LabElectronica/ProyectoQuadricoptero/QA3Fase1EstModYConArqRobMoviles/Balancin

Tabla de Contenidos

Modelo Balancín con Compensador PID
1. Modelo Continuo
2. Modelo Discreto

Modelo Balancín con Compensador PID

Modelo Continuo

$$ \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

$$ 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})$

$$ G_{planta(Z)} = 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}] = \frac{T^2k_\tau}{2J}\frac{z+1}{z^2-2z+1}$

$$ G_{PID(Z)} = K_P + \frac{K_I}{1-z^-1} + K_D(1-z^-1) = \frac{(K_P + K_I + K_D)z^2 + ( -2K_D - K_P )z + K_D }{ z^2 - z }$

$$ G_{LA(Z)} = G_{PID(Z)}G_{planta(Z)} = \frac{T^2k_\tau}{2J}\frac{(K_P + K_I + K_D)z^3 + (K_I - K_D)z^2 + (-K_P - K_D)z + K_D}{z^4-3z^3+3z^2-z}$

$$ G_{LC(Z)} = \frac{G_{LA(Z)}}{1+G_{LA(Z)}} = k_\tauT^2\frac{(K_P + K_I + K_D)z^3 + (K_I - K_D)z^2 + (-K_P - K_D)z + K_D}{ 2Jz^4 + ((K_P + K_I + K_D)k_\tauT^2 -6 )z^3 + ((K_I - K_D)k_\tauT^2 +6 )z^2 }$

+ ((-K_P - K_D)k_\tauT^{2 -2)z + K_Dk_\tauT}2} $$

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Modelo Balancín con Compensador PID

Modelo Continuo

Modelo Discreto