What Exactly are Systems?

Math & Math | Introducing Systems Engineering

image created by author

This part of engineering fundamentals is going to introduce you to Systems Engineering. I’m going to explain the basic concepts of how to describe a system in the time domain using examples to make things clearer. This story is not about modeling.

1. Abstract
2. The basic elements: P, I and D
3. Variations of basic elements
4. Summary

Abstract

To be very abstract, a system is something that gets an input and has an output. The numbers of inputs and outputs can vary so we will differentiate between a Single Input Single Output System (SISO-System) and a Multiple-Inputs Multiple-Output Systems (MIMO-System). The input is processed by the system. There are many different types of systems:

• mechanical
• electrical
• hydraulic
• biological
• chemical

Figure 1: SISO System (image by author)

Why is a system even interesting to describe in a theoretical manner? Because if we can describe it, we can approximate outputs for unknown inputs. It is important to differentiate between an exact calculation and an approximation here. There are two reasons why we approximate the behaviour of a system instead of calculating it:

1. We can’t describe a real-world system in all of its details, there are just too many
2. We can’t solve the mathematical equations describing the system analytically

Approximations these days are generated by computers through simulations where the mathematical equations are solved numerically. To describe a system, we need mathematics. Mathematic is the general tool used in Systems Engineering. I’m going to cover the most basic elements in System Engineering and their mathematical description in the following.

The basics: P, I and D¹

In systems engineering, everything can be broken down into 3 basic elements or any combination of them. The most simple element is the P-element. The P stands for “Proportional”. The output is directly proportional to the input, you can imagine it as an amplifier. A simple example for this is a spring. The deflection is proportional to the applied force, the stiffness of the spring is the proportionality factor.

In systems engineering the input is indicated with u and the output with y. It is a common technique to express the flow of a signal (from input to output) through a block diagram. The block element of a P-element can be seen in Fig. 1.

Figure 1: block of a P-element¹

Fig. 2 shows the block element of an IT1-element. I will go into more detail on variations of these basic block elements later. The right block in Fig. 2 shows an I-element block. While the output of a P-element is directly proportional to its input, the output of an I-element is proportional to the integral of the input.

The input is constantly added to the output, e.g. you can imagine an empty glass that you start filling with water. The water level inside the glass is increasing constantly, it is the integral of the volume flow of the water divided by the cross-sectional area of the glass. This is an example of an I-system.

Figure 2: block of an IT1-element¹

If the output of a system is proportional to the derivative of the input, it is a D-system. A D-element block is shown in Fig. 3. If we have a constant input, then our output is zero.

A common example of a D-element in electronics is the induced voltage which is proportional to the derivative of the magnetic flux. However, if neither the magnetic flux density nor the cross-sectional area of our system is changing we won’t measure any induced voltage.

Figure 3: block of a D-element¹

Fig. 4 sums up the mathematical formulations of our three basic elements:

1. P-element
2. I-element
3. D-element

Figure 4: basic equations (image by author)

Variations of basic elements¹

Real-world systems mostly incorporate a T-element in one of the basic elements to reflect a time delay . There are exceptions, but most systems will not respond to an input in an infinitely short amount of time. The delay is another parameter defining a system. Fig. 5 shows the block element of a PT1-system.

Figure 5: block of a PT1-element¹

A common example of a PT1-system is a first-order RC low-pass filter which is used in electronics. In Fig. 6 you can see the mathematical formulations of each basic system. Equation (4) shows a PT1-system, (5) an IT1-system and (6) a DT1-system. These basic systems have one thing in common: All of them are ordinary first-order differential equations. In Systems Engineering we can express a system via an ODE. If you are interested in how this is done exactly, have a look at the Author’s note at the end of the story.

Figure 6: basic first-order systems equations (image by author)

Last we take a look at one of the most common general formulations of a real-world system, the PT2-system shown in Fig. 7. In contrast to the PT1-system, this one can oscillate. The PT2-system is defined by its natural angular frequency and lehr’s damping measure. If D>1 the system won’t oscillate, it is aperiodic. Also, the PT2-system is described through a second-order ODE.

Figure 7: equation of a PT2 system (image by author)

Summary

This story introduced you to the basic concept of systems in the time domain. Basic elements like the P, I and D element were introduced and described mathematically. Also, some common variations of these basic elements like the PT1 and the PT2 system are covered in this story.

Author’s note

Thank you for reading this far! If you have any thoughts, annotations, or ideas about this story feel free to leave a comment. If you are interested in how to model systems in detail, have a look at my other stories and parts of engineering fundamentals.

A Beginner’s Guide to Modeling Technical Systems

References (S.200f.)

1. Lunze, J. (2016). Regelungstechnik 1: Systemtheoretische Grundlagen, Analyse und Entwurf einschleifiger Regelungen (11th edition). Berlin: Springer-Verlag Berlin Heidelberg. Retrieved on 7th of March 2021 from https://link.springer.com/book/10.1007/978-3-662-52678-1