Steady State Error In Control Systems
(Step Inputs)
Why Worry About Steady State Error?
What Is Steady State Errror (SSE)?
Systems With A Single Pole At The Origin
Problems
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Why Worry About Steady State Error?

        Control systems are used to control some physical variable.  That variable may be a temperature somewhere, the attitude of an aircraft or a frequency in a communication system.  Whatever the variable, it is important to control the variable accurately.

        If you are designing a control system, how accurately the system performs is important.  If it is desired to have the variable under control take on a particular value, you will want the variable to get as close to the desired value as possible.  Certainly, you will want to measure how accurately you can control the variable.  Beyond that you will want to be able to predict how accurately you can control the variable.

        To be able to measure and predict accuracy in a control system, a standard measure of performance is widely used.  That measure of performance is steady state error - SSE - and steady state error is a concept that assumes the following:


Goals For This Lesson

        Given our statements above, it should be clear what you are about in this lesson.  Here are your goals.

  Given a linear feedback control system,
  Be able to compute the SSE for standard inputs, particularly step input signals.
  Be able to compute the gain that will produce a prescribed level of SSE in the system.
Be able to specify the SSE in a system with integral control.
        In this lesson, we will examine steady state error - SSE - in closed loop control systems.  The closed loop system we will examine is shown below.
 
What  Is SSE?

        We need a precise definition of SSE if we are going to be able to predict a value for SSE in a closed loop control system.  Next, we'll look at a closed loop system and determine precisely what is meant by SSE.

        In this lesson, we will examine steady state error - SSE - in closed loop control systems.  The closed loop system we will examine is shown below.

        In our system, we note the following:     The signal, E(s), is referred to as the error signal.         A step input is often used as a test input for several reasons.         It helps to get a feel for how things go.  So, below we'll examine a system that has a step input and a steady state error.  That system is the same block diagram we considered above.

For the example system, the controlled system - often referred to as the plant - is a first order system with a transfer function:

G(s) = Gdc/(st + 1)

We will consider a system with the following parameters.

        Here is a simulation you can run to check how this works.  In this simulation, the system being controlled (the plant) and the sensor have the parameters shwon above.  You can adjust the gain up or down by 5% using the "arrow" buttons at bottom right.  You can also enter your own gain in the text box, then click the red button to see the response for the gain you enter.
The actual open loop gain is shown in the text box above the red button.

        Vary the gain.  You can set the gain in the text box and click the red button, or you can increase or decrease the gain by 5% using the green buttons.  You should see that the system responds faster for higher gain, and that it responds with better accuracy for higher gain. Try several gains and compare results.  The difference between the desired response (1.0 is the input = desired response) and the actual steady state response is the error.


Problem 1  For a proportional gain, Kp = 9, what is the value of the steady state output?

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:


Problem

P1  For a proportional gain, Kp = 9, what is the value of the steady state error?

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:


Problem

P2  For a proportional gain, Kp = 49, what is the value of the steady state output?

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:


Problem

P3  For a proportional gain, Kp = 49, what is the value of the steady state error?

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:



        When you do the problems above, you should see that the system responds with better accuracy for higher gain.  Try several gains and compare results using the simulation.  You need to understand how the SSE depends upon gain in a situation like this.  You need to be able to do that analytically.  That's where we are heading next.  Here is our system again.

       Now, we can get a precise definition of SSE in this system.  We have the following:

We get the Steady State Error (SSE) by finding the the transform of the error and applying the final value theorem.  To get the transform of the error, we use the expression found above.

        Since E(s) = 1 / s (1 + Ks Kp G(s)) applying the final value theorem

Multiply E(s) by s, and take the indicated limit to get:

Ess = 1/[(1 + Ks Kp G(0)]

We can draw a few conclusions from this expression.

You should also note that we have done this for a unit step input.  If we have a step that has another size, we can still use this calculation to determine the error.  If the step has magnitude 2.0, then the error will be twice as large as it would have been for a unit step.  The system is linear, and everything scales.  We can take the error for a unit step as a measure of system accuracy, and we can express that accuracy as a percentage error.
Problem 5  What loop gain - Ks Kp G(0) - will produce a system with 5% SSE?

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:


Problem

P4  What loop gain - Ks Kp G(0) - will produce a system with 1% SSE?

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:



Some Observations for Systems with Integrators

        This derivation has been fairly simple, but we may have overlooked a few items.

        Reflect on the conclusion above and consider what happens as you design a system.

Problems
Links To Related Lessons

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