Formula Sheet: Stability Analysis

Table of Contents
1. Laplace Transform and Transfer Functions
2. Poles and Zeros
3. Routh-Hurwitz Stability Criterion
4. Root Locus Method
5. Nyquist Stability Criterion
6. Bode Plot Analysis
7. State-Space Stability Analysis
8. Time-Domain Specifications
9. Relative Stability
10. Advanced Stability Topics
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Laplace Transform and Transfer Functions

Laplace Transform Fundamentals

• Laplace Transform Definition: \[F(s) = \mathcal{L}\{f(t)\} = \int_0^\infty f(t)e^{-st}\,dt\] where s = complex frequency (σ + jω), f(t) = time-domain function, F(s) = frequency-domain function
• Common Laplace Transform Pairs:
  • Unit step: \(\mathcal{L}\{u(t)\} = \frac{1}{s}\)
  • Ramp: \(\mathcal{L}\{t\} = \frac{1}{s^2}\)
  • Exponential: \(\mathcal{L}\{e^{-at}\} = \frac{1}{s+a}\)
  • Sine: \(\mathcal{L}\{\sin(\omega t)\} = \frac{\omega}{s^2 + \omega^2}\)
  • Cosine: \(\mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2 + \omega^2}\)
• Final Value Theorem: \[f(\infty) = \lim_{s \to 0} sF(s)\] Valid only if sF(s) has no poles in the right half-plane or on the imaginary axis (except possibly a simple pole at origin)
• Initial Value Theorem: \[f(0^+) = \lim_{s \to \infty} sF(s)\]

Transfer Function

• Transfer Function Definition: \[G(s) = \frac{Y(s)}{X(s)}\] where Y(s) = output (Laplace transform), X(s) = input (Laplace transform), assuming zero initial conditions
• General Transfer Function Form: \[G(s) = K\frac{(s-z_1)(s-z_2)\cdots(s-z_m)}{(s-p_1)(s-p_2)\cdots(s-p_n)}\] where K = gain constant, z_i = zeros, p_i = poles
• Standard Second-Order Form: \[G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\] where ω_n = natural frequency (rad/s), ζ = damping ratio (dimensionless)

Poles and Zeros

Pole and Zero Definitions

• Poles: Values of s that make the denominator of G(s) equal to zero (transfer function becomes infinite)
• Zeros: Values of s that make the numerator of G(s) equal to zero (transfer function becomes zero)
• Characteristic Equation: \[1 + G(s)H(s) = 0\] or equivalently, the denominator of the closed-loop transfer function equals zero
  Roots of the characteristic equation are the closed-loop poles

Pole Locations and System Behavior

• Stability Criterion (Pole Location):
  • System is stable if all poles are in the left half of the s-plane (Re(s) <>
  • System is unstable if any pole is in the right half of the s-plane (Re(s) > 0)
  • System is marginally stable if poles are on the imaginary axis with no poles in right half-plane
• Real Pole Response: For pole at s = -a (where a > 0):
  • Time response contains term: \(e^{-at}\)
  • Exponential decay with time constant τ = 1/a
• Complex Conjugate Poles: For poles at s = -σ ± jω_d:
  • Time response contains: \(e^{-\sigma t}\sin(\omega_d t)\) and \(e^{-\sigma t}\cos(\omega_d t)\)
  • Damped oscillatory response
  • σ = damping coefficient (determines decay rate)
  • ω_d = damped natural frequency (rad/s)

