sylvester_discrete.cpp

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 *  Copyright (c) 2020,
 *  TU Dortmund - Institute of Control Theory and Systems Engineering.
 *  All rights reserved.
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 *  GNU General Public License for more details.
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 *  Authors: Christoph Rösmann
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#include <corbo-core/value_comparison.h>
#include <corbo-numerics/matrix_utilities.h>
#include <corbo-numerics/sylvester_discrete.h>
 
#include <Eigen/Core>
#include <Eigen/Dense>
#include <Eigen/Eigenvalues>
#include <algorithm>
 
namespace corbo {
 
bool SylvesterDiscrete::solve(const Eigen::Ref<const Eigen::MatrixXd>& A, const Eigen::Ref<const Eigen::MatrixXd>& B,
                              const Eigen::Ref<const Eigen::MatrixXd>& C, Eigen::MatrixXd& X)
{
    assert(is_square(A));
    assert(is_square(B));
    assert(A.rows() == C.rows());
    assert(B.rows() == C.cols());
 
    Eigen::ComplexSchur<Eigen::MatrixXd> A_schur(A, true);
    Eigen::ComplexSchur<Eigen::MatrixXd> B_schur(B, true);
 
    if (A_schur.info() == Eigen::NoConvergence) return false;
    if (B_schur.info() == Eigen::NoConvergence) return false;
 
    const Eigen::MatrixXcd& T_A = A_schur.matrixT();
    const Eigen::MatrixXcd& U_A = A_schur.matrixU();
    const Eigen::MatrixXcd& T_B = B_schur.matrixT();
    const Eigen::MatrixXcd& U_B = B_schur.matrixU();
 
    const int n = C.rows();
    const int m = C.cols();
 
    // transform rhs
    Eigen::MatrixXcd F = U_A.adjoint() * C * U_B;
 
    Eigen::MatrixXcd Y(n, m);
    Y.setZero();
 
    Eigen::JacobiSVD<Eigen::MatrixXcd> lhs_svd(n, n, Eigen::ComputeThinU | Eigen::ComputeThinV);
 
    Eigen::MatrixXcd lhs(n, m);
    lhs.setIdentity();
 
    // solve T_A Y T_B - Y + F = 0
    // construct transformed solution by forward substitution
    const int mm = m - 1;
    for (int k = 0; k < m; ++k)
    {
        Eigen::VectorXcd rhs = F.col(k) + T_A * Y * T_B.col(k);
        // move non-zero diagonal elements to lhs and multiply by T
        // (this is the non-zero part of T_A Y T_B which depends only on Y.col(k)).
        const std::complex<double> factor = T_B(k, k);
        lhs -= T_A * factor;
        // compute SVD
        // TODO(roesmann): here is place for improving efficiency
        //                 and robustness by avoiding the SVD (see references).
        lhs_svd.compute(lhs);
        // solve linear system
        Y.col(k) = lhs_svd.solve(rhs);
        // revert rhs diagonal element for subsequent iteration
        if (k < mm) lhs.setIdentity();
    }
 
    // transform back
    X = (U_A * Y * U_B.adjoint()).real();
 
    return true;
}
 
bool SylvesterDiscrete::hasUniqueSolution(const Eigen::Ref<const Eigen::MatrixXd>& A, const Eigen::Ref<const Eigen::MatrixXd>& B)
{
    if (!is_square(A) || !is_square(B)) return false;
 
    Eigen::VectorXcd eigvals_A = A.eigenvalues();
    Eigen::VectorXcd eigvals_B = B.eigenvalues();
 
    for (int i = 0; i < eigvals_A.size(); ++i)
    {
        for (int j = 0; j < eigvals_B.size(); ++j)
        {
            std::complex<double> product = eigvals_A[i] * eigvals_B[j];
            if (approx_equal(product.real(), 1) && approx_equal(product.imag(), 0))  // TODO(roesman) is 1+i0 correct, or unit circle in general?
                return false;
        }
    }
    return true;
}
 
}  // namespace corbo