Definitions

Definition 1.1 [3]:

A system is a set of elements (parts, components, objects),

• Which mutually influence each other (interaction),

• Which are subjected to external influences and affection (input) and

• Which effect to the external (output).

Annotation 1.1:

Science provides access to various definitions of the term system, e.g.:

Definition 1.2 [4]:

An object is identified as a system as long as all particular elements and all attributes with their interdependencies (to outside the system also) are seen to be components of that whole in some logical sense. Additionally, there are axioms.

Elements are e.g. modules, components, objects, fractions.

Attributes are properties, qualities, features, characteristics and also interfaces between the system and its environment. The term state characterizes the constitution of the system at any given time.

Definition 1.3 [4]:

A system is a well-defined assembly which consists of interacting entities. This assembly is delimited by a cladding plain or boundary (a set of system elements with interfaces to outside the system). The cladding plain provides for an interface between the system and its environment. The relationships between attributes and states, transported by the interfaces mentioned above, are variables which describe the peculiar characteristics of the system.

A block is a way of illustrating a system. The borderline represents the cladding plain.

Fig. 1.4: Representation of a system as a block.

Interfaces are

• r input quantities ui,

• m output quantities yi and

• q interfering quantities zi, if applicable.

Closed systems are self-contained and maintain no connections with the environment, i.e.

outside events have no influence on the system. The total state of the system is defined by n state quantities xi.

Definition 1.4:

State variables are those variables in a system, which completely describe the system’s behaviour. State variables are time-dependent. The term system state refers to the entirety of all values of the state variables.

Annotation 1.2:

• State variables are “internal variables” of a system.

• There is no definite choice of state variables. But characteristics can be assigned to the chosen state variables, such as uniqueness, independence and freedom of redundancy.

From the latter characteristic it follows that the choice of state variables is absolutely arbitrary; however, the values are fixed.

• In practice, one is not interested in the entirety of all state variables, but only chooses the variables needed for the application; the so-called output quantities. Output quantities can also result from a combination of state variables.

Examples of state variables from different disciplines; cp Table 1.1.

Table 1.1: Examples of state variables in different disciplines.

Electrotechnology Mechanics Process engineering Environmental engineering

Current Position Temperature Population

Voltage Velocity Mass fraction Environmental state

Load Acceleration CO2 production

Kinetic energy Ozone value

Potential energy Example 1.3: Motor Vehicle System

In a motor vehicle system, the variables which emanate from the driver are e.g. the steering wheel angle and the brake and acceleration pedal travels which represent the input quantities. The variables describing the motion of the vehicle are defined as the state variables. Combinations or subsets of the quantities mentioned above are output quantities and the impulses resulting from the road are identified as intervening quantities (cp. Fig.

1.5).

Fig. 1.5: System of a motor vehicle [5].

A system which describes the vehicle, for example for vehicle dynamics analysis, is shown in Fig. 1.6.

Fig. 1.6: Degrees of freedom of the spatial twin-track model [6].

Example 1.4: Framework System

Elements of the system are specific bars of the framework. Internal interactions between the bars are nodal forces. Input quantities are external loads and reaction forces in the bearings can be defined as output quantities.

Example 1.5: Road Traffic System

Specific motor vehicles are the system elements in a delimited road traffic network. The driver and his motor vehicle can represent the subsystems. Internal interactions can be comparative distances and relative velocities of the vehicles to each other.

Definition 1.5:

A system with more than one input quantity and more than one output quantity is defined as a multi-quantities system (multi-input-multi-output system “MIMO”).

System elements from different technical disciplines are listed in Table 1.2.

Table 1.2: Examples of system elements from different disciplines.

Electrotechnology Mechanics Process engineering Population development

Resistance Mass Vessel Birth

Capacitor Spring Valve Death

Coil Damper Pipe line Disease

Transistor Beam Stirrer vessel Consumption

Amplifier Bearing Filter

Filter Guidance Reactor

Dead stop Position actuator

Force actuator

The form of interaction between the system elements defines the system structure. A system can be structured into subsystems (components).

The structuring process into subsystems has the advantage that we obtain a better overview of the behaviour of the system and that the subsystems can be constructed by different people. The structuring process should be carried out in such way that the subsystems can be reintegrated into the total system. This makes the clear definition of the interfaces inevitable, which also represents one of the major problems of industrial treatment of complex dynamical systems.

The system boundary (cladding plain) is arbitrary. But in general, limitations are introduced where a clear distinction to the environment exists, with a few uniquely defined relationships that can be observed and where the system is reactionless to its environment.

Definition 1.6:

The motion of dynamical systems is a change of the variables of the system with respect to time. Mechanical motion (changes referring to place) represents a special case of all general types of motions.

Furthermore, the term motion appears in literature a number of times, [7]:

System parameters are quantities which remain constant for the system during the observation period. Examples are given by natural constants, spring constants and damping factors, resistance, etc.

Environmental influences are quantities which affect the system from the outside, but which are not influenced by the system in return (reactionless). Examples are changes in temperature in big rooms, forced movements, etc.

Initial values of the state quantities.

Rate of change of a state quantity refers to the rate at which the value of the state quantity changes in course of time. Examples are given by velocity, acceleration, rate of birth and death of a population, mass flows, pressure changes, etc. According to the first example (velocity), state variables can be both rate of change of state quantities and state quantities themselves. This becomes important in mechanics.

In mathematics, dynamical systems are described by differential equations, algebraic equations of n-th order and differential-algebraic equations (cp. Chapter 3).