A Time-Freezing Approach for Numerical Optimal Control of Nonsmooth Differential Equations With State Jumps

Abstract

This paper introduces a novel reformulation and numerical methods for optimal control of complementarity Lagrangian systems with state jumps. The solutions of the reformulated system have jump discontinuities in the first time derivative instead of the trajectory itself, which is easier to handle theoretically and numerically. We cover not only the easier case of elastic impacts, but also the difficult case, when after the state jump the system evolves on the boundary of the dynamic's feasible set. In nonsmooth mechanics this corresponds to inelastic impacts. The main idea of the time-freezing reformulation is to introduce a clock state and an auxiliary dynamic system whose trajectory endpoints satisfy the state jump law. When the auxiliary system is active, the clock state is not evolving, hence by taking only the parts of the trajectory when the clock state was active, we can recover the original solution. We detail how to recover the solution of the original system, show how to select appropriate auxiliary dynamics and give practical numerical methods to handle discontinuous ODEs with nonunique sliding motions. Moreover, we introduce a novel auxiliary ODE for time-freezing for elastic impacts and overcome some drawbacks of [22]. The theoretical findings are illustrated on the nontrivial numerical optimal control example of a hopping one-legged robot.

Continuous Optimization for Control of Hybrid Systems with Hysteresis via Time-Freezing

“…In what follows, in the regions R 2 and R 3 we define auxiliary dynamic systems whose trajectory endpoints satisfy the state jump law of (4). In the regions R 1 and R 4 we will define so-called DAEforming ODE [13], which make sure that we obtain appropriate sliding modes on R A and R B , which are described by index 2 DAE [14] and witch match the dynamics of the original system.…”

“…The time-freezing reformulation transforms systems with state jumps into PSS and was first introduced in [12], [13]. This paper introduces a time-freezing reformulation to transform systems represented with the finite automaton in Fig.…”

“…As in [12], [13], we introduce auxiliary dynamic systems and a clock state. The auxiliary dynamic systems evolve in regions which are prohibited for the initial system and their trajectory endpoints satisfy the state jump law.…”

NOSNOC: A Software Package for Numerical Optimal Control of Nonsmooth Systems

“…of (1) is in general discontinuous in x and u is an externally chosen control function. Several classes of systems with state jumps can be brought into the form of (1) via the time-freezing reformulation [4], [8], [9]. Thus, the focus on PSS enables a unified treatment of many different kinds of nonsmooth systems.…”

“…It relies on the recently introduced Finite Elements with Switch Detection [3]which enables high accuracy optimal control and simulation of PSS. The time-freezing reformulation [4], which transforms several classes of systems with state jumps into PSS is supported as well. This enables the treatment of a broad class of nonsmooth systems in a unified way.…”