Routh-Hurwitz Stability Criterion

Routh Array Construction

• Characteristic Equation Form: \[a_n s^n + a_{n-1}s^{n-1} + \cdots + a_1 s + a_0 = 0\] where all coefficients a_i are real constants
• Necessary Condition for Stability: All coefficients a_i must be positive (or all negative) and non-zero
  If this condition is not met, the system is unstable
• Routh Array Structure: \[\begin{array}{c|cccc} s^n & a_n & a_{n-2} & a_{n-4} & \cdots \\ s^{n-1} & a_{n-1} & a_{n-3} & a_{n-5} & \cdots \\ s^{n-2} & b_1 & b_2 & b_3 & \cdots \\ s^{n-3} & c_1 & c_2 & c_3 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \\ s^1 & * & & & \\ s^0 & * & & & \end{array}\]
• Routh Array Calculation: \[b_1 = \frac{a_{n-1}a_{n-2} - a_n a_{n-3}}{a_{n-1}}\] \[b_2 = \frac{a_{n-1}a_{n-4} - a_n a_{n-5}}{a_{n-1}}\] \[c_1 = \frac{b_1 a_{n-3} - a_{n-1}b_2}{b_1}\] And so on for subsequent rows

Routh-Hurwitz Stability Criterion

• Stability Rule: The number of sign changes in the first column of the Routh array equals the number of roots with positive real parts (unstable roots)
  For stability: zero sign changes in the first column
• Special Case 1 - Zero in First Column: If a zero appears in the first column but other elements in the row are non-zero:
  • Replace the zero with a small positive number ε
  • Continue the array construction
  • Evaluate the limit as ε → 0
• Special Case 2 - Entire Row of Zeros: If an entire row becomes zero:
  • Form an auxiliary polynomial from the previous row
  • Take the derivative of the auxiliary polynomial
  • Use coefficients of the derivative to replace the row of zeros
  • Roots of auxiliary polynomial lie on the imaginary axis (symmetric about origin)
• Auxiliary Polynomial: If row s^k has coefficients a, b, c, ..., then: \[A(s) = as^k + bs^{k-2} + cs^{k-4} + \cdots\]

Root Locus Method

Root Locus Fundamentals

• Root Locus Definition: Graphical plot showing the path of closed-loop poles as a system parameter (typically gain K) varies from 0 to ∞
• Standard Feedback System: Closed-loop transfer function: \[T(s) = \frac{KG(s)}{1 + KG(s)H(s)}\] where K = variable gain, G(s) = forward path transfer function, H(s) = feedback path transfer function
• Root Locus Equation: \[1 + KG(s)H(s) = 0\] or equivalently: \[KG(s)H(s) = -1\] This yields two conditions:
  • Magnitude condition: \(|KG(s)H(s)| = 1\)
  • Angle condition: \(\angle G(s)H(s) = (2k+1) \times 180°\), k = 0, ±1, ±2, ...

Root Locus Construction Rules

• Rule 1 - Number of Branches: Number of branches = number of poles of G(s)H(s) = n
• Rule 2 - Symmetry: Root locus is symmetric about the real axis
• Rule 3 - Real Axis Segments: Root locus exists on a segment of the real axis if the total number of poles and zeros to the right of that segment is odd
• Rule 4 - Starting and Ending Points:
  • Branches start at open-loop poles (K = 0)
  • Branches end at open-loop zeros (K = ∞)
  • If n > m, then (n - m) branches approach infinity
  where n = number of poles, m = number of zeros
• Rule 5 - Asymptotes: For (n - m) branches going to infinity:
  • Angles of asymptotes: \[\theta_a = \frac{(2k+1) \times 180°}{n-m}\] where k = 0, 1, 2, ..., (n - m - 1)
  • Centroid (intersection point on real axis): \[\sigma_a = \frac{\sum \text{(real parts of poles)} - \sum \text{(real parts of zeros)}}{n - m}\]
• Rule 6 - Breakaway/Break-in Points: Points where multiple branches meet on the real axis
  Found by solving: \[\frac{dK}{ds} = 0\] or equivalently: \[\frac{d}{ds}[G(s)H(s)] = 0\] Breakaway point: branches leave the real axis
  Break-in point: branches enter the real axis
• Rule 7 - Imaginary Axis Crossings: Points where root locus crosses the imaginary axis (ω-axis)
  Find by:
  • Substitute s = jω into characteristic equation
  • Separate into real and imaginary parts
  • Solve for ω and corresponding K
  • Or use Routh-Hurwitz criterion to find critical gain
• Rule 8 - Departure/Arrival Angles:
  • Angle of departure from complex pole: \[\theta_d = 180° - \sum \angle \text{(from zeros)} + \sum \angle \text{(from other poles)}\]
  • Angle of arrival at complex zero: \[\theta_a = 180° + \sum \angle \text{(from poles)} - \sum \angle \text{(from other zeros)}\]

