Nonlinear MPC for Feedback-Interconnected Systems: a Suboptimal and Reduced-Order Model Approach

TL;DR

This work addresses the computational burden of nonlinear MPC for discrete-time, interconnected systems by merging a reduced-order model with a suboptimal optimization scheme, analyzed through a two-time-scale framework. The reduced dynamics $f_R(x,u,\delta)$ are coupled with a fast optimizer $z_t$ via $u_t=\Pi(z_t)$, and stability is established by exploiting a timescale separation tied to the sampling time $\delta$, ensuring global exponential convergence of the full interconnection. The main contributions include (i) a suboptimal, reduced-order MPC design, (ii) a boundary-layer and reduced-system stability analysis that leads to a global exponential stability guarantee for the closed-loop, and (iii) numerical validation on a pendulum actuated by a DC motor illustrating effective performance under appropriate $\delta$ values. These results offer a practical route to deploy MPC-like control in systems with fast inner dynamics while preserving rigorous stability guarantees, with relevance to mechatronics and robotics.

Abstract

In this paper, we propose a suboptimal and reduced-order Model Predictive Control (MPC) architecture for discrete-time feedback-interconnected systems. The numerical MPC solver: (i) acts suboptimally, performing only a finite number of optimization iterations at each sampling instant, and (ii) relies only on a reduced-order model that neglects part of the system dynamics, either due to unmodeled effects or the presence of a low-level compensator. We prove that the closed-loop system resulting from the interconnection of the suboptimal and reduced-order MPC optimizer with the full-order plant has a globally exponentially stable equilibrium point. Specifically, we employ timescale separation arguments to characterize the interaction between the components of the feedback-interconnected system. The analysis relies on an appropriately tuned timescale parameter accounting for how fast the system dynamics are sampled. The theoretical results are validated through numerical simulations on a mechatronic system consisting of a pendulum actuated by a DC motor.

Figures (7)

• Figure 1: Block diagram representation of the closed-loop system resulting from the interconnection of the full plant dynamics \ref{['eqp:plant']} and the reduced-order suboptimal MPC \ref{['eq:mpc_dyn']} for the pendulum-motor setup (cf. Example \ref{['ex:actuated_pendulum']}).
• Figure 2: Block diagram representation of the interconnected system.
• Figure 3: Block diagram representation of the boundary layer system.
• Figure 4: Block diagram representation of the reduced system.
• Figure 5: Closed-loop trajectories of the target states $\theta$ and $\omega$ (left) and angular position error in semilog scale (right) for $\delta = 0.01$ s.

Theorems & Definitions (5)

• Example 1
• Theorem 1
• Lemma 1
• Theorem 2
• proof