LQR Controller for Balancing an Inverted Pendulum on a Cart in a State-Space Model

Project info

• Field of studyRobotics and Mechatronics
• LevelDoctoral dissertation (S3)
• Method typeControl Systems
• Estimated build time± 7 working days
• Toolslqr, ss, ctrb, initial, eig (Control System Toolbox)
• PriceUS$38 - US$244

Research goal

Design and simulate LQR control so that the pendulum angle tracks its target through the cart force, accounting for the pendulum length.

Variables

Example result

• 0.537Gain K1
• 1.252Gain K2
• 3.368Settling time (s)
• 0.834Peak |u|
• 2.475Total cost J
• -1.025Max closed-loop Re(eig)

The LQR controller (K1=0.54, K2=1.25) settles Pendulum angle within a 2% band in 3.4 s with a peak control effort of 0.83 on Pendulum length and total cost J=2.48.

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

About State-Space LQR Controller

LQR designs a state feedback gain that minimises a quadratic cost on state deviation and control effort. It suits multivariable systems with a state-space model, such as pendulums, suspensions, or aircraft.

Steps

1. 1Build the state-space model A, B, C, D
2. 2Check controllability with the rank of ctrb
3. 3Choose weighting matrices Q and R
4. 4Compute the optimal gain K with lqr
5. 5Form the closed-loop system A-BK
6. 6Simulate the initial-condition response and control effort

Assumptions to check

• The pair (A, B) is controllable
• All states are measurable or estimated (with an observer)
• The linear model holds around the operating point

Reading and reporting the results

Report Q, R, the gain K, closed-loop poles, and state and control responses. Explain the effect of changing Q and R on response speed and control signal size.

Example MATLAB code

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

A = [0 1; -2 -0.5];
B = [0; 1];
C = eye(2);
D = zeros(2, 1);
sys = ss(A, B, C, D);
fprintf('Controllability rank: %d\n', rank(ctrb(A, B)));

cart_force = [1; 0];                               % initial state
t = 0:0.01:10;
figure; hold on;
for rr = [0.1 1 10]
    Q = diag([10 1]);
    R = rr;
    K = lqr(A, B, Q, R);
    Acl = A - B*K;
    syscl = ss(Acl, zeros(2,1), C, zeros(2,1));
    [pendulum_angle, tt, x] = initial(syscl, cart_force, t);
    u = -(K * x')';
    fprintf('R=%g  K=[%.3f %.3f]  poles: %s\n', rr, K, mat2str(eig(Acl), 3));
    subplot(2,1,1); hold on; plot(tt, x(:,1), 'DisplayName', sprintf('R=%g', rr));
    subplot(2,1,2); hold on; plot(tt, u, 'DisplayName', sprintf('R=%g', rr));
end
subplot(2,1,1); ylabel('Position'); legend show; grid on;
subplot(2,1,2); ylabel('Control u'); xlabel('Time (s)'); grid on;

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)

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