Stochastic Reachability Analysis of Hybrid Systems

A Unified Stochastic Hybrid System Approach to Aggregate Modeling of Responsive Loads

“…This phenomenon can affect the aggregate power output significantly. It can be modeled by the so-called random jump mechanism [20], [16], [15], which is determined by a transition intensity function λ(q, x) : Q × R n → R + , where R + denotes the non-negative real numbers. In particular, the probability of a random jump happening in an infinitesimal time interval (t, t + dt] is given by λ(q(t), x(t))dt .…”

“…Our result, though derived in the context of aggregate load modeling, can also be used to obtain forward equations of a general SHS with both deterministic and random switchings. Although SHS has been studied extensively in the literature [14], [15], [16], [17], [18], [19], its probability density evolution cannot be explicitly characterized using the existing results, especially when there are deterministic switchings with general switching surfaces in multi-dimensional spaces. It is well known that for a standard diffusion process, the adjoint of its (strong) generator determines the form of the forward equation and the domain of the generator (indirectly) affects the boundary conditions.…”

“…(GSHS extended generator[15, Theorem 4.11]) The extended generator (L, D(L)) of a GSHS process ξ t satisfies ∀ψ ∈ D(L),…”

Sign changes as a universal concept in first-passage-time calculations

“…Researchers can now draw on an extended suite of mathematical techniques to evaluate FPTs, ranging from analytical descriptions for special cases including asymptotic expansions to semi-analytical approaches based on integral equations to direct numerical simulations [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Notwithstanding these practical methods, the general solution to one of the most fundamental FPT problems is still unknown: diffusion to an arbitrary boundary [24,[32][33][34]. This is even more remarkable since diffusion processes are often used to approximate the dynamics of more complicated stochastic systems [16,35].…”

Safe autonomy under uncertainty : computation, control, and application

“…Seminal papers include [47], [48] on reachability and [32], [49], [50] on invariance. Representative textbooks on set-valued control are [51], [52].…”