How to Calculate Steady-state Error in Control Systems: a Step-by-step Approach

Steady-state error is an important parameter in control systems that indicates the difference between the desired output and the actual output after the system has settled. Calculating this error helps in designing systems that meet specific accuracy requirements. This article provides a step-by-step approach to determine steady-state error in various control system configurations.

Understanding Steady-State Error

Steady-state error is the residual error remaining after transient effects have dissipated. It is influenced by the system type, input signal, and the system’s transfer function. Common input signals include step, ramp, and parabolic inputs, each affecting the steady-state error differently.

Step-by-Step Calculation Method

Follow these steps to calculate the steady-state error:

• Identify the input signal type (step, ramp, etc.).
• Determine the system’s transfer function, G(s).
• Calculate the error constant based on the system type:
  • Position error constant (Kp) for step input.
  • Velocity error constant (Kv) for ramp input.
  • Acceleration error constant (Ka) for parabolic input.
• Use the appropriate formula:
  • For step input: e_ss = 1 / (1 + Kp)
  • For ramp input: e_ss = 1 / Kv
  • For parabolic input: e_ss = 1 / Ka
• Calculate the error constant using the system transfer function and input type.
• Compute the steady-state error using the formulas above.

Example Calculation

Consider a unity feedback control system with the transfer function G(s) = 10 / (s + 2). For a step input, the position error constant Kp is calculated as:

Kp = lim_s→0 G(s) = 10 / 2 = 5

The steady-state error is then:

e_ss = 1 / (1 + Kp) = 1 / (1 + 5) = 1/6 ≈ 0.167