Stiff System ode15s Simulation of a Robot Hydraulic Actuator with Fast Valve Dynamics

Project info

• Field of studyRobotics and Mechatronics
• LevelDoctoral dissertation (S3)
• Method typeNumerical Simulation
• Estimated build time± 7 working days
• Toolsode15s, ode45, odeset (MATLAB inti)
• PriceUS$34 - US$225

Research goal

Simulate the cylinder pressure using the ode15s solver and examine how the valve time constant and the fluid bulk modulus influence its behaviour.

Variables

Example result

• 68Stiffness parameter mu
• 7716ode45 steps
• 680ode15s steps
• 11.347Step ratio
• 49645ode45 f-evals
• 1716ode15s f-evals

For the stiff Cylinder pressure system (mu=68) ode15s needs only 680 steps versus 7716 for ode45 (a 11.3x ratio), with a mean solution difference of 0.001.

Computed from illustrative synthetic data, not from a real study.

About Stiff Systems with ode15s

ode15s is a solver for stiff differential equation systems, whose widely different time scales force ordinary solvers to take very small steps. It is used in chemical kinetics, electronic circuits, and process models with fast and slow dynamics.

Steps

1. 1Recognise stiffness: ode45 is very slow or takes tiny steps
2. 2Write the system as dy/dt = f(t,y)
3. 3Solve with ode15s and set tolerances
4. 4Also solve with ode45 as a comparison
5. 5Compare the number of steps and computation time
6. 6Plot the solutions and step counts

Assumptions to check

• The system is really stiff (large time-scale ratio)
• The Jacobian can be estimated or supplied for efficiency
• Tolerances are chosen according to accuracy needs

Reading and reporting the results

Report the number of steps and run time of ode15s versus ode45 as evidence of stiffness. Explain the fast and slow time scales in the model.

Example MATLAB code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

mu = 200;                                          % Van der Pol stiffness parameter
f = @(t, y) [y(2); mu*(1 - y(1)^2)*y(2) - y(1)];
tspan = [0 400];
y0 = [2; 0];
opts = odeset('RelTol', 1e-6, 'AbsTol', 1e-8);

tic; [t15, y15] = ode15s(f, tspan, y0, opts); t15s = toc;
tic; [t45, y45] = ode45(f, tspan, y0, opts); t45s = toc;
fprintf('ode15s: %d steps, %.3f s\n', numel(t15) - 1, t15s);
fprintf('ode45 : %d steps, %.3f s\n', numel(t45) - 1, t45s);

valve_time_constant = t15;                                      % time vector
cylinder_pressure = y15;                                       % states
figure;
subplot(2,1,1);
plot(valve_time_constant, cylinder_pressure(:,1), 'LineWidth', 1.2); xlabel('Time'); ylabel('y_1');
subplot(2,1,2);
bar([numel(t15) - 1, numel(t45) - 1]);
set(gca, 'XTickLabel', {'ode15s', 'ode45'}); ylabel('Steps taken');

Packages and prices

Choose how much help you need. Each higher package includes everything in the lower ones. Prices exclude 11% VAT.

• Payment: 50% down payment before work starts, balance after the deliverables are handed over
• Tax: Prices exclude 11% VAT
• Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
• Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

Frequently asked questions