Gain Calculation on Root Locus

• Gain K at any point s₀ on root locus: \[K = \frac{1}{|G(s_0)H(s_0)|} = \frac{\prod \text{(distances from } s_0 \text{ to zeros)}}{\prod \text{(distances from } s_0 \text{ to poles)}}\]

Nyquist Stability Criterion

Nyquist Plot Fundamentals

• Nyquist Plot: Polar plot of open-loop transfer function G(jω)H(jω) as ω varies from -∞ to +∞
  For practical purposes, ω varies from 0 to +∞, and the plot is symmetric about the real axis
• Nyquist Path: Closed contour in the s-plane that:
  • Follows the imaginary axis from -j∞ to +j∞
  • Semicircular path of infinite radius in the right half-plane
  • Encloses the entire right half of the s-plane

Nyquist Stability Criterion

• Nyquist Criterion: \[Z = N + P\] where:
  Z = number of zeros of (1 + GH) in right half-plane = number of closed-loop poles in RHP
  P = number of poles of GH in right half-plane = number of open-loop poles in RHP
  N = number of clockwise encirclements of -1 point by the Nyquist plot of GH
• Stability Condition: For a stable closed-loop system, Z = 0, therefore: \[N = -P\]
  • If P = 0 (open-loop stable): Nyquist plot must not encircle -1 point (N = 0)
  • If P > 0 (open-loop unstable): Nyquist plot must have P counterclockwise encirclements of -1 point
• Encirclement Rule:
  • Clockwise encirclement: N is positive
  • Counterclockwise encirclement: N is negative

Gain and Phase Margins

• Gain Margin (GM): \[GM = \frac{1}{|G(j\omega_{pc})H(j\omega_{pc})|}\] or in dB: \[GM_{dB} = -20\log_{10}|G(j\omega_{pc})H(j\omega_{pc})|\] where ω_pc = phase crossover frequency (frequency where phase = -180°)
  GM > 1 (or GM > 0 dB) indicates additional gain before instability
• Phase Margin (PM): \[PM = 180° + \angle G(j\omega_{gc})H(j\omega_{gc})\] where ω_gc = gain crossover frequency (frequency where |GH| = 1 or 0 dB)
  PM > 0° indicates additional phase lag before instability
• Typical Desired Margins:
  • Gain Margin: GM ≥ 6 dB
  • Phase Margin: 30° ≤ PM ≤ 60° (45° often considered optimal)
• Relationship to Damping Ratio: For second-order systems: \[PM \approx 100\zeta \text{ (degrees)}\] or more accurately: \[\zeta \approx \frac{PM}{100}\] Valid for ζ < 0.6="" approximately="">

Bode Plot Analysis

Bode Plot Fundamentals

• Bode Plot: Two separate plots versus frequency (ω):
  • Magnitude plot: 20log₁₀|G(jω)| in dB versus log(ω)
  • Phase plot: ∠G(jω) in degrees versus log(ω)
• Decibel Conversion: \[|G|_{dB} = 20\log_{10}|G|\]

