Transfer function technique VS State space technique

Question

What do you prefer best? Transfer functions or state space?

I know there is a lot of question who get the anser "None of them are best. They complementing each other".

But in control theory, I cannot find any way that a transfer function solves the problem better that a state space model.

If I list all the adventages and disadventages for transfer functions:

1. Adventage: Easy to use for small systems.
2. Adventage: Easy to estimate from data.
3. Adventage: Good for analysing frequency response.
4. Disadventage: Not for complex system.
5. Disadventage: Only for SISO systems
6. Disadventage: Cannot combine kalmanfilter with PID

List all the adventages and disadventages for state space models:

1. Adventage: Perfect for complex systems and small systems.
2. Adventage: For both SISO and MIMO systems.
3. Adventage: Can use kalmanfilters
4. Adventage: Can use PID
5. Adventage: Written on first grade ODE's - Perfect for analysis
6. Adventage: Working in time plane
7. Disadventage: Requires numerical simulation - Can be complex

Answer 1

Signals and disturbances are more often than not characterized in the frequency domain. Therefore, the frequency domain approach is often more powerful for robustness and disturbance analysis.

Some might also claim it helps with spelling ;-)

Answer 2

One of the main advantages of transfer functions is that they simplify analysis significantly. You can basically just multiply transfer functions that are in series and use very easy formulas for feedback loops. So you can perform calculations with transfer functions as if they were variables.

Take for example the standard disturbance observer structure. With transfer functions its almost trivial to derive how it achieves nominal tracking:

With $G$ (model), $G_N$ (nominal model), $u$ input, $y$ output, $d$ disturbance and $Q$ a filter we get

$$ y = \frac{GG_N(d + u - Q d)}{G_N + Q(G - G_N)} $$

so if $Q \rightarrow 1$ we get

$$ y = G_Nu $$

thus perfect nominal model tracking and disturbance rejection.

Very simple to derive with transfer functions. Try the same using state space and you will start to like transfer functions ;)

Answer 3

4. Disadventage: Not for complex system

5. Disadventage: Only for SISO systems

Both not true.

You can also have high order transfer functions.

In general you never go higher then second order. But you can split up a fourth order controller in two second order filters cascaded.

Adventage: Can use kalmanfilters

Why should that be an advantage? Kalman filters are highly overrated since they all have one very big flaw. You namely make the assumption that your disturbances are distributed according to a Gaussian distribution.

Frequency domain techniques are way more powerful then any state-space and any optimization techniques.

1. Optimization is never possible, because you will never ever have a model which is 100 percent accurate. Therefor you will never be able to optimize the problem.
2. When creating products which rely on control systems variation occurs. For one system the eigenfrequency is at 120 [Hz], for the other system it is at 110 [Hz]. Using frequency domain techniques you can design controllers which are robust for in both cases.
3. Controllers based on frequency domain are easily to understand.
4. Controllers based on transfer functions are easily implemented
5. Controllers based on transfer functions are very efficient (very few calculations needed, so few multiplications, additions, et cetera)
6. Controllers based on frequency domain techniques are easily debugged. Try to find out why a LQR based controller does not damp a certain frequency and how you can make sure that it does.
7. State-space techniques loose their physical interpretation.
8. ... many more.