Difference of convex functions in robust tube nonlinear MPC

Abstract

We propose a robust tube-based Model Predictive Control (MPC) paradigm for nonlinear systems whose dynamics can be expressed as a difference of convex functions. The approach exploits the convexity properties of the system model to derive convex conditions that govern the evolution of robust tubes bounding predicted trajectories. These tubes allow an upper bound on a performance cost to be minimised subject to state and control constraints as a convex program, the solution of which can be used to update an estimate of the optimal state and control trajectories. This process is the basis of an iteration that solves a sequence of convex programs at each discrete time step. We show that the algorithm is recursively feasible, converges asymptotically to a fixed point of the iteration and ensures closed loop stability. The algorithm can be terminated after any number of iterations without affecting stability or constraint satisfaction. A case study is presented to illustrate an application of the algorithm.

Robust trajectory optimisation for transitions of tiltwing VTOL aircraft

“…The algorithm can compute safe trajectories that are robust to model uncertainty for abrupt transitions at near constant altitude, extending the results in[4]. Another contribution of the present work is the extension of the robust tube optimisation paradigm presented in[5] to dynamic systems that are not convex, by means of a DC decomposition of the nonlinear dynamics. Limitations of the present approach are: i) to obtain a computationally tractable formulation, quadratic approximations of the DC polynomials are required; ii) the computation time, although relatively low compared to solving a NLP, is still too high to leverage the optimisation in a MPC setting.Future work will alleviate these problems by i) considering…”

“…Motivated by the fact that convex functions can be bounded tightly by convex and linear inequalities (as in [5]), we seek a decomposition of f as a Difference of Convex (DC) functions:…”

“…In what follows we will exploit the convexity properties of the functions g k , h k , f γ k = −mg cos γ k in (29) to approximate the dynamics by a set of convex inequalities with tight bounds on the state trajectories. To do so, we linearise the dynamics successively around feasible guessed trajectories and treat the linearisation error as a bounded disturbance [5]. We use the fact that the linearisation error of a convex (resp.…”

“…In this work, we propose a solution to this problem based on a DC decomposition of the nonlinear dynamics. This allows us to obtain tight bounds on the linearisation error and treat this error as a disturbance in a robust optimisation framework, exploiting an idea from tube MPC [5]. The main idea is to successively linearise the dynamics around predicted trajectories and treat the linearisation error as a bounded disturbance.…”

“…The contribution of this research is twofold: i) we solve an open problem in trajectory optimisation of VTOL aircraft by allowing aggressive transitions at high angle of attack while guaranteeing safety and computational tractability of the scheme; ii) we make a connection between DC decomposition and robust tube based optimisation and demonstrate the applicability and generalisability of the procedure in [5].…”