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HCS_Toolbox_ver01/demo_hpic.m

+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%  Example of Homogenenous Proportional Integral Control (HPIC) design
+%%   
+%%  System:      dx/dt=A*x+B*u  
+%%                
+%%        where  
+%%               x - system  state vector (n x 1)
+%%               u - control input (m x 1)
+%%               A - system matrix (n x n)
+%%               B - control matrix (m x m)
+%%      
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%
+%% Model of system
+%%%%%%%%%%%%%%%%%%%
+A= [0 1; -1 0]; % sysytem matrix (harmonic oscillator)
+
+B= [0;    1];  % control matrix
+
+n=2; m=1;
+ 
+
+%%%%%%%%%%%%%%%%%%
+%% HPIC design
+%%%%%%%%%%%%%%%%%%
+
+mu=-0.25;   %homogeneity degree
+
+[K0, K, Ki, Gd, P]=hpic_design(A,B,mu);
+
+%K0 - homogenization feedback gain
+%K  - proportional control gain
+%Ki - integral control gain 
+%Gd - generator of dilation
+%P  - shape matrix of the weighted Euclidean norm
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% Numerical Simulation
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+t=0; Tmax=5; 
+h=0.001; % sampling period
+
+x=[1;0];
+tl=[t];xl=[x];ul=[];
+
+
+alpha=0.00;%chattering attenuation parameter
+
+noise=0; %magnitude of measurement noises
+
+disp('Run numerical simulation...');
+
+p=1; %contstant perturbation with an unknown bound
+
+v=0; % for computation of integral term
+  
+while t<Tmax
+    xm=x+2*noise*(rand(n,1)-0.5);    
+    
+    %u=(K0+K)*xm;  ui=Ki*xm;                %linear PI control (for comparison)
+    
+    [u ui]=e_hpic(xm,K0,K,Ki,Gd,P,mu,alpha);%HPIC    
+    
+    u=u+v; v=v+h*ui;                     
+    
+    %u=e_hpic(xm,K0,K,Ki,Gd,P,mu,alpha);       %HPC (for comparison)
+
+
+    x=x+h*A*x+h*B*(u+p); % explicit Euler method 
+    t=t+h;
+    tl=[tl t];
+    xl=[xl x];
+    ul=[ul u];
+end;
+ul=[ul u];
+
+disp('Done!');
+
+%%norm of the state at the time instant Tmax
+disp(['||x(Tmax)||=',num2str(norm(x))])
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+% Plot simulation results
+%%%%%%%%%%%%%%%%%%%%%%%%%
+figure;
+axes1 = subplot(1,2,1);
+hold(axes1,'on');
+plot1 = plot(tl,xl,'LineWidth',2,'Parent',axes1);
+set(plot1(1),'DisplayName','$x_1$');
+set(plot1(2),'DisplayName','$x_2$');
+ylabel('$x$','Interpreter','latex');
+xlabel('$t$','Interpreter','latex');
+title({'n=2'});
+xlim(axes1,[0 Tmax]);
+ylim(axes1,[-2 2]);
+box(axes1,'on');
+hold(axes1,'off');
+set(axes1,'FontSize',30,'XGrid','on','YGrid','on');
+legend1 = legend(axes1,'show');
+set(legend1,'Interpreter','latex');
+
+
+axes2 = subplot(1,2,2);
+hold(axes2,'on');
+plot2 = plot(tl,ul,'LineWidth',2);
+ylabel('$u$','Interpreter','latex');
+xlabel('$t$','Interpreter','latex');
+title({'HPIC,m=1'});
+xlim(axes2,[0 Tmax]);
+ylim(axes2,[-2 2]);
+box(axes2,'on');
+hold(axes2,'off');
+set(axes2,'FontSize',30,'XGrid','on','YGrid','on');