**Analog ****con****trollers**

Examples of analog controls: speed control, positioning control, torque control, voltage control, etc. What do all these controls have in common? They control a fast reacting system!

Let us explain the function of a PID controller in this video:

__Analyze the control behavior__

Principle: A voltage jump of ideally 1V is applied to the input of the controller. The parameters of the controller are determined by means of the step response. We regard the P, I and D shares on their own:

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**P-controller** or **proportional controller**

A P-controller has an output signal u(t), that is proportional to the control error (u ~ e) without delay. As soon as the control error returns to the value 0, this controller does not have an output signal u any more.

Determine the parameter ** proportional gain factor K_{p}**:

**K**

_{p }= Δ u / ΔeAlthough a P-controller reacts immediately, a pure P-control has a permanent control deviation!

**I-Controller** or **Integral-controller**

The integral part is used for optimization because it can reduce a control deviation to zero. As you can see, the Integral force increases steadily as long as there is still a control deviation. When the control deviation is zero, the I-force remains at its constant value and reduces again only by a control deviation with opposite sign.

Determine of the parameters for the I-force:

Since the output signal of the I-force constantly changes during a control deviation, you would relate the slope of the I-force to the input quantity e:

*Integral gain* K_{I : }K_{I }= (Δ u / Δt) / Δ e = (u – u_{0}) / (Δt * Δe)

The unit of this integral gain K_{I} is 1/s. Because this parameter is hard to understand for the user you would use the so-called Integral time, which represents the reciprocal:

**Integral time T**_{I }= 1 / K_{I } = (Δ t * Δe ) / Δu *(without regarding the sign)*

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To tell it in a few words: T_{I} is the time until the output equals the level of the input signal.

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**D-controller **or** Differential controller**

The differential-part also serves for optimization. With its force, the controller reacts to a change of a control deviation very quickly. It also dampens the oscillation of the control.

The D controller reacts to a change in the control deviation! Problem: No parameters can be derived from the step response of a D controller. Trick: As input signal we use a ramp:

**Derivative gain K _{D}**

K_{D }= Δ__ u _ _ __ = Δ__ u * Δ____t__

. Δ e / Δt Δe

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**Exercise: Step response of a PI-Controller **

Empirically, you have set the PID controller as best as possible with the following parameters: K_{P} = 1.5, T_{i} = 0.002 s

A voltage jump of 1 volt is applied to the input of the controller. Complete the step response of the P- and I-force as well as the combined PI controller on the solution sheet.