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| == Modelo Balancín == === Continuo === {{attachment:use4651.png}} |
== Modelo Balancín con Compensador PID == === Modelo Continuo === {{attachment:planta_lc_continuo.png}} $$$ \sum{\tau_x} = \tau_2 - \tau_1 = J\frac{d^2\theta}{dt^2} $$ $$$ s^2\theta_{(s)}= \frac{\tau_{(s)}}{J} $$ $$$ G_{bal(s)} = \frac{1}{Js^2} $$ $$$ G_{torque(s)} = k_\tau $$ $$$ G_{PID(s)} = k_p\cdot(1 + \frac{1}{T_is} + T_ds}) $$ $$$ G_{LA(s)} = G_{bal(s)}G_{torque(s)}G_{PID(s)} = \frac{k_pk_\tau}{T_iJ}\frac{T_iT_ds^2 + T_is + 1}{s^3}$$ $$$ G_{LC(s)} = \frac{G_{LA(s)}}{1+G_{LA(s)}} = k_pk_\tau\frac{T_iT_ds^2 + T_is + 1}{T_iJs^3 + k_pk_\tauT_iT_ds^2 + k_pk_\tauT_is + k_pk_\tau}$$ === Modelo Discreto === {{attachment:planta_lc_discreto.png}} $$$ G_{bal(s)} = \frac{1}{Js^2} $$ $$$ G_{ROC(s)}=\frac{1-e^{-TS}}{S}$$ $$$ G_{torque(s)} = k_\tau $$ $$$ G_{PID(s)} = k_p\cdot(1 + \frac{1}{T_is} + T_ds}) $$ $$$ Z[G_{ROC(s)}G_{bal(s)}G_{torque(s)}] = Z[\frac{1 - e^{-TS}}{S}\frac{k_\tau}{JS^2}] = (1-z^-1)Z[\frac{2k_\tau}{2JS^3}] $$ $$$ Z[\frac{1 - e^{-TS}}{S} * \frac{1}{J*S^2}] = (1-z^-1) * Z[\frac{2}{2*J*S^3}] = \frac{1-z^{-1}}{2*J}*\frac{T^2*z^{-1}*(1+z^{-1})}{(1-z^{-1})^3}$$ $$$ Z[G_p, G_r] = \frac{T^2}{2*J} * \frac{z^{-1}*(1+z^{-1})}{(1-z^{-1})^{2}} $$ $$$ F(z) = G_p(z)*G_r(z)*G_c(z) = \frac{T^2}{2*J} * \frac{z^{-1}*(1+z^{-1})}{(1-z^{-1})^{2}} * Kp * [1 + \frac{1}{T_i*(1-z^{-1})} + T_d*(1-z^{-1})] $$ $$$ F(z) = \frac{T^2 * K_p}{2*J} * [\frac{z^{-2} + z^{-1}}{(1-z^{-1})^2} + \frac{z^{-2} + z^{-1}}{T_i*(1-z^{-1})^3} + \frac{T_d*z^ {-2} + T_d*z^{-1}}{(1-z^{-1})}] $$ $$$ F(z) = \frac{T^2 * K_p}{2*J} * \frac{T_i*T_d+(-T_i-T_i*T_d)*z+(1-T_i*T_d)*z^2+(1+T_i+T_i*T_d)*z^3}{z^4-3*z^3+3*z^2-z} $$ |
Tabla de Contenidos
Modelo Balancín con Compensador PID
Modelo Continuo
Modelo Discreto
