Transient and Steady State Response in a Control System

The transient and steady state responses of control systems describe how an output changes after an input or disturbance and what remains as time becomes large. Standard input signals make different systems easier to compare.
Standard Input Signals: Unit impulse, step, ramp, parabolic and sinusoidal inputs expose different dynamic properties. Their Laplace transforms let us calculate the response of a linear time-invariant system from its transfer function. The following definitions use the one-sided Laplace transform and zero initial time:

Unit Impulse Signal: In the time domain, the unit impulse is represented by δ(t). Its Laplace transformation is 1. This ideal signal has unit area and zero duration, as shown below.

Unit Step Signal: The unit step is represented by u(t). Its Laplace transform is 1/s. The signal changes from zero to one at time zero.

Unit Ramp Signal: A unit ramp is represented by r(t) = t for non-negative time. Its Laplace transform is 1/s², and its value increases at a constant rate.

Parabolic Type Signal: A unit parabolic input is t²/2 for non-negative time. Its Laplace transform is 1/s³.

Sinusoidal Type Signal: The signal sin(ωt) has the Laplace transform ω/(s² + ω²). Sinusoidal tests show how a system responds at a selected angular frequency ω.

Cosine Type of Signal: The signal cos(ωt) has the Laplace transform s/(s² + ω²). The numerator is s, unlike the numerator ω in the sine transform.

These test inputs let us separate the time response into transient and steady-state parts.

Transient Response of Control System

The transient response of control system is the part that changes after an input, initial condition or disturbance. For an asymptotically stable system, this component decays with time. Two common triggers are:

• Condition one: The system is switched on, or an input is applied or changed.
• Condition second: A disturbance changes the operating point, such as a sudden load change or a short circuit within the model’s safe limits.

Steady State Response of Control System

For an asymptotically stable system, the steady state response of control system is the long-term response after decaying modes have vanished. A sustained input and the system dynamics determine this response. It is often associated with forced response, although a forced response can also contain decaying terms.

The response of a control system can be separated into transient and steady-state parts when the relevant modes are stable. Step-response specifications quantify how the output moves between its initial and final values.
Delay Time: Represented by t_d, delay time is the first time the response reaches 50 percent of the change from its initial value to its final value.

Rise Time: Represented by t_r, the rise time formula depends on the stated thresholds. Two textbook conventions are:

1. For an underdamped response with damping ratio ζ less than one, rise time is often measured from 0 percent to the first 100 percent crossing of the final value.
2. For a non-oscillatory response, rise time is commonly measured from 10 percent to 90 percent of the initial-to-final change. The same stated thresholds can be used for any response when results are compared.

Peak Time: Represented by t_p, peak time is the time at which the response first reaches its maximum value.

Settling Time: Represented by t_s, the settling time formula estimates when the response enters and remains within a specified band around its final value. Common bands are 2 percent or 5 percent of the initial-to-final change.

Maximum Overshoot: Maximum overshoot is the largest positive amount by which the response exceeds its final value, usually expressed as a percentage of the initial-to-final change.
Steady state error: Steady-state error is the limiting difference between the reference input and the actual output as time tends to infinity, when that limit exists.

Transient State and Steady State Response of First Order Control System

Consider the first-order block diagram.

The stable first-order transfer function is 1/(sT+1), where T is positive. Its impulse, step and ramp responses each contain a decaying exponential whose rate is set by T. We will examine these three standard inputs.

1. Unit impulse.
2. Unit step.
3. Unit ramp.

Unit impulse response: The Laplace transform of the unit impulse is 1. Applying it to the first-order system gives:

The inverse Laplace transform is:

The exponential is the transient component. It decays at a rate set by T, and the steady state response of control system to the impulse is zero.
Unit Step Response: The unit-step transform is 1/s. Applying it to the first-order system gives:

Partial fractions and the inverse Laplace transform give:

The response approaches one as time tends to infinity, so this unity-gain model has zero final error for a unit step.
Unit Ramp Response: The unit-ramp transform is 1/s². Applying it to the first-order system gives:

Partial fractions and the inverse Laplace transform give:

The exponential decays as time tends to infinity. The output then follows the ramp with a constant lag T, so the limiting input-output error is T for this unity-gain model.

Transient State and Steady State Response of Second Order Control System

Consider the second-order unity-feedback block diagram.

The diagram shows the forward path ω²/{s(s + 2ζω)}. With unity feedback, the closed-loop transfer function becomes ω²/(s² + 2ζωs + ω²), a linear second-order model where ω is the undamped natural frequency and ζ is the damping ratio. We will examine the transient state response of control system for impulse and step inputs.

Unit Impulse Response: The unit-impulse transform is 1. Applying it to the second-order system gives:

Here, ω is natural frequency in rad/s and ζ is damping ratio.
Unit Step Response: The unit-step transform is 1/s. The step-output transform is therefore the impulse-output transform divided by s:

The damping ratio determines the pole locations and the form of the standard step response. Four important cases follow.

1. Under Damped System: For 0 < ζ < 1 and positive ω, the poles are a complex-conjugate pair in the left half-plane. The response is asymptotically stable and oscillatory, with overshoot in the standard zero-free model.
2. Critically Damped System: When ζ = 1 and ω is positive, a repeated negative real pole produces the fastest asymptotically stable, non-oscillatory response for fixed ω, without overshoot in the standard model.
3. Over Damped System: When ζ > 1 and ω is positive, two distinct negative real poles produce an asymptotically stable, non-oscillatory response that is slower than the critically damped case for fixed ω.
4. Sustained Oscillations: When ζ = 0, the poles lie on the imaginary axis. The ideal response oscillates without decay and is marginally stable, not asymptotically stable.

For the standard underdamped closed-loop model with a unit-step input, the following expressions describe the response when 0 < ζ < 1.
Rise Time: Set the response c(t) equal to one to find the first 100 percent crossing:

Solving gives the displayed rise-time expression:

Peak Time: Set dc(t)/dt to zero and select the first positive maximum to obtain:

Maximum Overshoot: Substitute the peak time into the standard step response to obtain:

Settling Time: The displayed expression is the common envelope approximation for a stated tolerance band, rather than an exact value for every second-order system:

Steady State Error: For this stable unity-gain closed-loop model and a unit-step input, the output approaches one, so the steady-state error is zero. Other loop types, inputs and gains can have a non-zero or undefined limiting error.