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Fixed-time Control under Spatiotemporal and Input Constraints: A Quadratic Program Based Approach

In this paper, we present a control synthesis framework for a general class of nonlinear, control-affine systems under spatiotemporal and input constraints. First, we study the problem of fixed-time convergence in the presence of input constraints. The relation between the domain of attraction for fixed-time stability with respect to input constraints and the required time of convergence is established. It is shown that increasing the control authority or the required time of convergence can expand the domain of attraction for fixed-time stability. Then, we consider the problem of finding a control input that confines the closed-loop system trajectories in a safe set and steers them to a goal set within a fixed time. To this end, we present a Quadratic Program (QP) formulation to compute the corresponding control input. We use slack variables to guarantee feasibility of the proposed QP under input constraints. Furthermore, when strict complementary slackness holds, we show that the solution of the QP is a continuous function of the system states, and establish uniqueness of closed-loop solutions to guarantee forward invariance using Nagumo's theorem. We present two case studies, an example of adaptive cruise control problem and an instance of a two-robot motion planning problem, to corroborate our proposed methods.