電機滑?？刂?LuGre摩擦模型

Simulation for A Sliding Mode Controller

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Abstract: In this paper we first briefly introduce sliding mode controllers. Then we mainly study a sliding mode controller which has appeared in the literature. A simulation of the sliding mode controller is conducted in Matlab 6.5 environment, a corresponding PID controller also is done for comparsion.

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1 Introduction

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Variable structure control (VSC) with sliding mode control was first proposed and elaborated in the early 1950’s in the Soviet Union by Emelyanov and several coresearchers[2]. In their pioneer works, the plant considered was a linear second-order system modeled in phase variable form. Since then, VSC has developed into a general design method being examined for a

wide spectrum of system types including nonlinear systems, multi-input/multi-output systems, discrete-time models, large-scale and infinite-dimensional systems, and stochastic systems. In addition, the objectives of VSC has been greatly extended from stabilization to other control

functions. The most distinguished feature of VSC is its ability to result in very robust control systems; in many cases invariant control systems result. Loosely speaking, the term “invariant” means that the system is completely insensitive to parametric uncertainty and external disturbances. Today, research and development continue to apply VSC to a wide variety of engineering systems.

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The main idea at the basis of SMC techniques is that of designing a sliding surface to which the controlled system trajectories must belong. On the sliding manifold the behaviour of the system is the expected one. In order to obtain the control aim a control must be designed with an authority sufficient to dominate the uncertainties and the disturbances acting on the system. The control

promptly reacts to any deviation, however small, from the prescribed behaviour steering the system back to the sliding manifold. An advantage of this approach is that the sliding behaviour is insensitive to model uncertainties and disturbances which do not steer the system outside from the chosen manifold.

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In this paper, We mainly studied one sliding mode controller for position and speed control of flight simulator servo system with large friction which was proposed by Liu Jin kun and Er Lian jie in 2003[1]. Then we simulated the sliding mode controller and a PID controller (for comparision) in Matlab 6.5 environment.

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2 Sliding Mode Controller design for position and speed control of flight simulator servo system with large friction[1]

???This part can refer to [1].

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Reference

[1] Liu Jin kun and Er Lian jie, “Sliding Mode Controller Design for Position and Speed Control of Flight Simulator Servo System with Large Friction”, Journal of System Engineering and Electronics, Vol. 14, No. 3, pp. 59-62, 2003

[2] John Y. Hung, Weibing Gao, and James C. Hung, “Variable Structure Control: A Survey”, IEEE Transactions on Industrial Electronics, Vol. 40, No. 1, February, 1993

