Software Oriented

Time Domain Analysis of First, Second and Third Order Systems

Aim

To study and analyze the time domain responses of first, second, and third order linear time-invariant (LTI) systems using MATLAB.

To analyze the time response of first, second, and third order LTI systems.
To study impulse, step, ramp, and parabolic responses.
To evaluate system stability and time domain specifications.
To verify theoretical results using MATLAB simulations.

Apparatus & Software

Sl. No.	Apparatus / Software	Technical Specification	Quantities
1	MATLAB	R2023 or compatible, with Control System Toolbox	1

Theory

The time domain response of a control system describes how the output of the system varies with time for a given input signal. To evaluate the dynamic performance of control systems, standard test inputs are commonly applied. These inputs are mathematically simple and represent practical physical situations, allowing a clear comparison of system behavior under different conditions.

Standard Test Inputs

Impulse Input: Represents a sudden shock or disturbance applied to the system. r(t) = δ(t)
Step Input: Represents a sudden change in position or reference value. r(t) = u(t)
Ramp Input: Represents a constant velocity input. r(t) = t·u(t)
Parabolic Input: Represents a constant acceleration input. r(t) = (t²/2)·u(t)

Part A: First Order System

A first order system contains a single energy storage element and is characterized by one pole. The standard transfer function of a first order system is:

G(s) = \frac{K}{\tau s + 1}

where K is the system gain and τ is the time constanttime constantA measure of how quickly a circuit responds to change. For RC circuits, τ = RC; for RL circuits, τ = L/R. After one time constant, the response reaches ~63% of its final value.. For an impulse input, the time response is:

c(t) = \frac{K}{\tau} e^{-t/\tau}

Time Domain Specifications of First Order Systems:

Delay Time (td): Time to reach 50% of final value. td ≈ 0.693τ
Rise Time (tr): Time to rise from 10% to 90% of final value. tr ≈ 2.2τ
Settling Time (ts): Time to remain within ±2% of final value. ts ≈ 4τ
Steady-State Value: lim(t→∞) c(t) = K
Steady-State Error (ess): ess = lim(t→∞) [r(t) − c(t)]

Part B: Second Order System

A second order system contains two energy storage elements and may exhibit oscillatory behavior depending on the damping ratio. The standard transfer function is:

G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}

where ωn is the natural frequency and ζ is the damping ratio. The impulse response is:

c(t) = \frac{\omega_n}{\sqrt{1-\zeta^2}} e^{-\zeta\omega_n t} \sin(\omega_d t)

where the damped natural frequency is:

\omega_d = \omega_n \sqrt{1 - \zeta^2}

Classification of Second Order Systems:

Undamped (ζ = 0): Sustained oscillations with constant amplitude.
Underdamped (0 < ζ < 1): Oscillations with exponentially decaying amplitude.
Critically Damped (ζ = 1): Fastest return to steady-state without oscillations.
Overdamped (ζ > 1): Returns to steady-state without oscillations but more slowly.

Time Domain Specifications of Second Order Systems:

t_r = \frac{\pi - \theta}{\omega_d}, \quad \theta = \tan^{-1}\left(\frac{\sqrt{1-\zeta^2}}{\zeta}\right)

t_p = \frac{\pi}{\omega_d}

M_p = e^{-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}} \times 100\%

t_s \approx \frac{4}{\zeta\omega_n} \quad (2\%\text{ criterion})

Part C: Third Order System

A third order system contains three energy storage elements and exhibits more complex dynamics. The transfer function of a typical third order system is:

G(s) = \frac{K}{(s + a)(s^2 + 2\zeta\omega_n s + \omega_n^2)}

The poles are s₁ = −a and s₂,₃ = −ζωn ± jωn√(1−ζ²). If the real pole −a lies far to the left of the imaginary axis (a ≫ ζωn), the complex conjugate poles dominate the system response and the third order system can be approximated as a dominant second order system.

Pre-Lab / Circuit Diagram (Not Applicable)

This section is not required for this experiment.

