Robustness of Orthogonal Eigenstructure Control to Actuators Failure

Abstract

Orthogonal eigenstructure control (OEC) is a feedback control method applicable to multi-input multi-output linear systems. While the available control design methodologies offer a large and complex design space of options that can often overwhelm a designer, this control method offers a significant simplification of the design task while still allowing some experience-based design freedom. In this chapter, the robustness of the method to the failure of the actuators was investigated. It was shown the control gain was capable of controlling the systems during an actuator failure, as OEC generates the control gain by maintaining the closed-loop eigenvectors within the achievable eigenvectors set. A system of lumped masses was used to explain the method; then, the problem of failed actuators in the vibration control of a plate was investigated. Finite element analysis was used for modeling the plate to simulate the dynamic behavior of the system. Five cases were considered and the suppression of the vibration in a plate with three working actuators was compared to the performance of a similar control system with a failed actuator. Also, the behaviors of the system with failed actuators were compared to the systems that were designed to operate with lesser control actuators. It was shown that the number of closed-loop eigenvalue pairs that moved from the cluster of the open-loop poles was equal to the number of working actuators. The closed-loop poles in all the systems were moved to the vicinity of one specific area, generating a break frequency with sufficient damping for robust active vibration control.

Key Symbols

$A$

Open-loop state matrix

${A_c}$

Closed-loop state matrix

$B$

Input matrix

$C$

Output matrix

$E$

Disturbance input matrix

${E_i}$

Modal energy of $i$th mode

$f$

Disturbance

$I$

Identity matrix

$K$

Gain matrix

$m$

Number of inputs (actuators/sensors)

${N^i}$

Matrix that spans the null space of $i$th mode

$n$

Dimension of second order system

${r^i}$

Vector of coefficients

${S_{{\lambda i}}}$

Augmented matrix associated with ${\lambda_i}$

$u$

Input vector

${U_i}$

Left unitary matrix of ${S_{{\lambda i}}}$

${{\bar{U}}^i}$

Eigenvalue matrix of $V_{12}^{i* }V_{12}^i$ and $V_{22}^{i* }V_{22}^i$

$\bar{U}_w^i$

Eigenvalue matrix of $V_{22}^{i* }V_{22}^i$, equals to ${{\bar{U}}^i}$

$\bar{U}_j^i$

Eigenvalue of $V_{12}^{i* }V_{12}^i$ associated with non-unity eigenvalues

$\bar{U}_J^i$

Eigenvalue of $V_{12}^{i* }V_{12}^i$ associated with unity eigenvalue

${V_i}$

Right unitary matrix of ${S_{{\lambda i}}}$

$V_{12}^i$

Upper part of ${N^i}$

$V_{22}^i$

Lower part of ${N^i}$

$V$

Appended matrix of $[V_{12}^i]{r^i}$

$W$

Appended matrix of $[V_{22}^i]{r^i}$

$x$

State vector

$\dot{x}$

Time derivative of state vector

$y$

Output vector

${\phi_i}$

$i$th closed-loop eigenvalue

$\phi_i^a$

Achievable eigenvector of $i$th mode

${\lambda_i}$

$i$th operating eigenvalue

$\bar{\lambda}_j^i$

Eigenvalues of $V_{12}^{i* }V_{12}^i$

${{\bar{\varLambda}}_i}$

Eigenvalue matrix of $V_{12}^{i* }V_{12}^i$

$\bar{\varLambda}_w^i$

Eigenvalue matrix of $V_{22}^{i* }V_{22}^i$

${\varSigma_i}$

Matrix of singular values of ${S_{{\lambda i}}}$

*

Conjugate transpose symbol