[3] Liu Jin Kun, “Matlab Simulation for Sliding Mode Control”, BeiJing, Publishing House for TsingHua University, 2005
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%%%%? Model.m %%%%function dx=Model(t,x)global A alfa J Ce R Km Ku a1 Fm Fc M kv c ep k delta persistent aa c=5.0; ep=30.6; k=30; a1=1.0; % Effect on the shape of friction curve Fm=20; Fc=15; kv=2.0; F=J*x(3); if t==0aa=0; enddF=J*aa; if abs(x(2))<=alfaif F>FmFf=Fm;dFf=0;elseif F<-FmFf=-Fm;dFf=0;elseFf=F;dFf=dF;end endif x(2)>alfaFf=Fc+(Fm-Fc)*exp(-a1*x(2))+kv*x(2);dFf=(Fm-Fc)*exp(-a1*x(2))*(-a1*x(3))+kv*x(3); elseif x(2)<-alfaFf=-Fc-(Fm-Fc)*exp(a1*x(2))+kv*x(2);dFf=-(Fm-Fc)*exp(a1*x(2))*(a1*x(3))+kv*x(3); endif x(2)>alfaFf=Fc+(Fm-Fc)*exp(-a1*x(2))+kv*x(2);dFf=(Fm-Fc)*exp(-a1*x(2))*(-a1*x(3))+kv*x(3); elseif x(2)<-alfaFf=-Fc-(Fm-Fc)*exp(a1*x(2))+kv*x(2);dFf=-(Fm-Fc)*exp(a1*x(2))*(a1*x(3))+kv*x(3); endA=1.0;F=1.0; r=A*sin(2*pi*F*t); dr=A*2*pi*F*cos(2*pi*F*t); ddr=-A*(2*pi*F)^2*sin(2*pi*F*t); e=r-x(1); de=dr-x(2); dde=ddr-x(3); s=c*e+de; M=2; if M==1u=200*(r-x(1))+40*(dr-x(2)); % PID elseif M==2delta=0.0003;kk=1/delta;if s>deltasats=1;elseif abs(s)<=delta % Saturated functionsats=kk*s;elseif s<-deltasats=-1;endslaw=-ep*sats-k*s;u=J*R/(Ku*Km)*(c*de+ddr+ep*sats+k*s+Km*Ce/(J*R)*x(2)+Ff/J); end du=200*de+40*dde; dx=zeros(3,1); dx(1)=x(2); dx(2)=-Km*Ce/(J*R)*x(2)+Ku*Km*u/(J*R)-Ff/J; dx(3)=-Km*Ce/(J*R)*x(3)+Ku*Km*du/(J*R)-dFf/J; aa=dx(3); % %%?? sim_smc.m?? %%%%%%%%% clear all; close all; global A alfa J Ce R Km Ku a1 Fm Fc M kv c ep k delta % Servo system Parameters J=0.6;Ce=1.2;Km=6; Ku=11;R=7.77; alfa=0.01; T=3.0; ts=0.001; % Sampling time TimeSet=[0:ts:T]; [t,x]=ode45('Model',TimeSet,[0,0,0],[],[]); x1=x(:,1); x2=x(:,2); x3=x(:,3);A=1.0;F=1.0; r=A*sin(2*pi*F*t); dr=A*2*pi*F*cos(2*pi*F*t); ddr=-A*(2*pi*F)^2*sin(2*pi*F*t); e=r-x1; de=dr-x2; s=c*e+de; F=J*x3; for i=1:1:T/ts+1time(i)=(i-1)*ts;if abs(x2(i))<=alfaif F(i)>FmFf(i)=Fm;elseif F(i)<-FmFf(i)=-Fm;elseFf(i)=F(i);endendif x2(i)>alfaFf(i)=Fc+(Fm-Fc)*exp(-a1*x2(i))+kv*x2(i);elseif x2(i)<-alfaFf(i)=-Fc-(Fm-Fc)*exp(a1*x2(i))+kv*x2(i);end endif M==1u=200*(r-x(1))+40*(dr-x(2)); % PID elseif M==2for i=1:1:T/ts+1kk=1/delta;if s(i)>deltasats(i)=1;elseif abs(s(i))<=delta % Saturated functionsats(i)=kk*s(i);elseif s(i)<-deltasats(i)=-1;endslaw(i)=-ep*sats(i)-k*s(i);u(i)=J*R/(Ku*Km)*(c*de(i)+ddr(i)+ep*sats(i)+k*s(i)+Km*Ce/(J*R)*x2(i)+Ff(i)/J);end endfigure(1); plot(t,r,'r',t,x(:,1),'b'); xlabel('time(s)');ylabel('position tracking'); figure(2); plot(t,dr,'r',t,x(:,2),'b'); xlabel('time(s)');ylabel('speed tracking'); figure(3) plot(t,e,'r'); xlabel('time(s)');ylabel('error'); figure(4); plot(x(:,2),Ff,'r'); xlabel('speed');ylabel('Friction'); figure(5); plot(t,s,'r'); xlabel('time(s)');ylabel('s'); figure(6); plot(time,u,'r'); xlabel('time(s)');ylabel('u'); figure(7); plot(r-x(:,1),dr-x(:,2),'r',r-x(:,1),-c*(r-x(:,1)),'b'); xlabel('e');ylabel('de');

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