A novel optimal fuzzy integrated control method of active suspension system

Abstract

A novel optimal fuzzy integrated control method is proposed and used to a half-car model with active suspension system. The cultural algorithm improved by the use of a dynamic acceptance function associated with niche algorithm is presented to optimize the fuzzy control rules with 4 input variables and 2 output variables. The vertical and rolling vibration accelerations of the bodywork are selected as an integrated control objective by using a fitness function. The simulation results of control system demonstrate that there are evident reductions in both vertical vibration acceleration and rolling angle acceleration if fuzzy control rules are optimized by the modified cultural algorithm and then applied to the active suspension system. It is simultaneously shown that the vertical displacement and rolling angle displacement also experience reduction processes. All of the numerical analyses give enormous supports to use of the proposed control scheme.

Appendix

E is a unit matrix. O is a zero matrix.

$${\kern 1pt} {\kern 1pt} {\text{A}} = \left[ {\begin{array}{*{20}c} \begin{gathered} {\text{O}}_{{4 \times 4}} \hfill \\ \cdots \cdots \cdots \hfill \\ \end{gathered} & \begin{gathered} \vdots \hfill \\ \vdots \hfill \\ \end{gathered} & \begin{gathered} \;{\text{E}}_{{4 \times 4}} \hfill \\ \cdots \cdots \cdots \hfill \\ \end{gathered} \\ {{\text{A}}_{{{\mathbf{1}}{\kern 1pt} {\kern 1pt} {\text{(}}4 \times 4{\text{)}}}} } & \vdots & {{\text{A}}_{{{\mathbf{2}}{\kern 1pt} {\kern 1pt} {\text{(}}4 \times 4{\text{)}}}} } \\ \end{array} } \right]$$

$${\text{B}} = \left[ {\begin{array}{*{20}c} {{\text{O}}_{{4 \times 4}} \begin{array}{*{20}c} \vdots \\ \vdots \\ \vdots \\ \vdots \\ \vdots \\ \vdots \\ \end{array} } & {\begin{array}{*{20}c} {\frac{{k_{{12}} }}{{m_{1} }}} & 0 & 0 & 0 \\ 0 & {\frac{{k_{{22}} }}{{m_{2} }}} & 0 & 0 \\ {\frac{1}{{m_{1} }}} & 0 & { - \frac{1}{M}} & {\frac{l}{J}} \\ 0 & {\frac{1}{{m_{2} }}} & { - \frac{1}{M}} & { - \frac{l}{J}} \\ \end{array} } \\ \end{array} } \right]^{T}$$

$${\mathbf{A}}_{{\mathbf{1}}} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{ = }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}c} { - \frac{{k_{{11}} + k_{{12}} }}{{m_{1} }}} & 0 & {\frac{{k_{{12}} }}{{m_{1} }}} & { - \frac{{l \cdot k_{{12}} }}{{m_{1} }}} \\ 0 & { - \frac{{k_{{21}} + k_{{22}} }}{{m_{2} }}} & {\frac{{k_{{22}} }}{{m_{2} }}} & {\frac{{l \cdot k_{{22}} }}{{m_{2} }}} \\ {\frac{{k_{{12}} }}{M}} & {\frac{{k_{{22}} }}{M}} & { - \frac{{k_{{12}} + k_{{22}} }}{M}} & 0 \\ { - \frac{{l \cdot k_{{12}} }}{J}} & {\frac{{l \cdot k_{{22}} }}{J}} & 0 & { - \frac{{l^{2} k_{{12}} + l^{2} k_{{22}} }}{J}} \\ \end{array} } \right]$$

$$A_{2} = \left[ {\begin{array}{*{20}c} { - \frac{{c_{1} }}{{m_{1} }}} & 0 & {\frac{{c_{1} }}{{m_{1} }}} & {\frac{{l \cdot c_{1} }}{{m_{1} }}} \\ 0 & { - \frac{{c_{1} }}{{m_{1} }}} & {\frac{{c_{2} }}{{m_{2} }}} & {\frac{{l \cdot c_{2} }}{{m_{2} }}} \\ {\frac{{c_{1} }}{M}} & {\frac{{c_{2} }}{M}} & { - \frac{{c_{1} + c_{2} }}{M}} & 0 \\ { - \frac{{lc_{1} }}{J}} & { - \frac{{lc_{2} }}{J}} & 0 & { - \frac{{l^{2} c_{1} + l^{2} c_{2} }}{J}} \\ \end{array} } \right]$$

$${\text{C}} = \left[ {\begin{array}{*{20}c} 0 & 0 & {\frac{k12}{M}} & { - \frac{l \cdot k12}{J}} \\ 0 & 0 & {\frac{k22}{M}} & {\frac{l \cdot k22}{J}} \\ 1 & 0 & { - \frac{k12 + k22}{M}} & 0 \\ 0 & 1 & 0 & { - \frac{{l^{2} (k12 + k22)}}{J}} \\ 0 & 0 & { - \frac{c1}{M}} & { - \frac{lc1}{J}} \\ 0 & 0 & {\frac{c2}{M}} & { - \frac{lc2}{J}} \\ 0 & 0 & { - \frac{c1 + c2}{M}} & 0 \\ 0 & 0 & 0 & { - \frac{{l^{2} (c1 + c2)}}{J}} \\ \end{array} } \right]^{T}$$

$$\begin{aligned} {\text{D}} = {\text{O}}4 \times 4 \hfill \\ u = \left[ {z10,\,z20,F1,F2} \right]^{T} \hfill \\ \end{aligned}$$