**Nikola Šerman,**prof. emeritus

*Linear Theory*nserman@fsb.hr

*Transfer Function Concept Digest*

**0. INTRODUCTION**

Controlling various physical variables most often relies on negative feedback concept.

Controller detects a deviation between set up and current value and acts back to the controlled process by trying to change the current value in a direction opposite to the deviation. Through this an action of a circular flow influences within loop* controller – controlled process – controller* is established. Such an arrangement is commonly referred to as *closed loop*.

Dynamic behavior of closed loop significantly differ from what the controlled process and the controller behaves individually. In situations when dynamic behavior of a closed loop is technically unacceptable, e.g. due to large oscillations, it is not easy to find a remedy. Quite often even a mighty “common sense” might be misleading.

There are plenty of fine textbooks and tutorials on the subject. The following web pages are not intended to substitute them. They contain a compressed digest of the matter. Might be used either as an introductory “first glance” or it can be a refresher to the readers already dealt with the subject.

The following content offers a good foundation for the qualitative analysis approach in controls study and design. Such an approach is most often the best available tool in troubleshooting control loops problems.

A basic assumption here is that closed loop system components behave as linear, lumped-parameters, time-invariant dynamic systems^{(1)} within certain limits and acceptable level of accuracy.

Mathematical model, describing such a system dynamics ends up in a form of *an ordinary linear differential equation with constant coefficients* or as a set of such equations.

There are two main concepts for analyzing such models:

- State Space Concept
- Transfer Function Concept

**State Space Concept** relies on a differential equations matrix formulation to represent the system dynamics. It requires a full knowledge of system structure and parameters. The analysis is performed by computer-added matrix calculations. State Space is a powerful mean for quantitative analysis of complex dynamic systems. However, it offers a little ground, if any, for troubleshooting and resolving real world control engineering problems.

**Transfer Function Concept, **although based on a bit abstract notion of the Laplace transform it presents a firm ground for qualitative analysis approach in solving feedback related control problems.

The following pages offer a brief digest of Transfer Function Concept from a closed loop dynamics point of view.

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- Brief description of linear, lumped-parameters and time-invariant dynamic systems is given in chapter 13.