Chapter 5. Internet based measurement of the servo motor

$$v_a=v_{R_a}+v_{L_a}+v_i$$

( 5.1 )

$$v_{R_a}=R_a{i}_a$$

( 5.2 )

$$v_{L_a}=L_a\frac{\text{d}{i}_a}{\text{d}t}$$

( 5.3 )

$$v_i=(k\phi )\Omega$$

( 5.4 )

$$i_a={(k\phi )}^{-1}{T}_E$$

( 5.5 )

$$J\frac{\text{d}\Omega }{\text{d}t}=T_E-T_L$$

( 5.6 )

$$J=J_M+J_L$$

( 5.7 )

$$v_a(s)=v_{R_a}(s)+v_{L_a}(s)+v_i(s)$$

(5.8)

$$v_{R_a}(s)=R_a{i}_a(s)$$

(5.9)

$$v_{L_a}(s)=L_{a}s{i}_a(s)$$

(5.10)

$$v_i(s)=(k\phi )\Omega (s)$$

(5.11)

$$i_a(s)={(k\phi )}^{-1}{T}_E(s)$$

(5.12)

$$Js\Omega (s)=T_E(s)-T_L(s)$$

(5.13)

$$J=J_M+J_L$$

(5.14)

$$W_{V_a\to \Omega }=\frac{\Omega (s)}{V_a(s)}=\frac{1}{(k\phi )}\frac{1}{1+T_{m}s+T_m{T}_e{s}^2}$$

(5.15)

$$W_{T_L\to \Omega }=\frac{\Omega (s)}{T_L(s)}=\frac{R_a}{{(k\phi )}^2}\frac{1+T_{e}s}{1+T_{m}s+T_m{T}_e{s}^2}$$

(5.16)

$$T_e=\frac{L_a}{R_a}$$

(5.17)

$$T_m=\frac{R_{a}J}{{(k\phi )}^2}$$

(5.18)

$$\frac{\text{d}}{\text{d}t}\left[\begin{array}{c}{i}_a\\ \Omega \end{array}\right]=\left[\begin{array}{cc}-\frac{R_a}{L_a}& -\frac{(k\phi )}{L{}_a}\\ \frac{(k\phi )}{J}& 0\end{array}\right]\left[\begin{array}{c}{i}_a\\ \Omega \end{array}\right]+\left[\begin{array}{cc}\frac{1}{L_a}& 0\\ 0& -\frac{1}{J}\end{array}\right]\left[\begin{array}{c}{v}_a\\ T_L\end{array}\right]$$

(5.19)

$$\dot{x}=Ax+Bu$$

(5.20)

$$\frac{\text{d}()}{\text{d}t}=0$$

(5.21)

($$T=T_E=T_L$$)

(5.22)

$$\Omega (T)=\frac{V_a}{(k\phi )}-\frac{R_a}{{(k\phi )}^2}T$$

(5.23)

$$\Omega (T)=\frac{V_a}{(k\phi )}-\frac{R_a}{{(k\phi )}^2}T$$

(5.24)

($$P=11\text{W}$$)

( 5.25 )

($$V_a=15\text{V}$$)

( 5.26 )

($$n=7950\frac{\text{1}}{\text{min}}$$)

( 5.27 )

($$I_{starting}=4.19\text{A}$$)

( 5.28 )

($$R_a=3.58\Omega$$)

( 5.29 )

($$L_a=0.33\text{mH}$$)

( 5.30 )

($$I_{continuous\max }=1.07\text{A}$$)

( 5.31 )

($$T_{E\max }=18.9\text{mNm}$$)

( 5.32 )

($$P_{\max }=14.6\text{W}$$)

( 5.33 )

($$k_s=541\frac{\text{rpm}}{\text{V}}$$)

( 5.34 )

($$k_e=k_m=k\phi =17.6\cdot {10}^{-3}\frac{\text{Vs}}{\text{rad}}$$)

( 5.35 )

($$T_m=14.5\text{ms}$$)

( 5.36 )

($$J_M=1.26\cdot {10}^{-6}{\text{kgm}}^{\text{2}}$$)

( 5.37 )

$u\left(t\right)={\text{Ω}}_{measured}\left(t\right), y\left(t\right)={\text{Ω}}_{filtered}\left(t\right)$

(5.46)

$\frac{{\text{Ω}}_{filtered}\left(s\right)}{{\text{Ω}}_{measured}\left(s\right)}=\frac{1}{1+s{T}_c}�\frac{1}{1+s{T}_c}�\frac{1}{1+s{T}_c}=\frac{1}{1+3s{T}_c+3s{T}_c^2+s{T}_c^3}$

(5.47)

$\left(1+3s{T}_c+3s^2{T}_c^2+s^3{T}_c^3\right){\text{Ω}}_{filtered}\left(s\right)={\text{Ω}}_{measured}\left(s\right)$

(5.48)

$$T_c^3{\stackrel{⃛}{\Omega }}_{filtered}^{}(t)+3T_c^2{\ddot{\Omega }}_{filtered}^{}(t)+3T_c^{}{\dot{\Omega }}_{filtered}^{}(t)+{\Omega }_{filtered}(t)={\Omega }_{measured}^{}(t)$$

