GATE EE

Control Systems: Stability and Frequency Response â€“ GATE EE Detailed Notes

Lecture Notes

SEC 01

Introduction

1Stability and Frequency Response

1Key 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

1Objective

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

SEC 02

Stability Concepts

1Stability Concepts

1BIBO Stability

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

1Necessary Condition

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

1Types 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).

SEC 03

Routh-Hurwitz Criterion

1Routh-Hurwitz Stability Criterion

1Characteristic Equation

For a system with characteristic equation:

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

1Routh 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 \(b_1 = \dfrac{a_1 a_2 - a_3}{a_1}\)

1Stability Condition

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

1Special Cases

Zero in first column: Replace with small \(\epsilon > 0\)
Entire row zero: Use auxiliary equation

1Routh-Hurwitz Special Cases - Detailed

1Case 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

1Case 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

1Example: 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 \([4, 0]\)

1Routh-Hurwitz Example

1Example

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

1Routh Table

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

1Stability Conditions

For stability, all first column elements must be positive:

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

Therefore: \(0 < K < 6\)

1Critical 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)

SEC 04

Nyquist Criterion

1Nyquist Stability Criterion

1Fundamental 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 \(Z = 0\).

1Example

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: \((-1, j0)\)

1Nyquist Criterion - Special Cases

1Poles on \(j\omega\)-axis

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

1Example

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

1Stability Assessment

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

1Nyquist Criterion - Marginal Stability

1Critical Cases

Nyquist plot passes through \((-1, j0)\): Marginally stable
Multiple crossings: Count net encirclements
Tangent to unit circle: Critical gain condition

1Gain 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 = \(a\)

1Phase Margin from Nyquist

PM = Angular distance from \((-1, j0)\) to unit circle crossing
Measured from negative real axis
Positive PM indicates stable system

SEC 05

Bode Plots

1Bode Plots and Frequency Response

1Construction 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

1Standard 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

1Bode Plot - Quadratic Factors

1Second-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: \(\omega_r = \omega_n\sqrt{1-2\zeta^2}\)

1Damping Effects

\(\zeta > 0.707\): No resonance
\(\zeta = 0.707\): Maximally flat response
\(\zeta < 0.707\): Resonant peak present
\(\zeta \to 0\): Sharp resonance

1Asymptotic Behavior

Low frequency: \(0\) dB/decade
High frequency: \(-40\) dB/decade
Phase: \(0^{\circ}\) to \(-180^{\circ}\)
Steepest slope at \(\omega_n\)

1Bode Plot Examples

1Example

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

1Analysis

\(K = 100 \Rightarrow 20\log(100) = 40\) dB
Pole at origin: \(-20\) dB/dec, \(-90^{\circ}\)
Pole at \(s = -10\): corner at \(\omega = 10\)
Overall: \(-40\) dB/dec after \(\omega = 10\)

1Key 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 \(-180^{\circ}\)

1Minimum Phase Systems

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

Magnitude uniquely determines phase
Bode plots are sufficient for stability analysis

1Non-minimum Phase

Systems with RHP zeros:

Additional phase lag
Magnitude alone insufficient
Examples: Systems with time delay

SEC 06

Stability Margins

1Gain and Phase Margins

1Definitions

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)

1Design 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

1Stability Margins from Bode Plots

1Reading 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

1Relationship Between Margins and Transient Response

1Phase 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

1Bandwidth 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

1Design 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

SEC 07

Root Locus

1Root Locus and Stability

1Root 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

1Stability Analysis

System stable when all poles in LHP
Critical gain: when poles cross \(j\omega\)-axis
Marginal stability: poles on \(j\omega\)-axis

1Root Locus Construction Rules

1Basic 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: \(\sigma_a = \dfrac{\sum \text{poles} - \sum \text{zeros}}{n-m}\)

1Advanced Rules

Breakaway points: Solve \(\dfrac{dK}{ds} = 0\)
Angle condition: \(\sum \theta_{\text{zeros}} - \sum \theta_{\text{poles}} = 180^{\circ}(2k+1)\)
Magnitude condition: \(K = \dfrac{\prod |s-z_i|}{\prod |s-p_i|}\)

1Design Implications

Poles moving toward RHP indicate decreasing stability
Complex poles indicate oscillatory behavior
Dominant poles determine system response

SEC 08

Polar and Nichols

1Polar Plots and Nichols Charts

1Polar Plot Characteristics

Plot of \(G(j\omega)\) in complex plane
Frequency parameter along curve
Direct application of Nyquist criterion
Shows both magnitude and phase

1Nichols Chart

Magnitude (dB) vs Phase (degrees)
Constant \(M\) and \(N\) circles
Closed-loop frequency response
Design tool for gain-phase relationship

1Applications

Polar: Nyquist stability, gain/phase margins
Nichols: Closed-loop design, bandwidth determination

1Closed-Loop Frequency Response

1M 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)

1Bandwidth 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

1Design Guidelines

\(M_r < 1.3\) (2.3 dB) for good transient response
Avoid intersection with critical circles
Higher bandwidth requires careful noise consideration

SEC 09

Frequency Response Specifications

1Frequency Response Specifications

1Key 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

1Design Relationships

Higher bandwidth \(\Rightarrow\) faster response
Lower \(M_r\) \(\Rightarrow\) better stability
Trade-off between speed and stability

1Disturbance Rejection and Noise Characteristics

1Low-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

1High-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

1Design Compromise

Low frequencies: High gain for disturbance rejection
High frequencies: Low gain for noise rejection
Crossover region: Critical for stability margins

SEC 10

GATE Questions

1GATE Practice Questions

1Example

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:

\(0 < K < 20\)
\(0 < K < 5\)
\(K > 20\)
\(K > 5\)

1Solution

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

1GATE Practice Questions (Contd.)

1Example

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

\(45^{\circ}\)
\(53^{\circ}\)
\(60^{\circ}\)
\(\infty\)

1Solution

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

1GATE Practice Questions (Contd.)

1Example

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:

1Solution

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

1GATE Practice Questions (Contd.)

1Example

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:

\(\zeta = 0.4\), \(\omega_n = 5\) rad/s
\(\zeta = 0.2\), \(\omega_n = 5\) rad/s
\(\zeta = 0.4\), \(\omega_n = 25\) rad/s
\(\zeta = 0.2\), \(\omega_n = 25\) rad/s

1Solution

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

SEC 11

Summary

1Summary

1Key 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

1Design 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