Procedure

Define the transfer function of each system using MATLAB's tf() function.
Apply impulse, step, ramp, and parabolic inputs to each system.
Plot the time responses using MATLAB's impulse(), step(), and lsim() functions.
Extract time domain specifications (rise timerise timeThe time required for a system's transient response to rise from 10% to 90% (typically for overdamped systems) or 0% to 100% (for underdamped systems) of its final steady-state value., peak time, settling timesettling timeThe time required for a system's response curve to reach and permanently stay within a specified tolerance band (usually 2% or 5%) of the final steady-state value., maximum overshootmaximum overshootThe maximum peak value of the response curve measured from the desired steady-state value, typically expressed as a percentage. It indicates the relative stability of the system., steady-state value) using MATLAB's stepinfo() function.
Identify system parameters (K, τ for first order; ωn, ζ for second order) by comparing with standard forms.
For the third order system, identify dominant poles and compare with second order approximation.
Verify the MATLAB results against theoretical calculations.

The following six systems are analyzed:

System 1 (First Order): G(s) = 3 / (0.6s + 1)
System 2 (Second Order — Undamped): G(s) = 121 / (s² + 121)
System 3 (Second Order — UnderdampedunderdampedA system response that oscillates with decreasing amplitude before reaching steady state. Occurs when damping is less than the critical value (ζ < 1).): G(s) = 144 / (s² + 19.2s + 144)
System 4 (Second Order — Critically Dampedcritically dampedA system response that returns to steady state as fast as possible without oscillating. Occurs when the damping ratio exactly equals one (ζ = 1).): G(s) = 144 / (s² + 24s + 144)
System 5 (Second Order — OverdampedoverdampedA system response that returns to steady state slowly without oscillating. Occurs when damping exceeds the critical value (ζ > 1).): G(s) = 144 / (s² + 72s + 144)
System 6 (Third Order): G(s) = 1 / [(s + 36)(s² + 14s + 196)]

Simulation / Execution

MATLAB simulations were performed for all six systems to obtain their impulse, step, ramp, and parabolic responses, and to compute time domain specifications.

Observations

First Order System — G(s) = 3 / (0.6s + 1)

Impulse Response for first order system

Step Response for first order system

Ramp Response for first order system

Parabolic Response for first order system

Step Response Specifications for first order system

Parameter	Value
Gain K	3
Time Constant τ	0.6
Rise Time	1.3182 s
Peak Time	4.3933 s
Maximum Overshoot	0 %
Settling Time	2.3472 s
Steady-State Value	3

Second Order System — Undamped (ζ = 0) — G(s) = 121 / (s² + 121)

The undamped second order system exhibits sustained oscillations with constant amplitude. No energy dissipation is observed, and the system does not reach steady-state.

Impulse Response for undamped second order system

Step Response for undamped second order system

Ramp Response for undamped second order system

Parabolic Response for undamped second order system

Second Order System — Underdamped (0 < ζ < 1) — G(s) = 144 / (s² + 19.2s + 144)

The underdamped system exhibits oscillatory behavior with exponentially decaying amplitude. The response overshoots the steady-state value before settling.

Impulse Response for underdamped second order system

Step Response for underdamped second order system

Ramp Response for underdamped second order system

Parabolic Response for underdamped second order system

Step Response Specifications for underdamped system

Parameter	Value
Natural Frequency ωn	12
Damping Ratio ζ	0.8
Rise Time	0.20566 s
Peak Time	0.43653 s
Maximum Overshoot	1.5165 %
Settling Time	0.31302 s
Steady-State Value	1

Second Order System — Critically Damped (ζ = 1) — G(s) = 144 / (s² + 24s + 144)

The critically damped system returns to steady-state in the minimum possible time without oscillations. It represents the fastest non-oscillatory response.