( 5.49 )

$$\left(\begin{array}{c}{\dot{\Omega }}_{filtered}^{}\\ {\ddot{\Omega }}_{filtered}^{}\\ {\stackrel{⃛}{\Omega }}_{filtered}^{}\end{array}\right)=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -\frac{1}{T_c^3}& -\frac{3}{T_c^2}& -\frac{3}{T_c^{}}\end{array}\right)\left(\begin{array}{c}{\Omega }_{filtered}^{}\\ {\dot{\Omega }}_{filtered}^{}\\ {\ddot{\Omega }}_{filtered}^{}\end{array}\right)+\left(\begin{array}{c}0\\ 0\\ \frac{1}{T_c^3}\end{array}\right){\Omega }_{measured}$$

( 5.50 )

$$\left(\begin{array}{c}{\Omega }_{filtered}^k\\ {\Omega }_1^k\\ {\Omega }_2^k\end{array}\right)=\left(\begin{array}{ccc}{a}_{d11}& a_{d12}& a_{d13}\\ a_{d21}& a_{d22}& a_{d23}\\ a_{d31}& a_{d32}& a_{d33}\end{array}\right)\left(\begin{array}{c}{\Omega }_{filtered}^{k-1}\\ {\Omega }_1^{k-1}\\ {\Omega }_2^{k-1}\end{array}\right)+\left(\begin{array}{c}{b}_{d1}\\ b_{d2}\\ b_{d3}\end{array}\right){\Omega }_{measured}^k$$

(5.51)

$$A=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -\frac{1}{T_c^3}& -\frac{3}{T_c^2}& -\frac{3}{T_c^{}}\end{array}\right);B=\left(\begin{array}{c}0\\ 0\\ \frac{1}{T_c^3}\end{array}\right);$$

(5.52)

$$Ad=\left(\begin{array}{ccc}{a}_{d11}& a_{d12}& a_{d13}\\ a_{d21}& a_{d22}& a_{d23}\\ a_{d31}& a_{d32}& a_{d33}\end{array}\right);Bd=\left(\begin{array}{c}{b}_{d1}\\ b_{d2}\\ b_{d3}\end{array}\right);$$

(5.53)

${\omega }_{noise}\approx 2\text{*}\pi \text{*}\frac{1}{2\text{*}{T}_{sample}}=3141.6\left[\frac{rad}{sec}\right]$

(5.54)

$T_c=10\frac{1}{{\omega }_{zaj}}=0.0032$

(5.55)

$\frac{{\text{Ω}}_{filtered}\left(s\right)}{{\text{Ω}}_{measured}\left(s\right)}=\frac{15}{s^{3}T{c}^3+6s^{2}T{c}^2+15sTc+15}$

(5.56)

$\frac{{\text{Ω}}_{filtered}\left(s\right)}{{\text{Ω}}_{measured}\left(s\right)}=\frac{945}{s^{5}T{c}^5+15s^{4}T{c}^4+105s^{3}T{c}^3+420s^{2}T{c}^2+945sTc+945}$

(5.57)

($$P:W_C=A_P$$)

(5.58)

($$PI:W_{PI}=A_P+\frac{1}{T_{I}s}=\frac{1+T_I{A}_{P}s}{T_{I}s}=A_{PI}\frac{1+T_{PI}s}{T_{PI}s}$$)

(5.59)

($$A_{PI}=A_P,T_{PI}=T_I{A}_P$$)

(5.60)

($$PID:W_{PID}=A_P+\frac{1}{T_{I}s}+T_{D}s=\frac{1+T_I{A}_{P}s+T_I{T}_P{s}^2}{T_{I}s}=A_{PID}\frac{1+T_{PI}s+T_{PI}{T}_{PD}{s}^2}{T_{PI}s}$$)

(5.61)

($$A_{PID}=A_P,T_{PI}=T_I{A}_P,T_{PD}=\frac{T_D}{A_P}$$)

(5.62)

$$W_x\approx A_{PI}\frac{1+T_{PI}s}{T_{PI}s}\frac{1}{(1+T_{m}s)(1+T_{e}s)}$$

(5.63)

$$W_x\approx A_{PI}\frac{1}{T_{m}s(1+T_{e}s)}$$

(5.64)

$$W_x\approx A_P\frac{1}{T_{m}s(1+T_{e}s)}$$

(5.65)

$$W_c\approx \frac{A_P\frac{1}{T_{m}s(1+T_{e}s)}}{1+A_P\frac{1}{T_{m}s(1+T_{e}s)}}=\frac{A_P}{T_{m}s(1+T_{e}s)+A_P}=\frac{1}{1+\frac{T_{m}s}{A_P}(1+T_{e}s)}$$

(5.66)

$$W_{\Omega }=A_{\Omega }\frac{1+T_{\Omega }s}{T_{\Omega }s}$$

(5.67)

$$W_i=A_i\frac{1+T_{i}s}{T_{i}s}$$

(5.68)

$$\frac{\omega }{{\omega }_e}=A_{\Omega }\frac{1+T_{\Omega }s}{T_{\Omega }s}\frac{1}{1+\frac{R_a}{A_i}{T}_{e}s}(k\phi )\frac{1}{Js}=A_{\Omega }\frac{1+T_{\Omega }s}{T_{\Omega }s}\frac{1}{1+\frac{1}{A_i}{T}_{e}s}\frac{1}{(k\phi )}\frac{1}{T_{m}s}$$

(5.69)

AP

TI

TD

P control

AU/2

PI control

AU/2.2

TU/1.2

PID control

AU/1.7

TU/2

TU/8