Steady State Error in control system
Table of Contents
What is Steady State Error?
As time approaches infinity (i.e. when the response of the control system has reached steady-state), steady-state error is defined as the discrepancy between the desired value and the actual value of a system output. A characteristic of the input/output response for a linear system is steady-state error. A good control system will typically have a small steady-state error.
By examining a first-order transfer function’s steady state response, we will first talk about its steady-state error. Let us think about the following transfer function:
This first order transfer function is straightforward, with a gain of 1 and a time constant of 0.7 seconds. Because the denominator’s s has the highest power of 1, it is important to note that this function is referred to as a first-order transfer function. If it were instead 0.7s^2+1, it would be a second order transfer function instead.
Figure 1 illustrates how this transfer function responds to an input with a steady-state value. The output is exactly equal to the input in steady-state, as can be seen. It follows that the steady-state error is zero.
Figure 2 displays this function’s response to a unit ramp input. It is clear that there is a difference between input and output in steady-state. As a result, there is a steady-state error for a unit ramp input.
Note that a first-order transfer function’s steady-state error against ramp input is equal to the time constant in many control system textbooks. Figure 2 above demonstrates that this is the case. The input is 3 and the output is 2.3 at time t=3. As a result, the steady-state error for this first-order transfer function is 0.7, which is equal to the time constant.
Please note the following important tips:
- A ramp input typically has a lower steady-state error than a step input, which is even lower when the input is parabolic. The steady-state error is zero against step input, 0.7 against ramp input, and it can be determined that it is ∞ against parabolic input, just as in the explanation above.
- It should be noted that stability does not depend on input, whereas steady-state error does.
Let’s consider a closed loop control system having transfer function
where symbols are used as intended. The system’s stability is dependent on the denominator, or “1+G(s)H(s)”. Characteristics equation is defined as “1+G(s)H(s) = 0”. The stability of the system is indicated by its roots. Depends on R(s) for steady-state error.
In a closed loop control system the error signal can be calculated as
Steady state error can be found as
where steady-state error is the value of the error signal in steady state. From this we can see that the steady-state error depends on R(s).
- As was already mentioned, the stability is determined by the denominator, which is 1 + G(s)H(s). Since ‘1’ in this equation is constant, the stability depends on G(s)H(s), the variable part of the equation. So, you can understand the Bode plot, Nyquist plot is drawn with the help of G(s)H(s), but they indicate the stability of C(s)/R(s) .
- G(s)H(s) is called an open-loop transfer function and C(s)/R(s) is called a closed-loop transfer function. The stability of a closed-loop transfer function can be determined using the Bode plot and Nyquist plot by analyzing the open-loop transfer function, or G(s)H(s).