Impulse Response for critically damped second order system

Step Response for critically damped second order system

Ramp Response for critically damped second order system

Parabolic Response for critically damped second order system

Step Response Specifications for critically damped system

Parameter	Value
Natural Frequency ωn	12
Damping Ratio ζ	1
Rise Time	0.27986 s
Peak Time	0.99395 s
Maximum Overshoot	0 %
Settling Time	0.48618 s
Steady-State Value	1

Second Order System — Overdamped (ζ > 1) — G(s) = 144 / (s² + 72s + 144)

The overdamped system does not exhibit oscillations and returns to steady-state slowly compared to the critically damped case due to the presence of two real and distinct poles.

Impulse Response for overdamped second order system

Step Response for overdamped second order system

Ramp Response for overdamped second order system

Parabolic Response for overdamped second order system

Parameter	Value
Natural Frequency ωn	12
Damping Ratio ζ	3
Rise Time	1.0675 s
Peak Time	3.5564 s
Maximum Overshoot	0 %
Settling Time	1.9147 s
Steady-State Value	1

Third Order System — G(s) = 1 / [(s + 36)(s² + 14s + 196)]

The third order system consists of an additional real pole (s₁ = −36) along with complex conjugate poles (s₂,₃ ≈ −7 ± j12.12). The complex poles are dominant. The dominant pole approximation was used to analyze the transient response.

Impulse Response for third order LTI system

Step Response for third order LTI system

Ramp Response for third order LTI system

Parabolic Response for third order LTI system

Step Response Specifications for third order system

Parameter	Value
Natural Frequency ωn (dominant)	84.2378
Damping Ratio ζ (dominant)	2.9915
Rise Time	0.11707 s
Peak Time	0.25657 s
Maximum Overshoot	16.2929 %
Settling Time	0.57685 s
Steady-State Value	0.00014172

Calculations

System 1 — First Order

G_1(s) = \frac{3}{0.6s + 1} \Rightarrow K = 3,\; \tau = 0.6

Key results: td = 0.693 × 0.6 = 0.4158 s, tr = 2.2 × 0.6 = 1.32 s, ts = 4τ = 2.4 s, steady-state value = 3, steady-state error = −4 (for ramp input).

System 2 — Undamped Second Order

G_2(s) = \frac{121}{s^2 + 121} \Rightarrow \omega_n^2 = 121 \Rightarrow \omega_n = 11,\; 2\zeta\omega_n = 0 \Rightarrow \zeta = 0

System 3 — Underdamped Second Order

G_3(s) = \frac{144}{s^2 + 19.2s + 144} \Rightarrow \omega_n = 12,\; 2\zeta\omega_n = 19.2 \Rightarrow \zeta = 0.8

System 4 — Critically Damped Second Order

G_4(s) = \frac{144}{s^2 + 24s + 144} \Rightarrow 2\zeta\omega_n = 24 \Rightarrow \zeta = 1

System 5 — Overdamped Second Order

G_5(s) = \frac{144}{s^2 + 72s + 144} \Rightarrow 2\zeta\omega_n = 72 \Rightarrow \zeta = 3

Poles: s = −2.06, s = −69.94 (factored as (s + 2.06)(s + 69.94)).

System 6 — Third Order

G_6(s) = \frac{1}{(s+36)(s^2+14s+196)}

Poles: s₁ = −36, s₂,₃ = −7 ± j√147 ≈ −7 ± j12.12. The complex conjugate poles s₂, s₃ are dominant (closer to imaginary axis).

Results & Analysis

The time domain specifications obtained from MATLAB for all systems are summarized below.

