Sequential Action Control: Closed-Form Optimal Control for Nonlinear and Nonsmooth Systems

Abstract

This paper presents a new model-based algorithm that computes predictive optimal controls on-line and in closed loop for traditionally challenging nonlinear systems. Examples demonstrate the same algorithm controlling hybrid impulsive, underactuated, and constrained systems using only high-level models and trajectory goals. Rather than iteratively optimize finite horizon control sequences to minimize an objective, this paper derives a closed-form expression for individual control actions, i.e., control values that can be applied for short duration, that optimally improve a tracking objective over a long time horizon. Under mild assumptions, actions become linear feedback laws near equilibria that permit stability analysis and performance-based parameter selection. Globally, optimal actions are guaranteed existence and uniqueness. By sequencing these actions on-line, in receding horizon fashion, the proposed controller provides a min-max constrained response to state that avoids the overhead typically required to impose control constraints. Benchmark examples show the approach can avoid local minima and outperform nonlinear optimal controllers and recent, case-specific methods in terms of tracking performance, and at speeds orders of magnitude faster than traditionally achievable.3 are nonlinear in state x : R → R n . Though these methods apply more broadly, we derive controls for the case where (1) is linear with respect to the control, u : R → R m , satisfying control-affine form, f (t, x(t), u(t)) = g(t, x(t)) + h(t, x(t)) u(t) .(2)The time dependence in (1) and (2) will be dropped for brevity. The prediction phase simulates motion resulting from some choice of nominal control, u = u 1 . Thus, the nominal predicted motion corresponds to, u 1 (t)).Although the nominal control may be chosen arbitrarily, all examples here use a null nominal control, u 1 = 0. Hence, in the SLIP example, SAC seeks actions that improve performance relative to doing nothing, i.e., letting the SLIP fall.With l 1 : R n → R and m : R n → R, the cost functional,

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“…A model predictive controller (MPC) similar to [18] is used. However, any model predictive controller that is capable of completing the task can be used.…”

Controllers as filters: Noise-driven swing-up control based on Maxwell's demon

“…. , m} (see [9] for proof). By iterating on this process (Section II-A until Section II-D), SAC rapidly synthesizes piecewisecontinuous, constrained control laws for nonlinear systems.…”

“…If the inner product is positive and the angle between the two vectors is small enough, the noise vector should move the system towards the desired control direction. The controller/filter we use is a recently formulated algorithm for control of nonlinear systems called sequential action control (SAC) presented in [9], [10], [11] and summarized in Section II. Nevertheless, it must be noted that the implementation of this method is not SAC-specific; any controller that can successfully control the system of interest can be used instead.…”

“…As a test platform, we use the swing-up problem of the cart-pendulum, pendubot and acrobot systems ( [9], [12]- [18]), benchmark examples for nonlinear control methods and techniques. The objective of swing-up control is to swing the system to a small neighborhood W s of the upright unstable equilibrium, where, often, control authority is ceded to a locally stabilizing controller, e.g.…”

Utilizing Sequential Action Control Method in GaN-Based High-Speed Drive for BLDC Motor

“…When measured up against its alternatives, it is observed that SAC computes higher bandwidth (1 KHz) closed-loop trajectories at a lower final cost in lesser time (milliseconds/seconds versus hours). Therefore, it can be concluded that SAC provides a model-based response to state capably even for saturated, underactuated, nonlinear, and high dimensional systems [8,9].…”