Introduction
Stability and Frequency Response
Key Topics
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Stability concepts and definitions
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Routh-Hurwitz stability criterion
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Nyquist stability criterion
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Relative stability: Gain and phase margins
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Bode plots and frequency response
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Polar plots and Nichols charts
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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
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Absolute: Determines if the system is stable (no RHP poles).
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Relative: Quantifies how stable using gain and phase margins.
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Marginal: Poles on \(j\omega\)-axis (oscillatory response).
Routh-Hurwitz Criterion
Routh-Hurwitz Stability Criterion
Characteristic Equation
For a system with characteristic equation:
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
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Zero in first column: Replace with small \(\epsilon > 0\)
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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.
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Sign changes in first column indicate RHP poles
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Number of sign changes = Number of RHP poles
Case 2: Entire Row of Zeros
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Form auxiliary equation from row above zeros
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Differentiate auxiliary equation to get next row
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Auxiliary equation roots are also characteristic equation roots
Example: Row of Zeros
For \(s^4 + 2s^3 + 3s^2 + 2s + 1 = 0\):
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If \(s^2\) row becomes all zeros
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Auxiliary equation: \(2s^2 + 2 = 0\)
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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:
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\(K > 0\)
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\(\dfrac{6-K}{2} > 0 \Rightarrow K < 6\)
Therefore: \(0 < K < 6\)
Critical Values
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\(K = 0\): One pole at origin (marginally stable)
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\(K = 6\): Two poles on \(j\omega\)-axis (marginally stable)
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\(K > 6\): Unstable (poles in RHP)
Nyquist Criterion
Nyquist Stability Criterion
Fundamental Principle
For a closed-loop system with loop transfer function \(G(s)H(s)\):
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\(Z\): Number of closed-loop poles in the right half-plane (RHP)
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\(N\): Net number of clockwise encirclements of \(-1+j0\) by \(G(s)H(s)\)
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\(P\): Number of open-loop poles in the RHP
Stable if \(Z = 0\).
Example
For \(G(s)H(s) = \dfrac{K}{s(s+1)(s+2)}\):
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Poles: \(s = 0, -1, -2 \Rightarrow P = 0\)
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For stability: \(N = 0 \Rightarrow Z = 0\)
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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 \(G(s)H(s) = \dfrac{K}{s(s+1)}\):
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Pole at origin requires semicircular detour
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Contribution: \(-90^{\circ}\) to \(-270^{\circ}\) (clockwise)
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Net effect: \(-180^{\circ}\) phase shift
Stability Assessment
Count encirclements carefully considering the direction (clockwise = positive N).
Nyquist Criterion - Marginal Stability
Critical Cases
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Nyquist plot passes through \((-1, j0)\): Marginally stable
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Multiple crossings: Count net encirclements
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Tangent to unit circle: Critical gain condition
Gain Margin from Nyquist
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GM = Distance from origin to point where Nyquist plot crosses negative real axis
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GM in dB = \(20\log(\text{GM})\)
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If plot crosses at \(-1/a\), then GM = \(a\)
Phase Margin from Nyquist
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PM = Angular distance from \((-1, j0)\) to unit circle crossing
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Measured from negative real axis
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Positive PM indicates stable system
Bode Plots
Bode Plots and Frequency Response
Construction Rules
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Magnitude: \(20 \log |G(j\omega)|\) in dB
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Phase: \(\angle G(j\omega)\) in degrees
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Slope: \(\pm 20n\) dB/decade for \(n\) poles/zeros
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Corner frequency: At pole/zero locations
Standard Factors
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Constant \(K\): \(20\log K\) dB, \(0^{\circ}\)
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Pole at origin: \(-20\log\omega\) dB, \(-90^{\circ}\)
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Simple pole: \(-20\log\sqrt{1+(\omega/\omega_p)^2}\) dB
Bode Plot - Quadratic Factors
Second-Order Systems
For \(G(s) = \dfrac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\):
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Corner frequency: \(\omega_n\)
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Resonant peak: \(M_r = \dfrac{1}{2\zeta\sqrt{1-\zeta^2}}\) for \(\zeta < 0.707\)
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Resonant frequency: \(\omega_r = \omega_n\sqrt{1-2\zeta^2}\)
Damping Effects
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\(\zeta > 0.707\): No resonance
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\(\zeta = 0.707\): Maximally flat response
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\(\zeta < 0.707\): Resonant peak present
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\(\zeta \to 0\): Sharp resonance
Asymptotic Behavior
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Low frequency: \(0\) dB/decade
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High frequency: \(-40\) dB/decade
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Phase: \(0^{\circ}\) to \(-180^{\circ}\)
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Steepest slope at \(\omega_n\)
Bode Plot Examples
Example
Draw Bode plot for \(G(s) = \dfrac{100}{s(s+10)}\)
Analysis
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\(K = 100 \Rightarrow 20\log(100) = 40\) dB
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Pole at origin: \(-20\) dB/dec, \(-90^{\circ}\)
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Pole at \(s = -10\): corner at \(\omega = 10\)
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Overall: \(-40\) dB/dec after \(\omega = 10\)
Key Points
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At \(\omega = 1\): \(40 - 20\log(1) = 40\) dB
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At \(\omega = 10\): \(40 - 20\log(10) = 20\) dB
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Phase starts at \(-90^{\circ}\), ends at \(-180^{\circ}\)
Minimum Phase Systems
Systems with all poles and zeros in LHP are minimum phase.
