##Borrar esta linea y dejar la siguiente que permite que ésta página sea pública
#format inline_latex
#acl BecariosGrupo:read,write,admin,delete All:read
<<TableOfContents(6)>>

== 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}} 

$$$ G_{bal(s)} = \frac{1}{Js^2} $$

$$$ G_{ROC(s)}=\frac{1-e^{-TS}}{S}$$

$$$ G_{torque(s)} = k_\tau $$

$$$ G_{PID(s)} = k_p\cdot(1 + \frac{1}{T_is} + T_ds}) $$

$$$ G_{planta(Z)} = Z[G_{ROC(s)}G_{bal(s)}G_{torque(s)}] = Z[\frac{1 - e^{-TS}}{S}\frac{k_\tau}{JS^2}] = (1-z^-1)Z[\frac{2k_\tau}{2JS^3}] = \frac{T^2k_\tau}{2J}\frac{z+1}{z^2-2z+1}$$

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

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

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