Finite Horizon Backward Reachability Analysis and Control Synthesis for Uncertain Nonlinear Systems

Abstract

A method is proposed to compute robust innerapproximations to the backward reachable set for uncertain nonlinear systems. It also produces a robust control law that drives trajectories starting in these sets to the target set. The method merges dissipation inequalities and integral quadratic constraints (IQCs) with both hard and soft IQC factorizations. Computational algorithms are presented using the generalized S-procedure and sum-of-squares techniques. The use of IQCs in backward reachability analysis allows for a variety of perturbations including parametric uncertainty, unmodeled dynamics, nonlinearities, and uncertain time delays. The method is demonstrated on two examples, including a 6state quadrotor with actuator uncertainties.

Sum-of-Squares Flight Control Synthesis for Deep-Stall Recovery

“…In Algorithm 1, as discussed in the beginning, the choice of the shape P proves to be crucial for the form of the synthesized feedback law and thus for the shape of the resulting provable invariant set. Alternatively proposed iteration schemes, such as Algorithm 1b, aimed to increase the volume of the region-of-attraction estimate directly rather than enlarging an inscribing polynomial shape function [26,27,29,30]. In particular, Algorithm 1b replaces the ellipsoidal shape by a constraint enforcing that each prior estimate is nested in the next estimate, thus removing the necessity of choosing a shape function.…”

“…The thus modified algorithm, to which we will refer as Algorithm 1b, does not need an ellipsoidal shape as input. Similar approaches have recently been employed for analysis of reachable sets of polynomial systems [26] and the region of attraction of hybrid systems [27]. While Algorithm 1b allows the region-of-attraction estimate to grow in all directions rather than the one given by E, we will see later that is unsuitable for the purpose of deep-stall recovery.…”

“…In this section, we modify the control synthesis into a simplified backwards reachability scheme, where the ellipsoidal shape is replaced by a target state whence recovery is to be ensured. An alternative approach for a finite-horizon backwards reachability analysis was presented in [26]. As Algorithm 1b, this approach lacks any directional information for enlarging the closed-loop reachable set under the control law to be synthesized.…”

Backward Reachability for Polynomial Systems on A Finite Horizon

“…The conference version [13] of this paper decomposes the control synthesis process into two steps: constructing storage functions first, and then computing control laws using the obtained storage functions through quadratic programs. The current paper presents a single-step design and accommodates control saturation, which is not addressed in [13]. In addition, [13] considers only a terminal target set, whereas this paper addresses a target tube.…”

“…The current paper presents a single-step design and accommodates control saturation, which is not addressed in [13]. In addition, [13] considers only a terminal target set, whereas this paper addresses a target tube. In a separate publication [14], we have studied forward reachable sets without control design.…”