GATE EE Control Systems: Time & Frequency Response, Stability Analysis Notes

Introduction

Analysis of LTI Systems

Key Topics

  • Time response analysis (transient and steady-state)

  • Standard test signals and system responses

  • Steady-state error analysis

  • Routh-Hurwitz stability criterion

  • Root locus technique

  • Impulse and frequency response

  • Pole-zero analysis and system behavior

Objective

Analyze the behavior of Linear Time-Invariant (LTI) systems for GATE EE preparation.

System Representation

Transfer Function and System Representation

Transfer Function

\[G(s) = \dfrac{Y(s)}{X(s)} = \dfrac{b_m s^m + b_{m-1} s^{m-1} + \dots + b_0}{a_n s^n + a_{n-1} s^{n-1} + \dots + a_0}\]

Poles and Zeros

  • Poles: Values of \(s\) that make denominator zero

  • Zeros: Values of \(s\) that make numerator zero

  • Order: Highest power of \(s\) in denominator

Example

\[G(s) = \dfrac{s+2}{(s+1)(s+3)}\]
  • Zero at \(s = -2\)

  • Poles at \(s = -1, -3\)

  • Second-order system

Stability Condition

System is stable if and only if all poles lie in the left half of s-plane (negative real parts).

Impulse Response

Impulse Response Analysis

Definition

Impulse response \(h(t)\) is the output when input is unit impulse \(\delta(t)\).

\[h(t) = \mathcal{L}^{-1}\{G(s)\}\]

Properties

  • Convolution: \(y(t) = x(t) * h(t)\)

  • Laplace: \(Y(s) = X(s) \cdot G(s)\)

  • System characterization

Example

For \(G(s) = \dfrac{1}{s+a}\):

\[h(t) = e^{-at}u(t)\]
where \(u(t)\) is unit step function.

Stability from Impulse Response

System is BIBO stable if \(\int_0^{\infty} |h(t)| dt < \infty\)

Time Response Analysis

Time Response Analysis

Components

  • Transient response: Initial dynamic behavior

  • Steady-state response: Long-term behavior

Standard Test Signals

  • Impulse: \(\delta(t)\) \(\leftrightarrow\) \(1\)

  • Step: \(u(t)\) \(\leftrightarrow\) \(\dfrac{1}{s}\)

  • Ramp: \(tu(t)\) \(\leftrightarrow\) \(\dfrac{1}{s^2}\)

  • Parabolic: \(\dfrac{t^2}{2}u(t)\) \(\leftrightarrow\) \(\dfrac{1}{s^3}\)

Example

First-order system:

\[\dfrac{C(s)}{R(s)} = \dfrac{1}{Ts + 1}\]
Step response:
\[c(t) = 1 - e^{-t/T}, \quad t \geq 0\]
where \(T\) is the time constant.

First Order System Parameters

System: \(G(s) = \dfrac{K}{Ts + 1}\)

  • DC Gain: \(K\)

  • Time constant: \(T\) (time to reach 63.2% of final value)

  • Settling time: \(t_s = 4T\) (for 2% criterion)

Step Response

\[c(t) = K(1 - e^{-t/T})u(t)\]

Ramp Response

\[c(t) = K(t - T + Te^{-t/T})u(t)\]

Key Point

First-order systems never exhibit overshoot for step input.

Second Order Systems

Second Order System Analysis

Standard Form

\[\dfrac{C(s)}{R(s)} = \dfrac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\]
where:
  • \(\zeta\): Damping ratio

  • \(\omega_n\): Natural frequency (rad/s)

  • \(\omega_d = \omega_n\sqrt{1-\zeta^2}\): Damped frequency

Response Characteristics
Parameter Formula
Peak overshoot (\(M_p\)) \(e^{-\pi\zeta/\sqrt{1-\zeta^2}} \times 100\%\)
Peak time (\(t_p\)) \(\dfrac{\pi}{\omega_n\sqrt{1-\zeta^2}}\)
Rise time (\(t_r\)) \(\dfrac{\pi - \cos^{-1}\zeta}{\omega_n\sqrt{1-\zeta^2}}\)
Settling time (\(t_s\)) \(\dfrac{4}{\zeta\omega_n}\) (for 2% criterion)

Damping Cases for Second Order Systems

Response Types Based on Damping Ratio

  • \(\zeta > 1\): Overdamped - No oscillations, slow response

  • \(\zeta = 1\): Critically damped - Fastest response without overshoot

  • \(0 < \zeta < 1\): Underdamped - Oscillatory response with overshoot

  • \(\zeta = 0\): Undamped - Sustained oscillations

Step Response Forms

Underdamped (\(0 < \zeta < 1\)):

\[c(t) = 1 - \dfrac{e^{-\zeta\omega_n t}}{\sqrt{1-\zeta^2}} \sin(\omega_d t + \phi)\]
where \(\phi = \cos^{-1}\zeta\)

Pole Locations

\[s_{1,2} = -\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2}\]
  • Real part: \(-\zeta\omega_n\)

  • Imaginary part: \(\pm\omega_d\)

Important Relations

  • \(\zeta = \cos\theta\) where \(\theta\) is angle from negative real axis

  • \(M_p\) decreases as \(\zeta\) increases

  • \(t_s\) is independent of \(\omega_d\)

Higher Order Systems

Dominant Pole Concept

For systems with multiple poles:

  • Dominant poles: Closest to imaginary axis

  • Determine the main characteristics of transient response

  • Other poles should be at least 5-10 times farther from imaginary axis

Example

\[G(s) = \dfrac{100}{(s+1)(s+2)(s+10)}\]
Dominant poles: \(s = -1, -2\)

Approximate as second-order system for initial analysis.

Approximation Guidelines

  • Neglect poles that are 5+ times farther than dominant poles

  • Use dominant pole approximation for quick analysis

  • Verify with complete system analysis if needed

Steady-State Error

Steady-State Error Analysis

Error Signal and Final Value Theorem

For unity feedback system: \(E(s) = \dfrac{R(s)}{1 + G(s)}\)

Steady-state error: \(e_{ss} = \lim_{t \to \infty} e(t) = \lim_{s \to 0} sE(s)\)

System Type and Error Constants

System type is determined by the number of integrators (\(1/s\)) in \(G(s)\):

  • Position constant: \(K_p = \lim_{s\to 0} G(s)\)

  • Velocity constant: \(K_v = \lim_{s\to 0} sG(s)\)

  • Acceleration constant: \(K_a = \lim_{s\to 0} s^2G(s)\)

Steady-State Error for Unity Feedback Systems
Input Type 0 Type 1 Type 2
Step (\(\dfrac{1}{s}\)) \(\dfrac{1}{1+K_p}\) 0 0
Ramp (\(\dfrac{1}{s^2}\)) \(\infty\) \(\dfrac{1}{K_v}\) 0
Parabolic (\(\dfrac{1}{s^3}\)) \(\infty\) \(\infty\) \(\dfrac{1}{K_a}\)

Generalized Error Analysis

Non-Unity Feedback Systems

For system with feedback \(H(s)\):

\[E(s) = \dfrac{R(s)}{1 + G(s)H(s)}\]
Error constants are calculated using \(G(s)H(s)\).

Disturbance Rejection

For disturbance \(D(s)\) at plant input:

\[E(s) = \dfrac{-G(s)D(s)}{1 + G(s)H(s)}\]
Higher loop gain reduces disturbance effects.

Position Error Constant Formula

For \(G(s) = \dfrac{K \prod (s + z_i)}{s^N \prod (s + p_j)}\):

\[K_p = \lim_{s \to 0} G(s) = \begin{cases} \dfrac{K \prod z_i}{\prod p_j} & \text{if } N = 0 \\ \infty & \text{if } N \geq 1 \end{cases}\]

Key Insight

  • Increasing system type reduces steady-state error

  • Higher gain improves accuracy but may affect stability

  • Trade-off between accuracy and stability

Stability Analysis

Routh-Hurwitz Criterion

Procedure

For characteristic equation \(a_n s^n + a_{n-1} s^{n-1} + \dots + a_0 = 0\):

  1. All coefficients must be positive (necessary condition)

  2. Construct Routh array

  3. Count sign changes in the first column

  4. Number of sign changes = Number of RHP poles

Example

For \(s^3 + 4s^2 + 6s + 4 = 0\):

\[\begin{array}{c|cc} s^3 & 1 & 6 \\ s^2 & 4 & 4 \\ s^1 & 5 & 0 \\ s^0 & 4 & \\ \end{array}\]
No sign changes \(\Rightarrow\) stable.

Special Cases

  • Zero in first column: Replace with small \(\epsilon\)

  • Row of zeros: Use auxiliary equation

  • Marginal stability: Poles on imaginary axis

Advanced Stability Concepts

Marginal Stability

System has poles on imaginary axis. From row of zeros:

  • Form auxiliary equation from previous row

  • Differentiate to get next row

  • Marginal frequency from auxiliary equation

Example

For \(s^4 + 2s^3 + 3s^2 + 2s + 2 = 0\):

\[\begin{array}{c|ccc} s^4 & 1 & 3 & 2 \\ s^3 & 2 & 2 & 0 \\ s^2 & 2 & 2 & 0 \\ s^1 & 0 & 0 & 0 \\ \end{array}\]
Auxiliary equation: \(2s^2 + 2 = 0 \Rightarrow s = \pm j1\)

System is marginally stable with oscillation at \(\omega = 1\) rad/s.

Note

When row of zeros occurs, replace that row with derivative of auxiliary equation.

Root Locus

Root Locus Technique

Basic Concept

Root locus shows the path of closed-loop poles as gain \(K\) varies from 0 to \(\infty\).

For unity feedback: \(1 + KG(s)H(s) = 0\)

Construction Rules

  • Locus starts at open-loop poles (\(K=0\))

  • Locus ends at zeros or infinity (\(K=\infty\))

  • Real-axis segments: Exist where total #poles + #zeros to the right is odd

  • Asymptotes: \(n-m\) branches to infinity

  • Asymptote angles: \(\dfrac{(2q+1)\pi}{n-m}\), \(q = 0,1,2,\ldots,(n-m-1)\)

  • Centroid: \(\sigma_a = \dfrac{\sum \text{poles} - \sum \text{zeros}}{n-m}\)

Root Locus - Advanced Rules

Breakaway/Break-in Points

Solve \(\dfrac{dK}{ds} = 0\) where \(K = -\dfrac{1}{G(s)H(s)}\)

Alternative: \(\sum_{i=1}^n \dfrac{1}{s-p_i} = \sum_{j=1}^m \dfrac{1}{s-z_j}\)

Angle and Magnitude Conditions

For any point \(s\) on root locus:

  • Angle condition: \(\sum \angle(s-z_i) - \sum \angle(s-p_j) = (2k+1)\pi\)

  • Magnitude condition: \(K = \dfrac{\prod |s-p_j|}{\prod |s-z_i|}\)

Example

For \(G(s) = \dfrac{K}{s(s+4)}\), breakaway point:

\[\dfrac{1}{s} + \dfrac{1}{s+4} = 0 \Rightarrow s = -2\]
At \(s = -2\): \(K = \dfrac{|-2| \cdot |2|}{1} = 4\)

Root Locus - Additional Rules

Departure/Arrival Angles

For complex poles/zeros:

  • Departure angle from complex pole:

    \[\theta_d = 180^{\circ} - \sum \angle(\text{pole to other poles}) + \sum \angle(\text{pole to zeros})\]
  • Arrival angle at complex zero:

    \[\theta_a = 180^{\circ} + \sum \angle(\text{zero to poles}) - \sum \angle(\text{zero to other zeros})\]

Intersection with Imaginary Axis

  • Use Routh-Hurwitz criterion

  • Find value of \(K\) that makes system marginally stable

  • Solve auxiliary equation for intersection points

Pole-Zero Analysis

Effect of Poles and Zeros on Response

Pole Effects

  • Real poles: Exponential terms \(e^{-\sigma t}\)

  • Complex poles: Oscillatory terms \(e^{-\sigma t}\cos(\omega t + \phi)\)

  • Dominant poles: Closest to imaginary axis (slowest modes)

  • Distance from origin: Related to natural frequency

Zero Effects

  • LHP zeros: Reduce overshoot, speed up response

  • RHP zeros: Cause undershoot, non-minimum phase behavior

  • Near pole-zero cancellation: Reduces effect of both

  • Zeros at origin: Derivative action

Design Guidelines

  • Place poles in LHP for stability

  • Use zeros to shape transient response

  • Avoid pole-zero cancellation in RHP

  • Consider sensitivity to parameter variations

Performance Specifications

Time Domain Performance Specifications

Standard Specifications

  • Rise time (\(t_r\)): Time to go from 10% to 90% of final value

  • Peak time (\(t_p\)): Time to reach maximum overshoot

  • Settling time (\(t_s\)): Time to stay within 2% or 5% of final value

  • Maximum overshoot (\(M_p\)): Maximum deviation from final value

Relationships for Second-Order Systems

  • Faster rise time \(\Rightarrow\) Higher \(\omega_n\)

