# Steady State Error and Error Coefficients

**Introduction**

Steady-state error is defined as the difference between the input (command) and the output of a system in the limit as time goes to infinity (i.e. when the response has reached steady state). The steady-state error will depend on the type of input (step, ramp, etc.) as well as the system type (0, I, or II).

We can calculate the steady-state error for this system from either the open- or closed-loop transfer function using the Final Value Theorem.

- Step Input (
*R*(*s*) = 1 /*s*) - Ramp Input (
*R*(*s*) = 1 /*s*^2) - Parabolic Input (
*R*(*s*) = 1 /*s*^3)

**Static Error Coefficients**

If you refer back to the equations for calculating steady-state errors for unity feedback systems, you will find that we have defined certain constants (known as the static error constants). These constants are the position constant (*Kp*), the velocity constant (*Kv*), and the acceleration constant (*Ka*). Knowing the value of these constants, as well as the system type, we can predict if our system is going to have a finite steady-state error.

Follow the given steps to determine the static error coefficients and steady state error of a given control system.

1. Open MATLAB.

2. Define the transfer function of the system you want to analyze. State its numerator and denominator explicitly as shown below.

The transfer function formulated is illustrated below.

3. Determine the position constant and steady state error with the help of the commands shown for step input.

4. In order to determine the error for ramp input, multiply the transfer function by ‘s’ as displayed.

5. Now, calculate the velocity constant and the corresponding steady state error.

6. Lastly, multiply the previous transfer function by ‘s’ to obtain the response for a parabolic input.

7. The acceleration constant and its corresponding steady state error is as follows.

**Conclusion**

The static error coefficients and their corresponding steady state errors of a given control system were obtained in MATLAB.

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