If it were possible to place a zero at EXACTLY the same location as a right half plane pole then the pole would be cancelled. However, exact placement is not achievable practically. The result is a pole and zero very close together on the positive real axis of the s-plane. Consequently, the root locus will have a branch on the positive real axis, going from the pole to the zero, which will give an unstable closed-loop. The zero, essentially, traps the pole in the right half plane.
If, instead, the zero were placed at an appropriate position on the real NEGATIVE axis, the root locus would have the capacity of moving the unstable pole into the left half plane and hence provide a stable closed loop.
Consider the very simple case of an open-loop with a single pole at s=1, i.e. an OLTF:
G(s) = 1/(s-1)
Placing a zero at, say, s=0.9 would give an OLTF:
G(s)=(s-0.9)/(s-1) and a CLTF = (s-0.9)/(2s-1.9), which is unstable.
However, placing a zero at s=-2 would give an OLTF:
G(s)=(s+2)/(s-1) and a CLTF = (s+2)/(2s+1) which has a stable closed-loop pole; ie, the unstable open loop pole at s=1 has been dragged into the left-hand plane by the zero, and now lies at s=-0.5
In a former life, I designed and built automated test stands for engineered fluid handling components and systems. PID control is the way to go, I always started with a manual method and tweaked later and of course we did plenty of system testing to validate before turning the thing loose.
I'd stick with one active PID loop at a time, though depending in the state of your system, you might switch between them. Say one for warmup and another for operation.
I think you're going to have a hard time with the system as it's set up now, I have some questions you might consider.
- Is there a reason the fluid circuit is open loop?
- could you recirculate?
- could you have a bypass loop and when ready to run, simply switch a valve?
Best Answer
A micro-PLC (programmable logic controller) is ideal for this. They are very flexible, come in a variety of voltages (24 V DC, 110/230 V AC), relay or transistor outputs, etc., are industrially hardened and user programmable. Many allow writing of the program from a keypad / LCD on the unit or via PC software.
Figure 1. A typical DIN-rail mount micro-PLC.
Your program can be written in Ladder Logic - a diagramatic representation of an electrical circuit. In your case it might look like this.
Notes:
Examine the options carefully before you buy and try the demo software. Once you get started on this you will find many more applications so you should choose a brand and device that will meet your future needs.