algebraic_riccati_discrete.cpp

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 *  Copyright (c) 2020,
 *  TU Dortmund - Institute of Control Theory and Systems Engineering.
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 *  Authors: Christoph Rösmann
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#include <corbo-numerics/algebraic_riccati_discrete.h>
 
#include <corbo-core/value_comparison.h>
#include <corbo-numerics/matrix_utilities.h>
#include <corbo-numerics/schur.h>
 
#include <corbo-core/console.h>
 
#include <Eigen/Cholesky>
#include <Eigen/Core>
#include <Eigen/Dense>
#include <algorithm>
 
namespace corbo {
 
bool AlgebraicRiccatiDiscrete::solve(const Eigen::Ref<const Eigen::MatrixXd>& A, const Eigen::Ref<const Eigen::MatrixXd>& B,
                                     const Eigen::Ref<const Eigen::MatrixXd>& Q, const Eigen::Ref<const Eigen::MatrixXd>& R, Eigen::MatrixXd& X,
                                     Eigen::MatrixXd* G)
{
    assert(is_square(A));
    assert(have_equal_size(A, Q));
    assert(A.rows() == B.rows());
    assert(R.rows() == B.cols());
    assert(is_square(R));
    const int p = A.rows();  // dim_states
 
    Eigen::LDLT<Eigen::MatrixXd> R_cholesky(R);
    Eigen::MatrixXd R_inv_B_t   = R_cholesky.solve(B.transpose());
    Eigen::MatrixXd A_inv_trans = A.inverse().transpose();  // A must be invertible
 
    // Construct Hamiltonian matrix
    // H = [ A+B R^-1 B^T (A^-1)^T Q      -B R^-1 B^T (A^-1)^T]
    //     [-(A^-1)^T Q                     (A^-1)^T       ]
    Eigen::MatrixXd H(2 * p, 2 * p);
    H << A + B * R_inv_B_t * A_inv_trans * Q, -B * R_inv_B_t * A_inv_trans, -A_inv_trans * Q, A_inv_trans;
 
    bool success = solveRiccatiHamiltonianSchur(H, X);
 
    if (G && success)
    {
        Eigen::LDLT<Eigen::MatrixXd> G_cholesky(R + B.transpose() * X * B);
        *G = G_cholesky.solve(B.transpose()) * X * A;
    }
    return success;
}
 
bool AlgebraicRiccatiDiscrete::solveRiccatiHamiltonianSchur(const Eigen::MatrixXd& H, Eigen::MatrixXd& X)
{
    assert(is_square(H));
    assert(H.rows() % 2 == 0);
 
    // perform schur to find an orthogonal transformation that reduces H to real Schur form
    Eigen::RealSchur<Eigen::MatrixXd> schur(H);
 
    Eigen::MatrixXd T = schur.matrixT();
    Eigen::MatrixXd U = schur.matrixU();
 
    // reorder Schur form such that all negative eigenvalues are located on the upper left blocks.
    int subspace_dim;
    bool success = reorder_schur_blocks(T, U, AlgebraicRiccatiDiscrete::isInsideUnitCircle, &subspace_dim, false);
 
    const int p = H.rows() / 2;
 
    if (!success || p != subspace_dim) return false;
 
    // U = [U1; U2
    //      U2; U3]
 
    // solution X can be obtained by X = U2 * U1^-1
    // since U1.inverse() is not recomended due to numerical effects
    // we must rewrite the equation in order to facilitate solving with Eigen:
    // X^T = (U2 * U1^-1)^T = (U1^T)^(-1) U2^T
    Eigen::JacobiSVD<Eigen::MatrixXd> U1_svd(U.block(0, 0, p, p).transpose(), Eigen::ComputeThinU | Eigen::ComputeThinV);
    X = U1_svd.solve(U.block(p, 0, p, p).transpose());  // X is symmetric, hence no final transpose
    return true;
}
 
bool AlgebraicRiccatiDiscrete::isClosedLoopStable(const Eigen::Ref<const Eigen::MatrixXd>& A, const Eigen::Ref<const Eigen::MatrixXd>& B,
                                                  const Eigen::Ref<const Eigen::MatrixXd>& G)
{
    assert(is_square(A));
    assert(A.rows() == B.rows());
    assert(A.rows() == G.cols());
    assert(G.rows() == B.cols());
 
    Eigen::VectorXcd eig = (A - B * G).eigenvalues();
    for (int i = 0; i < eig.size(); ++i)
    {
        if (std::abs(eig[i]) >= 1) return false;
    }
    return true;
}
 
}  // namespace corbo