Computing Viable Sets and Reachable Sets to Design Feedback Linearizing Control Laws Under Saturation

Abstract

Abstract-We consider feedback linearizable systems subject to bounded control input and nonlinear state constraints. In a single computation, we synthesize 1) parameterized nonlinear controllers based on feedback linearization, and 2) the set of states over which this controller is valid. This is accomplished through a reachability calculation, in which the state is extended to incorporate input parameters. While we use a HamiltonJacobi formulation, a viability approach is also feasible. The result provides a mathematical guarantee that for all states within the computed set, there exists a control law that will simultaneously satisfy two separate goals: envelope protection (no violation of state constraints), and stabilization despite saturation. We apply this technique to two real-world systems: the longitudinal dynamics of a civil jet aircraft, and a twoaircraft, planar collision avoidance scenario. The result, in both cases, is a feasible range of input parameters for the nonlinear control law, and a corresponding controlled invariant set.

Online Safe Flight Envelope Prediction for Damaged Aircraft: A Database-driven Approach

“…envelope protection and stabilization) simultaneously. 41 In such case, physical constraints in conventional flight envelopes may not be enough for the control design, and simulations may also be inadequate to help predict the unanticipated problems with all possible initial conditions. Alternatively, reachability analysis can provide a new set-valued insight into the safety and control design of dynamic systems.…”

“…If the target set K is defined as the safe set where the system is aimed to reach, the viability set is often computed for 'safety-preserving' controllers that keep the trajectories of system within the safe region, so it is also called "the largest controlled invariant set". 41 However, the viability set is hardly useful when the aircraft has flown out of the pre-defined safe region or trim set. Alternatively we should calculate the boundary of maximum reachable set, which includes the states that have the potential of returning to the safe region.…”

Modeling and Formal Analysis of Human-Machine Interaction

“…Tomlin, Mitchell, Bayen, and Oishi (2003) provide an overview of current methods for modeling and verifying hybrid systems, using a commercial jet autopilot as a case study. Oishi, Mitchell, Tomlin, & Saint-Pierre (2006) show how to prove simultaneous satisfaction of envelope protection and stability requirements, illustrating their method with a two-aircraft collision avoidance scenario.…”

Aircraft Accident Prevention: Loss-of-Control Analysis

“…These questions have been raised in the literature, for example. 8,9,43 Furthermore, suppose that C is defined by…”

“…Questions like this have been considered in the control literature. 8,9,[17][18][19][20] In Section V below we discuss the problem of identifying the largest set within a prescribed envelope which can be made positively invariant and of characterizing the control strategy necessary to do so. This set will be called the safe set.…”