Practically Safe Extremum Seeking

Abstract

We demonstrate the recent designs of Safe Extremum Seeking (Safe ES) on the 1 kilometer-long charged particle accelerator at the Los Alamos Neutron Science Center (LANSCE). Safe ES is a modification of ES which, in addition to minimizing an analytically unknown cost, also employs a safety filter based on an analytically unknown control barrier function (CBF) safety metric.Accelerator tuning is necessitated by the accelerators being large, with many drifting parameters due to thermal effects and degradation. At the same time, safe operation (the maintenance of state constraints) is crucial, as damage brings astronomical costs, both financially and in operation downtime.Our measured (but analytically unknown) safety metric is the beam current. We perform multivariable Safe ES on three accelerator applications, in which we adapt 4, 6, and 3 magnet strength parameters, respectively. Two of the three applications are for validated simulation models of beamlines at LANSCE: the first for the Proton Radiography (pRad) beamline of 800 MeV protons for spot size tuning; the second on a high performance code, HPSim, for tuning the low energy beam transport (LEBT) region of of 750 keV protons. The third is an experimental tuning of the steering magnets in the LEBT at LANSCE.

Safe Model-Free Optimal Voltage Control via Continuous-Time Zeroth-Order Methods

“…In the context of ES, to allow for enough exploration via dithering, the projection maps are applied to a shrunken feasible set that can be made arbitrarily close to the nominal feasible set by decreasing the amplitude of the dithers. In this way, the algorithms are able to provide suitable evolution directions near the boundary of the feasible set, achieving a property of "practical safety", similar in spirit to the one studied in [26]. In addition to the hard constraints, the proposed controllers are also able to simultaneously handle soft constraints via primaldual ES vector fields, thus achieving safety and optimality in a variety of model-free optimization problems.…”

“…Switching ES algorithms that emulate sliding-mode techniques were also presented in [25] to handle hard constraints in time-varying problems. More recently, an innovative approach that combines safety filters and ES was introduced in [26] using control barrier functions and quadratic programming. To handle soft constraints, ES approaches based on saddle flows have also been studied in [27]- [30].…”

“…As we will show below in Lemma 1, the projection map in (10a) guarantees that x remains always in the feasible set X , and thus the actual decision input x always remains in a small tunable O(ε a )inflation of X . This property defines a notion of "practical" safety, similar to those studied in using safety filters and smooth orthogonal projections [20], [26]. However, in contrast to other constrained model-free algorithms that use barrier functions [17], orthogonal projections [20], or safety filters [26], the state x in (10a) can actually hit the boundary of X in a finite time, a situation that emerges in problems with saturation constraints.…”

“…This property defines a notion of "practical" safety, similar to those studied in using safety filters and smooth orthogonal projections [20], [26]. However, in contrast to other constrained model-free algorithms that use barrier functions [17], orthogonal projections [20], or safety filters [26], the state x in (10a) can actually hit the boundary of X in a finite time, a situation that emerges in problems with saturation constraints. On the other hand, for safety-critical applications where the decision input x must stay within X , the projection map can be applied to a shrunk feasible set X εa satisfying X εa +ε a B ⊆ X .…”

“…where the state τ is a timer indicating when the signal q is allowed to switch via (26b). In (26), τ d > 0 is called the dwell-time, and N 0 is the chatter bound. As shown in [36, Ch.2], the hybrid system (26) guarantees that every switching signal q satisfies an average dwell-time (ADT) constraint.…”