Instruction to simulate pendulum example are in the following video:

Notice that this example uses the simulink HyEQ simulator. Thus, you may create the files from scratch, or download the files here. Please, change the extension of the downloaded file from -zip.doc to .zip, then you can unzip and browse the files for the example.

Code for simulations

Flow map: f.m

function xdot = f(x) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matlab M-file Project: HyEQ Toolbox @ Hybrid Dynamics and Control % Lab, http://www.u.arizona.edu/~sricardo/index.php?n=Main.Software % % Project: Simulation of a hybrid system % % Name: f.m % % Description: Flow map % % Version: 1.0 % Required files: - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % flow map: xdot=f(x,u,parameters); x1=x(1); x2=x(2); length=1; gamma=-9.81; xdot=[x(2);gamma/length*sin(x(1))];

Flow set: C.m

function v = C(x) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matlab M-file Project: HyEQ Toolbox @ Hybrid Dynamics and Control % Lab, http://www.u.arizona.edu/~sricardo/index.php?n=Main.Software % % Project: Simulation of a hybrid system % % Name: C.m % % Description: Flow set % % Version: 1.0 % Required files: - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Check on flow conditions % E.g., % if (x(1) >= u(1)) % flow condition % v = 1; % report flow % else % v = 0; % do not report flow % end x1=x(1); x2=x(2); if x1 >= 0; v = 1; else v=0; end

Jump map: g.m

function xplus = g(x) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matlab M-file Project: HyEQ Toolbox @ Hybrid Dynamics and Control % Lab, http://www.u.arizona.edu/~sricardo/index.php?n=Main.Software % % Project: Simulation of a hybrid system % % Name: g.m % % Description: Jump map % % Version: 1.0 % Required files: - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % jump map: xplus = g(x,u,parameters); x2=x(2); xplus = [0.001;-.8*x2];

Jump set: D.m

function v = D(x) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matlab M-file Project: HyEQ Toolbox @ Hybrid Dynamics and Control % Lab, http://www.u.arizona.edu/~sricardo/index.php?n=Main.Software % % Project: Simulation of a hybrid system % % Name: D.m % % Description: Jump set % % Version: 1.0 % Required files: - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Check on flow conditions % % E.g., % if (x(1) <= u(1)) && (x(2) <= 0) % jump condition % v = 1; % report jump % else % v = 0; % do not report jump % end x1=x(1); x2=x(2); if x1 <= 0 && x2 <= 0 v=1; else v = 0; end

Initialization script: initialization_pendulum.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matlab M-file Project: HyEQ Toolbox @ Hybrid Dynamics and Control % Lab, http://www.u.arizona.edu/~sricardo/index.php?n=Main.Software % % Filename: initialization.m % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % initialization for bouncing ball example % clear all clc % initial conditions x0 = [pi/4;0]; % physical variables global gamma lambda gamma = -9.81; % gravity constant lambda = 0.8; % restitution coefficent % simulation horizon T = 10; J = 20; % rule for jumps % rule = 1 -> priority for jumps % rule = 2 -> priority for flows % rule = 3 -> no priority, random selection when simultaneous conditions rule = 1; %solver tolerances RelTol = 1e-6; MaxStep = 1e-3;

Plot script: postprocessing_pendulum.m

%postprocessing for the bouncing ball example % plot solution figure(1) clf subplot(2,1,1),plotflows(t,j,x(:,1)) grid on ylabel('x1') subplot(2,1,2),plotjumps(t,j,x(:,1)) grid on ylabel('x1') figure(2) clf subplot(2,1,1),plotflows(t,j,x(:,2)) grid on ylabel('x2') subplot(2,1,2),plotjumps(t,j,x(:,2)) grid on ylabel('x2') % plot hybrid arc plotHybridArc(t,j,x(:,1)) xlabel('j') ylabel('t') zlabel('x1')