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An Introduction To System Dynamics – First Order Systems Introduction Goals The System Impulse Response Problems You are at Basic Concepts – Time Response – 1st Order Step and Impulse Response

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Why Worry About Time Response Behavior?

Time behavior of a system is important. When you design a system, the time behavior may well be the most important aspect of its’ behavior. Points you might worry about include the following.

- How quickly a system responds is important. If you have a control system that’s controlling a temperature, how long it takes the temperature to reach a new steady state is important.
- Say you’re trying to control a temperature, and you want the temperature to be 200 o C. If the temperature goes to 250
**o**C before it settles out, you’ll want to know that. Control systems designers worry about overshoot and how close a system comes to instability . - If you’re trying to control speed of an automobile at 55mph and the speed keeps varying between 50mph and 60mph, your design isn’t very good. Oscillations in a system are not usually desirable .
- If you are trying to control any variable, you want to control it accurately. so you will need to be able to predict the steady state in a system.

These examples are intended to show you that the ability to predict details of how a system responds is important when you design systems. These are but a few of many different aspects of time behavior of a system that are important in control system design. The examples above really are talking about aspects like:

- Speed of response
- Relative stability of the system
- S tability of the system

When you design systems or circuits you often need to worry about these aspects of the system’s time behavior. Before you build your system, you want to know how it will perform. You need to make predictions.

In this lesson, we will begin to examine how it is possible to predict aspects of the time behavior of a system. We’ll do that by starting with a first order system and examining the parameters of that kind of system that control their time behavior. We’ll do that in ways that will let us generalize concepts to more complex systems – and there are lots of more complex systems you’ll be worrying about. With that under your belt, you will have the knowledge you need to predict how first order systems behave. That will set the stage for learning about more complex systems. Goals For This Lesson

There are a number of goals for you in this lesson.

First, if you have a first order system, you need to be able to predict and understand how it responds to an input, so you need to be able to do this. Given a first order system, Determine the impulse and step response of the system. Secondly, you may go into a lab and measure a system, and if it is first order, you need to be able to do this. Given the step response of a first order system, Determine the parameters – DC gain and time constant of the system. This second goal is considerably different from the first. In the first goal, you are given information about the system and the input to the system, and have to determine how the system responde. In the second goal, you are given information about the input and output of a system and have to determine what the system is. That’s a completely different kind of problem, but in both cases you will need to learn the material in the rest of this lesson. There is a separate lesson on system identification where you look at input and output and work to determine a model for the system. Click here to go to that lesson.

The System – And Some Examples

The simplest possible dynamic system is one which satisfies a first order, linear, differential equation. Here’s a generic form of the differential equation. A block diagram representation of the system is given at the right. **t** dx/dt + x(t) = G** _{dc}** u(t)

x(t) = Response of the System,

u(t) = Input to the System, **t** = The System Time Constant.

G** _{dc}** = The DC Gain of the System. The parameters you find in a first order system determine aspects of various kinds of responses. Whether we are talking about impulse response, step response or response to other inputs, we will still find the following relations.

**t**. the time constant, will determine how quickly the system moves toward steady state.- G
. the DC gain of the system, will determine the size of steady state response when the input settles out to a constant value._{dc}

You also need to note that a system that satisfies the differential equation above has a transfer function of the form: G1(s) = X(s)/U(s) = G** _{dc}** /(s

**t**+ 1)

Some Examples of First Order Systems

The differential equation describes many different systems of many different types. We can look at some of the systems whose behavior is described by a first order differential equation like the one above.

The parameters you find in a first order system determine aspects of various kinds of responses. Whether we are talking about impulse response, step response or response to other inputs, we will still have the following quantities and system parameters. x(t) = Response of the System,

u(t) = Input to the System, **t** = The System Time Constant.

G** _{dc}** = The DC Gain of the System. Every system will have an input which we can call u(t), and a response we will denote by x(t). Each system will also have a time constand and a DC gain.

**t**. the time constant. will determine how quickly the system moves toward steady state.- G
. the DC gain of the system, will determine the size of steady state response when the input settles out to a constant value._{dc}

Now, let’s look at some example systems. The first system is not entirely whimsical.

A Cartoon Biplane

Some Observations on First Order Systems

There are some important points to note about the step response of a first order linear system.

- When the step is applied, the derivative of the output changes immediately .
- To check this observation, move back up to the videos and note how the derivative changes when the step is applied.
- The size of the derivative change depends upon the size of the step, but as long as the step is non-zero, the derivative will have a jump.

- To get the steady state value, multiply the input step size by the DC Gain.
- If the input is not a step but if it does reach a steady state value, the output will be the DC Gain multiplied by the steady state value of the input.

That’s pretty much it for the step response of a first order system. Now that you know what it looks like it’s time to start looking at how you can use this concept.

Encountering First Order Systems

Once you know how a first order system responds to impulse and step inputs, there are several different ways you can use that information.

- If you have a first order system, with either a step or impulse input, you can compute the output response of the system. That is an analysis problem.
- If you have an unknown system, and you have input and output data, and your data set resembles an impulse input and a first order impulse response, or a step input and a first order step response, then you can use what you know to determine what the system is. That is a system identification problem.

Problems

- Problem SysDynP01 – Impulse Response
- Problem SysDynP02 – Impulse Response
- Problem SysDynP03 – Impulse Response
- Problem SysDynP04 – Step Response
- Problem SysDynP05 – Step Respons e
- Problem SysDynP06 – Step Response

You might also want to examine this problem.

- Problem IntroP02

Links to Related Lessons

- System Dynamics – First Order Linear Systems
- System Dynamics – Second Order Linear Systems
- System Dynamics – Time Constants Measurements

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