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Magnitude uniquely determines phase
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Bode plots are sufficient for stability analysis
Non-minimum Phase
Systems with RHP zeros:
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Additional phase lag
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Magnitude alone insufficient
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Examples: Systems with time delay
Stability Margins
Gain and Phase Margins
Definitions
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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
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Good design: PM \(> 45^{\circ}\), GM \(> 6\) dB
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Minimum acceptable: PM \(> 30^{\circ}\), GM \(> 3\) dB
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Excellent design: PM \(= 60^{\circ}\), GM \(= 10\) dB
Stability Margins from Bode Plots
Reading from Bode Plot
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GM: Negative of magnitude at phase crossover frequency
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PM: \(180^{\circ} +\) phase at gain crossover frequency
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Easier to read than Nyquist plot
Relationship Between Margins and Transient Response
Phase Margin and Damping
For second-order systems:
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Higher PM \(\Rightarrow\) better damping
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PM \(= 60^{\circ}\) corresponds to \(\zeta \approx 0.6\)
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PM \(< 30^{\circ}\) leads to excessive overshoot
Bandwidth and Speed
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Bandwidth: Frequency where closed-loop gain drops by 3 dB
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Relation: \(\omega_{BW} \approx \omega_{gc}\) for well-designed systems
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Higher bandwidth \(\Rightarrow\) faster response
Design Trade-offs
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Stability vs Speed: Higher gain increases speed but reduces margins
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Bandwidth vs Noise: Wider bandwidth admits more noise
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Steady-state vs Transient: Different requirements for each
Root Locus
Root Locus and Stability
Root Locus Method
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Plot of closed-loop pole locations as gain \(K\) varies
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Characteristic equation: \(1 + KG(s)H(s) = 0\)
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Direct visualization of stability boundaries
Stability Analysis
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System stable when all poles in LHP
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Critical gain: when poles cross \(j\omega\)-axis
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Marginal stability: poles on \(j\omega\)-axis
Root Locus Construction Rules
Basic Rules
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Starting points: Open-loop poles (K=0)
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Ending points: Open-loop zeros (K=\(\infty\))
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Asymptotes: \(\dfrac{180^{\circ}(2k+1)}{n-m}\) for \(k = 0, 1, \ldots, n-m-1\)
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Centroid: \(\sigma_a = \dfrac{\sum \text{poles} - \sum \text{zeros}}{n-m}\)
Advanced Rules
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Breakaway points: Solve \(\dfrac{dK}{ds} = 0\)
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Angle condition: \(\sum \theta_{\text{zeros}} - \sum \theta_{\text{poles}} = 180^{\circ}(2k+1)\)
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Magnitude condition: \(K = \dfrac{\prod |s-z_i|}{\prod |s-p_i|}\)
Design Implications
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Poles moving toward RHP indicate decreasing stability
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Complex poles indicate oscillatory behavior
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Dominant poles determine system response
Polar and Nichols
Polar Plots and Nichols Charts
Polar Plot Characteristics
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Plot of \(G(j\omega)\) in complex plane
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Frequency parameter along curve
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Direct application of Nyquist criterion
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Shows both magnitude and phase
Nichols Chart
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Magnitude (dB) vs Phase (degrees)
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Constant \(M\) and \(N\) circles
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Closed-loop frequency response
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Design tool for gain-phase relationship
Applications
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Polar: Nyquist stability, gain/phase margins
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Nichols: Closed-loop design, bandwidth determination
Closed-Loop Frequency Response
M and N Circles
For unity feedback system with \(T(j\omega) = \dfrac{G(j\omega)}{1 + G(j\omega)}\):
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M-circles: Constant magnitude loci
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N-circles: Constant phase loci
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Critical M-circle: \(M = 1\) (0 dB)
Bandwidth from Nichols Chart
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Bandwidth: Frequency where \(|T(j\omega)| = -3\) dB
