Error Analysis 2.5New

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In the previous tutorial, we understood several time response specifications and how they are crucial in the design of control systems. Error in a system is also one such crucial parameter and needs a deeper look. We already have been introduced to this term in the tutorial on first order systems and in this tutorial, we shall march ahead to explore more on this.

In a control system, we are mostly concerned with the error that exists when the system appears to be settled. Hence, we shall restrict ourselves to the error at steady state. Here again, you can take the example of a door damper. If the door closes perfectly, the error is zero. But, if a small opening exists for a long time, then this is what bothers us and this is the steady state error that exists in the system. For this, we first need to know the final value to which a system settles when it is excited by a certain input.

Formally defining, the final value of the system is that value which is obtained as time approaches infinity.

Let us look at a few basic responses. Try this – For which of these responses does a final value exist?

System Basic Responses

For the response A, the final value that the system attains is clearly 0. For the response B, the final value attained by the system is 1. What about the response C? It keeps oscillating and never settles, and a final value does not exist. In the case of the response D, the response keeps increasing forever and again we cannot define the final value in this case either. So, the basic conclusion here is – for a system response to have a final value, it must settle. We can also synonymously tell that for a system to have a final value, the system must be fully stable (all poles in the left half of the s-plane). You can ignore the latter statement for now as we haven’t really dealt with stability yet but it will be important to know eventually.

Now, to find the final value of given system (which settles or is stable), we make use of the final value theorem, according to which –

Before we proceed further, we shall learn what does “type of a system” mean.

If you can recall from the transfer function tutorial, the open loop transfer function in its pole zero form looks something like

Here z1, z2, are the zeros and p1, p2 are the poles. The term sn represents the number of poles at origin. And,

Type of the system = number of poles at origin = n

This means that if there is one pole at origin, then it’s a type 1 system, if there are 2, then it’s a type 2 system and if there are none, then it’s a type 0 system

Steady State Error (ess)

We’ve had a glimpse of what steady state error is in the beginning of this tutorial. Now, we shall go about it mathematically.

The steady state error is the difference between the desired output (input) and the actual output of the system during steady state (as time approaches infinity).

The illustrations below will make it clear:

Steady State Error 1
Steady State Error 2

Consider a basic closed loop system as shown:

Basic Closed Loop System Block Diagram

It is clearly seen from the block diagram above that,

Also,

Hence,

According to the definition of the steady state error,

By using the final value theorem,

For a unity feedback system where H(s) = 1

You may wonder why there should be any steady state error in the first place. The steady state error exists due to the non-linear behaviour of certain elements in our system like an inductor or a motor which we linearize (assume to be linear) and use it while modelling.

Now, we shall see the steady state error of systems subjected to standard test inputs.

  • Unit Step Input
  • In this case,
    Hence, the steady state error becomes,
    Here, Kp is called the position error constant.
  • Unit Ramp Input
  • In this case,
    Hence, the steady state error becomes,
    Here, Kv is called the velocity error constant.
  • Unit Parabolic Input
  • In this case,
    Hence, the steady state error becomes,
    Here, Ka is called the acceleration error constant.

Kp, Kv and Ka are called the static error coefficients and their names – position, velocity and acceleration respectively are due to the fact that a step input represents a constant position, a ramp input represents a constant velocity and a parabolic input represents a constant acceleration. These error constants describe the ability of the system to minimize the steady state error. The values of these constants depend on the type of the system which we shall see soon.

Let’s have a second look at the value of these constants.

We shall conclude a few points here before we move ahead.

  1. Steady state error is a criterion to judge the system accuracy. Ideally, the system should match the reference input at steady state, and this means that for practical systems, steady state error should be as low as possible and hence is one of the measures of the performance of the system.
  2. The steady state error depends on two main factors – the type of input R(s) and the type of the system G(s).
  3. Steady state error makes sense only for closed loop, stable systems.

We shall now see the steady state error for different systems one by one.

  • Type – 0 system
  • As an example, consider
    Now,
    Hence for a type-0 system, position error is constant while velocity and acceleration errors are infinite at steady state.
  • Type – 1 system
  • As an example, consider
      Now,
      Hence for a type-1 system, position error is zero while velocity is constant and acceleration error is infinite at steady state.
    • Type – 2 system
    • As an example, consider
      Now,
      Hence for a type-2 system, position and velocity errors are zero while acceleration error is infinite at steady state.

    We can summarize these results with the table below,

    Steady State Error System Types

    As we can notice, when the type of the system is increased for a specific input, the steady state error reduces. This is an important observation to be made.

    We can also consider their error constants to be the specifications during the design of control systems as these constants itself speak a lot of information regarding the system. Don’t believe me? Check this –

    Suppose say a given system has Kv = 100.

    First, since Kv is defined and this implies steady state error exists and the system is stable. Next, as Kv is constant, it means that the system is of type 1. Next, we can tell that the input being applied is a ramp. Finally, we can tell the error between the input ramp and the output ramp is 1/Kv = 0.01 at a steady state.

    To summarize, in this tutorial, we started with a brief introduction to steady state error. Then, we moved to the final value of the responses and checked when the final value exists for a response. We then had a glimpse of the final value theorem and then we discussed the type of the system. We next learned more about steady state error and the steady state error for different standard test inputs by introducing the static error coefficients and finally, we used these error constants to obtain steady state error for different types of systems for standard test inputs.

    With this, we come to the end of the second chapter - Time response analysis as a part of the control systems tutorial series. In the next tutorial, we shall start with the new chapter on stability of control systems.

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