Steady State Error

In any closed-loop control system, the output may not perfectly equal the desired reference input even after all transient oscillations have died out. The difference that persists between the desired and actual output as time approaches infinity is called the steady state error. Analyzing and minimizing this error is a central problem in control system design and a key topic in GATE.

Core Concept Explanation

Steady state error is evaluated using the final value theorem of Laplace transforms. If E(s) is the error signal in the s-domain, then the steady state error is ess = lim(s->0) s*E(s), provided the closed-loop system is stable. This theorem converts the complex time-domain limiting behavior into a simple algebraic limit evaluation.

For a unity feedback system where H(s) = 1, the error signal is E(s) = R(s) / [1 + G(s)]. Applying the final value theorem and substituting different input types (step, ramp, parabola) gives the standard steady state error expressions in terms of the error constants Kp, Kv, and Ka.

The position error constant Kp applies to step inputs. The velocity error constant Kv applies to ramp inputs. The acceleration error constant Ka applies to parabolic inputs. These names come from position control terminology: a step in position, a ramp (constant velocity), and a parabola (constant acceleration) represent the three standard test inputs.

Mathematical Expression

For a step input R(s) = A/s, the steady state error is ess = A / (1 + Kp) where Kp = lim(s->0) G(s)H(s). For a ramp input R(s) = A/s^2, the steady state error is ess = A/Kv where Kv = lim(s->0) s*G(s)H(s). For a parabolic input R(s) = A/s^3, the steady state error is ess = A/Ka where Ka = lim(s->0) s^2*G(s)H(s).

An important point is that these error formulas are valid only for stable closed-loop systems. If the system is unstable, the final value theorem does not apply and the error grows unboundedly. Therefore, stability analysis must always precede steady state error analysis.

For non-unity feedback systems where H(s) is not 1, the analysis is more involved. The error signal is E(s) = R(s) - H(s)*C(s). The same final value theorem applies, but the error constants are computed using G(s)*H(s), the open-loop transfer function including the feedback element.

Practical Understanding

In a temperature control system maintaining 200 degrees Celsius, a steady state error of 5 degrees means the system settles at 195 degrees permanently. For many industrial processes, this is unacceptable. A higher loop gain or an integrator in the forward path would reduce this error.

Steady state error due to disturbances is equally important in practice. A motor speed controller may track the reference speed accurately, but when a load disturbance is applied, the output drops and the steady state error caused by disturbance depends on where in the loop the disturbance enters and the loop gain at that point.

Given:
G(s) = 50 / [s(s+10)],  H(s) = 1 (unity feedback)
Input: Unit ramp r(t) = t,  so R(s) = 1/s^2

Why this formula applies:
System is Type 1 (one pole at s=0), so Kv is finite.
Ramp steady state error = 1/Kv

Formula:
Kv = lim(s->0) s * G(s) * H(s)
eSS = 1 / Kv

Substitution:
Kv = lim(s->0) [s * 50 / (s*(s+10))]
Kv = lim(s->0) [50 / (s+10)]

Calculation:
Kv = 50 / 10 = 5
eSS = 1 / 5

Final Answer with units:
Kv = 5 sec^-1,  Steady state ramp error = 0.2 units

Exam Tip: For GATE, always check system stability before applying final value theorem. If any closed-loop pole is in the right half s-plane, ess formula gives wrong answer. Also note: for a ramp input to a Type 0 system, ess = infinity (not 1/Kp). Never apply the wrong error constant to the wrong input type.

Quick Revision

• Steady state error: ess = lim(t->inf) e(t) = lim(s->0) s*E(s) using final value theorem.
• Error constants: Kp = lim(s->0) G(s)H(s), Kv = lim(s->0) s*G(s)H(s), Ka = lim(s->0) s^2*G(s)H(s).
• Step error = A/(1+Kp), Ramp error = A/Kv, Parabola error = A/Ka. A is input magnitude.
• Final value theorem valid only for stable systems. Check stability first.
• Type 1 system: Kp = inf (zero step error), Kv = finite, Ka = 0 (infinite parabola error).
• Increasing open-loop gain K reduces steady state error but may reduce stability margins.
• Exam trap: Applying step error formula 1/(1+Kp) to a ramp input is a common mistake. Match input type to correct error constant.