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Resonant peak: Maximum value of \(|T(j\omega)|\)
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Resonant frequency: Frequency of maximum magnitude
Design Guidelines
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\(M_r < 1.3\) (2.3 dB) for good transient response
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Avoid intersection with critical circles
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Higher bandwidth requires careful noise consideration
Frequency Response Specifications
Frequency Response Specifications
Key Parameters
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Bandwidth (\(\omega_B\)): Frequency where \(|G(j\omega)|\) drops to \(-3\) dB
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Resonant frequency (\(\omega_r\)): Peak magnitude frequency
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Resonant peak (\(M_r\)): Maximum magnitude
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Cut-off rate: Roll-off slope beyond bandwidth
Design Relationships
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Higher bandwidth \(\Rightarrow\) faster response
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Lower \(M_r\) \(\Rightarrow\) better stability
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Trade-off between speed and stability
Disturbance Rejection and Noise Characteristics
Low-Frequency Disturbance Rejection
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Disturbance rejection: Inversely proportional to \(|1 + G(j\omega)H(j\omega)|\)
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High loop gain: Better disturbance rejection at low frequencies
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Steady-state error: Related to loop gain at DC
High-Frequency Noise
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Noise amplification: Depends on closed-loop bandwidth
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Roll-off rate: Steeper roll-off reduces noise
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Sensor noise: Amplified by feedback at high frequencies
Design Compromise
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Low frequencies: High gain for disturbance rejection
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High frequencies: Low gain for noise rejection
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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:
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\(0 < K < 20\)
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\(0 < K < 5\)
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\(K > 20\)
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\(K > 5\)
Solution
For stability, all elements in first column must be positive:
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\(K > 0\)
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\(\dfrac{20-K}{4} > 0 \Rightarrow K < 20\)
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:
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\(45^{\circ}\)
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\(53^{\circ}\)
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\(60^{\circ}\)
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\(\infty\)
Solution
Gain crossover frequency (\(|G(j\omega_g)| = 1\)):
Phase at \(\omega_g\):
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:
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0
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1
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2
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3
Solution
Using Nyquist criterion: \(Z = N + P\)
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\(P = 0\) (no open-loop poles in RHP)
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\(N = +1\) (one clockwise encirclement)
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Therefore: \(Z = 1 + 0 = 1\)
Correct answer: B
GATE Practice Questions (Contd.)
Example
A second-order system has the closed-loop transfer function:
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\(\zeta = 0.4\), \(\omega_n = 5\) rad/s
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\(\zeta = 0.2\), \(\omega_n = 5\) rad/s
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\(\zeta = 0.4\), \(\omega_n = 25\) rad/s
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\(\zeta = 0.2\), \(\omega_n = 25\) rad/s
Solution
Comparing with standard form \(T(s) = \dfrac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\):
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\(\omega_n^2 = 25 \Rightarrow \omega_n = 5\) rad/s
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\(2\zeta\omega_n = 4 \Rightarrow 2\zeta(5) = 4 \Rightarrow \zeta = 0.4\)
Correct answer: A
Summary
Summary
Key Concepts Covered
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Stability Definitions: BIBO stability, pole locations
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Routh-Hurwitz: Algebraic stability test, special cases
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Nyquist Criterion: Graphical stability analysis
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Bode Plots: Frequency response, gain/phase margins
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Root Locus: Pole placement, parameter variation
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Polar/Nichols: Alternative frequency domain tools
Design Guidelines
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Phase margin \(> 45^{\circ}\), Gain margin \(> 6\) dB
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Proper balance between speed and stability
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Consider noise and disturbance rejection
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Use multiple analysis tools for verification