# 2. Controlled systems

# Controlled systems

For a control loop to function well, the properties of the controlled system must be known. We differentiate between static parameters such as gain and operating point and dynamic parameters such as rise time and frequency response. The controlled system can be defined in the frequency range or the time range. Mathematical techniques for analyzing and designing controllled systems fall into two different categories: **frequency domain and time domain. **

For example, in the **frequency domain** we know the **Bode plot**, in which the amplitude gain and phase shift between input and output are plotted over the frequency (frequency response). The best known description in the ** time domain** is the **step response**, where controlled variable x is plotted over time.

There is a wide range of other representations (such as the Nvyquist diagram, pulse response etc.), but we will not describe them in detail in this brief introduction.

The acquisition and evaluation of a step response are described below. The acquisition of a step response is easy to perform; usually the already existing possibilities of the control system can be used.

## 2.1 Parameters of the controlled system and step response

In this experiment, the unknown response of the controlled system to a known **manipulated variable step (step input)** is plotted.

The controlled variable is measured at the output of the controlled system. In the time domain its representation is called **step response**. The characteristics of the **step response reflect the dynamic and static behaviour **in the time domain of the controlled system. Depending on the characteristics of the step response, two basic types of control loop are differentiated: controlled systems with or without self-regulation.

Acquisition of the step response

## 2.2 Controlled systems with self-regulation

In controlled systems** with** self-regulation, the controlled variable x reaches a steady-state final value after a step change in the manipulated variable has been applied. In the steady state, the controlled system shows proportional behaviour with a certain gain.

Step response of a controlled system **with** self-regulation

In controlled systems** **with self-regulation, a **steady state** is established after a certain time. The controlled system output reaches the final value **x**_{S} of the controlled variable **x**.

This time response can be used to determine the **parameters of the controlled system **.

**proportional gain K**_{S}**dead time T**_{t}**dwell time T**_{u}**compensating time T**_{g}

Parameters of the controlled system with self-regulation

The **proportional gain K**_{S} reflects the static transmission behaviour.

The **dead time T**_{t} specifies the delay with which the controlled variable x responds to the manipulating variable step.

The **dwell time T**_{u} records the influence of a higher order of the controlled system.

The **compensating time T**_{g } is a measure of the inertia of the controlled system.

Dwell time **T**_{u} and compensating time ** T**_{g} are determined by drawing a tangent (inflection tangent) at the point of the greatest slope of the step response (inflection point). The distance from the first measurable response of the control loop to the tangent / zero line intersection specifies the dwell time **T**_{u}. The distance from the tangent / zero line intersection to the tangent / stationary final value intersection is the compensating time **T**_{g}.

## 2.3 Controlled systems without self-regulation

If the controlled system output does not move towards a stationary final value but towards infinity after a step input, we refer to this type as a **controlled system without self-regulation**. It shows integral behaviour.

For controlled systems without self-regulation, we distinguish two cases:

The steady-state final value of the controlled variable lies outside the permitted limits and strives towards infinity. In this case, the controlled system shows

**integral control action**. Even with very low manipulating variables y the controlled variable x will ultimately be infinite. For level control, this behaviour can be achieved by closing the drain valve.The steady-state final value of the controlled variable lies outside the permitted limits but has a finite magnitude. In this case, the control loop has

**PT**_{1}**behaviour**with a high proportional gain**K**_{S}. For level control, this behaviour can be achieved by opening the drain valve in an appropriately dosed manner.

To determine the** parameters **of a controlled systems without self-regulation, the slopes of the step response are determined at two different times with **Δ****t** in between. Assuming PT_{1} behaviour:

Step response of a controlled system **without** self-regulation with** integral control action**

Step response of a controlled system **without** self-regulation with **PT**_{1}** ****behaviour**

For two measurements, we obtain the relationship:

This gives us the **compensating time**:

The **proportional gain** is calculated as:

Determination of substitute parameters for a controlled systems without self-regulation

## 2.4 Characteristic diagram and operating point

By investigating the steady-state behaviour of a controlled system with self-regulation, we can create the characteristic diagram. For a constant manipulating variable y the steady-state controlled variable x is measured and plotted in a diagram. The **slope of the characteristic curve **gives the proportional gain **K**_{S} .

In practice, the relationship is usually non-linear, so that the proportional gain **K**_{S} is not constant for different manipulated variables y. If the disturbance variable z can also be set, this characteristic curve is recorded for different values of the disturbance variable z, thus obtaining the characteristic diagram of the controlled system.

Characteristic diagram of a controlled system with different disturbance variables z

Knowledge of the proportional gain **K**_{S } is important for the subsequent dimensioning of the controller. As the controller can be of a different kind depending on the operating point, strictly speaking the optimum setting for the controller is only valid for a very narrow controlled variable range. If the controller needs to have a stable function over a broad controlled variable range, it must be designed for the range with the highest proportional gain. In other ranges, the control loop then responds more slowly.

If, on the other hand, the control is designed for a range with a low proportional gain, the control loop can become extremely unstable in other ranges.

To determine the parameters of a control loop, the manipulating variable step used to determine the step response should not be too large should be within the subsequent operating point range.

Non-linear characteristic curve with different proportional gains

This video covers a few interesting things about the step response. It explains what a step response is and some of the ways it can be used to specify design requirements for closed loop control systems. It also shows why design requirements like rise time, overshoot, settling time, and steady state error are popular and how they are related to natural frequency and damping ratio for a second order system with no finite zeros.