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I am currently taking a course in control theory.

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In control theory the system or impulse response is usually an algebraic expression in Laplace domain. Much of the work done in school is to find this transfer function or shift the pole of this transfer function around to make the system stable.

But this all feels a bit unrealistic. How so? Suppose I wanted to find the transfer function of my cell phone, how would I ever go about doing this? How do I know that the input-output relationship would be characterized by a simple differential equation? And how would I ever go about "shifting" the poles in real life so to achieve stable or some desired operation?

Can someone provide me with a realistic example of how a real device can be characterized using control theory (i.e. so that the system transfer function is known) and what it physically means to design the controller or shift the poles. 100 points for most real life example. Thanks!

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  • \$\begingroup\$ You are trying to model so many systems if you think about just the cell phone... when each component is designed this is done, but not the whole phone. \$\endgroup\$ – Jeff Wurz Mar 16 '15 at 19:28
  • \$\begingroup\$ Your model is terrible. How do you find the transfer function of a human life? You need to start with a system where the inputs and outputs are known, then you can do math on it. \$\endgroup\$ – Samuel Mar 16 '15 at 19:30
  • \$\begingroup\$ The "most real life example" is only valid until a better one comes up and this means there is no best answer to this and answers will tend to become out of date. Now how about you think of a decent proposal for a system - maybe a motor that turns to a preset position following a demand from a potentiometer? \$\endgroup\$ – Andy aka Mar 16 '15 at 19:45
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    \$\begingroup\$ Orbital mechanics and spacecraft control are a good example. The system is most of the time well defined (gravity!) and the disturbance forces small. Actually, it was one of or the very first application of control theory. \$\endgroup\$ – pserra Mar 16 '15 at 19:59
  • \$\begingroup\$ You need to determine the relationship(s) between the input(s) and output(s) of your cell phone and express these in the form output/input. If the input is text and the output is the same text delayed by T secs then the transfer function is a pure time delay, e^-sT. Same procedure with any system, but determining the I/O relationship is not always so simple. \$\endgroup\$ – Chu Mar 16 '15 at 23:27
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I see two elements to what you're asking here :

a) We have no idea about the system transfer function of the plant. How could we find it?

b) We know the strucure of the plant. How do we determine the parameters?

(the plant being the thing you're trying to control).

The difference between a) and b) is that for b) we know the model or can derive the model from the circuit or system, but for a) we do not.

So, a) needs a system model that we can then find the parameters of. For a) we understand that all linear systems can be modelled as MA (Moving Average, Zeros only), or AR (Auto-regressive, poles only). Yes, an MA system can be approximated by and AR and vice versa. So a very common model to fit all linear systems is an ARMAX model which incorporate AR, MA and an eXogenous input (i.e. disturbance, offset etc.).

Now we have an appropriate model that brings us to b). How to find the parameters. That can be done using system identification.

See the diagram below (source). Once you've chosen the appropriate model type, then an adaptive system like this can ID the parameters of that model. The idea is that the adaptive model adjusts so that it matches the unknown system, driving e to zero.

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Now if you want to go further and use this in a control system; this is an adaptive controller. Basically a system ID block and a controller designer. This Model Identification Adaptive Controller is very typical (source).

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In real life it is common to use offline (i.e. on your PC) sys ID using an ARMAX model to identify an unkown plant. Then use pole-placement techniques to design the controller. You can apply this to any linear system.

In my experience, it's far more common to derive the model of a system (e.g. a Buck Converter) and use that for compensation.

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Can someone provide me with a realistic example of how a real device can be characterized using control theory (i.e. so that the system transfer function is known) and what it physically means to design the controller or shift the poles. 100 points for most real life example.

Some typical examples "of a real device" (from the field of electronics) which are designed/optimized using the rules of control theory are:

1.) X-Y plotter: In order to make the plotter reaction fast and exact (without too much overshoot) control theory shows that we should use a controller with a PD-T1 characteristic.

2.) Automatic Gain Control (AGC): Detailed investigations of the relevant control loop show that it is necessary to use an amplifier that is "linear in dB". This means: The gain must NOT linearly depend on the control voltage but it should follow an exponential law. Why? Because only in this case the LOOP GAIN (relevant for closed-loop behaviour) is independent on the varying input voltage.

3.) Phase-locked loop (PLL): The PLL is a highly non-linear system which is designed under some specific conditions which allow applying the rules of the linear control theory. This applies, in particular, to the loop filter which primarily determines the dynamic behaviour of the closed loop - also unde non-linear conditions.

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