Standard Bode Plot Elements

• Constant Gain K:
  • Magnitude: 20log₁₀K dB (horizontal line)
  • Phase: 0° if K > 0, -180° if K <>
• Integrator (1/s):
  • Magnitude: -20 dB/decade slope, passes through 0 dB at ω = 1
  • Phase: -90° (constant)
• Differentiator (s):
  • Magnitude: +20 dB/decade slope, passes through 0 dB at ω = 1
  • Phase: +90° (constant)
• First-Order Zero (1 + s/ω_z):
  • Magnitude: 0 dB for ω <>_z, +20 dB/decade for ω > ω_z
  • Phase: 0° for ω <>_z, +45° at ω = ω_z, +90° for ω >> ω_z
  • Corner frequency: ω_z
• First-Order Pole (1/(1 + s/ω_p)):
  • Magnitude: 0 dB for ω <>_p, -20 dB/decade for ω > ω_p
  • Phase: 0° for ω <>_p, -45° at ω = ω_p, -90° for ω >> ω_p
  • Corner frequency: ω_p
• Second-Order Terms: For \(\frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\)
  • Magnitude: 0 dB for ω <>_n, -40 dB/decade for ω >> ω_n
  • Peak magnitude (for ζ < 0.707):="" \(m_p="20\log_{10}\left(\frac{1}{2\zeta\sqrt{1-\zeta^2}}\right)\)" db="" at="" \(\omega_r="">
  • Phase: 0° for ω <>_n, -90° at ω = ω_n, -180° for ω >> ω_n

Stability from Bode Plots

• Gain Crossover Frequency (ω_gc): Frequency where |G(jω)H(jω)| = 0 dB
• Phase Crossover Frequency (ω_pc): Frequency where ∠G(jω)H(jω) = -180°
• Gain Margin from Bode: \[GM_{dB} = -|G(j\omega_{pc})H(j\omega_{pc})|_{dB}\] Read the magnitude (in dB) at ω_pc and change sign
• Phase Margin from Bode: \[PM = 180° + \angle G(j\omega_{gc})H(j\omega_{gc})\] Read the phase at ω_gc and add 180°
• Minimum Phase Systems: Systems with all zeros in left half-plane
  For minimum phase systems:
  • Magnitude plot uniquely determines phase plot
  • If GM > 0 dB and PM > 0°, system is stable
• Non-Minimum Phase Systems: Systems with zeros in right half-plane or time delays
  Each RHP zero contributes additional -90° phase at high frequencies

State-Space Stability Analysis

State-Space Representation

• State Equations: \[\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}\] \[\mathbf{y} = \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{u}\] where:
  x = state vector (n × 1)
  u = input vector (r × 1)
  y = output vector (m × 1)
  A = system matrix (n × n)
  B = input matrix (n × r)
  C = output matrix (m × n)
  D = feedforward matrix (m × r)
• Transfer Function from State-Space: \[G(s) = \mathbf{C}(s\mathbf{I} - \mathbf{A})^{-1}\mathbf{B} + \mathbf{D}\] where I = identity matrix

Eigenvalue Analysis

• Characteristic Equation: \[\det(s\mathbf{I} - \mathbf{A}) = 0\] Eigenvalues of A are the roots of this equation and are the system poles
• Stability Criterion (Eigenvalues):
  • System is asymptotically stable if all eigenvalues of A have negative real parts
  • System is unstable if any eigenvalue has positive real part
  • System is marginally stable if eigenvalues are on imaginary axis with no positive real parts
• Eigenvalue Equation: \[\mathbf{A}\mathbf{v} = \lambda\mathbf{v}\] where λ = eigenvalue, v = corresponding eigenvector

Controllability and Observability

• Controllability Matrix: \[\mathcal{C} = \begin{bmatrix} \mathbf{B} & \mathbf{AB} & \mathbf{A}^2\mathbf{B} & \cdots & \mathbf{A}^{n-1}\mathbf{B} \end{bmatrix}\] System is controllable if rank(ℂ) = n
• Observability Matrix: \[\mathcal{O} = \begin{bmatrix} \mathbf{C} \\ \mathbf{CA} \\ \mathbf{CA}^2 \\ \vdots \\ \mathbf{CA}^{n-1} \end{bmatrix}\] System is observable if rank(𝒪) = n

