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$$$ c(t) = k[e(t) + \frac{1}{T_i}\int_0^t \! e(t) dt \ + T_d \frac{de(t)}{dt} ] $$

$$$ C(s) = k( 1 + \frac{1}{T_i s} + T_d s ) E(s) $$

$$$ C(z) = k[1-\frac{T}{2T_i} + \frac{T}{T_i}\frac{1}{1-z^{-1}} + \frac{T_d}{T}(1-z^{-1}) ]E(z) $$$$$ C(z) = [K_p + \frac{K_i}{1-z^{-1}} + K_d(1-z^{-1}) ]E(z) $$

$$$ c(k) = c(k-1) + K_p[e(k)-e(k-1)] + K_i e(k) + K_d[e(k)-2e(k-1)+e(k-2)] $$

planta_lc_continuo.png $$$ \sum{\tau_x} = \tau_2 - \tau_1 = J\frac{d^2\theta}{dt^2} $$$$$ s^2\theta_{(s)}= \frac{\tau_{(s)}}{J} $$$$$ G_{bal(s)} = \frac{1}{Js^2} $$$$$ G_{torque(s)} = k_\tau $$$$$ G_{PID(s)} = k_p\cdot(1 + \frac{1}{T_is} + T_ds}) $$$$$ G_{LA(s)} = G_{bal(s)}G_{torque(s)}G_{PID(s)} = \frac{k_pk_{\tau}}{T_iJ}\frac{T_iT_ds^2 + T_is + 1}{s^3} $$$$$ G_{LC(s)} = \frac{G_{LA(s)}}{1+G_{LA(s)}} = k_p k_{\tau} \frac{T_i T_d s^2 + T_i s + 1}{T_i J s^3 + k_p k_{\tau} T_i T_d s^2 + k_p k_{\tau} T_i s + k_p k_{\tau}}$$

Simulacion de la planta a lazo abierto con el PID. PID_LA_continuo.m Respuesta en el tiempo de la planta a lazo cerrado con el PID. PID_LC_continuo.m

planta_lc_discreto.png $$$ G_{bal(s)} = \frac{1}{Js^2} $$$$$ G_{ROC(s)}=\frac{1-e^{-TS}}{S}$$$$$ G_{torque(s)} = k_\tau $$$$$ G_{PID(s)} = k_p\cdot(1 + \frac{1}{T_is} + T_ds}) $$$$$ G_{planta(Z)} = Z[G_{ROC(s)}G_{bal(s)}G_{torque(s)}] = Z[\frac{1 - e^{-TS}}{S}\frac{k_\tau}{JS^2}] = (1-z^-1)Z[\frac{2k_\tau}{2JS^3}] = \frac{T^2k_\tau}{2J}\frac{z+1}{z^2-2z+1}$$$$$ G_{PID(Z)} = K_P + \frac{K_I}{1-z^-1} + K_D(1-z^-1) = \frac{(K_P + K_I + K_D)z^2 + ( -2K_D - K_P )z + K_D }{ z^2 - z } $$$$$ G_{LA(Z)} = G_{PID(Z)}G_{planta(Z)} = \frac{T^2k_\tau}{2J}\frac{(K_P + K_I + K_D)z^3 + (K_I - K_D)z^2 + (-K_P - K_D)z + K_D}{z^4-3z^3+3z^2-z} $$$$$ G_{LC(Z)} = \frac{G_{LA(Z)}}{1+G_{LA(Z)}} = \frac{k_{\tau}T^2((K_P + K_I + K_D)z^3 + (K_I - K_D)z^2 + (-K_P - K_D)z + K_D)}{ 2Jz^4 + ((K_P + K_I + K_D)k_{\tau}T^2 -6J )z^3 + ((K_I - K_D)k_{\tau}T^2 +6J )z^2 + ( ( -K_P - K_D ) k_{\tau} T^2 -2J )z + K_D k_{\tau} T^2 } $$

Simulacion de la planta a lazo abierto con el PID. PID_LA_discreto.m Respuesta en el tiempo de la planta a lazo cerrado con el PID. PID_LC_discreto.m