Development of Optimization Algorithms for the Parametric Synthesis of Control Systems

The article elucidates the issue of increasing the efficiency in designing systems for stretching synthetic sewing thread using automated computational procedures of a guided random search for controlled parameters.

Appendix

$$ \left[{T}_2^2{Lp}^3+\left({T}_2^2R+{T}_1L\right){p}^2+\left({T}_1R+L\right)p+\left(R+\frac{2{c}_d{k}_{ng}}{cr}\right)\right]X=\frac{2{c}_d{k}_{ng}}{cr}{X}_{\mathrm{spec}}+\frac{2{c}_d}{cr}{U}_r-\frac{2{c}_d}{c}\upomega, $$

(2)

where c – spring stiffness; c_d = c_m= c_e – motor constant; R, L – active and inductive resistance of the motor windings; k_d – damping coefficient; k_ng – gain factor; r – bobbin radius; X – thread tension; U_r – reference voltage, supplied to the motor winding at a constant linear speed; M_m, ω – torque and angular speed of the motor.

$$ \left({A}_0{p}^5+{A}_1{p}^4+{A}_2{p}^3+{A}_3{p}^2+{A}_4p+{A}_5\right)X={b}_3\left({L}_0{p}^2+{L}_1p+1\right){x}_{\mathrm{spec}}+{c}_3\left({L}_0{p}^2+{L}_1p+1\right){U}_r+{d}_3\frac{R+ Lp}{c_d^2} cr(t){X}_{\mathrm{spec}}. $$

(5)

$$ {\displaystyle \begin{array}{l}\left\{\frac{JL^2{T}_2^2}{c_d^2}{p}^5+\frac{JL}{c_d^2}\left[2{T}_2^2\left(R+{T}_1L\right){p}^4+\frac{JL}{c_d^2}\left({T}_1R+L+\frac{\left({T}_1 RL+R{T}_2^2\right)R}{L}\right)+{T}_2^2L\right]{p}^3+\right.\\ {}+\left[\frac{J}{c_d^2}\left( LR+\frac{k_0{c}_d{k}_{ng}}{r}+{T}_1{R}^2+ LR\right)+{T}_1L+{T}_2^2\right]{p}^2+\\ {}\begin{array}{l}\left.+\left[\frac{J}{c_d^2}\left(R+\frac{k_0{c}_d{k}_{ng}}{r}\right)+{T}_1R+L\right]p+\left(\frac{k_0{c}_d{k}_{ng}}{r}+R\right)\right\}X=\\ {}=\left[\frac{JL}{c_d}\bullet \frac{k_0{k}_{ng}}{r}{p}^2+\left(\frac{JL}{C_d}\bullet \frac{k_0{k}_{ng}}{r}+2L\right)p+\left(\frac{k_0{c}_d{k}_{ng}}{r}+2R\right)\right]{X}_{\mathrm{spec}}+\left(\frac{JL}{c_d}\bullet \frac{k_0}{r}+\frac{JL}{c_d}\bullet \frac{k_0}{r}\right){U}_r.\end{array}\end{array}} $$

(6)

$$ {\displaystyle \begin{array}{c}{m}_2=0.17{\Delta }_2,\\ {}{m}_3=0.17{\Delta }_3\left[1+9{\Delta }_3+2{\Delta }_2^2+0.4{\Delta }_2^3\right],\\ {}\begin{array}{c}{m}_4=0.17{\Delta }_4\left[1+9{\Delta }_3+2\left(1+6{\Delta }_2\right){\Delta }_3^2+0.4\left(1+100{\Delta }_2\right){\Delta }_3^3\right],\\ {}{m}_5=0.17{\Delta }_5\left[1+9{\Delta }_4+2\left(1+6{\Delta }_3\right){\Delta }_4^2+0.4\left(1+100{\Delta }_3\right){\Delta }_4^3\right],\\ {}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \end{array}\end{array}} $$

(8)

$$ {m}_n=0.17{\Delta }_n\left[1+9{\Delta }_{n-1}+2\left(1+6{\Delta }_{n-2}\right){\Delta }_n^2+0.4\left(1+100{\Delta }_{n-2}\right){\Delta }_{n-1}^3\right], $$

where ${\Delta }_2={a}_0{a}_2/{a}_1^2,{\Delta }_3={a}_3{a}_1/{a}_2^2,{\Delta }_4={a}_4{a}_2/{a}_3^2,{\Delta }_5={a}_5{a}_3/{a}_4^2,\dots, {\Delta }_n={a}_n{a}_{n-2}/{a}_{n-1}^2$.