Discrete-Time Stochastic (dt-SS)

Functions for computing barrier certificates for discrete-time stochastic systems over a finite time horizon with noise.

dt_SS

from src.functions.dt_SS import dt_SS

result = dt_SS(b_degree, dim, L_initial, U_initial, L_unsafe, U_unsafe,
               L_space, U_space, x, varsigma, f, t, noise_type="normal",
               optimize=False, solver="mosek", confidence=None, gam=None,
               lam=None, c_val=None, mean=None, sigma=None, rate=None,
               a=None, b=None, l_degree=None)

Parameters:

Parameter	Type	Description
b_degree	int	Degree of the barrier polynomial.
dim	int	Dimension of the state space.
L_initial	np.ndarray	Lower bounds of the initial set. Shape: (dim,).
U_initial	np.ndarray	Upper bounds of the initial set. Shape: (dim,).
L_unsafe	np.ndarray	Lower bounds of unsafe set(s). Shape: (n_regions, dim).
U_unsafe	np.ndarray	Upper bounds of unsafe set(s). Shape: (n_regions, dim).
L_space	np.ndarray	Lower bounds of the state space. Shape: (dim,).
U_space	np.ndarray	Upper bounds of the state space. Shape: (dim,).
x	tuple	SymPy symbols for state variables.
varsigma	tuple	SymPy symbols for noise variables.
f	np.ndarray	Array of SymPy expressions defining the dynamics.
t	int	Finite time horizon.
noise_type	str	Type of noise: "normal", "exponential", or "uniform".
optimize	bool	If True, optimize the barrier certificate (requires gam and c_val). Default: False.
solver	str	"mosek" (default) or "cvxopt".
confidence	float or None	Minimal confidence threshold for the feasible solution.
gam	float or None	Fixed value for \(\gamma\).
lam	float or None	Fixed value for \(\lambda\).
c_val	float or None	Value for \(c\) (multiplied by time horizon in the confidence bound).
mean	list or None	Mean values for normal noise (assumed zero if not provided).
sigma	list or None	Standard deviations for normal noise per dimension.
rate	list or None	Rate parameter(s) for exponential noise.
a	list or None	Lower bounds for uniform noise.
b	list or None	Upper bounds for uniform noise.
l_degree	int or None	Degree of Lagrangian multipliers. Defaults to b_degree.

Noise-specific parameters:

• Normal: provide sigma (and optionally mean)

• Exponential: provide rate

• Uniform: provide a and b

Returns: dict with barrier certificate details, or None if infeasible.

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parallel_dt_SS

from src.functions.parallel_dt_SS import parallel_dt_SS

result = parallel_dt_SS(b_degree, dim, L_initial, U_initial, L_unsafe,
                        U_unsafe, L_space, U_space, x, varsigma, f, t,
                        noise_type="normal", optimize=False, solver="mosek",
                        confidence=None, gam=None, lam=None, c_val=None,
                        mean=None, sigma=None, rate=None, a=None, b=None,
                        l_degree=None)

Searches barrier degrees 2, 4, …, b_degree in parallel. All parameters are identical to dt_SS except b_degree is the maximum degree.

Note

Must be called inside if __name__ == '__main__': due to Python multiprocessing.

Returns: The result dict from the first successful degree, or None.