Time-Domain Specifications

Second-Order System Response

• Standard Second-Order Transfer Function: \[G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\] where ω_n = undamped natural frequency (rad/s), ζ = damping ratio
• Damping Classification:
  • Underdamped: 0 < ζ="">< 1="" (oscillatory="">
  • Critically damped: ζ = 1 (fastest response without overshoot)
  • Overdamped: ζ > 1 (slow, no overshoot)
  • Undamped: ζ = 0 (sustained oscillations)
• Poles for Second-Order System: \[s_{1,2} = -\zeta\omega_n \pm \omega_n\sqrt{\zeta^2 - 1}\] For underdamped case (0 < ζ="">< 1):="" \[s_{1,2}="-\zeta\omega_n" \pm="" j\omega_n\sqrt{1="" -="" \zeta^2}="-\sigma" \pm="" j\omega_d\]="" where="">σ = ζω_n = damping coefficient, ω_d = ω_n√(1 - ζ²) = damped natural frequency

Step Response Specifications (Underdamped Systems)

• Peak Time (T_p): Time to reach first peak \[T_p = \frac{\pi}{\omega_d} = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}}\] Units: seconds
• Percent Overshoot (PO or %OS): \[PO = e^{-\frac{\zeta\pi}{\sqrt{1-\zeta^2}}} \times 100\%\]
• Damping Ratio from Percent Overshoot: \[\zeta = \frac{-\ln(PO/100)}{\sqrt{\pi^2 + \ln^2(PO/100)}}\]
• Rise Time (T_r): Time to rise from 10% to 90% of final value (or 0% to 100%)
  For 0% to 100% definition (underdamped): \[T_r = \frac{1}{\omega_d}\left(\pi - \arctan\left(\frac{\omega_d}{\sigma}\right)\right) = \frac{1}{\omega_d}\left(\pi - \arctan\left(\frac{\sqrt{1-\zeta^2}}{\zeta}\right)\right)\] Approximation for 10% to 90%: \(T_r \approx \frac{1.8}{\omega_n}\)
• Settling Time (T_s): Time for response to settle within ±2% (or ±5%) of final value
  • 2% criterion: \(T_s = \frac{4}{\zeta\omega_n} = \frac{4}{\sigma}\)
  • 5% criterion: \(T_s = \frac{3}{\zeta\omega_n} = \frac{3}{\sigma}\)
  Units: seconds
• Delay Time (T_d): Time to reach 50% of final value
  Approximation: \(T_d \approx \frac{1 + 0.7\zeta}{\omega_n}\)

Relative Stability

Resonant Peak and Bandwidth

• Resonant Peak (M_r): Maximum value of closed-loop frequency response magnitude \[M_r = \frac{1}{2\zeta\sqrt{1-\zeta^2}}\] Valid for ζ < 0.707;="" otherwise="">_r = 1 (no peaking)
• Resonant Frequency (ω_r): Frequency at which resonant peak occurs \[\omega_r = \omega_n\sqrt{1 - 2\zeta^2}\] Valid for ζ < 0.707;="" for="" ζ="" ≥="" 0.707,="" no="" resonant="" peak="" exists="">
• Bandwidth (ω_BW): Frequency range where closed-loop gain is within -3 dB of DC gain
  For second-order system: \[\omega_{BW} = \omega_n\sqrt{(1-2\zeta^2) + \sqrt{4\zeta^4 - 4\zeta^2 + 2}}\] Approximation for small ζ: \(\omega_{BW} \approx \omega_n\)

