Set-point control and local stability for flat nonlinear systems using model-following control

TL;DR

This work analyzes model-following control (MFC) for nonlinear flat systems subject to locally Lipschitz perturbations. By coupling a nonlinear model control loop (MCL) with a high-gain processed loop (PCL) and leveraging a quadratic Lyapunov-based stability analysis, the authors derive a robustness bound and methods to estimate the region of attraction (ROA) for set-point tracking. The case study on a mass-spring-damper demonstrates that MFC can achieve the robustness and ROA comparable to or exceeding that of a high-gain single-loop design while reducing initial control effort and preserving steady-state accuracy. Overall, MFC offers a practical, two-loop alternative that enhances stability and robustness for flat nonlinear systems and can be extended to systems with internal dynamics.

Abstract

We consider the set-point control problem for nonlinear systems with flat output that are subject to perturbations. The nonlinear dynamics as well as the perturbations are locally Lipschitz. We apply the model-following control (MFC) approach which consists of a model control loop (MCL) for a feedforward generation and a process control loop (PCL) that compensates the perturbations using high-gain feedback. We analyse the resulting closed-loop system and discuss its relation to a standard flatness-based high-gain approach. In particular we analyse the estimated region of attraction provided by a quadratic Lyapunov function. A case study illustrates the approach and quantifies the region of attraction obtained for each control approach. Using the initial condition of the model control loop as tuning parameter for the MFC design, provides that a significantly larger region of attraction can be guaranteed compared to a conventional single-loop high-gain design.

Figures (8)

• Figure 1: Block diagram model-following control (MFC)
• Figure 2: Sketch of the mass-springer-damper system.
• Figure 3: Estimated regions of attraction for $\bm{x}_{\mathrm{d}}=(0.75 \; 0)^\top$ and $\bm{x}_0=\bm{0}$ and various control designs.
• Figure 4: Phase portrait and estimated region of attraction for $\bm{x}_{\mathrm{d}}=(0.75 \; 0)^\top$ and $\bm{x}_0=\bm{0}$. For the MFC scheme we have $\bm{x}^\star_0=\bm{0}$ with three different initial values of the process state: $\bm{x}_0=\bm{0}$ (solid blue line), $\bm{x}_0=(0.1 \enspace -8)^\top$ (dotted blue line), and $\bm{x}_0=(-0.25 \enspace 6)^\top$ (dashed-dotted blue line).
• Figure 5: Time-responses of the states and input signal for $\bm{x}_{\mathrm{d}}=(0.75 \; 0)^\top$ and $\bm{x}_0=\bm{0}$. For the MFC scheme we have $\bm{x}^\star_0=\bm{0}$ with three different initial values of the process states: $\bm{x}_0=\bm{0}$ (solid blue line), $\bm{x}_0=(0.1 \enspace -8)^\top$ (dotted blue line), and $\bm{x}_0=(-0.25 \enspace 6)^\top$ (dashed-dotted blue line).

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4