Composite learning backstepping control with guaranteed exponential stability and robustness

TL;DR

This work addresses adaptive backstepping control for strict-feedback nonlinear systems with mismatched uncertainties, where achieving exponential stability traditionally requires persistent excitation ($PE$). It proposes Composite Learning Backstepping Control (CLBC), which combines modular backstepping with a composite learning high-order tuner and staged exciting strength maximization to exploit interval excitation ($IE$) and partial IE, while introducing dual prediction errors to improve transient performance without nonlinear damping. Theoretical guarantees show parameter convergence and closed-loop exponential stability under $IE$ or partial IE, plus robustness to additive disturbances. Simulations demonstrate that CLBC outperforms state-of-the-art modular backstepping methods in parameter estimation accuracy, tracking performance, and transient behavior, while avoiding high gains and nonlinear damping terms.

Abstract

Adaptive backstepping control provides a feasible solution to achieve asymptotic tracking for mismatched uncertain nonlinear systems. However, the closed-loop stability depends on high-gain feedback generated by nonlinear damping terms, and closed-loop exponential stability with parameter convergence involves a stringent condition named persistent excitation (PE). This paper proposes a composite learning backstepping control (CLBC) strategy based on modular backstepping and high-order tuners to compensate for the transient process of parameter estimation and achieve closed-loop exponential stability without the nonlinear damping terms and the PE condition. A novel composite learning mechanism is designed to maximize the staged exciting strength for parameter estimation, such that parameter convergence can be achieved under a condition of interval excitation (IE) or even partial IE that is strictly weaker than PE. An extra prediction error is employed in the adaptive law to ensure the transient performance without nonlinear damping terms. The exponential stability of the closed-loop system is proved rigorously under the partial IE or IE condition. Simulations have demonstrated the effectiveness and superiority of the proposed method in both parameter estimation and control compared to state-of-the-art methods.

Figures (5)

• Figure 1: An illustration of the current maximal exciting strength $\sigma_{\rm c}$ in Algorithm 1. Note that the black solid line denotes $\sigma_{\rm c}$, the blue and green dash lines are $\sigma_{\min}(\Psi_\zeta)$ in two partial IE stages, and the red dotted line is $\sigma_{\min}(\Psi)$.
• Figure 2: An illustration of the relationship between the channels $\bm\phi_{{\rm s},i}$ and the excitation matrix $\Psi$ with $N=7$. Note that the blue modules denote active channels, the white modules are inactive channels, and $\bm p$ is defined in \ref{['eq09']}.
• Figure 3: Performance comparisons of three controllers for the tracking problem under the PE condition. (a) The absolute tracking errors $|e_1|$. (b) The estimation error norms $\|\tilde{\bm\theta}\|$. (c) The control inputs $u$. (d) The exciting strengths $\sigma_{\rm c}$.
• Figure 4: Performance comparisons of three controllers for the regulation problem under partial IE or IE condition. (a) The partial estimation errors $\|\tilde{\bm\theta}_\zeta\|$. (b) The estimation errors $\|\tilde{\bm\theta}\|$. (c) The absolute tracking error norms $|e_1|$. (d) The control inputs $u$. (e) The exciting strengths $\sigma_{\rm c}$.
• Figure 5: Performance comparisons of three controllers for the tracking problem under different values of the damping parameters $k_{\rm{d} \it 1}$ and $k_{\rm{d} \it 2}$. (a) The absolute tracking errors $|e_1|$. (b) The absolute estimation errors $|\tilde{\theta}|$. Note that the arrows indicate the increasing direction $k_{\rm{d} \it 1}$ and $k_{\rm{d} \it 2}$.

Theorems & Definitions (3)

• proof
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