Mathematical Models of Systems

As we mentioned in the previous section the first step in designing a control system is to understand the original system you have at hand. You have to first analyze your original system to see how it behaves without a control system, and then be able to design the correct control system to augment the original system with.

In order to analyze your overall original system (plant) you basically need equations which describe the mechanics of the different parts. The equations for the mechanics of the different parts come from the physics behind them. As we previously said the parts can be either mechanical or electrical or a combination of both. Therefore, using knowledge we have from physics, like mass-spring relations, circuit analysis, motor dynamics, etc we can derive mathematical equations for each part of our system.

We then combine all these equations into a single equation which describes how the system acts from the input to the output. Basically, this equation will be a differential equation and will relate the input to the output.

This differential equation is the center of all our future analyses. We will be using it to examine the different outputs of the plant (without a control system) to different inputs we give it. We can then determine its current performance parameters of stability, transient response, and steady-state response. By knowing this information we will be able to design the correct control system.

In the next two sections we will be talking about different types of inputs and outputs of a system.

Feedback Systems

Feedback systems are comprised of several parts. In this section we will introduce the different components of an overall feedback system and explain what each of them do:

The two main components of a feedback system are the original system and the compensator.

The original system, is also called the plant.

The central component of the control system is the compensator. This is the brain of the control system. Given a plant, our overall goal is to design a compensator, to command the plant to perform as we desire.

The reference input is the desired output we want. For example, a certain position. Sometimes the system cannot understand the reference input. Therefore, it needs to be “translated”. For example, a rover robot’s computer does not understand what a distance of 5m means. Therefore, this reference input is converted to a certain voltage, which the computer is able to understand. This “conversion” is done through an input transducer. The same is true for the output signal; the actual output sensed is “converted” to a system-readable format through an output transducer. It is important to note that the scale of conversion is not always unity; in some cases 5m means 5v. This is called unity feedback. However, in some cases we have non-unity feedback; in these cases there exists a certain scaling factor (gain) between the actual signal and the system-readable signal. It is important to note which case we have, as the analysis for each is different.

The actual output is the actual behavior of the plant.

The feedback path is the actual output signal being fed back to the system.

The summing junction subtracts the actual output (which was fed back) from the reference input (desired output).

The result is called the error.

The error is fed into the compensator. The compensator then generates and commands a control signal which is based on the error. This control signal then drives the plant.

This overall process is repeated until the plant performs our desired output.

Now that we have become familiar with a control system’s components (and reasons for having a control system) we would like to see how we can design one?! The first step in designing a control system is understanding your plant dynamics. This means you have to have mathematical models of your system’s parts. We will talk about this important first step in the next section.

Introduction to Control Systems

In the previous section we talked about systems which are constructed to accomplish tasks.

However, not always does a system behave like we expect it to do; the actual output is not the desired output (the steady-response is not what we want it to be). Reasons for this can be intrinsic system mechanics, disturbances, or computation delays. Even in systems which give the desired steady-state output, the transient response of the system might not be as fast as we would like it to be. And in some cases, the system might not be stable at all. Given these different scenarios (individually or a combination of them), we are inspired to design control systems and augment our systems with them. The control system will make sure the system performs what it is expected of it.

In general, there are two types of control systems: open-loop and closed-loop.

Open-loop control systems are very simple. Given a reference input, they only command a certain “non-changing” control input to the system and do not make any corrections if the system experiences any disturbances.

Closed-loop control systems use feedback to make sure the system is performing according to specifications. A closed-loop control system, senses the actual output, and through feedback, compares it to the reference input (desired output). According to the difference, which is called the error, the control system commands a corresponding control input to the system, in order to decrease the error. The control system repeats this process in real-time until it drives the error to zero.

In the next section we will dissect a system along with its closed-loop control system, to examine their different components.

Introduction to Systems

When you have multiple parts joined together to perform a certain task you have a system. In different systems, these parts can be purely mechanical, purely electrical, or they can be a combination of both. For example, a bicycle is a purely mechanical system; a calculator is a purely electrical system; modern day cars are a combination of electrical and mechanical parts and form an electro-mechanical system. Regardless of the type, all these systems are governed by the physics and mechanics of their parts. The combination of the unique features of these parts allow you to create systems which are able to accomplish tasks. Some examples of parts are: masses, dampers, springs, inductors, capacitors and resistors.

In all systems there are two common things which stand out. All of them have an input and an output. You turn a knob on your home’s heater to a certain temperature setting, as the input, and you get a certain temperature from your heater as the output. You give a rover robot a certain voltage (which represents a distance to be traveled), as the input, and it traverses a distance as the output. Basically, the input is your desired output. We will talk about inputs and outputs later in depth to see how they affect the design of control systems.

In addition, there are three parameters which define the performance of the system. The transient response, steady-state response, and stability.

Stability is the ability of the system to be stable when given a certain input. If you give a rover robot a voltage input and it accelerate infinitely, then the robot is unstable. However, if the robot moves to position x=5 and stops there, then the robot is said to be stable. Stability is very important in the analysis of systems and in the design of control systems.

The transient response of the system is the output of the system while it is transitioning to its final state. When you tell an airplane to pitch down 10 degrees, it can pitch down fast, pitch down slow, pitch down with wild oscillations, or pitch down smoothly, until it reaches the 10 degree mark. What goes on during this period is known as the transient response of the system. When we design a system, our goal is for it to operate smoothly as possibly. Therefore transient response analysis is a key step in designing control systems.

The steady-state response of the system is the output of the system when it reaches its final state. Basically, what remains after all the transients have decayed and the system has halted, is the steady-state response. Analyzing a system for its steady-state response is very important, as we want to make sure the system actually accomplishes its goal; when given a command to a rover robot to go to position x=5, we want to make sure it doesn’t go and stop at x=3.

In this introductory section we have introduced the concept of systems and their performance parameters. In the next section we will introduce the concept of control systems and why we need to design them.