Stabilization of an Axially Moving Accelerated/Decelerated System via an Adaptive Boundary Control

Abstract

In this chapter, an adaptive boundary control is developed for vibration suppression of an axially moving accelerated/decelerated belt system. The dynamic model of the belt system is represented by partial-ordinary differential equations with consideration of the high acceleration/deceleration and unknown distributed disturbance. By utilizing adaptive technique and Lyapunov-based backstepping method, an adaptive boundary control is proposed for vibration suppression of the belt system, a disturbance observer is introduced to attenuate the effects of unknown boundary disturbance, the adaptive law is developed to handle parametric uncertainties and the S-curve acceleration/deceleration method is adopted to plan the belt’s speed. With the proposed control scheme, the well-posedness and stability of the closed-loop system are mathematically demonstrated.

Appendix 1: Simulation Program

close all; clear all; clc; nx=50; nt=100*10^3; L= 1; tmax =10; % --- Compute mesh spacing and time step dx = L/(nx-1); dt = tmax/(nt-1); % --- Create arrays to save data for export x = linspace(0,L,nx); t = linspace(0,tmax,nt); % --- parameters m = 1 ; T = 4900; pi = 3.1415926; V0=0; c = 1; % --------------velocity----------------- V = zeros(1,nt); V(1,1) = V0; J=3.5*9.8; for j=2:10*10^3-1     V(1,j) = V(1,1)+0.5*J*(j*dt)^2; end V(1,10*10^3)=V(1,1)+0.5*J; for j=(10*10^3+1):20*10^3-1 V(1,j)=V(1,10*10^3)+0.5*J*(j*dt-1); end V(1,20*10^3)=V(1,10*10^3)+J; for j=(20*10^3+1):30*10^3-1     V(1,j) = V(1,20*10^3)+J*(j*dt-2)-0.5*J*(j*dt-2)^2; end V(1,30*10^3)=V(1,20*10^3)+0.5*J; for j=(30*10^3+1):70*10^3-1     V(1,j)=V(1,30*10^3); end V(1,70*10^3)=V(1,30*10^3); for j=(70*10^3+1):80*10^3-1     V(1,j)=V(1,70*10^3)-0.5*J*(j*dt-7)^2; end V(1,80*10^3)=V(1,70*10^3)-0.5*J; for j=(80*10^3+1):90*10^3-1     V(1,j)=V(1,80*10^3)-J*(j*dt-8); end V(1,90*10^3)=V(1,80*10^3)-J; for j=(90*10^3+1):nt-1     V(1,j)=V(1,90*10^3)-J*(j*dt-9)+0.5*J*(j*dt-9)^2; end V(1,100*10^3)=V(1,90*10^3)-0.5*J; % -------------disburtance------------- d = zeros(1,nt); f = zeros(nx,nt); for j=1:nt     d(1,j)=10+cos(1*(j-1)*dt)+cos(2*(j-1)*dt)+cos(3*(j-1)*dt); end for j = 1 : nt     for i = 1 : nx        f(i,j) = 0.001*(10*(i-1)*dx+sin(pi*(i-1)*dx*(j-1)*dt)+sin(2*pi*(i-1)*dx*(j-1)*dt)+sin(3*pi*(i-1)*dx*(j-1)*dt));     end end %**************************************** % uncontrolled %**************************************** ds=0.25; mc=5; w1 = zeros(nx,nt); w1(1,:) = 0; for i=2:nx     w1(i,1) = 0;     w1(i,2) = 0; end for j = 3 : nt     for i = 2 : nx - 1        w1(i,j) = 2*w1(i,j-1)-w1(i,j-2) +((T-m*V(1,j-1)^2)*(w1(i+1,j-1)-2*w1(i,j-1)+w1(i-1,j-1))/dx^2 -c*(w1(i,j-1)-w1(i,j-2))/dt - c*V(1,j-1)*(w1(i,j-1)-w1(i-1,j-1))/dx -2*V(1,j-1)*m*(w1(i,j-1)-w1(i,j-2)-w1(i-1,j-1)+w1(i-1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w1(i,j-1)-w1(i-1,j-1))/dx + f(i,j) ) / (m/dt^2);     end     w1(nx,j) = ( (2*mc/dt^2 + ds/dt )*w1(nx,j-1) - (mc/dt^2)*w1(nx,j-2) + d(1,j) + (T/dx)*w1(nx-1,j-1) )/ (mc/dt^2 + ds/dt + T/dx); end tshort = linspace(0,tmax,(nt/400+1)); xshort = linspace(0,L,nx); for j=1:nt/400     for i=1:nx        w1short(i,j)=w1(i,j*400);     end end w1short=[w1(:,1),w1short]; figure (1); surf(tshort,xshort,w1short); colormap('jet') ; xlabel('Time [s]','Fontsize',14); ylabel('x [m]','Fontsize',14); zlabel('w (x,t) [m]','Fontsize',14); set(gca,'YDir','reverse') %**************************************** % backstepping adaptive + controlled + disturbance observer %**************************************** ds = 0.25; mc = 5; k1 = 100; k2 = 100; k3 = 10; gamma = 0.1; lambda = 0.001; G=diag([1 1 1 1]); w2 = zeros(nx,nt); u1 = zeros(1,nt); z_e = zeros(1,nt); d_e1 = zeros(1,nt); phi_1 = zeros(1,nt); error_1 = zeros(1,nt); T_e = zeros(1,nt); mc_e = zeros(1,nt); m_e = zeros(1,nt); ds_e = zeros(1,nt); err_T = zeros(1,nt); err_mc = zeros(1,nt); err_ds = zeros(1,nt); err_m = zeros(1,nt); zeta=1; for j = 1 : 2     T_e(1,j) = 0;     mc_e(1,j) = 0;     ds_e(1,j) = 0;     m_e(1,j) = 0; end w2(1,:) = 0; for i=2:nx     w2(i,1) = 0;     w2(i,2) = 0; end for j = 3 : nt     for i = 2 : nx - 1        w2(i,j) = 2*w2(i,j-1)-w2(i,j-2) +((T-m*V(1,j-1)^2)*(w2(i+1,j-1)-2*w2(i,j-1)+w2(i-1,j-1))/dx^2 -c*(w2(i,j-1)-w2(i,j-2))/dt - c*V(1,j-1)*(w2(i,j-1)-w2(i-1,j-1))/dx -2*V(1,j-1)*m*(w2(i,j-1)-w2(i,j-2)-w2(i-1,j-1)+w2(i-1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w2(i,j-1)-w2(i-1,j-1))/dx + f(i,j) ) / (m/dt^2);     end     w2(nx,j) = 2*w2(nx,j-1)-w2(nx,j-2)+dt^2/mc*(u1(1,j-1)+d(1,j)-T*(w2(nx,j-1)-w2(nx-1,j-1))/dx-ds*(w2(nx,j-1)-w2(nx,j-2))/dt);     z_e(1,j) = (w2(nx,j)-w2(nx,j-1))/dt + k1*(w2(nx,j) - w2(nx-1,j))/dx ;     m_e(1,j) = m_e(1,j-1)+ dt*(-zeta*G(1,1)*m_e(1,j-1) + z_e(1,j)*G(1,1)*(k1*gamma*V(1,j)-gamma*V(1,j)^2-k1*lambda*L)*(w2(nx,j)-w2(nx-1,j))/dx );     mc_e(1,j) = mc_e(1,j-1)+dt*(-zeta*G(2,2)*mc_e(1,j-1)+z_e(1,j)*G(2,2)*k1*((w2(nx,j) - w2(nx-1,j) - w2(nx,j-1) + w2(nx-1,j-1))/(dx*dt)));     ds_e(1,j) = ds_e(1,j-1) +dt*(-zeta*G(3,3)*ds_e(1,j-1)+z_e(1,j)*G(3,3)*k1*(w2(nx,j) - w2(nx-1,j))/dx);     T_e(1,j) = T_e(1,j-1) +dt*(-zeta*G(4,4)*T_e(1,j-1)+z_e(1,j)*G(4,4)*(gamma-1)*(w2(nx,j)-w2(nx-1,j))/dx );     err_T = T - T_e(1,j);     err_mc = mc - mc_e(1,j);     err_m = m - m_e(1,j);     err_ds = ds - ds_e(1,j);     u1(1,j) = -d_e1(1,j-1) - ds_e(1,j)*k1*(w2(nx,j)-w2(nx-1,j))/dx -(gamma-1)*T_e(1,j)*(w2(nx,j)-w2(nx-1,j))/dx-mc_e(1,j)*k1*(w2(nx,j) -w2(nx-1,j)-w2(nx,j-1)+w2(nx-1,j-1))/(dt*dx)+m_e(1,j)*( gamma*V(1,j)^2 - k1*gamma*V(1,j) + k1*lambda*L)*(w2(nx,j)-w2(nx-1,j))/dx -k2*(w2(nx,j)-w2(nx,j-1))/dt - k1*k2*(w2(nx,j)-w2(nx-1,j))/dx;     phi_1(1,j) = phi_1(1,j-1) - dt*(k3*phi_1(1,j-1) - k3*ds*(w2(nx,j)-w2(nx,j-1))/dt - k3*T*(w2(nx,j)-w2(nx-1,j))/dx +k3*u1(1,j) + (k3)^2*mc*(w2(nx,j)-w2(nx,j-1))/dt );     d_e1(1,j) = phi_1(1,j) + k3*mc*(w2(nx,j) - w2(nx,j-1))/dt;     error_1(1,j) = d(1,j) - d_e1(1,j); end for j=1:nt/400     for i=1:nx        w2short(i,j)=w2(i,j*400);     end end w2short=[w2(:,1),w2short]; figure (2); surf(tshort,xshort,w2short); colormap('jet') ; xlabel('Time [s]','Fontsize',14); ylabel('x [m]','Fontsize',14); zlabel('w(x,t) [m]','Fontsize',14); set(gca,'YDir','reverse') figure (3); hold on subplot(2,1,1); plot(tshort,w1short(50,:),'b',tshort,w2short(50,:),':r'); title('(a)'); xlabel('Time [s]','Fontsize',14); ylabel('w(1,t) [m]','Fontsize',14); legend('uncontrolled','controlled'); box on grid on subplot(2,1,2); plot(tshort,w1short(25,:),'b',tshort,w2short(25,:),':r'); title('(b)'); xlabel('Time [s]','Fontsize',14); ylabel('w(0.5,t) [m]','Fontsize',14); legend('uncontrolled','controlled'); box on grid on hold off figure (4); plot(tshort,w2short(50,:),':r',tshort,w2short(25,:),'b'); legend('x = 1m','x = 0.5m'); xlabel('Time [s]','Fontsize',14); ylabel('w(x,t) [m]','Fontsize',14); figure(5); plot(t,d_e1,'b',t,d,'--r'); xlabel('Time [s]','Fontsize',14); ylabel('d(t) [N]','Fontsize',14); legend('estimate of disturbance','actual disturbance'); box on grid on figure(6); plot(t,u1,'b'); xlabel('Time [s]','Fontsize',14); ylabel('U(t) [N]','Fontsize',14); box on grid on