is the integral time and derivative time (Ts and Td) defined by the
clock period of the circuit that perform these computations?
The integral time and the derivative time are related to any associated strobes used around those block.
Digital electronics take a finite time to perform the required operations. Take a trapezoid discrete integrator who's difference equation looks like this:
\$y_n = y_{n-1} + K_i*T_s*\frac{x_n + x_{n-1}}{2} \$
There is two additions, two multiplications and one division (although a cheap shift right division) for one integration step. This must be completed before the next sample \$x_n\$ arrives. Now to save on a multiplication that \$K_i * T_s\$ could be combined into one scalar, which is standard practice & this results on the gain variable being related to the sample time \$T_s\$
What sets your \$T_s\$ ? . Those multiplications and additions take a finite time & they MUST be completed by the time the next sample is available \$x_n\$
if my error signal is sampled by the same clock that computes the
integral and derivative parts of my law, Ti=Td=Ts?
Yes it would be, if Ts is the data acquisition or interpolator rate, of one is used)
If Ti=Td=Ts, Kp = Ki = Kd? It sounds really odd.
No because Ts is only part of the discrete integral equation as there is then the gain factor. As previously mentioned the gain factor AND the sample period can be pre-calculated to save on a multiplication step
--edit--
To help clarify. Below is a piece of python code that implements a discrete integrator (choice of 3, but Trap more often than not is the one you want). The Sample time, \$T_s\$ here is 1us (as well as 100us to prove a point). In this instance \$T_i = T_s = 1us (or 100us)\$. Likewise the "clock" of the processor, the python virtual machine clock speed, for simplicity, can be considered to be 2.9GHz. This is a reasonable example for a FPGA/uP running at a higher speed and the ADC's are sampling at a slower rate BUT a rate that is slow enough for the main processor to complete its computational steps.
#!/usr/bin/env python
import numpy as np
from matplotlib import pyplot as plt
def fwdEuler(X=0, dt=0):
'''y(n) = y(n-1) + [t(n)-t(n-1)]*x(n-1)'''
x[0] = X*Ki # apply gain prior to integration
y[0] = self.y[-1] + dt*x[-1]
x[-1] = float(x[0])
y[-1] = float(y[0])
return y[0]
def revEuler(X=0, dt=0):
'''y(n) = y(n-1) + [t(n)-t(n-1)]*x(n)'''
x[0] = X*Ki # apply gain prior to integration
y[0] = y[-1] + dt*x[0]
x[-1] = float(x[0])
y[-1] = float(y[0])
return y[0]
def Trap(X=0, dt=0):
'''y(n) = y(n-1) + K*[t(n)-t(n-1)]*[x(n)+x(n-1)]/2'''
x[0] = X*Ki # apply gain prior to integration
y[0] = y[-1] + dt*(x[0]+x[-1])/2
x[-1] = float(x[0])
y[-1] = float(y[0])
return y[0]
plt.hold(True)
plt.grid(True)
f,p = plt.subplots(4)
x = [0,0]
y = [0,0]
Ki = 1
t = np.arange(0,1,1e-6) # 1million samples at 1us step
data = np.ones(t.size)*1 #
p[0].plot(t,data)
p[0].set_title('Simple straight line to integrate')
int_1 = np.zeros(t.size)
for i in range(t.size):
int_1[i] = Trap(data[i],t[1]-t[0])
p[1].set_title('Trap integration with Ki=1 and Ti=1u')
p[1].plot(t,int_1)
x = [0,0]
y = [0,0]
Ki = 1
t = np.arange(0,1,100e-6) # 1million samples at 1us step
data = np.ones(t.size)*1 #
int_2 = np.zeros(t.size)
for i in range(t.size):
int_2[i] = Trap(data[i],t[1]-t[0])
p[2].set_title('Trap integration with Ki=1 and Ti=100u')
p[2].plot(t,int_2)
x = [0,0]
y = [0,0]
Ki = 2
t = np.arange(0,1,1e-6) # 1million samples at 1us step
data = np.ones(t.size)*1 #
int_3 = np.zeros(t.size)
for i in range(t.size):
int_3[i] = Trap(data[i],t[1]-t[0])
p[3].set_title('Trap integration with Ki=2 and Ti=1u')
p[3].plot(t,int_3)
plt.show()
As you can see, for a valid integral difference algorith, the actual integral gain is unity and is thus can be ... tuned, via an additional gain, \$K_i\$. This is equally true for differential terms.
Best Answer
The real problem seems to be your mechanical setup, not the PID control detail. PID control may not even be what you want at all.
I'd start by putting the container of yogurt culture inside a water bath, then attempt to regulate the water bath temperature. The extra thermal mass of the water will lengthen the dominant time constant, and won't depend on what is going on inside the container much.
With a longer time constant, simple thermostat on/off temperature control should work well enough. You want the heater power to be low enough so that the water bath only changes a little, like maybe 2-4 degrees, within one time constant. Simple threshold-detecting on/off control will then work very well.
Instead of adding hysteresis to prevent rapid oscillations at the setpoint, just hold the output fixed for a short time, like 1/50 of the time constant or so.
A long time ago, I made a temperature controlled water bath for photographic film processing. The mechanical setup was one of those plastic "Rubbermade" style tubs, a fish tank pump to keep the water circulating, and a 300 W off the shelf immersion heater meant for a coffee cup. Two thermistors in series placed on opposite sides of the tub provided the temperature feedback. This was before the age of microcontrollers, so a flip-flop sampled and held the on/off signal every 64 line cycles. That controlled a relay, which switched the immersion heater on and off. It worked really well. Once it got to regulation, the temperature stayed fixed to a fraction of a degree F.
You are over-thinking this with a PID controller.