Steady State Error: What is it? (And How To Calculate It)

Contents

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Key learnings:

Steady State Error Defined: Steady state error is the discrepancy between a control system’s desired and actual output values once the output has stabilized.
Influence of Input Types: The size of the steady state error varies with different types of inputs—zero for step inputs, a constant for ramp inputs, and infinite for parabolic inputs.
System Stability: Unlike steady state error, the stability of a control system does not depend on the type of input but on the system’s transfer function parameters.
Role of PI Controllers: PI controllers help reduce steady state error but can compromise the stability of the system, demonstrating a critical balance in control system design.
Computational Formulas: Steady state error calculations involve using specific coefficients like Positional error coefficient (Kp), Velocity error coefficient (Kv), and Acceleration error coefficient (Ka) to determine error based on system response to different inputs.

What is Steady State Error?

Steady-state error is the limiting difference between a reference input and the corresponding output, provided that this difference approaches a finite value as time tends to infinity.

This error is one performance measure, not a complete measure of control quality. A design must also meet stability, transient response, disturbance rejection, noise, actuator and safety requirements.

Start with the steady state response of the stable first-order transfer function below:

$\begin{equation*} \frac {C(s)}{R(s)}=\frac {1}{0.7s+1} \end{equation*}$

This first order transfer function has DC gain one, one pole at -1/0.7 and a time constant of 0.7 seconds. System order follows the degree of the denominator after pole-zero cancellation, not the appearance of one isolated term. The protected inline example $0.7s^2 + 1$ is second order but has imaginary-axis poles, so it is not a stable comparison model.

Figure 1 shows its unit-step response. Because the model is stable and its DC gain is one, the output tends to the unit input and the limiting tracking error is zero.

Time response of First order Transfer Function against step Input. — **Figure-1: It is the time response of First order Transfer Function against step Input. It can be seen that steady state error is zero**

Figure 2 shows the same model driven by a unit ramp. After the transient decays, the output follows the ramp with a constant lag of 0.7.

Time response of First order Transfer Function against Ramp Input. — **Figure-2: It is the time response of First order Transfer Function against ramp Input. It can be seen that steady state error exists in this case**

For this specific unity-DC-gain model T(s)=1/(τs+1), the unit-ramp tracking error tends to τ. Thus the limiting error is 0.7, although at t = 3 seconds the exact transient output is about 2.31 rather than precisely 2.3. This result is not a rule for every first-order transfer function.

Keep these conditions in view:

Step, ramp and parabolic errors depend on both the input and the system type. For this particular model they are zero, 0.7 and unbounded, respectively. Type 1 and type 2 feedback systems have different results.
For a linear time-invariant closed loop, internal stability follows its characteristic roots. A bounded input does not move those roots, but an unstable system may make steady-state error undefined.

Now consider a negative-feedback closed loop with this transfer function:

$\begin{equation*} \frac{C(s)}{R(s)}= \frac{G(s)}{1+G(s)H(s)} \end{equation*}$

Here G(s) is the forward-path transfer function and H(s) is the feedback-path transfer function. The characteristic equation is 1 + G(s)H(s) = 0 after accounting for any cancellations and internal modes. Its roots determine closed-loop stability. Tracking error also depends on the reference and loop structure.

For the summing-junction signal in the stated block diagram, $E(s)= \frac{R(s)}{1+G(s)H(s)}.$ The protected legacy expression following e_ss, $\lim_{s \rightarrow 0 } E(s)$ , omits the final-value factor s. The correct theorem is ess = lim as s tends to zero of sE(s), and it applies only when the poles of sE(s) lie in the open left half-plane.

The loop transfer L(s)=G(s)H(s) is used for a Bode plot or Nyquist plot. These analyses can determine or assess stability of the closed-loop transfer $\frac{C(s)}{R(s)}$ under their stated assumptions, including the number of open-loop right-half-plane poles; a plot does not establish stability from “1” being constant.
G(s)H(s) is the loop transfer function, often called the open-loop transfer function, while $\frac{C(s)}{R(s)}$ is one closed-loop transfer function. Bode margins describe relative stability under their usual conditions; the Nyquist criterion can determine closed-loop stability when open-loop pole count and encirclements are included.

Steady State Error Examples

Steady State Error for a Unit Step Input

The next example calculates steady-state error for a stable unity-feedback system with a unit-step reference.

