Why Worry About Steady State Error?

What Is Steady State Errror (SSE)?

Systems With A Single Pole At The Origin

Problems

        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:

• The system under test is stimulated with some standard input.  Typically, the test input is a step function of time, but it can also be a ramp or other polynomial kinds of inputs.
• The system comes to a steady state, and the difference between the input and the output is measured.
• The difference between the input - the desired response - and the output - the actual response is referred to as the error.

        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.

• The system to be controlled has a transfer function G(s).
• There is a sensor with a transfer function K_s.
• There is a controller with a transfer function K_p(s) - which may be a constant gain.

        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.

• The system to be controlled has a transfer function G(s).
• There is a sensor with a transfer function K_s.
• There is a controller with a transfer function K_p(s).
• A controller like this, where the control effort to the plant is proportional to the error, is called a proportional controller.

• The input is often the desired output.  In other words, the input is what we want the output to be.  If the input is a step, then we want the output to settle out to that value.
• The output is measured with a sensor.  Often the gain of the sensor is one. That is especially true in computer controlled systems where the output value - an analog signal - is converted into a digital representation, and the processing - to generate the error, E(s) - takes place inside a computer.

• The error signal is the difference between the desired input and the measured input.
• The error signal is a measure of how well the system is performing at any instant.  When the error signal is large, the measured output does not match the desired output very well.

• A step input is really a request for the output to change to a new, constant value.
• If the system is well behaved, the output will settle out to a constant, steady state value.
• The difference between the measured constant output and the input constitutes a steady state error, or SSE.

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

We will consider a system with the following parameters.

• G_dc = 1
• t = 1
• K_s = 1.
• K_p  can be set to various values in the range of 0 to 10,
• The input is always 1.

        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.

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

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

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

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

• The input is assumed to be a unit step.  If the input is a step, but not a unit step, the system is linear and all results will be proportional.
• We can calculate the output, Y(s), in terms of the input, U(s) and we can determine the error, E(s).  Later we will interpret relations in the frequency (s) domain in terms of time domain behavior.
• We have:
• E(s) = U(s) - K_s Y(s) since the error is the difference between the desired response, U(s),
• The measured response, =  K_s Y(s).
• And we know:  Y(s) = K_p G(s) E(s).
• Combine our two relations:
• E(s) = U(s) - K_s Y(s) and:
• Y(s) = K_p G(s) E(s), to get:
• E(s) = U(s) - K_s  K_p G(s) E(s)
• Since E(s) = U(s) - K_s  K_p G(s) E(s) we can combine all the E(s) terms on the left side of this equation:
• E(s) [1 + K_s K_p G(s)] = U(s)
• or E(s) = U(s) / [1 + K_s K_p G(s)]
• Now, we know that:
• E(s) = U(s) / [1 + K_s K_p G(s)]
• This relation is in the frequency - s - domain, but has implications for time response behavior.
• Assume a unit step input.  The transformed input, U(s), will then be given by:
• U(s) = 1/s
• With U(s) = 1/s, the transform of the error signal is given by:
• E(s) = 1 / s [1 + Ks Kp G(s)]
• The final value theorem can be applied to get the steady state error.

        Since E(s) = 1 / s (1 + K_s K_p G(s)) applying the final value theorem

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

We can draw a few conclusions from this expression.

• The steady state error depends upon the loop gain - K_s K_p G(0).
• The term, G(0), in the loop gain is the DC gain of the plant.
• To make SSE smaller, increase the loop gain.
• And, the only gain you can normally adjust is the gain of the proportional controller, K_p.

• If the response to a unit step is 0.9 and the error is 0.1, then the system is said to have a 10% SSE.

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

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

• If the system has an integrator - as it would with an integral controller - then G(0) would be infinite.  That would imply that there would be zero SSE for a step input.
• The pole at the origin can be either in the plant - the system being controlled - or it can also be in the controller - something we haven't considered until now.  However, it should be clear that the same analysis applies, and that it doesn't matter where the pole at the origin occurs physically, and all that matters is that there is a pole at the origin in the controller or the plant.

• You may have a requirement that the system exhibit very small SSE.
• You can get SSE of zero if there is a pole at the origin.
• If there is no pole at the origin, then add one in the controller.  You will have reinvented integral control, but that's OK because there is no patent on integral control.  Click here to learn more about integral control.
• If you want to add an integrator, you may need to review op-amp integrators or learn something about digital integration.  Those are the two common ways of implementing integral control.  You can click here to see how to implement integral control.

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