In engineering, it’s recommended to use **models of systems**, instead of real hardware, in order to perform an analysis of the system or to design a control system. The usage of models instead of real hardware has several advantages:

- lower cost
- possibility to test extreme scenarios
- re-usability

In our approach we consider that a system is a **mathematical model** which describes the behavior of a particular real (hardware) system.

A graphical representation of a system is in the image below.

We consider a system being made up from a **plant** and a **sensor**. The plant is the **mathematical description** of the behavior of the system, the sensor is a **measuring device** that outputs the observable states of the plant.

Generally speaking, we can summarize the behavior of the system and its interactions with the surroundings with the following variables/signals:

Signal | Type | Properties |

u | input | – external signal – can be measured – set by the user |

w | disturbance | – external signal – probably can be measured – can not be manipulated – depends on external factors, environment – has impact on the behaviour of the system |

x | state | – for a dynamic system, is determined by the mathematical model (differential equation) of the system; it also depends on the previous input values and disturbances of the system – for a static system is determined by an algebraic relationship between input, disturbance and state |

v | disturbance | – external signal – can not be manipulated – is usually the noise introduced by the sensor |

y | output | – is the signal/variable/state measured by the sensor – is measured |

For a better understanding, let’s consider our system as being a **vehicle driving on a road at constant speed**. The speed is controller by the driver (not by an automatic system) through the position of the accelerator pedal (which translates into engine torque).

The system is this case is the vehicle. It contains the **plant**, represented by the differential equations of the vehicles longitudinal dynamics, and the wheel speed **sensor**. The vehicle speed is directly linked to the wheel speed. What we measure is the wheel speed but the driver is aware of the vehicle speed.

When the driver wants to accelerate or decelerate the vehicle, it will manipulate the **input (engine torque)** through the accelerator pedal. Higher values of the engine torque will increase the vehicle speed while lower values of the engine torque will decrease the vehicle speed.

The **disturbances** can be various. The **road gradient** is on type of disturbance. Imagine that you are driving on a flat road (0% gradient), for a given constant input engine torque, the vehicle will have a constant speed. If a hill comes up (e.g. 10% gradient) for the same input torque, the vehicle speed will decrease, because of the higher road load. In this case, the driver, in order to maintain the same speed, has to **compensate for the disturbance** and press more on the accelerator pedal, which will result in a **higher input engine torque**. Another type of resistance is the air drag loss, which is directly linked to the intensity and direction of the wind.

The **state** of the plant is the wheel speed which is calculated based on the differential equations of the longitudinal vehicle dynamics.

The **sensor** measures the wheel speed and provides the information to the driver. Being an electrical signal (voltage), the wheel speed sensor is subject to electromagnetic interference (disturbances) which can modify the measured value.

In systems theory there are **three basic problems** that can be distinguished: **simulation**, **control** and **identification**.

When we have the system description (governing laws) and we set the inputs as we need, we are dealing with a **simulation problem**. This is also called the **direct problem**.

When we know the system description and also what outputs we expect from the system, we are dealing with a **control problem**. The solution of this problem is the design of a control system which is positioning the outputs at the desired values. The control problem is also called the **inverse problem**.

When we know the input of the system and the expected output but we don’t know the structure of the system and/or its parameters, we are dealing with a **system identification problem**. If the structure of the system is know but not the parameters, we are dealing with a **state estimation problem**.

The system identification is the problem around which the whole **system identification theory** was developed.

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