dkpy.SsvLmiBisection

class dkpy.SsvLmiBisection(bisection_atol=1e-05, bisection_rtol=0.0001, max_iterations=100, initial_guess=10, lmi_strictness=None, solver_params=None, n_jobs=-1, objective='constant')

Bases: StructuredSingularValue

Structured singular value using an LMI approach with bisection.

Synthesis method based on Section 4.25 of [CF24].

Examples

Structured singular value computation from [SP06]

>>> P, n_y, n_u, K = example_skogestad2006_p325
>>> block_structure = [
...     dkpy.ComplexFullBlock(1, 1),
...     dkpy.ComplexFullBlock(1, 1),
...     dkpy.ComplexFullBlock(2, 2),
... ]
>>> omega = np.logspace(-3, 3, 61)
>>> N = P.lft(K)
>>> N_omega = N(1j * omega)
>>> mu_omega, D_l_omega, D_r_omega, info = dkpy.SsvLmiBisection(n_jobs=None).compute_ssv(
...     N_omega,
...     block_structure,
... )
>>> float(np.max(mu_omega))
5.77
Parameters:
__init__(bisection_atol=1e-05, bisection_rtol=0.0001, max_iterations=100, initial_guess=10, lmi_strictness=None, solver_params=None, n_jobs=-1, objective='constant')

Instantiate SsvLmiBisection.

Solution accuracy depends strongly on the selected solver and tolerances. Setting the solver and its tolerances in solver_params and setting lmi_strictness manually is recommended, rather than relying on the default settings.

Parameters:

  • bisection_atol (float) – Bisection absolute tolerance.

  • bisection_rtol (float) – Bisection relative tolerance.

  • max_iterations (int) – Maximum number of bisection iterations.

  • initial_guess (float) – Initial guess for bisection.

  • lmi_strictness (Optional[float]) – Strictness for linear matrix inequality constraints. Should be larger than the solver tolerance. If None, then it is automatically set to 10x the solver’s largest absolute tolerance.

  • solver_params (Optional[Dict[str, Any]]) – Dictionary of keyword arguments for cvxpy.Problem.solve(). Notable keys are 'solver' and 'verbose'. Additional keys used to set solver tolerances are solver-dependent. A definitive list can be found at [1].

  • n_jobs (Optional[int]) – Number of processes to use to parallelize the bisection. Set to None for a single thread, or set to -1 (default) to use all CPUs. See [2].

  • objective (str) – Set to 'constant' to solve a feasibility problem at each bisection iteration. Set to 'minimize' to minimize the trace of the slack variable instead, which may result in better numerical conditioning.

References

Methods

__init__([bisection_atol, bisection_rtol, ...])

Instantiate SsvLmiBisection.

compute_ssv(N_omega[, block_structure])

Compute structured singular value.
compute_ssv(N_omega, block_structure=None)

Compute structured singular value.

Parameters:

  • N_omega (np.ndarray) – Closed-loop transfer function evaluated at each frequency.

  • block_structure (Union[List[uncertainty_structure.UncertaintyBlock], List[List[int]], np.ndarray]) – Uncertainty block structure representation. Returns

Returns:

Structured singular value at each frequency, D-scales at each frequency, and solution information. If the structured singular value cannot be computed, the first two elements of the tuple are None, but solution information is still returned.

Return type:

Tuple[np.ndarray, np.ndarray, Dict[str, Any]]