Online Optimal Parameter Compensation method of High-dimensional PID Controller for Robust stability

TL;DR

The paper tackles online robust tuning of high-dimensional PID controllers for nonlinear MIMO systems with disturbances. It introduces a velocity-form error dynamic and reframes parameter regulation as two-stage eigenvalue problems solved online via interior-point methods to obtain $K$ and a compensator $\Delta K$. The approach yields guaranteed robust stability near the origin and improved finite-time convergence at the current state, demonstrated through a fixed-wing aircraft simulation where dynamic compensation reduces oscillations and enhances stability across channels. The results offer a practical method for online, coupled-parameter tuning in complex MIMO systems with strong inter-channel interactions, faults, and disturbances.

Abstract

Classical PID control is widely applied in an engineering system, with parameter regulation relying on a method like Trial - Error Tuning or the Ziegler - Nichols rule, mainly for a Single - Input Single - Output (SISO) system. However, the industrial nonlinear Multiple - Input Multiple - Output (MIMO) system demands a high - robustness PID controller due to strong state coupling, external disturbances, and faults. Existing research on PID parameter regulation for a nonlinear uncertain MIMO system has a significant drawback: it's limited to a specific system type, the control mechanism for a MIMO nonlinear system under disturbances is unclear, the MIMO PID controller over - relies on decoupled control, and lacks dynamic parameter compensation. This paper theoretically analyzes a high - dimensional PID controller for a disturbed nonlinear MIMO system, providing a condition for online dynamic parameter regulation to ensure robust stability. By transforming the parameter regulation into a two - stage minimum eigenvalue problem (EVP) solvable via the interior point method, it enables efficient online tuning. The experiment proves that the designed dynamic compensation algorithm can achieve online robust stability of system errors considering multi - channel input coupling, addressing the key limitation in the field.

Figures (4)

• Figure 1: The convergence process of the elliptical form $\sqrt{e^T(t)Pe(t)}$ of error $e$, with respect to time $t$.
• Figure 2: Comparison of the time-series profiles of the states $e_{\gamma}$, $e_{\chi}$ and their differentials $\dot{e}_{\gamma}$, $\dot{e}_{\chi}$ before and after adopting the original parameters $K$ and the parameters $K + \Delta K$ after optimal compensation. (a): Comparison of the $e_{\gamma}$; (b): Comparison of the $e_{\chi}$; (c): Comparison of the $\dot{e}_{\gamma}$; (d): Comparison of the $\dot{e}_{\chi}$.
• Figure 3: A comparison is made of the eigenvalue distributions of the matrices before compensation $\tilde{J}_K(0)$ and the matrices after compensation $\tilde{J}_{K+\Delta K}(0)$ at the origin.
• Figure 4: Quantitative comparison of differences in high-dimensional PID controller before and after compensation using ITAE, PT, and MO indictors from the perspectives of the states $e_\gamma$, $e_\chi$, $\dot{e}_\gamma$, and $\dot{e}_\chi$. (a): $e_{\gamma}$; (b): $e_{\chi}$; (c): $\dot e_{{\gamma}}$; (d): $\dot e_{{\chi}}$.

Theorems & Definitions (12)

• Definition 1
• Lemma 1
• proof
• Lemma 2
• Lemma 3
• proof
• Theorem 1
• proof
• Theorem 2
• proof