Robust nonlinear control synthesis by using centre manifold-based reduced models for the mitigating of friction-induced vibration

Abstract

This paper is concerned with the synthesis of reduced-order robust nonlinear controllers for more efficient robust mitigating of friction-induced vibration (FIV) issued from the mode-coupling mechanism. A novel scheme is proposed and developed. It consists of the centre manifold approach which is first proposed for reducing the dimension of the system model to control. Then, the obtained reduced model is exploited to synthesize a sliding-mode-based reduced-order controller which is then applied on the full-order original model for suppressing or at least mitigating the mode-coupling-based FIV. The main objective of the proposed study is to analyse performances of the proposed reduced-order controller in terms of its capacities to efficiently mitigate mode-coupling-based vibrations while ensuring suitable robustness levels with respect to the centre manifold-based reduced model inaccuracy and the friction coefficient uncertainty.

Appendix

As presented previously in Sect. 2, the centre manifold $\varvec{\phi }$ which is the solution of the algebraic Eq. (8) is in practice approximated by a polynomial with a predefined order. The dynamical systems considered are with only cubic nonlinearities. They can be expressed in a compact general (6)-like equation with a zero input by using the Kronecker product operator.

$$\begin{aligned} \left\{ \begin{array}{llll} \dot{{\varvec{z}}}_{c}(t)={\varvec{A}}_{c}({\tilde{\mu }}){\varvec{z}}_{c}(t)+[G_{(3)}^{ij}]. {\varvec{z}}_{a}(t)\otimes {\varvec{z}}_{a}(t)\otimes {\varvec{z}}_{a}(t)\\ \dot{{\varvec{z}}}_{s}(t)={\varvec{A}}_{s}({\tilde{\mu }}){\varvec{z}}_{s}(t)+[H_{(3)}^{ij}]. {\varvec{z}}_{a}(t)\otimes {\varvec{z}}_{a}(t)\otimes {\varvec{z}}_{a}(t)\\ \dot{{\tilde{\mu }}}=0 \end{array} \right. \end{aligned}$$

(33)

where ${\varvec{z}}_{a}=[{\varvec{z}}_{c};{\varvec{z}}_{s};{\tilde{\mu }}]$, $G_{3}^{ij}$ and $H_{3}^{ij}$ (with $1\le i \ge 2, 1 \le k \ge (n+3)^{3}$) are the coefficients of the cubic monomials in the state variables with $\otimes$ being the Kronecker operator.

Then, the coefficients of the second-order centre manifolds for both systems (Hultèn and Hoffman-like systems) are null. Then, third centre manifolds are searched for. The associated polynomials contain only the third-order monomials in the centre variables, which lets the stable variable to be approximated by the following third-order polynomial in the centre variables ${\varvec{z}}_{c}$ and ${\tilde{\mu }}$

$$\begin{aligned} {\varvec{z}}_{s}= \varvec{\phi }({\varvec{z}}_{c},{\tilde{\mu }})=\sum _{p=i+j+l=2}^{3}\sum _{j=0}^{p}\sum _{l=0}^{p}{\varvec{c}}_{ijl}z_{c_{1}}^{i}z_{c_{i}}^{j}{\tilde{\mu }}^{l} \end{aligned}$$

(34)

and equivalently:

$$\begin{aligned} {\varvec{z}}_{s}= & {} {\varvec{c}}_{300}z_{c_{1}}^{3}+{\varvec{c}}_{210}z_{c_{1}}^{2}z_{c_{2}}\nonumber \\&+{\varvec{c}}_{120}z_{c_{1}}z_{c_{2}}^{2}+{\varvec{c}}_{030}z_{c_{2}}^{3}+{\varvec{c}}_{201}z_{c_{1}}{\tilde{\mu }}\nonumber \\&+{\varvec{c}}_{111}z_{c_{1}}z_{c_{2}}{\tilde{\mu }}+....+{\varvec{c}}_{003}{\tilde{\mu }}^{3} \end{aligned}$$

(35)

Then, by replacing the previous polynomial in the algebraic equation (8), the centre manifold coefficients for the $k^{\text {th}}$-stable variable can be identified. For the sake of brevity, only the first coefficients are given, while the expressions of the remaining coefficients and also the higher-order centre manifold coefficients can be found in [47].

$$\begin{aligned} c_{k,300}= & {} \frac{H_{(3)}^{k,1}}{3A_{c_{1}}-A_{s_{k}}} \end{aligned}$$

(36)

$$\begin{aligned} c_{k,210}= & {} \frac{H_{(3)}^{k,2}+H_{(3)}^{k,n+2}+H_{(3)}^{k,(n+1)^{2}}}{2A_{c_{1}}+A_{c_{2}}-A_{s_{k}}} \end{aligned}$$

(37)

$$\begin{aligned} c_{k,210}= & {} \frac{H_{(3)}^{k,2}+H_{(3)}^{k,n+2}+H_{(3)}^{k,(n+1)^{2}}}{2A_{c_{1}}+A_{c_{2}}-A_{s_{k}}} \end{aligned}$$

(38)

$$\begin{aligned} c_{k,120}= & {} \frac{H_{(3)}^{k,n+3}+H_{(3)}^{k,(n+1)^{2}+2}+H_{(3)}^{k,(n+1)^{2}+n+2}}{2A_{c_{1}}+A_{c_{2}}-A_{s_{k}}}\nonumber \\ \end{aligned}$$

(39)

$$\begin{aligned} c_{k,030}= & {} \frac{H_{(3)}^{k,(n+1)^{2}+n+3}}{3A_{c_{2}}-A_{s_{k}}} \end{aligned}$$

(40)

where $A_{c_{1}}$ and $A_{c_{2}}$ are the first and second terms of the matrix ${\varvec{A}}_{c}$, respectively, while $A_{s_{k}}$ is the $k^{\text {th}}$-diagonal term of the matrix ${\varvec{A}}_{s}$.