Improved residual mode separation for finite-dimensional control of PDEs: application to the Euler-Bernoulli beam

TL;DR

This work tackles spillover in finite-dimensional $H_\infty$ control of a PDE system by integrating residue effects into the design. It introduces a time-domain, mode-decomposition approach that treats the neglected modes as disturbances and computes their $L^2$ gains via the bounded real lemma, then adds these gains to the cost for the dominating modes to prevent spillover. The method is demonstrated on a damped Euler-Bernoulli beam with a piezoelectric actuator, showing that with $N=8$ modes the $L^2$ gain and performance improve significantly and spillover is avoided, whereas accounting for the residue is essential to guarantee stability and performance as more modes are included. The approach is extensible to other PDEs (heat, wave, Kuramoto–Sivashinsky) and to related control frameworks (guaranteed cost, regional stability, delays, sampled-data).

Abstract

We consider a simply-supported Euler-Bernoulli beam with viscous and Kelvin--Voigt damping. Our objective is to attenuate the effect of an unknown distributed disturbance using one piezoelectric actuator. We show how to design a suitable $H_\infty$ state-feedback controller based on a finite number of dominating modes. If the remaining (infinitely many) modes are ignored, the calculated $L^2$ gain is wrong. This happens because of the spillover phenomenon that occurs when the effect of the control on truncated modes is not accounted for in the feedback design. We propose a simple modification of the $H_\infty$ cost that prevents spillover. The key idea is to treat the control as a disturbance in the truncated modes and find the corresponding $L^2$ gains using the bounded real lemma. These $L^2$ gains are added to the control weight in the $H_\infty$ cost for the dominating modes, which prevents spillover. A numerical simulation of an aluminum beam with realistic parameters demonstrates the effectiveness of the proposed method. The presented approach is applicable to other types of PDEs, such as the heat, wave, and Kuramoto-Sivashinsky equations, as well as their semilinear versions. While this work focuses on $H_\infty$ control, the same methodology can be applied to guaranteed cost control, regional stability analysis, input-to-state stability, and systems with time-varying delays, including sampled-data systems.

Figures (6)

• Figure 1: The eigenvalues of $A_n$ given in \ref{['eigenvalues']} for $n=1,\ldots,50$. Red dots --- no damping ($c_1=0=c_2$); blue dots --- viscous damping ($c_1=1.4\times 10^{-3}$, $c_2=0$); green dots --- viscous and Kelvin--Voigt damping ($c_1=1.4\times 10^{-3}$, $c_2=1.3\times 10^{-3}$).
• Figure 2: The value of $J_N(t)$, defined in \ref{['J(t)']}, for $N=5$ (blue), $N=6$ (green), and $N=50$ (red). The black line is $J_N(t)$ for $N=50$ and $u\equiv 0$. A controller designed for the first $5$ modes cannot guarantee \ref{['HinfObjective']} for the original system because of the spillover phenomenon.
• Figure 3: The $L^2$ gain of the Euler--Bernoulli beam \ref{['EB_beam']} for different numbers of controlled modes $N$.
• Figure 4: Euler--Bernoulli beam without and with control. The red dashed lines show the ends of the piezoelectric actuator.
• Figure 5: The value of $\|z(\cdot,t)\|_J$, defined in \ref{['Jnorm']}, without (black) and with (blue) control.

Theorems & Definitions (11)

• Proposition 1
• Remark 1: Solution existence
• Corollary 1: Bounded Real Lemma
• Remark 2: Damping model
• Remark 3: Performance index
• Proposition 2: $L^2$ gain without control
• Theorem 1
• Remark 4: Internal stability
• Remark 5: Solution existence
• Remark 6: Number of modes and the $L^2$ gain