Subsystem-Based Control with Modularity for Strict-Feedback Form Nonlinear Systems

TL;DR

This work addresses the challenge of designing a modular, globally stabilizing controller for nth-order nonlinear systems in strict-feedback form (SFF) under parametric uncertainty. It introduces adaptive subsystem-based control (SBC) with per-subsystem stability connectors to manage inter-SS interactions and a smooth projection function to bound adaptive estimates, providing Lyapunov-based guarantees of global asymptotic stability (GAS) with $e_k \to 0$ as $t \to \infty$. Key contributions include a single generic modular control form for each SS, a dedicated stability connector to cancel cross-SS effects, and a projection-based adaptation strategy that preserves modularity and scales to high dimensions, all validated by a 3rd-order numerical example. The approach offers a scalable alternative to backstepping/DSC with provable GAS and practical adaptability for high-dimensional SFF systems, enabling localized design and analysis while mitigating complexity growth.

Abstract

This study proposes an adaptive subsystem-based control (SBC) for systematic and straightforward nonlinear~control of nth-order strict-feedback form (SFF) systems.~By decomposing the SFF system to subsystems, a generic~term (namely stability connector) can be created to address dynamic interactions between the subsystems. This 1) enables modular control design with global asymptotic stability, 2) such that both the control design and the stability analysis can be performed locally at a subsystem level, 3) while avoiding an excessive growth of the control design complexity when the system order n increases. The latter property makes the method suitable especially for high-dimensional systems. We also design a smooth projection function for addressing system parametric uncertainties. Numerical simulations demonstrate the efficiency of the method.

Figures (6)

• Figure 1: Diagram of the proposed adaptive SBC (highlighted in light blue). The desired variables (and the control output $u$) are shown in red, the feedback signals are in green, the adaptive control is in blue, and the system output states are in black. The bold lines are vectors and the thin lines are scalar variables.
• Figure 2: Control performance in C1 with inaccurate parameter values $\theta_{12}$ = 6 and $\theta_{22}$ = 4 in relation to the actual plant parameters $a_1$ = 5 and $a_2$ = 5. The desired trajectories are shown in black and their controlled variables in gray (plots 1--3). The last plot shows the control output $u$.
• Figure 3: Tracking errors $e_k$, $\forall k \in \{1,2,3\}$, in C1 (adaptive control disabled).
• Figure 4: Control performance in C2 with initial parameter values $\widehat{\theta}_{12}(0)$ = 6 and $\widehat{\theta}_{22}(0)$ = 4, while $a_1$ = 5 and $a_2$ = 5 hold for the respective plant parameters. The desired trajectories are shown in black and their controlled variables in gray (plots 1--3). The last plot shows the control output $u$.
• Figure 5: Tracking errors $e_k$, $\forall k \in \{1,2,3\}$, in C2 and C3 (adaptive control enabled). The results in C2 are in black and the results in C3 are in gray.

Theorems & Definitions (16)

• Remark 3.1
• Definition 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Definition 4.1
• Lemma 4.1
• Remark 4.1
• Lemma 4.2
• Remark 4.2