LQR Control of a Damped Oscillator

In this example, we will take an existing damped oscillator model, linearize the (already linear) dynamics around the steady state using Auto-Differentiation with Code Generation and finally design an LQR controller based on the linearized model.

#include <ct/optcon/optcon.h>  // also includes ct_core
#include "exampleDir.h"

int main(int argc, char** argv)
{
    // get the state and control input dimension of the oscillator
    const size_t state_dim = ct::core::SecondOrderSystem::STATE_DIM;
    const size_t control_dim = ct::core::SecondOrderSystem::CONTROL_DIM;

    // create an auto-differentiable instance of the oscillator dynamics
    ct::core::ADCGScalar w_n(50.0);
    std::shared_ptr<ct::core::ControlledSystem<state_dim, control_dim, ct::core::ADCGScalar>> oscillatorDynamics(
        new ct::core::tpl::SecondOrderSystem<ct::core::ADCGScalar>(w_n));

    // create an Auto-Differentiation Linearizer with code generation on the quadrotor model
    ct::core::ADCodegenLinearizer<state_dim, control_dim> adLinearizer(oscillatorDynamics);

    // compile the linearized model just-in-time
    adLinearizer.compileJIT();

    // define the linearization point around steady state
    ct::core::StateVector<state_dim> x;
    x.setZero();
    ct::core::ControlVector<control_dim> u;
    u.setZero();
    double t = 0.0;

    // compute the linearization around the nominal state using the Auto-Diff Linearizer
    auto A = adLinearizer.getDerivativeState(x, u, t);
    auto B = adLinearizer.getDerivativeControl(x, u, t);

    // load the weighting matrices
    ct::optcon::TermQuadratic<state_dim, control_dim> quadraticCost;
    quadraticCost.loadConfigFile(ct::optcon::exampleDir + "/lqrCost.info", "termLQR");
    auto Q = quadraticCost.stateSecondDerivative(x, u, t);    // x, u and t can be arbitrary here
    auto R = quadraticCost.controlSecondDerivative(x, u, t);  // x, u and t can be arbitrary here

    // design the LQR controller
    ct::optcon::LQR<state_dim, control_dim> lqrSolver;
    ct::core::FeedbackMatrix<state_dim, control_dim> K;

    std::cout << "A: " << std::endl << A << std::endl << std::endl;
    std::cout << "B: " << std::endl << B << std::endl << std::endl;
    std::cout << "Q: " << std::endl << Q << std::endl << std::endl;
    std::cout << "R: " << std::endl << R << std::endl << std::endl;

    lqrSolver.compute(Q, R, A, B, K);

    std::cout << "LQR gain matrix:" << std::endl << K << std::endl;

    return 1;
}

You can run this example with the following command

rosrun ct_optcon ex_LQR

Note

Make sure you have built the examples before trying to run it.