Robust Nonlinear Optimal Control via System Level Synthesis

Abstract

This paper addresses the problem of finite horizon constrained robust optimal control for nonlinear systems subject to norm-bounded disturbances. To this end, the underlying uncertain nonlinear system is decomposed based on a firstorder Taylor series expansion into a nominal system and an error (deviation) described as an uncertain linear time-varying system. This decomposition allows us to leverage system level synthesis to optimize an affine error feedback while planning the nominal trajectory and ensuring robust constraint satisfaction for the nonlinear system. The proposed approach thereby results in a less conservative planning compared with state-of-theart techniques. A tailored sequential quadratic programming strategy is proposed to solve the resulting nonlinear program efficiently. We demonstrate the benefits of the proposed approach to control the rotational motion of a rigid body subject to state and input constraints.

Robust Optimal Control for Nonlinear Systems with Parametric Uncertainties via System Level Synthesis

“…Besides the controller and the emphnonlinear trajectory, a convex overbound of the parametric uncertainties and linearization errors is also jointly optimized, leveraging SLS. This presents an advantage over [22], where the over-bound is neither convex nor accounts for affine parametric uncertainties. The convex over-bounding enables features such as convex constraints to guarantee robust performance.…”

“…The resulting nonlinear optimization problem is detailed in Section III-C. Compared to related work [22], the proposed approach considers parametric uncertainties and achieves a convex over-bound of the uncertainties, reducing conservatism in handling uncertainties.…”

“…for any (x, u) ∈ C, (z, v) ∈ C, θ ∈ Θ. Proof: See [22]. The evaluation of the constants µ i represents our proposed approach's only offline design requirement, distinguishing it from other prominent strategies, such as those presented in [11], [12], which necessitate the design of a (nonlinear) incrementally stabilizing controller.…”

“…with Z, a block-downshift operator commonly used in SLS (see, e.g., [22]), i.e., a matrix with identity matrices along its first block sub-diagonal and zeros elsewhere, ∆x := ∆x 1:T , ∆ũ := ∆ũ 1:T , d := d0:T −1 , and à := blkdiag( Ã1 , . .…”

“…Because the error feedback is optimized online, the offline systemspecific design is reduced, and SLS-based RMPC typically exhibits reduced conservatism [15], [16], especially with parametric uncertainties [17]. While SLS has been extended to nonlinear systems [18]- [21], including a formulation with robust constraint satisfaction [22], none of the previous approaches have considered parametric uncertainties. In this paper, we fill this gap by treating parametric uncertainties similarly as in linear SLS [17], all while optimizing the nonlinear trajectory.…”