Safety Control Synthesis with Input Limits: a Hybrid Approach

Abstract

We introduce a hybrid (discrete-continuous) safety controller which enforces strict state and input constraints on a system-but only acts when necessary, preserving transparent operation of the original system within some safe region of the state space. We define this space using a Min-Quadratic Barrier function, which we construct along the equilibrium manifold using the Lyapunov functions which result from linear matrix inequality controller synthesis for locally valid uncertain linearizations. We also introduce the concept of a barrier pair, which makes it easy to extend the approach to include trajectory-based augmentations to the safe region, in the style of LQR-Trees. We demonstrate our controller and barrier pair synthesis method in simulation-based examples.

A Barrier Pair Method for Safe Human-Robot Shared Autonomy

“…In general, a larger value of 0 makes the LMI problem in (28) more solvable. However, we also need to have 11 of Algorithm 2 can be fulfilled. How to select the value of 1 is also a complicated problem.…”

“…B. Barrier Pair Rapidly-Exploring Random Trees Definition 1 [11]: A barrier pair is a pair consisting of a barrier function and a controller (B, k) with the following properties (a) −1 < B(q, q) ≤ 0, u = k(q, q) =⇒ Ḃ(q, q) < 0, (b) B(q, q) ≤ 0 =⇒ [q , q ] ∈ Z, k(q, q) ∈ U, where [q , q ] ∈ Z and u ∈ U are the state and input constraints.…”

“…where B is a quadratic barrier function with a positive definite matrix Q and k is a full state feedback controller, the barrier pair synthesis becomes a linear matrix inequality (LMI) optimization problem [11].…”

“…Based on the norm bound LDI model expressed in (10) and (11), we can formulate the LMIs for creating our barrier pair synthesis problem. In order to ensure that the ellipsoidal sublevel set E(1) of a barrier pair contains a desired polytopic region a d , we sample a number of points from all edges of a d and let E(1) contain the joint space projections of these Cartesian space samples using the following set of LMIs…”

“…For state-space constraints, controllers can be synthesized simultaneously with barrier functions using back-stepping [6] or quadratic programming methods [7], [8], [9]. Input constraints can also be enforced using semi-definite programming methods [10], [11]. By incorporating samplingbased methods into the synthesis of barrier functions and controllers [12], robots can also guarantee safe operation with non-convex state-space constraints.…”