Example-1:

Consider the following control system (system-1) as shown in Figure-3:

**Figure-3: Closed Loop Control System**

The reference R_s is a unit step.

Various steady-state values of System-1 are shown in Figure-4.

Steady State Value Block Diagram — **Figure-4: Various Steady State Values in a Control System**

Figure 4 gives a limiting error of 0.5. The following final-value calculations are valid because the relevant closed-loop signals converge.

For a stable proper transfer function with no unresolved pole at the origin, its DC gain is the limit as $s\rightarrow 0$ . A final output value still requires the final value theorem applied to the complete output transform.

You can calculate the output as follows:

$\begin{equation*} \frac{C(s)} {R(s)}= \frac{4}{s+8} \end{equation*}$

For a unit step, $R(s)$ = $\frac{1}{s}$ , so:

$\begin{equation*} C(s)= \frac{4}{s+8} \times R(s)= \frac{4}{s(s+8)} \end{equation*}$

The steady state value of the output is:

$\begin{equation*} \lim_{s \rightarrow 0 } sC(s) = s\frac{4}{s+8}\frac{1}{s} =\frac{1}{2} \end{equation*}$

The method applies to a signal only when the final value exists and the final-value theorem’s pole condition is satisfied. For this input:

Input is $R(s)= \frac{1}{s}$ (Input is unit step input)

Its steady state value= $\lim_{s \rightarrow 0 }\ sR(s)=s \frac{1}{s}$ = 1.

Similarly, the error signal can be calculated as:

$\begin{equation*} E(s)= \frac{R(s)}{1+G(s)H(s)}= \frac{s+4}{s(s+8)} \end{equation*}$

The steady state value of the error signal (i.e. the steady state error) is:

$\begin{equation*} \lim_{s \rightarrow 0} sE(s)= s\frac{s+4}{s(s+8)}= \frac{1}{2} \end{equation*}$

Figure 4 agrees: the limiting input-output difference is 0.5.

For a stable unity-feedback system and standard polynomial inputs, static error constants give the same result:

The position constant K_p = $\lim_{s \rightarrow 0 } G(s)H(s)$ is 1 for this unity-feedback example. Thus K_p = 1 and e_ss= $\frac{1}{1+Kp}$ = 0.5.

For a step of amplitude 3, $R(s)=\frac{3}{s}$ , the stable unity-feedback result is e_ss= $\frac{3}{1+Kp}$ .

For a unit ramp, define the velocity constant K_v= $\lim_{s \rightarrow 0 }s G(s)H(s)$ ; then e_ss= $\frac{1}{Kv}$ when the assumptions hold.

For a unit parabolic input, define the acceleration constant Ka= $\lim_{s \rightarrow 0 } s^2G(s)H(s)$ ; then e_ss= $\frac{1}{Ka}$ when the assumptions hold.

The constants K_p, K_v and K_a show how system type interacts with step, ramp and parabolic references. They are not universal formulas for non-unity feedback without first forming the true reference-to-output error.

PI Controller And Steady State Error

A PI controller (i.e. a proportional controller plus integral controller) reduces the steady state error (e_ss), but has a negative effect on the stability.

The protected orange summary overstates the trade-off. Integral action can reduce or eliminate steady-state error for specified inputs if the resulting closed loop is stable. Its effect on stability and transient response depends on the plant, gains and implementation.

A PI controller adds an integrator pole and a controller zero. Those additions reshape the closed-loop characteristic equation. Depending on tuning, poles may move toward or away from the imaginary axis; overshoot and settling time may rise or fall. Increased order alone does not determine stability.

For the two examples, s³+ s²+ 3s+20=0 is unstable by the Routh-Hurwitz test because 1×3 is less than 20, while s²+3s+20=0 has roots with negative real parts. This comparison follows from their coefficients and roots, not from third order always being less stable than second order.

Figure 5 adds a PI controller to system 1. For this stable example and a constant reference, the integrator raises the system type and the output tends to the reference, giving zero limiting error. This result depends on closed-loop stability and the absence of sustained saturation or other nonlinear limits.

PI Controller Block Diagram — **Figure-5: Effect of PI Controller can be seen in this diagram**

The ideal PI transfer function is Kp+\frac{Ki}{s} or \frac{Kps+Ki}{s}. In time domain, its output is Kp times the present error plus Ki times the accumulated error. When the present error becomes zero, the integrator state can retain a nonzero constant that supplies the plant input needed at equilibrium.