Steady-State Error

• Steady-State Error Definition: \[e_{ss} = \lim_{t \to \infty} e(t) = \lim_{s \to 0} sE(s)\] where E(s) = error signal in Laplace domain
• Unity Feedback System Error: For input R(s) and open-loop transfer function G(s): \[E(s) = \frac{R(s)}{1 + G(s)}\] \[e_{ss} = \lim_{s \to 0} \frac{sR(s)}{1 + G(s)}\]
• System Type: Number of integrators (poles at origin) in open-loop transfer function G(s)
  Type 0: no poles at s = 0
  Type 1: one pole at s = 0
  Type 2: two poles at s = 0
• Static Error Constants:
  Position Error Constant: \(K_p = \lim_{s \to 0} G(s)\)
  Velocity Error Constant: \(K_v = \lim_{s \to 0} sG(s)\)
  Acceleration Error Constant: \(K_a = \lim_{s \to 0} s^2G(s)\)
• Steady-State Error for Step Input (R(s) = A/s): \[e_{ss} = \frac{A}{1 + K_p}\]
  • Type 0: \(e_{ss} = \frac{A}{1 + K_p}\)
  • Type 1 or higher: e_ss = 0
• Steady-State Error for Ramp Input (R(s) = A/s²): \[e_{ss} = \frac{A}{K_v}\]
  • Type 0: e_ss = ∞
  • Type 1: \(e_{ss} = \frac{A}{K_v}\)
  • Type 2 or higher: e_ss = 0
• Steady-State Error for Parabolic Input (R(s) = A/s³): \[e_{ss} = \frac{A}{K_a}\]
  • Type 0 or 1: e_ss = ∞
  • Type 2: \(e_{ss} = \frac{A}{K_a}\)
  • Type 3 or higher: e_ss = 0

Advanced Stability Topics

Lyapunov Stability

• Lyapunov Stability Theorem: For system \(\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x})\), if there exists a scalar function V(x) (Lyapunov function) such that:
  • V(x) > 0 for all x ≠ 0 (positive definite)
  • V(0) = 0
  • \(\dot{V}(\mathbf{x}) \leq 0\) (negative semidefinite) → stable
  • \(\dot{V}(\mathbf{x}) < 0\)="" for="" all="" x="" ≠="" 0="" (negative="" definite)="" →="" asymptotically="">
• Lyapunov Function for Linear Systems: For \(\dot{\mathbf{x}} = \mathbf{A}\mathbf{x}\), choose: \[V(\mathbf{x}) = \mathbf{x}^T\mathbf{P}\mathbf{x}\] where P is a positive definite matrix
  Time derivative: \(\dot{V} = \mathbf{x}^T(\mathbf{A}^T\mathbf{P} + \mathbf{P}\mathbf{A})\mathbf{x}\)
• Lyapunov Equation: \[\mathbf{A}^T\mathbf{P} + \mathbf{P}\mathbf{A} = -\mathbf{Q}\] System is asymptotically stable if, for any positive definite Q, there exists a positive definite solution P

Nichols Chart

• Nichols Chart: Plot of open-loop frequency response with:
  • x-axis: phase angle ∠G(jω)H(jω) in degrees
  • y-axis: magnitude |G(jω)H(jω)| in dB
  • Overlaid with closed-loop M and N contours
• M-Circles: Contours of constant closed-loop magnitude ratio \[M = \frac{|G|}{|1 + GH|}\]
• Stability Assessment:
  • Critical point: (0 dB, -180°)
  • System stable if plot does not enclose critical point
  • Gain margin: vertical distance from plot to critical point at -180°
  • Phase margin: horizontal distance from plot to critical point at 0 dB

Jury Stability Test

• Jury Test: Stability criterion for discrete-time systems
  For discrete transfer function with characteristic equation: \[a_0z^n + a_1z^{n-1} + \cdots + a_{n-1}z + a_n = 0\]
• Jury Table Construction: Similar to Routh array but for z-domain
  System is stable if all roots lie inside unit circle in z-plane
• Necessary Conditions:
  • P(1) > 0
  • (-1)ⁿP(-1) > 0
  • |a_n| <>₀
  where P(z) is the characteristic polynomial

Describing Function Method

• Describing Function: For nonlinear element with sinusoidal input: \[x(t) = A\sin(\omega t)\] Output fundamental component: \[y_1(t) = N(A)A\sin(\omega t + \phi)\] where N(A) = describing function (depends on amplitude)
• Stability Prediction: Plot of:
  • -1/N(A) locus in complex plane
  • G(jω) locus (linear part)
  • Intersection indicates limit cycle