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## Overview Controlled Systems

**Descripe and classify controlled systems**

In order to check and classify the behavior of the controlled systems, the input signal is preferably a voltage change from 0V to 1V. The step response can be used to determine parameters of the controlled system:

**Proportional Controlled systems**

Controlled systems with proportional behavior or P-systems have an end value or steady state value and are therefore also referred to as controlled systems. They can be further distinguished in P- systems with and without delay time and dead time:

**PT0-system**

A controlled system without delay exists when the controlled variable follows the manipulated variable without a measurable time delay and thus suddenly reaches its steady state. In practice, there is practically no pure P-system. Even a transistor has a delay. However, for simplicity, if the time delay is very small or barely measurable, then the reaction can be idealized.

Parameter to describe this system: **Transfer coefficient Kps **also referred as **gain**:** K _{PS} = Δx/Δy**

**P-T1-element (1**^{st}** order system)**

^{st}

The behavior of a single storage element system (1^{st} order system) can be described by an exponential function:

Parameter: **Transfer coefficient K _{s }, Time constant T**

The e-function is described by: x = Ks * y * (1 – e ^{–t/T})

The time constant T is a measure for how fast the steady-state condition can be reached. Graphically it can be determined using the slope tangent at the starting point. This tangent cuts the straight line of the maximum value. Now draw a straight line vertically to the time axis.

**The time constant is the time that the controlled value x needs to reach 63,2% of the final value (maximum, steady-state). Practically, the steady-state is reached after about 5T.**

**Multiple storage element ( P-T**_{n}**)**

_{n}

If a controlled system has **sever****al stages which store energy**, then you call it a **PTn - system**. Here you see an example: A tin bath (PT2-system) consisiting of 2 separate elements: the heating resistors and the tin itself - both elements store energy in form of heat.

How to determine the parameter via the step-response you see here:

Parameter: **T _{g} balancing time, T_{u} delay time**

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