Please note the following important tips;. Now, we will explain, steady-state error in a closed loop control system with few numerical examples:. Consider the following control system system-1 as shown in Figure It is not necessary in all cases. For example, if ramp input is applied in this system, then conditions may be different.
Various steady state values of system-1 is shown in Figure It can be viewed that the steady-state value of the error signal is 0. If the system is stable then various steady-state values can be obtained as follows:. You can calculate the value of output as follows:. Steady state value of any signal can be calculated with the above method. For examples.
Its Steady state value is. Similarly, the error signal can be calculated as:. Steady state value of error signal steady-state error is. Another method to calculate steady-state error is as follows:. If the input is unit ramp input, then.
Calculate, Velocity error coefficient K v is. If the input is unit parabolic input, than. Calculate, Acceleration error coefficient K a is. PI controller Proportional plus integral controller reduces the steady state error e ss , but have negative effect on the stability.
We can find steady state error using the final value theorem as follows. Let us discuss how to find steady state errors for unity feedback and non-unity feedback control systems one by one. Consider the following block diagram of closed loop control system, which is having unity negative feedback. The given input signal is a combination of three signals step, ramp and parabolic.
The following table shows the error constants and steady state error values for these three signals. In order to get a better view, we must zoom in on the response. We choose to zoom in between time equals In essence we are not distinguishing between the controller and the plant in our feedback system. Now we want to achieve zero steady-state error for a ramp input. From our tables, we know that a system of type 2 gives us zero steady-state error for a ramp input.
Therefore, we can get zero steady-state error by simply adding an integrator a pole at the origin. As you can see, there is initially some oscillation you may need to zoom in. However, at steady state we do have zero steady-state error as desired. Let's zoom in around seconds trust me, it doesn't reach steady state until then.
As you can see, the steady-state error is zero. Feel free to zoom in on different areas of the graph to observe how the response approaches steady state. Tutorials Contact. In this section, however, we shall not discuss errors due to imperfections in the system components. Rather, we shall investigate a type of steady-state error that is caused by the incapability of a system to follow particular types of inputs.
Steady-state error is the difference between the input and the output for a prescribed test input as time tends to infinity. Test inputs used for steady-state error analysis and design are summarized in Table 7. In order to explain how these test signals are used, let us assume a position control system, where the output position follows the input commanded position.
Step inputs represent constant position and thus are useful in determining the ability of the control system to position itself with respect to a stationary target. An antenna position control is an example of a system that can be tested for accuracy using step inputs. Ramp inputs represent constant-velocity inputs to a position control system by their linearly increasing amplitude. For example, a position control system that tracks a satellite that moves across the sky at a constant angular velocity.
Parabolas inputs , whose second derivatives are constant, represent constant acceleration inputs to position control systems and can be used to represent accelerating targets, such as a missile. Any physical control system inherently suffers steady-state error in response to certain types of inputs. A system may have no steady-state error to a step input, but the same system may exhibit nonzero steady-state error to a ramp input.
The only way we may be able to eliminate this error is to modify the system structure. Whether a given system will exhibit steady-state error for a given type of input depends on the type of open-loop transfer function of the system.
The steady-state errors of linear control systems depend on the type of the reference signal and the type of system. Before undertaking the error in steady state, it must be clarified what is the meaning of the system error. The error can be seen as a signal that should quickly be reduced to zero, if this is possible. Consider the system of Figure Where r t is the input signal, u t is the acting signal, b t is the feedback signal and y t is the output signal. The error e t of the system can be defined as:.
We must remember that r t and y t do not necessarily have the same dimensions.
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