Robustness Quantification of MIMO-PI Controller From the Perspective of \(γ\)-Dissipativity

TL;DR

This work proposes a $\gamma$-dissipativity framework to quantify and enhance the robustness of MIMO-PI controllers for disturbed nonlinear MIMO systems. By integrating dissipativity theory with the Hamilton-Jacobi-Isaacs inequality, it derives sufficient conditions for $\gamma$-dissipativity and quantifies it in a local region via a $P$-matrix Lyapunov condition, linking $\gamma$ to the $L_2$-gain. It then formulates optimal parameter tuning as an eigenvalue problem and LMIs, enabling online computation of the MIMO-PI gains $K_P$ and $K_I$ that minimize $\gamma$ while maintaining dissipativity. Simulation on a disturbed nonlinear UAV-like model validates the approach, demonstrating that larger dissipativity domains (quantified by $W_K$) correlate with lower ITAE and smoother convergence, and introduces the $\gamma_K(\Omega)$ index as a region-based measure of robustness and energy attenuation.

Abstract

The proportional-integral-derivative (PID) controller and its variants are widely used in control engineering, but they often rely on linearization around equilibrium points and empirical parameter tuning, making them ineffective for multi-input-multi-output (MIMO) systems with strong coupling, intense external disturbances, and high nonlinearity. Moreover, existing methods rarely explore the intrinsic stabilization mechanism of PID controllers for disturbed nonlinear systems from the perspective of modern robust control theories such as dissipativity and $\mathcal{L}_2$-gain. To address this gap, this study focuses on $γ$-dissipativity (partially equivalent to $\mathcal{L}_2$-gain) and investigates the optimal parameter tuning of MIMO-PI controllers for general disturbed nonlinear MIMO systems. First, by integrating dissipativity theory with the Hamilton-Jacobi-Isaacs (HJI) inequality, sufficient conditions for the MIMO-PI-controlled system to achieve $γ$-dissipativity are established, and the degree of $γ$-dissipativity in a local region containing the origin is quantified. Second, an optimal parameter tuning strategy is proposed, which reformulates the $γ$-dissipativity optimization problem into a class of standard eigenvalue problems (EVPs) and further converts it into linear matrix inequality (LMI) formulations for efficient online computation. Comprehensive simulation experiments validate the effectiveness and optimality of the proposed approach. This work provides a theoretical basis for the robust stabilization of general disturbed nonlinear MIMO systems and enriches the parameter tuning methods of PID controllers from the perspective of dissipativity.

Figures (5)

• Figure 1: Two-dimensional phase portraits of error trajectories under different initial conditions $\chi(0)$ and $\gamma(0)$, which illustrate the error origin, $\mathcal{L}_{K^*}(e)$ at every point $e$, and the corresponding boundary of $\mathcal{L}_{K^*}(\Omega)=0$ (zero line).
• Figure 2: Dissipativity indices $\gamma_K(\Omega_i)$ over regions $\Omega_i,i=1,2,3,4$ for different controller parameters $K$. (a): For the $K_1$; (b): For the $K_2$; (c): For the $K_3$; (d): For the $K_4$; (e): For the $K_5$; (f): For the $K_6$.
• Figure 3: Time curves of the amplitudes of the error $e(t)$ and its derivative $\dot{e}(t)$ at the six different regions in Figure \ref{['Fig:1']}. (a): Time curves of the amplitudes of the error $e(t)$; (b): Time curves of the amplitudes of the error $\dot{e}(t)$.
• Figure 4: Relationships between indictor $W_K$ (different $K$) and ITAE/Standard deviation of $e_{\chi}, e_{\gamma}, \dot{e}_{\chi}, \dot{e}_{\gamma}$. (a): Relationships between $W_K$ and ITAE of $e_{\chi}$ and $e_{\gamma}$; (b): Relationships between $W_K$ and ITAE of $\dot{e}_{\chi}$ and $\dot{e}_{\gamma}$; (c): Relationships between $W_K$ and standard deviation of $e_{\chi}$ and $e_{\gamma}$; (d): Relationships between $W_K$ and standard deviation of $\dot{e}_{\chi}$ and $\dot{e}_{\gamma}$;
• Figure 5: Curves of the relationships between the robustness index $R_K$ and $\gamma_K(\Omega_i)$ as well as $W_K$ respectively. (a): Relationship between $R_K$ and $\gamma_K(\Omega_i)$, $i=1,2,3,4$; (b): Relationship between $R_K$ and $W_K$.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof