Control Systems: Stability and Frequency Response – GATE EE Detailed Notes

Introduction

Stability and Frequency Response

Key Topics

  • Stability concepts and definitions

  • Routh-Hurwitz stability criterion

  • Nyquist stability criterion

  • Relative stability: Gain and phase margins

  • Bode plots and frequency response

  • Polar plots and Nichols charts

  • Root locus and stability analysis

Objective

Understand frequency-domain and time-domain stability analysis for GATE EE preparation.

Stability Concepts

Stability Concepts

BIBO Stability

A system is Bounded-Input Bounded-Output (BIBO) stable if every bounded input produces a bounded output.

Necessary Condition

All poles of the transfer function must lie in the left half of the s-plane (\(\text{Re}(s) < 0\)).

Types of Stability

  • Absolute: Determines if the system is stable (no RHP poles).

  • Relative: Quantifies how stable using gain and phase margins.

  • Marginal: Poles on \(j\omega\)-axis (oscillatory response).

Routh-Hurwitz Criterion

Routh-Hurwitz Stability Criterion

Characteristic Equation

For a system with characteristic equation:

\[s^n + a_1 s^{n-1} + a_2 s^{n-2} + \cdots + a_{n-1} s + a_n = 0\]

Routh Table

\(s^n\) 1 \(a_2\)
\(s^{n-1}\) \(a_1\) \(a_3\)
\(s^{n-2}\) \(b_1\) \(b_2\)
\(\vdots\) \(\vdots\) \(\vdots\)
\(s^1\) \(c_1\) 0
\(s^0\) \(d_1\)

where \(b_1 = \dfrac{a_1 a_2 - a_3}{a_1}\)

Stability Condition

System is stable if all elements in the first column have the same sign (usually positive).

Special Cases

  • Zero in first column: Replace with small \(\epsilon > 0\)

  • Entire row zero: Use auxiliary equation

Routh-Hurwitz Special Cases - Detailed

Case 1: Zero in First Column

Replace zero with small positive number \(\epsilon\) and continue.

  • Sign changes in first column indicate RHP poles

  • Number of sign changes = Number of RHP poles

Case 2: Entire Row of Zeros

  • Form auxiliary equation from row above zeros

  • Differentiate auxiliary equation to get next row

  • Auxiliary equation roots are also characteristic equation roots

Example: Row of Zeros

For \(s^4 + 2s^3 + 3s^2 + 2s + 1 = 0\):

  • If \(s^2\) row becomes all zeros

  • Auxiliary equation: \(2s^2 + 2 = 0\)

  • Differentiate: \(4s\), so next row is \([4, 0]\)

Routh-Hurwitz Example

Example

Determine stability of \(s^3 + 2s^2 + 3s + K = 0\) for different values of \(K\).

Routh Table

\(s^3\) 1 3
\(s^2\) 2 \(K\)
\(s^1\) \(\dfrac{6-K}{2}\) 0
\(s^0\) \(K\)

Stability Conditions

For stability, all first column elements must be positive:

  • \(K > 0\)

  • \(\dfrac{6-K}{2} > 0 \Rightarrow K < 6\)

Therefore: \(0 < K < 6\)

Critical Values

  • \(K = 0\): One pole at origin (marginally stable)

  • \(K = 6\): Two poles on \(j\omega\)-axis (marginally stable)

  • \(K > 6\): Unstable (poles in RHP)

Nyquist Criterion

Nyquist Stability Criterion

Fundamental Principle

For a closed-loop system with loop transfer function \(G(s)H(s)\):

\[Z = N + P\]
where:

  • \(Z\): Number of closed-loop poles in the right half-plane (RHP)

  • \(N\): Net number of clockwise encirclements of \(-1+j0\) by \(G(s)H(s)\)

  • \(P\): Number of open-loop poles in the RHP

Stable if \(Z = 0\).

Example

For \(G(s)H(s) = \dfrac{K}{s(s+1)(s+2)}\):

  • Poles: \(s = 0, -1, -2 \Rightarrow P = 0\)

  • For stability: \(N = 0 \Rightarrow Z = 0\)

  • Critical point: \((-1, j0)\)

Nyquist Criterion - Special Cases

Poles on \(j\omega\)-axis

When open-loop has poles on \(j\omega\)-axis, use small semicircular detour in RHP.

Example

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

  • Pole at origin requires semicircular detour

  • Contribution: \(-90^{\circ}\) to \(-270^{\circ}\) (clockwise)

  • Net effect: \(-180^{\circ}\) phase shift

Stability Assessment

Count encirclements carefully considering the direction (clockwise = positive N).

Nyquist Criterion - Marginal Stability

Critical Cases

  • Nyquist plot passes through \((-1, j0)\): Marginally stable

  • Multiple crossings: Count net encirclements

  • Tangent to unit circle: Critical gain condition

Gain Margin from Nyquist

  • GM = Distance from origin to point where Nyquist plot crosses negative real axis

  • GM in dB = \(20\log(\text{GM})\)

  • If plot crosses at \(-1/a\), then GM = \(a\)

Phase Margin from Nyquist

  • PM = Angular distance from \((-1, j0)\) to unit circle crossing

  • Measured from negative real axis

  • Positive PM indicates stable system

Bode Plots

Bode Plots and Frequency Response

Construction Rules

  • Magnitude: \(20 \log |G(j\omega)|\) in dB

  • Phase: \(\angle G(j\omega)\) in degrees

  • Slope: \(\pm 20n\) dB/decade for \(n\) poles/zeros

  • Corner frequency: At pole/zero locations

Standard Factors

  • Constant \(K\): \(20\log K\) dB, \(0^{\circ}\)

  • Pole at origin: \(-20\log\omega\) dB, \(-90^{\circ}\)

  • Simple pole: \(-20\log\sqrt{1+(\omega/\omega_p)^2}\) dB

Bode Plot - Quadratic Factors

Second-Order Systems

For \(G(s) = \dfrac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\):

  • Corner frequency: \(\omega_n\)

  • Resonant peak: \(M_r = \dfrac{1}{2\zeta\sqrt{1-\zeta^2}}\) for \(\zeta < 0.707\)

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

Damping Effects

  • \(\zeta > 0.707\): No resonance

  • \(\zeta = 0.707\): Maximally flat response

  • \(\zeta < 0.707\): Resonant peak present

  • \(\zeta \to 0\): Sharp resonance

Asymptotic Behavior

  • Low frequency: \(0\) dB/decade

  • High frequency: \(-40\) dB/decade

  • Phase: \(0^{\circ}\) to \(-180^{\circ}\)

  • Steepest slope at \(\omega_n\)

Bode Plot Examples

Example

Draw Bode plot for \(G(s) = \dfrac{100}{s(s+10)}\)

Analysis

  • \(K = 100 \Rightarrow 20\log(100) = 40\) dB

  • Pole at origin: \(-20\) dB/dec, \(-90^{\circ}\)

  • Pole at \(s = -10\): corner at \(\omega = 10\)

  • Overall: \(-40\) dB/dec after \(\omega = 10\)

Key Points

  • At \(\omega = 1\): \(40 - 20\log(1) = 40\) dB

  • At \(\omega = 10\): \(40 - 20\log(10) = 20\) dB

  • Phase starts at \(-90^{\circ}\), ends at \(-180^{\circ}\)

Minimum Phase Systems

Systems with all poles and zeros in LHP are minimum phase.

  • Magnitude uniquely determines phase

  • Bode plots are sufficient for stability analysis

Non-minimum Phase

Systems with RHP zeros:

  • Additional phase lag

  • Magnitude alone insufficient

  • Examples: Systems with time delay

Stability Margins

Gain and Phase Margins

Definitions

  • Gain Margin (GM):

    \[GM = 20 \log \left( \dfrac{1}{|G(j\omega_{pc})|} \right) \text{ dB}\]
    where \(\angle G(j\omega_{pc}) = -180^{\circ}\) (phase crossover)

  • Phase Margin (PM):

    \[PM = 180^{\circ} + \angle G(j\omega_{gc})\]
    where \(|G(j\omega_{gc})| = 1\) (gain crossover)

Design Requirements

  • Good design: PM \(> 45^{\circ}\), GM \(> 6\) dB

  • Minimum acceptable: PM \(> 30^{\circ}\), GM \(> 3\) dB

  • Excellent design: PM \(= 60^{\circ}\), GM \(= 10\) dB

Stability Margins from Bode Plots

Reading from Bode Plot

  • GM: Negative of magnitude at phase crossover frequency

  • PM: \(180^{\circ} +\) phase at gain crossover frequency

  • Easier to read than Nyquist plot

Relationship Between Margins and Transient Response

Phase Margin and Damping

For second-order systems:

\[PM \approx 100 \times \zeta \text{ (degrees)}\]

  • Higher PM \(\Rightarrow\) better damping

  • PM \(= 60^{\circ}\) corresponds to \(\zeta \approx 0.6\)

  • PM \(< 30^{\circ}\) leads to excessive overshoot

Bandwidth and Speed

  • Bandwidth: Frequency where closed-loop gain drops by 3 dB

  • Relation: \(\omega_{BW} \approx \omega_{gc}\) for well-designed systems

  • Higher bandwidth \(\Rightarrow\) faster response

Design Trade-offs

  • Stability vs Speed: Higher gain increases speed but reduces margins

  • Bandwidth vs Noise: Wider bandwidth admits more noise

  • Steady-state vs Transient: Different requirements for each

Root Locus

Root Locus and Stability

Root Locus Method

  • Plot of closed-loop pole locations as gain \(K\) varies

  • Characteristic equation: \(1 + KG(s)H(s) = 0\)

  • Direct visualization of stability boundaries

Stability Analysis

  • System stable when all poles in LHP

  • Critical gain: when poles cross \(j\omega\)-axis

  • Marginal stability: poles on \(j\omega\)-axis

Root Locus Construction Rules

Basic Rules

  • Starting points: Open-loop poles (K=0)

  • Ending points: Open-loop zeros (K=\(\infty\))

  • Asymptotes: \(\dfrac{180^{\circ}(2k+1)}{n-m}\) for \(k = 0, 1, \ldots, n-m-1\)

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

Advanced Rules

  • Breakaway points: Solve \(\dfrac{dK}{ds} = 0\)

  • Angle condition: \(\sum \theta_{\text{zeros}} - \sum \theta_{\text{poles}} = 180^{\circ}(2k+1)\)

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

Design Implications

  • Poles moving toward RHP indicate decreasing stability

  • Complex poles indicate oscillatory behavior

  • Dominant poles determine system response

Polar and Nichols

Polar Plots and Nichols Charts

Polar Plot Characteristics

  • Plot of \(G(j\omega)\) in complex plane

  • Frequency parameter along curve

  • Direct application of Nyquist criterion

  • Shows both magnitude and phase

Nichols Chart

  • Magnitude (dB) vs Phase (degrees)

  • Constant \(M\) and \(N\) circles

  • Closed-loop frequency response

  • Design tool for gain-phase relationship

Applications

  • Polar: Nyquist stability, gain/phase margins

  • Nichols: Closed-loop design, bandwidth determination

Closed-Loop Frequency Response

M and N Circles

For unity feedback system with \(T(j\omega) = \dfrac{G(j\omega)}{1 + G(j\omega)}\):

  • M-circles: Constant magnitude loci

  • N-circles: Constant phase loci

  • Critical M-circle: \(M = 1\) (0 dB)

Bandwidth from Nichols Chart

  • Bandwidth: Frequency where \(|T(j\omega)| = -3\) dB

  • Resonant peak: Maximum value of \(|T(j\omega)|\)

  • Resonant frequency: Frequency of maximum magnitude

Design Guidelines

  • \(M_r < 1.3\) (2.3 dB) for good transient response

  • Avoid intersection with critical circles

  • Higher bandwidth requires careful noise consideration

Frequency Response Specifications

Frequency Response Specifications

Key Parameters

  • Bandwidth (\(\omega_B\)): Frequency where \(|G(j\omega)|\) drops to \(-3\) dB

  • Resonant frequency (\(\omega_r\)): Peak magnitude frequency

  • Resonant peak (\(M_r\)): Maximum magnitude

  • Cut-off rate: Roll-off slope beyond bandwidth

Design Relationships

  • Higher bandwidth \(\Rightarrow\) faster response

  • Lower \(M_r\) \(\Rightarrow\) better stability

  • Trade-off between speed and stability

Disturbance Rejection and Noise Characteristics

Low-Frequency Disturbance Rejection

  • Disturbance rejection: Inversely proportional to \(|1 + G(j\omega)H(j\omega)|\)

  • High loop gain: Better disturbance rejection at low frequencies

  • Steady-state error: Related to loop gain at DC

High-Frequency Noise

  • Noise amplification: Depends on closed-loop bandwidth

  • Roll-off rate: Steeper roll-off reduces noise

  • Sensor noise: Amplified by feedback at high frequencies

Design Compromise

  • Low frequencies: High gain for disturbance rejection

  • High frequencies: Low gain for noise rejection

  • Crossover region: Critical for stability margins

GATE Questions

GATE Practice Questions

Example

The Routh table for the characteristic equation \(s^3 + 4s^2 + 5s + K = 0\) is:

\(s^3\) 1 5
\(s^2\) 4 \(K\)
\(s^1\) \(\dfrac{20-K}{4}\) 0
\(s^0\) \(K\)

The range of \(K\) for stability is:

  1. \(0 < K < 20\)

  2. \(0 < K < 5\)

  3. \(K > 20\)

  4. \(K > 5\)

Solution

For stability, all elements in first column must be positive:

  • \(K > 0\)

  • \(\dfrac{20-K}{4} > 0 \Rightarrow K < 20\)

Therefore: \(0 < K < 20\) Correct answer: A

GATE Practice Questions (Contd.)

Example

A unity feedback system has \(G(s) = \dfrac{10}{s(s+2)}\). The phase margin is:

  1. \(45^{\circ}\)

  2. \(53^{\circ}\)

  3. \(60^{\circ}\)

  4. \(\infty\)

Solution

Gain crossover frequency (\(|G(j\omega_g)| = 1\)):

\[\left| \dfrac{10}{j\omega_g (j\omega_g + 2)} \right| = \dfrac{10}{\omega_g \sqrt{\omega_g^2 + 4}} = 1\]
Solving: \(\omega_g^2 (\omega_g^2 + 4) = 100 \Rightarrow \omega_g \approx 3.1\) rad/s

Phase at \(\omega_g\):

\[\angle G(j\omega_g) = -90^{\circ} - \tan^{-1}\left(\dfrac{\omega_g}{2}\right) \approx -90^{\circ} - \tan^{-1}(1.55) \approx -127^{\circ}\]
Phase margin: \(PM = 180^{\circ} + (-127^{\circ}) = 53^{\circ}\) Correct answer: B

GATE Practice Questions (Contd.)

Example

The Nyquist plot of \(G(s)H(s) = \dfrac{K}{s(s+1)(s+2)}\) encircles the \((-1, j0)\) point once in the clockwise direction. The number of closed-loop poles in the RHP is:

  1. 0

  2. 1

  3. 2

  4. 3

Solution

Using Nyquist criterion: \(Z = N + P\)

  • \(P = 0\) (no open-loop poles in RHP)

  • \(N = +1\) (one clockwise encirclement)

  • Therefore: \(Z = 1 + 0 = 1\)

Correct answer: B

GATE Practice Questions (Contd.)

Example

A second-order system has the closed-loop transfer function:

\[T(s) = \dfrac{25}{s^2 + 4s + 25}\]
The damping ratio \(\zeta\) and natural frequency \(\omega_n\) are:

  1. \(\zeta = 0.4\), \(\omega_n = 5\) rad/s

  2. \(\zeta = 0.2\), \(\omega_n = 5\) rad/s

  3. \(\zeta = 0.4\), \(\omega_n = 25\) rad/s

  4. \(\zeta = 0.2\), \(\omega_n = 25\) rad/s

Solution

Comparing with standard form \(T(s) = \dfrac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\):

  • \(\omega_n^2 = 25 \Rightarrow \omega_n = 5\) rad/s

  • \(2\zeta\omega_n = 4 \Rightarrow 2\zeta(5) = 4 \Rightarrow \zeta = 0.4\)

Correct answer: A

Summary

Summary

Key Concepts Covered

  • Stability Definitions: BIBO stability, pole locations

  • Routh-Hurwitz: Algebraic stability test, special cases

  • Nyquist Criterion: Graphical stability analysis

  • Bode Plots: Frequency response, gain/phase margins

  • Root Locus: Pole placement, parameter variation

  • Polar/Nichols: Alternative frequency domain tools

Design Guidelines

  • Phase margin \(> 45^{\circ}\), Gain margin \(> 6\) dB

  • Proper balance between speed and stability

  • Consider noise and disturbance rejection

  • Use multiple analysis tools for verification