tdcpy.roots module#

Implementation of tds_roots

tdcpy.roots.roots(tds, r=0.0, **kwargs)#

Computes the characteristic roots of a time-delay system in a given right half-plane or rectangular region.

Parameters:

tds (TDS) – instance of time-delay system, i.e., RDDE, NDDE or DDAE
r (int or list) – region, specify r as number on real axis or rectangular region via 4 coordinates [Re_min, Re_max, Im_min, Im_max], default r=0.0
kwargs –

discretization (int): discretization for discretizing DDAE into DAE,
optional, default None, if not specified, heuristic from [1] will be used to obtain sufficient discretization

max_size_evp (int): maximum allowed size of eigenvalue problem (EVP)
optional, default 600

base_delay (float): base delay in case delays are commensurate,
optional, default None, used in discretization heuristic

Returns:

tuple containing

cr (array): vector of obtained roots
roots_info (RootsInfo): Roots metadata containing:
discretization (int): discretization used for obtaining EVP gamma_r_exceeds_one (bool): flag indicating gamma(r) > 1 index_exceeds_one (bool): flag that index exceeds one discretization_eigenvalues (array): eigenvalues of EVP max_size_evp_enforced (bool): flag if maximum size of EVP was enforced newton_inital_guesses (array): roots before newton corrections newton_final_values (array): roots after newton corrections newton_residuals (array): newton residuals newton_unconverged_initial_guesses (array): mask of unconverged newton initial guesses newton_large_corrections (array): mask of “large” corrections

Return type:

References:

[1] Wu, Z. and Michiels, W. (2012). Reliably computing all: characteristic roots of delay differential equations in a given right half plane using a spectral method. Journal of Computational and Applied Mathematics, 236(9), pp. 2499-2514.
[2] Michiels, W. (2011). Spectrum-based stability analysis and: stabilisation of systems described by delay differential algebraic equations. IET control theory & applications, 5(16), pp. 1829-1842.