System	Transfer Function	Type	ωn / K	ζ / τ	Rise Time (s)	Peak Time (s)	Max Overshoot (%)	Settling Time (s)	SS Value
System 1	3 / (0.6s+1)	1st Order	K=3	τ=0.6	1.3182	4.3933	0	2.3472	3
System 2	121 / (s²+121)	2nd — Undamped	11	0	—	—	—	∞	1 (oscillating)
System 3	144 / (s²+19.2s+144)	2nd — Underdamped	12	0.8	0.20566	0.43653	1.5165	0.31302	1
System 4	144 / (s²+24s+144)	2nd — Critically Damped	12	1	0.27986	0.99395	0	0.48618	1
System 5	144 / (s²+72s+144)	2nd — Overdamped	12	3	1.0675	3.5564	0	1.9147	1
System 6	1 / [(s+36)(s²+14s+196)]	3rd Order	84.24	2.99	0.11707	0.25657	16.2929	0.57685	0.000142

For second order systems, the effect of the damping ratio on system behavior was clearly observed. As ζ increases from 0 to values greater than 1, the system transitions from sustained oscillations (undamped) → oscillatory decay (underdamped) → fastest non-oscillatory response (critically damped) → sluggish non-oscillatory response (overdamped). The third order system analysis demonstrated the dominant pole concept, where the overall transient responsetransient responseThe temporary behavior of a circuit immediately after a switching event or disturbance, before it reaches its new steady-state condition. was primarily governed by the poles closest to the imaginary axis.

Conclusion

In this experiment, the time domain responses of first, second, and third order linear time-invariant (LTI) systems were analyzed using standard test inputs such as impulse, step, ramp, and parabolic signals. The theoretical behavior of each system was studied using standard transfer function models, and key time domain specifications including rise time, peak time, settling time, and maximum overshoot were evaluated. The responses clearly demonstrated how system order and pole locations influence transient and steady-state characteristics.

For second order systems, the effect of the damping ratio on system behavior was clearly observed. The undamped system exhibited sustained oscillations, while the underdamped system showed oscillatory behavior with decaying amplitude and measurable overshoot. The critically damped system provided the fastest non-oscillatory response, and the overdamped system returned to steady-state more slowly without oscillations. The third order system analysis further illustrated the dominant pole concept, where the overall transient response was primarily governed by the poles closest to the imaginary axis.

The MATLAB simulations closely matched the theoretical predictions, validating the analytical calculations and reinforcing the relationship between system parameters and time domain performance. Overall, the experiment provided a comprehensive understanding of how system order and damping characteristics affect dynamic response, stability, and performance in control systems.

Post-Lab / Viva Voce

Q: What is the effect of damping ratio (

\zeta

) on the maximum overshoot and settling time of a second-order system?

A: As the damping ratio

\zeta

decreases from 1 towards 0, the maximum overshoot increases significantly. However, a highly underdamped system may have a longer settling time due to sustained oscillations, whereas a heavily overdamped system (

\zeta > 1

) approaches the steady state too slowly, also increasing settling time. The optimal damping ratio for minimal settling time without excessive overshoot is typically around

\zeta = 0.7

Q: Why is a critically damped system preferred in applications like analog meter movements or door closers?

A: A critically damped system provides the fastest possible response without any overshoot or oscillation. This is strictly preferred when an overshoot could cause mechanical damage, reading inaccuracies (in meters), or slamming (in door closers).

Q: Explain the relationship between the location of closed-loop poles and the nature of the time response (underdamped vs. overdamped).

A: When closed-loop poles are complex conjugates located in the left half of the s-plane, the system response is underdamped and exhibits oscillations. If the poles are purely real, distinct, and negative, the system is overdamped with a sluggish response. If the two real poles are identical (repeated root), the system is critically damped.

Q: How does adding a zero to a second-order system affect its step response? How does adding a pole affect it?

A: Adding a zero to the forward path of a second-order system generally increases peak overshoot, decreases rise time, and makes the system 'faster' but less stable. Conversely, adding a pole increases rise time, slows down the response, and usually increases stability while damping out overshoots.

Q: In a third-order system, what condition allows it to be accurately approximated as a dominant second-order system?

A: A third-order system can be approximated as a dominant second-order system if the third pole is located significantly further to the left in the s-plane than the pair of dominant second-order poles. Typically, if the real part of the third pole is at least 5 to 10 times larger (more negative) than the real part of the dominant poles, its transient effect decays very rapidly and can be neglected.

References & Resources (Not Applicable)

This section is not required for this experiment.

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