Difference Between Integral Time and Ki(Integral Gain)

It sounds like you have some things there that makes it hard to find a ready device that can do the work (but I'm not sure).

It sounds like you have to either write a PID yourself, or use a PID in some library (like the Arduino).

I am unsure about the stability of the Arduino solutions, will this solution still be stable after a year? Or do you need to reboot it every day? But maybe it is good enough? (Test it for a month a see what it can do)

If you decide to write a PID yourself, the paper called PID-without-a-PhD is a good start:

• http://www.eetimes.com/design/embedded/4211211/PID-without-a-PhD or
• http://www.eetimes.com/ContentEETimes/Documents/Embedded.com/2000/f-wescot.pdf

And you also have application notes on this topic from more or less every MCU maker out there.

The figure below shows the steps in order to find the \$K_{cr}\$(or \$K_u\$) and \$P_{cr}\$ (or \$P_u)\$, by changing the proportional gain only (with \$T_d=0\$ and \$T_i=\infty\$) - an example for temperature control:

The time unit to be used should be consistent with its response curve. The relationship with the sample period \$\Delta t\$ can be obtained, after the discretization of the PID controller. Using the standard form, in contrast with other implementations (for example, when the derivative term is taken from output):

$$u(t)=K_pe(t)+\frac{K_p}{T_i}\int_0^t{e(\tau)d\tau+K_pT_d\frac{de(t)}{dt}}$$

Taking the derivative of \$u(t)\$: $$u'(t)=K_pe'(t)+\frac{K_p}{T_i}e(t)+K_pT_de''(t)$$ A possible approach: To approximate the first and second derivatives using finite differences (eg backward), where \$k\$ is the sample id: $$x'(t)\approx\frac{x_k-x_{k-1}}{\Delta t}$$ $$x''(t)\approx\frac{x_k-2x_{k-1}+x_{k-2}}{\Delta t^2}$$ So, the discrete PID controller takes the form (velocity algorithm): $$u_k=u_{k-1}+K_p[(1+\frac{\Delta t}{T_i}+\frac{T_d}{\Delta t})e_k+(-1-\frac{2T_d}{\Delta t})e_{k-1}+\frac{T_d}{\Delta t}e_{k-2}]$$

Alternative definitions can include \$K_i=\frac{K_p}{T_i}\$ and \$K_d=K_pT_d\$. Also, the derivative term can be modified in order to reduce issues with high frequency noise - eg a low pass filter. Other discretization methods exist as well, such as Tustin, ZOH.

FURTHER EXPLANATION:

1. Choose a sample time \$\Delta t\$ consistent with your process. There is extensive literature on the subject. For example, to avoid aliasing \$F_S > 2F_{BW}\$. In practice: Sampling frequency should be 10 to 30 times the bandwidth freq.

2. Set \$K_p\$ to some low value (with \$T_i=\infty\$ and \$T_d=0\$ at this stage). So, the above equation is simplified to: $$u_k=u_{k-1}+K_p(e_k-e_{k-1})$$

3. Implement the previous equation (a P controller) in your digital system along with that suitable \$\Delta t\$, testing \$K_p\$ to see if it causes continuum oscillation (marginally stable). If the oscillations decay, keep increasing \$K_p\$. If the oscillations increase in amplitude (unstable system), reduce \$K_p\$. Do this until the system is marginally stable. When you arrive at this point, you have found \$K_{cr}=K_p\$ and \$P_{cr}=\$oscillation period (see the figure above).

4. Using the table you have provided (also above), determine the \$K_p\$, \$T_i\$ and \$T_d\$ values from the \$K_{cr}\$ and \$P_{cr}\$ ones.

5. Implement the complete PID controller: $$u_k=u_{k-1}+K_p[(1+\frac{\Delta t}{T_i}+\frac{T_d}{\Delta t})e_k+(-1-\frac{2T_d}{\Delta t})e_{k-1}+\frac{T_d}{\Delta t}e_{k-2}]$$