  • Lower overshoot \(\Rightarrow\) Higher \(\zeta\)

  • Faster settling \(\Rightarrow\) Higher \(\zeta\omega_n\)

  • Trade-off between speed and overshoot

Typical Design Requirements

  • \(M_p < 20\%\) \(\Rightarrow\) \(\zeta > 0.45\)

  • \(t_s < 4\) sec \(\Rightarrow\) \(\zeta\omega_n > 1\)

GATE Questions

GATE Practice Questions

Example 1

A unity feedback system has \(G(s) = \dfrac{K}{s(s+3)}\). The value of \(K\) for damping ratio \(\zeta = 0.5\) is:

  1. 1

  2. 3

  3. 9

  4. 27

Solution

Characteristic equation: \(s^2 + 3s + K = 0\)
Standard form: \(s^2 + 2\zeta\omega_n s + \omega_n^2 = 0\)
Compare: \(\omega_n^2 = K\), \(2\zeta\omega_n = 3\)
For \(\zeta = 0.5\): \(\omega_n = \dfrac{3}{2 \times 0.5} = 3\)
Thus, \(K = \omega_n^2 = 9\). Answer: C

GATE Practice Questions (Contd.)

Example 2

The steady-state error of a unity feedback system with \(G(s) = \dfrac{10}{s(s+2)}\) to a unit ramp input is:

  1. 0

  2. 0.2

  3. 0.5

  4. \(\infty\)

Solution

System type = 1 (one integrator)
Velocity constant: \(K_v = \lim_{s\to 0} s \cdot \dfrac{10}{s(s+2)} = \dfrac{10}{2} = 5\)
Steady-state error for ramp: \(e_{ss} = \dfrac{1}{K_v} = \dfrac{1}{5} = 0.2\)
Answer: B

GATE Practice Questions (Contd.)

Example 3

For the characteristic equation \(s^3 + 4s^2 + Ks + 4 = 0\), the range of \(K\) for stability is:

  1. \(K > 0\)

  2. \(0 < K < 4\)

  3. \(0 < K < 16\)

  4. \(K > 4\)

Solution

Routh array:

\[\begin{array}{c|cc} s^3 & 1 & K \\ s^2 & 4 & 4 \\ s^1 & \dfrac{4K-4}{4} = K-1 & 0 \\ s^0 & 4 & \\ \end{array}\]
For stability: All first column elements must be positive
\(K-1 > 0 \Rightarrow K > 1\) and \(4 > 0\) (always satisfied)
Also need \(K > 0\). Therefore: \(K > 1\). Answer: None exactly matches, but closest is A

GATE Practice Questions (Contd.)

Example 4

The peak overshoot of a second-order system with \(\zeta = 0.6\) is approximately:

  1. 9.5%

  2. 15.2%

  3. 25.4%

  4. 30.8%

Solution

Peak overshoot: \(M_p = e^{-\pi\zeta/\sqrt{1 -\zeta^2}} \times 100\%\)

For \(\zeta = 0.6\): \(M_p = e^{-\pi \times 0.6/\sqrt{1-0.6^2}} \times 100\%\) \(M_p = e^{-\pi \times 0.6/\sqrt{0.64}} \times 100\%\) \(M_p = e^{-\pi \times 0.6/0.8} \times 100\%\) \(M_p = e^{-2.356} \times 100\% = 0.095 \times 100\% = 9.5\%\)

Answer: A

Summary

Summary

Key Takeaways

  • Transfer function analysis provides complete system characterization

  • Pole locations determine stability and transient response nature

  • Time domain specifications help in controller design

  • Steady-state error analysis ensures accuracy requirements

  • Routh-Hurwitz criterion is essential for stability analysis

  • Root locus technique shows parameter sensitivity

GATE Preparation Tips

  • Practice standard forms and their responses

  • Master Routh-Hurwitz for all special cases

  • Understand pole-zero effects on system behavior

  • Focus on steady-state error calculations

  • Learn root locus construction rules

Important Formulas to Remember

  • Second-order parameters: \(M_p\), \(t_p\), \(t_s\), \(t_r\)

  • Error constants: \(K_p\), \(K_v\), \(K_a\)

  • Stability conditions and Routh array