Steady-State Error Formulae

This section discusses steady-state error and the formulae used to calculate it based on different types of inputs.

In this section, we explore the concept of steady-state error, including how it differs for step, ramp, and parabolic inputs. It highlights the relevant error constants and provides specific formulae for calculating steady-state error.

Steady-State Error Formulae

In control systems, the steady-state error quantifies the difference between the desired output and the actual output as time approaches infinity. This section primarily focuses on the steady-state error associated with various input types: step, ramp, and parabolic inputs.

Key Concepts Covered:

• Steady-State Error: The value that remains constant once the system settles after transient dynamics fade.
• Error Constants: These constants (Kp, Kv, Ka) characterize the system's performance for specific input types:
  • Position Error Constant (Kp): It dictates the steady-state error for a step input.
  • Velocity Error Constant (Kv): It influences the steady-state error for a ramp input.
  • Acceleration Error Constant (Ka): It determines the steady-state error for a parabolic input.

Steady-State Error Formulae:

1. Step Input Error Formula:

   • Given a step input, the steady-state error formula is:

     $e_{ss} = \frac{1}{1 + K_p}$

2. Ramp Input Error Formula:

   • For a ramp input, the steady-state error formula is:

     $e_{ss} = \frac{1}{K_v}$

3. Parabolic Input Error Formula:

   • Lastly, for a parabolic input, the formula is:

     $e_{ss} = \frac{1}{K_a}$

Example:

For a system with a position error constant Kp = 10, the steady-state error for a step input is calculated as follows:

$e_{ss} = \frac{1}{1 + 10} = 0.09$

This section is significant in understanding how different inputs affect a control system's accuracy and performance, allowing engineers to design systems with desirable properties.