(1) The earlier numerical argument compared an error of 2x10^-3 with a supposed value s = 2x10^-3. That is invalid: s is the complex Laplace variable, not a time-varying signal that can be assigned the same instantaneous value as the error.

Figure 6 shows another closed-loop system with PI control:

**Figure-6: An Example of Closed Loop Control System with PI Controller**

The former explanation again paired a hypothetical error of 2x10^-3 with “s = 4×10⁻³”. Their ratio has no instantaneous physical meaning. The finite controller output comes from the integrator’s stored state and the trajectory of past error.

Steady-state analysis uses a limit as t tends to infinity. Under the final value theorem this corresponds to s tending to zero. The protected legacy formula after the line break incorrectly states t tends to zero:
$s\rightarrow 0, t\rightarrow 0.$

(2) Substituting s = 0 directly into $\frac{Kps+Ki}{s}$ is not a time-domain equilibrium calculation. Also, $\frac{0}{0}$ is indeterminate, not “any finite value”. The relevant limit depends on the complete closed-loop transfer functions and their poles.

PI Controller — **Figure-7: Input to Transfer Function is zero but output is a finite value**

(3) $\frac{1}{s}$ represents an ideal integrator under zero initial conditions in transfer-function form. In time domain, integrating zero after a given time preserves the value accumulated before that time. The integral is defined; its constant output depends on prior error and initial state.

Parameter sensitivity in open-loop and closed-loop systems

The remaining diagrams compare how open-loop and feedback systems respond to a plant-parameter change. Feedback can reduce output sensitivity within its bandwidth, but it does not make the output invariant.

An open loop control system can be represented as follows:

A closed loop control system (feedback control system) can be represented as follows:

A nominal plant model is fixed for analysis, while the physical plant can vary with operating point, environment, ageing and load. These examples assume H(s)=1. The designer selects controller parameters such as K_p, K_d and K_i; an operator should change them only through an authorised tuning procedure.

Controller choices include P, PI, PD, PID controller and other strategies. Requirements are application-specific. A damping ratio of 0.7 to 0.9 and zero steady-state error are not universal targets; tracking, regulation, robustness, control effort, noise and safety can impose different trade-offs.

Changing controller gains can affect both transient and steady-state behaviour, but increasing damping does not inherently increase steady-state error. Design and tuning should verify the complete set of requirements across the expected plant range.

A PI controller can reduce the steady-state error (e_ss) for specified references or disturbances. It can stabilise or destabilise a particular loop depending on plant dynamics and gain selection, so stability must be checked rather than assumed to worsen.

The final examples illustrate sensitivity to a plant-gain change.

Figure 10 is an open-loop system.

With a unit-step input, the reference and plant input remain 1 because no feedback path adjusts them. If the plant’s DC gain changes, the limiting output changes in direct proportion, provided the open-loop plant remains stable.

Figures 11 and 12 show the feedback example before and after a plant-parameter change.

Closed loop system — **Figure-12: Closed Loop System, Plant output is same but plant input is changed due to change in Transfer Function**

Both are closed loop control systems. Feedback changes the plant input in response to output error. With adequate loop gain, the output variation caused by a plant change can be smaller than in open loop, but it is not generally zero.

Figure 12’s stated numbers show the plant input changing from 0.5 to 0.476 while its rounded output remains at the reference. The PI controller can hold a different constant output at zero present error because its integral state accumulated a different history during the transition.

The useful distinction is sensitivity: open loop cannot correct from measured output error, while feedback modifies the plant input to reduce that error. Closed-loop output can still change when loop gain is finite or the controller saturates.

In the books of the control system, you can find the following statement:

For standard negative feedback, output sensitivity to multiplicative plant variation is reduced by the sensitivity function 1/(1+L) where loop gain L is large. The benefit is frequency-dependent and requires a stable, robust loop.

The diagrams illustrate that principle, while the sensitivity function supplies the general quantitative statement.

Controller design must verify steady-state accuracy together with stability and transient performance.

The worked examples use ideal linear models. Real controllers also need checks for actuator saturation, integral windup, sampling, sensor noise, model uncertainty and operating limits.

Steady State Error: What is it? (Steady-State Gain, Value & Formula)

What is Steady State Error?

Steady State Error Examples

Steady State Error for a Unit Step Input

PI Controller And Steady State Error

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