Implementation of model and design of controllers

The model derived in accordance with section 4.2, was implemented into the Scilab/Xcos simulation environment. It allowed to perform the simulation tests necessary to design the most

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reliable wave maker controller. The fuzzy-logic controller of the Mamdani type with the flap-position feedback and flap-velocity feedback was considered. It was chosen as the target one due to the prime robustness, greatest stiffness, finest stability and satisfying step-response parameters, investigated along the dissertation.

The fuzzy-logic controller FLS of the flap velocity and flap position, has been considered.

This type of controller is successfully applied for improving the control performance of various types of objects, including the electrohydraulic actuators [41]-[43].

The FLS applied to the wave maker actuators has been established with two inputs and one output. The structure uses the flap velocity signal AX1 and the flap position signal AX2. The structural diagram of the proposed control system is shown in Fig. 4.5. The first FLS input is the flap position error FPE, calculated as the difference between the reference flap position signal AX2r and measured flap position signal AX2. The second FLS input is the flap velocity error FVE, calculated as the difference between the reference flap velocity signal taken as FPE and the measured flap velocity signal AX1. The FLS output is the S signal, given to the input of flap velocity module (I1), that acts the flap position module (I2) to move the flap. The scaling of the input and output signals is implemented in hardware with use of the signal matching circuits. The scaling parameters were selected to match the range of the FLS process variables with the thresholds of the sensors and actuators.

Fig. 4.5. Structural diagram of the flap velocity and flap position fuzzy-logic control system

The following fuzzy sets for input variable FVE have been formulated: Negative Fast NF, Negative Medium NM, Zero ZO, Positive Medium PM, Positive Fast PF.

Afterward, the following fuzzy sets for input variable FPE have been formulated: Negative Large NL, Negative Medium NM, Zero ZO, Positive Medium PM, Positive Large PL.

Wherefore, the following fuzzy sets for output variable S have been formulated: Positive Large PL, Positive Large PL, Zero ZO, Positive Medium PM, Negative Large NL, Negative Medium NM.

The membership functions of the fuzzy sets: mu(FPE), mu(FVE), mu(S), determine the grade of membership with a fuzzy set formulated for given variable: FPE, FVE or S, respectively.

Flap position module Flap velocity module

Flap velocity and flap position fuzzy-logic controller

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The Λ-, Γ- and L-type membership functions of fuzzy set have been formulated in accordance with [27]. The singleton-type membership functions of fuzzy set have been formulated in accordance with [28].

The membership functions of the input variables – FVE and FPE – are graphically presented in Fig. 4.6 and Fig. 4.7, respectively.

Fig. 4.6. Graph of the membership functions: Λ-type, Γ-type and L-type, determined to grade the membership mu of the FVE input variable with the fuzzy sets: NF, NM, ZO, PM, PF

Fig. 4.7. Graph of the membership functions: Λ-type, Γ-type and L-type, determined to grade the membership mu of the FPE input variable with the fuzzy sets: NL, NM, ZO, PM, PL

The membership functions of the output variable S are graphically presented in Fig. 4.8.

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Fig. 4.8. Graph of the membership functions singleton-type, determined to grade the membership mu of the S output variable with the fuzzy sets: PL, PM, ZO, NM, NL

The Mamdani fuzzy inference system has been established with a rule base presented in the Tab. 4.1. The FLS was intended to real-time computing in the embedded system applied to the wave maker. Thus, the calculations had to be simple and fast. Consequently, the output membership function and all the operations, were selected to be a plain addition or multiplication, as follows. The fuzzy implication of the values of membership with the fuzzy sets is performed in accordance with the algebraic product method. The aggregation of the active rules is performed in accordance with the algebraic sum method. The defuzzification is performed in accordance with the output membership function shown in the Fig. 4.8 and with the centre of gravity (CoG) method [27].

Tab. 4.1. Table of the rules base of the fuzzy inference system

FVEFPE NL NM ZO PM PL

PF PM PM PM PL PL

PM PM PM PM PM PL

ZO NM NM ZO PM PM

NM NL NM NM NM NM

NF NL NL NM NM NM

The simulation of work has been run and the output control surface of the modelled fuzzy-controller has been plotted as shown in Fig. 4.9.

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Fig. 4.9. Output control surface of the fuzzy-logic controller considered

The stability of the proposed system has been examined in the Lyapunov sense. It has been done using the procedure of state variable trajectory tracking, that is recommended for fuzzy-control systems [27], [29]. According to this procedure, a system is stable, if the state variables tend to the origin from any start state on the phase plan. The examination in Xcos/Scilab environment has been carried out for the closed-loop system presented in Fig. 4.5. The origin for the system considered is coordinated as (0.5;0.5). The trajectories have been tracked for given initial states of FPE and FVE on the phase plan as presented in Fig. 4.10. According to the results presented in Fig. 4.10, the state variables tend to the origin from all given initial states – thus the proposed system is stable in Lyapunov sense.

Fig. 4.10. Trajectories for given initial states of FPE and FVE on the phase plane – simulation in Xcos/Scilab

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Finally, the quality of regulation has been verified for the model of the closed loop-system with the established fuzzy-logic controller, under the parameters of step response: rise time tn, setting time tN, settling time tR, overshoot D, oscillating d/D.

The step response is shown in Fig. 4.11. It has been registered for the model of closed-loop system (Fig. 4.5) with AX2r given as input signal and AX2 given as output signal scaled to launch the step of 1 m stroke of the flap X2.

Fig. 4.11. Step response of the closed-loop system with fuzzy-logic controller considered – simulation in Xcos/Scilab

The current model has been chosen as the most reliable due to fine time-parameters of the step response and acceptable overshoot with oscillating. The fuzzy-logic controller has been intended for implementation to the real plant.