development of technology in the last few years, robots are seen as advanced mechatronic systems which require knowledge from mechanics, actuators, and control in order to perform very complex tasks. Different kinds of servo-systems, especially electrohydraulic, could be met at the executive level of the robots. Taking into account the most advanced control approaches, this paper deals with the implementation of advanced controllers besides conventional ones which are used in an electrohydraulic system. The considered electrohydraulic system is one of the axes of a robot. These robots possess three or more electrohydraulic axes, which are identical with the axis studied in this chapter. An electrohydraulic axis whose mathematical model (MM) is described in this chapter presents a multitude of nonlinearities. Conventional controllers are becoming increasingly inappropriate to control the systems with an imprecise model where many nonlinearities are manifested. Therefore, advanced techniques such as neural networks and fuzzy algorithms are deeply involved in the control of such systems. Neural networks, initially proposed by McCulloch and Pitts, Rosenblatt, Widrow, had several Florin Ionescu University of Applied Sciences Crina Vlad Politeknica University of Bucharest Dragos Arotaritei Aalborg University Esbjerg ©2002 CRC Press LLC . 0066_Frame_C32.fm Page 2 Wednesday, January 9, 2002 7:54 PM 2002 CRC Press LLC 33 Advanced Control of an Electrohydraulic Axis 33 .1 Introduction 33 .2 Generalities Concerning ROBI _3, a Cartesian. CA (a) +1 +1 +2 A B C T = 1.5 AB + C (b) 0066_Frame_C32.fm Page 2 Wednesday, January 9, 2002 7:54 PM 2002 CRC Press LLC 32 .2 Neuron Cell A biological neuron is a complicated structure,. Controllers 33 .5 Control of Electrohydraulic Axis with Fuzzy Controllers 33 .6 Neural Techniques Used to Control the Electrohydraulic Axis Neural Control Techniques 33 .7 Neuro-Fuzzy