Structural vibration control using delayed state feedback via LMI approach: with application to chatter stability problems

Abstract

This paper considers the problem of active vibration control in machining processes to prevent dynamic instability and chatter. A method for robust delayed-state feedback control is presented based on Lyapunov–Krasovskii functionals (LKFs). For fast computation and realizability, a partial state feedback approach based on a reduced order model of structural vibration is considered that matches available sensors and captures the main vibratory modes that impact on stability limits. Robustness to neglected dynamics and other model errors are accounted for by a frequency domain description of model uncertainty, augmented with the basic LKF stability condition. The controller synthesis problem is formulated as a set of linear matrix inequality equations that can be solved numerically using convex optimization methods. Evaluations on a hardware-in-the-loop flexible structure test-bed show that feedback controllers optimized via the LKF stability condition provide significant improvements in stability regions compared with optimized PD control and quadratic regulation. The results confirm the methodology as a useful approach for robust chatter control based on combined flexible structure and cutting process models.

Appendix

Given negative definite matrices, $Q = Q^{T}$ and $R = R^{T}$ for which a bilinear matrix inequality (BMI) is defined as $Q - S^{T} R^{ - 1} S < 0$ then an equivalent LMI can be obtained according to [22]:

$$ R < 0,Q - S^{T} R^{ - 1} S < 0 \Leftrightarrow \left[ {\begin{array}{*{20}c} Q & {S^{T} } \\ S & R \\ \end{array} } \right] < 0 $$

Thus, considering Eq. (17) in the form

$$ \begin{aligned} {{\Phi }} & = \left[ {\begin{array}{*{20}c} {P\left( {\widetilde{A}_{0} + \widetilde{B}_{u}^{T} K} \right) + \left( {\widetilde{A}_{0} + \widetilde{B}_{u}^{T} K} \right)^{T} P + Q} & {P\widetilde{A}_{1} + P\widetilde{B}_{u} K_{\tau } } & {P\widetilde{B}_{u} } \\ * & { - Q} & 0 \\ * & * & { - \gamma^{2} I} \\ \end{array} } \right] \\ & \quad - \left[ {\begin{array}{*{20}c} {\widetilde{C}_{r} + \widetilde{D}_{r} K} \\ {\widetilde{D}_{r} K_{\tau } } \\ 0 \\ \end{array} } \right]^{T} \left[ { - I} \right]\left[ {\begin{array}{*{20}c} {\widetilde{C}_{r} + \widetilde{D}_{r} K} & {\widetilde{D}_{r} K_{\tau } } & 0 \\ \end{array} } \right] < 0 \\ \end{aligned} $$

leads to the equivalent condition in Eq. (18):

$$ {{\varPsi }} = \left[ {\begin{array}{*{20}c} {P\left( {\widetilde{A}_{0} + \widetilde{B}_{u}^{T} K} \right) + \left( {\widetilde{A}_{0} + \widetilde{B}_{u}^{T} K} \right)^{T} P + Q} & {P\widetilde{A}_{1} + P\widetilde{B}_{u} K_{\tau } } & {P\widetilde{B}_{u} } & {\widetilde{C}_{r}^{T} + K^{T} \widetilde{D}_{r}^{T} } \\ * & { - Q} & 0 & {K_{\tau }^{T} \widetilde{D}_{r}^{T} } \\ * & * & { - \gamma^{2} I} & 0 \\ * & * & * & { - I} \\ \end{array} } \right] < 0. $$