The method of uniform monotonous approximation of the reachable set border for a controllable system

Abstract

A numerical method of a two-dimensional non-linear controllable system reachable set boundary approximation is considered. In order to approximate the boundary right piecewise linear closed contours are used: a set of broken lines on a plane. As an application of the proposed technique a method of finding linear functional global extremum is described, including its use for systems with arbitrary dimensionality.

Chance-Constrained Sequential Convex Programming for Robust Trajectory Optimization

“…Implementation: We use T = 10s, discretize the problem with a Runge-Kutta (RK4) scheme with ∆t = 1 s, and enforce the constraints in (19) at each timestep t k = k∆t, which is justified by the continuity of the state trajectories. The feedback gain K is given by a standard linear-quadratic regulator (LQR).…”

“…We evaluate Lt and Ht using 10 5 samples 4 and directly use the error bounds t predicted in (10). This choice of t ensures that the problem with the approximated constraints (19) gives solutions that satisfy the original constraints in (17), see also [29,Corollary 5.3]. We only parameterize the nominal state and control trajectories (x 0 , ū) and evaluate (19) and its gradient as a function of ū.…”

“…However, in the nonlinear setting, the reachable sets may be non-convex, and finding the extremal disturbance trajectories that generate boundary states requires solving boundary-value problems (BVPs, see BVP d ). This approach was explored in the threedimensional case [19], but solving BVPs remains computationally challenging. Although sophisticated tech-niques exist to efficiently solve BVPs [20], [21], additional non-trivial analysis is still required for efficient implementation.…”

“…, M , t ∈ [0, T ]. The conservatism of (19) follows from the convexity of ( 17) and Corollary 2, see [29,Corollary 5.3]. Given a reference x ref = (1, 0, .…”

Polyhedral approximation method for reachable sets of linear delay systems

“…Gornov et al . [11] proposed a numerical method based on the Pontryagin maximum principle of optimal control theory to construct approximations of reachable sets. Xiang et al .…”