# Transfer function technique VS State space technique

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
• you forgot to mention the most important thing , i.e transfer function is evaluated without considering any initial conditions or for a initially relaxed system , which is a big disadvantage , and it can be overcomed using SS – zeno-san Jul 4 '17 at 17:58
• Yes. I see no use for transfer funtions anymore. I have totaly changed to SS from TF. – Daniel Mårtensson Jul 4 '17 at 18:06
• I don't really understand most of your points... "Not for complex systems" - Why not? You can also have high order transfer functions. "Only for SISO systems" - You can use transfer function matrices "Can use PID" - You can use PID also with transfer functions. "Requires numerical simulation" - ??? Transfer functions are also simulated using numeric methods (except maybe for the most simple cases) "Easy to estimate from data" - How is that easier for transfer functions than in state space? – SampleTime Jul 4 '17 at 20:14
• If TF was so good that it can handledare conplex systems, why was SS invented then? – Daniel Mårtensson Jul 5 '17 at 5:36
• Please see my answer. It's not about more information, it's about simplifying analysis. It is by the way still unclear what you mean with "basic" and/or "complex" system. Please provide a definition. – SampleTime Jul 5 '17 at 18:09

## 3 Answers

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 ;-)

• Give me some working areas about TF who winns over SS? – Daniel Mårtensson Jul 5 '17 at 16:33
• Noise filtering. And spelling ;-) – Pait Jul 6 '17 at 13:23

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 ;)

• But multiplying TF will not give you serial system dynamics, just another new system. – Daniel Mårtensson Jul 5 '17 at 18:59
• Of course it does. If you have two TF in series you can just multiply them and that gives you the serial dynamics. These are basic properties of transfer functions. – SampleTime Jul 5 '17 at 19:28
• Well, OK. I think it is better to describe the whole system in just one big matrix. Not several transfer functions. You can view the states to in SS without using Simulink or numerical simulation. – Daniel Mårtensson Jul 5 '17 at 21:47
• And why do you think so? Did you try out the example I gave? And what do you mean with "viewing the states"? Of course you have to perform a numerical simulation with SS if you want to see the systems state trajectories... Which representation is "better" depends on what you are trying to achieve. – SampleTime Jul 6 '17 at 18:27
1. Disadventage: Not for complex system

2. 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.
• Have you ever heard about LQG/LTR ? That's LQG with robust control. LQG/LTR is not optimal control. – Daniel Mårtensson Jul 9 '17 at 15:19
• Yes, but those techniques rely on state-space representation and furthermore are not intuitive in their working, e.g., to complex. When you are working in a production environment things need to be simple. For instance, customer support needs to be able to help a customer when the controller does not work and he needs to retune it for instance. – WG- Jul 9 '17 at 15:24
• Yes. I know that most of the time, a PID controller are used in the industry. But I'm working with R&D. But I agree with you that transfer functions is better when it comes to knowing the amplitude for every frequency. I'm going to learn LQG/LTR, but if you have better suggestions for robustness for MIMO systems (e.g LQR/LQG), you can convince me to learn H2 and H-infinity control. I would like to learn that too, but I don't know which one is the best for me. I want optimal, and both robust. LQG is optimal + filtering. LTR is robust. Seems OK for me...perhaps. – Daniel Mårtensson Jul 9 '17 at 15:53
• Which techniques I would suggest depends on the problem you are trying to solve. However, we only discussed feedback controllers (FB)... In general I would advice to put a lot of effort in obtaining a very good model of your plant dynamics. This you'll use to create a feedforward controller (FF). To give a comparison, the FB delivers like 1% of the control output and the FF 99%. The FB goal is to minimize the impact of distrubances, the FF will give you performance. Several aspects are important in a good controller: 1) model, 2) FB, 3) FF, 4) feedthroughs, 5) setpoint design. – WG- Jul 10 '17 at 14:18
• The comparison, "To give a comparison, the FB delivers like 1% of the control output and the FF 99%" is not competly valid because that totally depends. But it is valid for the machines I am working on. We design machines which make chips on high volume levels and have accelerations up to 50 m/s^2. All our machines have (sub)nanometre accuracy. We do a lot of research, also to LQG, Kalman, Feedforward, Hinfinity. But in the end we always impement controllers with simple filters and PID's. At one time we even fitted an Hinfinity designed controller onto a PID controller :'D – WG- Jul 10 '17 at 14:33