Real-time MPC - Stability through robust MPC design

Abstract

Abstract-Recent results have suggested that online ModelPredictive Control (MPC) can be computed quickly enough to control fast sampled systems. High-speed applications impose a hard real-time constraint on the solution of the MPC problem, which generally prevents the computation of the optimal controller. In current approaches guarantees on feasibility and stability are sacrificed in order to achieve a real-time setting. In this paper we develop a real-time MPC scheme based on robust MPC design that recovers these guarantees while allowing for extremely fast computation. We show that a simple warmstart optimization procedure providing an enhanced feasible solution guarantees feasibility and stability for arbitrary time constraints. The proposed method can be practically implemented and efficiently solved for dynamic systems of significant problem size. Implementation details for a real-time robust MPC method are provided that achieves computation times equal to those reported for methods without guarantees. A 12-dimensional problem with 3 control inputs and a prediction horizon of 10 time steps is solved in 2msec with a performance deterioration less than 1% and thereby allows for sampling rates of 500Hz.

Efficient interior point methods for multistage problems arising in receding horizon control

“…To show the code performance for a more complex problem, we add a quadratic terminal cost V f (x N ) := x T N P x N and a quadratic terminal constraint x T N P x N ≤ α to (19), where P solves the discrete-time Riccati equation related to LQR control and α determines the maximum level set of V f such that no constraints are violated. In addition, we add a real-time constraint V N (x) ≤ τ to ensure that the system is stable even in case of early termination of the solver, see [5] for details. The parameter τ is the cost of the previous solution minus a small multiple of the stage cost x T 0 Qx 0 .…”

“…Among these are formulations providing stability guarantees via ellipsoidal terminal sets [4], real-time stability guarantees [5] in face of early termination of the solver or robustness against modeling errors and noise in a tube MPC setting [6]. Moreover, some systems have inherent quadratic system constraints for which polytopic sets provide only a coarse approximation, such as power electronics [7].…”

Suboptimal Predictive Control for Satellite Detumbling

“…5-10) perform surprisingly well. The reference [26], however, imposes an extra constraint on the cost reduction. With the availability of a feasible solution at the start, the imposed constraint ensures stability for any available computational time for linear systems, when no disturbances are considered.…”

“…This feasible point, which ensures a monotonic decrease in the cost, also ensures closed-loop stability. The references [26,27] discuss sub-optimality in terms of early termination of the optimizer based on the available time or maximum number of iterations. Both references treat linear systems and propose using primal barrier interior-point methods to solve the optimization problem.…”

Certification of fixed computation time first-order optimization-based controllers for a class of nonlinear dynamical systems

“…Augmenting the optimization problem by a constraint enforcing the Lyapunov decrease in conjunction with an optimization method ensuring primal feasibility after each iteration warm-started by the shifted sequence then guarantees stability; see [20] for details. This approach, however, assumes that the MPC problem is designed in such a way that the closed-loop system is stable when solved to the full accuracy and that the shifted sequence is feasible and ensures a Lyapunov decrease.…”