# Systems theory (engineering sciences)

The system theory is an interdisciplinary discipline of engineering , especially in the field of electrical engineering , with applications in the areas such as intelligence and high-frequency engineering and control engineering . Systems theory deals with the mathematical description and calculation of physical systems on an abstract level. Such physical systems can be filters or a control loop , for example .

## General

The most important concepts in systems theory are the signal and the system .

To describe it, the physical excitation and reaction of the system is reduced by its physical units and the system is expressed as mathematical functions of independent variables of time or place. The system's excitations are called the input signal and the system's reactions are called the output signal. The system is described abstractly in the context of a mathematical model and defined by an operator who maps the input signals to the output signals , as shown in the adjacent block diagram. It thus establishes a relationship between the input and the output signal, which in linear time-invariant systems is also described by the transfer function. ${\ displaystyle T (\ cdot)}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ## System classification

In systems theory, the different systems are classified according to various criteria, such as

• the definition and value ranges of the input and output signals,
• whether the systems are discrete or continuous,
• whether the system is linear or non-linear , i.e. whether the operator is a linear operator or not,${\ displaystyle T}$ • causal or not causal,
• deterministic or stochastic , d. H. whether the system reacts predictably or randomly,
• time-invariant time-variant or or shift invariant or shift variant
• concentrated or spatially distributed, d. H. whether the system description requires partial differential equations and derivatives according to the "location"

Many systems that are important in engineering can be described as a so-called dynamic system , a system whose time-dependent processes depend on the initial state but not on the absolute starting time.

Some important classes of systems are shown below.

### Discrete-time and continuous systems

Discrete-time systems are characterized by the fact that internal states are only defined at individual points in time and discrete-time signals occur at the inputs and outputs . They play an important role in information technology and digital signal processing and are described in the form of sequences . The modeling takes place with the help of difference equations .

Continuous systems are characterized by a continuous course of their states, are described in the form of smooth functions and differential equations and play a role in the modeling of physical systems. An example of continuous systems are electrical lines within the framework of line theory.

Combinations of time-discrete and continuous systems are called hybrid systems.

### Linear and time-invariant systems

Linear time-invariant systems, abbreviated to LZI systems or LTI systems (Linear Time Invariant), play an important role in technology such as control technology or communications technology , because they are simple and often sufficient. Continuous LZI systems are accessible with the mathematical means of the Fourier transformation and Laplace transformation . In the case of discrete systems, the discrete Fourier transformation and the Z transformation are used accordingly .

Linear systems with concentrated memories are particularly simple - these are described in the time domain by linear ordinary differential equations or differential equations with constant coefficients. The Laplace transformation or z-transformation allows the description and the closed analytical representation of the transfer function in the form of a rational function as the usual form of representation.

### State space representation

Dynamic systems that cannot be described as LTI systems can be modeled using the state space representation, among other things. Here are differential equations of order n in a system of n-coupled state first order differential equations transferred and all relations of the state variables, of the input variables and output variables in the form of matrices and vectors shown.

The state space representation is considered a method of analysis and synthesis of dynamic systems in the time domain and is particularly efficient in the control-technical treatment of multi-variable systems, non-linear and time-variable transmission systems.

### Causal systems

All physically feasible systems are causal systems, which means that the output value of a system only depends on the current and past input values, but not on future input values. In clear terms, an effect occurs at the earliest at the time of the cause, but not earlier.

In the field of modeling there are acausal systems in which this cause-effect principle is broken; this simplifies u. U. the consideration of the system. It is also not a necessary prerequisite for some problems, especially in digital signal processing . If the inaccuracies that arise during the technical implementation can be tolerated, the causality can be neglected. An example of an acausal system is the ideal low-pass filter , which in practice can only be implemented approximately as a causal system in the form of low-pass filters , or the Hilbert transformation .

Mathematically, a system that is described by a transfer function is called causal if its output values ​​only depend on the current and past input values. The step response of a causal system disappears for negative times; assuming linearity , this means that an effect A (t) and its cause B (t) must be related as follows:

${\ displaystyle A (t) = \ int _ {t '= - \ infty} ^ {t} \, X_ {A; B} (t-t') B (t ') \, {\ rm {d} } t '\ ,.}$ The function is also referred to as the influence function; it represents the step response. Your Fourier transform , the frequency spectrum, contains all the information about the system behavior. It is known as generalized susceptibility ; it is only well-defined for a positive imaginary part of . This corresponds to the assumption that for negative t vanishes. ${\ displaystyle \, X_ {A, B} (t)}$ ${\ displaystyle \ textstyle \ chi _ {A; B} (\ omega) = \ int _ {- \ infty} ^ {\ infty} X_ {A; B} (t) \, \ exp (-i \ omega t ) \, {\ rm {d}} t}$ ${\ displaystyle \ omega}$ ${\ displaystyle \, X_ {A; B} (t)}$ A system in which the output values ​​only depend on the current and future input values ​​is called anti-causal . The impulse response disappears for positive times.

## literature

• Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to system theory, signals and systems in electrical engineering and information technology . 4th edition. Teubner-Verlag, 2007, ISBN 978-3-8351-0176-0 .
• Thomas Frey, Martin Bossert: Signal and System Theory . 2nd Edition. Vieweg-Teubner, 2008, ISBN 978-3-8351-0249-1 .
• Rolf Unbehauen : System Theory, Vol. 1: General Basics, Signals and Linear Systems in the Time and Frequency Domain . 8th edition. Oldenbourg, 2002, ISBN 3-486-25999-7 .
• Rolf Unbehauen: System Theory, Vol. 2: Multidimensional, Adaptive and Nonlinear Systems . 7th edition. Oldenbourg, 1998, ISBN 3-486-24023-4 .
• Martin Werner: Signals and Systems, textbook and workbook . 3. Edition. Vieweg-Teubner, 2008, ISBN 978-3-8348-0233-0 .