UNIT 4: CONTROL SYSTEM - EXAM-FOCUSED SHORT NOTES

1.0 SYSTEM MODELING & REPRESENTATION

1.1 Block Diagram Representation

• Components:

  • Block: Represents a system/component with transfer function G(s).

  • Summing Point: Algebraic sum (+ or -) of input signals.

  • Take-off Point: Point where signal is sampled for feedback or measurement.

  • Forward Path: Path from input R(s) to output C(s) containing blocks.

  • Feedback Path: Path from output to summing point, transfer function H(s).

• Reduction Rules (Apply sequentially):

  • Series: G₁G₂...Gₙ

  • Parallel: G₁ ± G₂ ± ... ± Gₙ

  • Feedback: T(s) = \frac{G(s)}{1 ± G(s)H(s)} (Negative feedback: 1 + G(s)H(s)).

• Open Loop vs. Closed Loop:

  | Feature | Open Loop | Closed Loop | | :--- | :--- | :--- | | Feedback | Absent | Present | | Accuracy | Low, depends on calibration | High, corrects errors | | Stability | Inherently stable | Needs design for stability | | Complexity | Simple | Complex | | Sensitivity | High to parameter changes | Low to parameter changes |

[!TIP] Exam Focus: Block diagram reduction is foundational. Always simplify to a single forward path and single feedback loop before writing T(s).

1.2 Signal Flow Graph (SFG) & Mason's Gain Formula

• SFG: Graphical representation of linear algebraic equations.

  • Node: System variable (signal).

  • Branch: Directed edge with gain G(s).

  • Forward Path Gain: Product of branch gains along a path from input to output.

  • Loop Gain: Product of branch gains in a closed path.

  • Non-Touching Loops: Loops with no common nodes.

• Mason's Gain Formula:

$$T(s) = \frac{C(s)}{R(s)} = \frac{\sum_{k=1}^{N} P_k \Delta_k}{\Delta}$$

Where:

*   `P_k` = k-th forward path gain.

*   `Δ` = `1 - (sum of all individual loop gains) + (sum of gain products of all possible two non-touching loops) - ...`

*   `Δ_k` = Value of `Δ` for that part of the graph **not** touching the k-th forward path.

• Advantages over Block Diagram: Direct application, no need for successive reduction, handles complex interconnections easily.

[!TIP] Common Pitfall: While calculating Δ_k, remove all nodes and branches of the k-th forward path, then compute Δ for the remaining subgraph.

1.3 Analogous Systems

• Force-Voltage (F-V) / Direct Analogy:

  | Mechanical (Translational) | Electrical (Voltage) | | :--- | :--- | | Force (F) | Voltage (V) | | Velocity (v) | Current (I) | | Displacement (x) | Charge (q) | | Mass (M) | Inductance (L) | | Damping (B) | Resistance (R) | | Spring (K) | Inverse Capacitance (1/C) |

• Force-Current (F-I) / Inverse Analogy:

  | Mechanical (Translational) | Electrical (Current) | | :--- | :--- | | Force (F) | Current (I) | | Velocity (v) | Voltage (V) | | Displacement (x) | Flux (φ) | | Mass (M) | Capacitance (C) | | Damping (B) | Conductance (1/R) | | Spring (K) | Inverse Inductance (1/L) |

[!TIP] Exam Tip: For rotational systems, replace Force F with Torque T, Velocity v with Angular velocity ω, Displacement x with Angular displacement θ, Mass M with Moment of Inertia J, Damping B with Viscous friction coefficient B.

1.4 System Components & Actuators

• AC Servomotor:

  • Principle: Two-phase induction motor. Two stator windings (90° apart): reference winding (voltage V_r) and control winding (voltage V_c). Rotor is squirrel-cage.

  • Assumptions:

    1. Constant field flux (linear magnetization).

    2. Small torque angle (linear torque-speed relation).

    3. Negligible rotor inductance (L_r ≈ 0).

  • Transfer Function Derivation:

    Torque T_e ∝ V_r V_c sin(θ) ≈ V_r V_c θ (for small θ).

    Mechanical equation: J d²θ/dt² + B dθ/dt = T_e.

    After Laplace and linearization:

$$\frac{\Theta(s)}{V_c(s)} = \frac{K}{s(T_M s + 1)}$$

    Where `K` = motor constant, `T_M = J/B` = mechanical time constant.

• Stepper Motor:

  • Principle: Converts digital pulses into discrete angular rotations. Rotor moves in fixed step angles (e.g., 1.8°/step).

  • Types: Variable Reluctance (VR), Permanent Magnet (PM), Hybrid (HB).

  • Applications: Printers, plotters, CNC machines, robotics (open-loop position control).

• Tacho-Generator:

  • Principle: DC generator where output voltage V_t ∝ ω (angular speed).

  • Use in Feedback: Provides a voltage proportional to shaft speed for speed control systems (velocity feedback). Transfer function: V_t(s)/ω(s) = K_t.

2.0 TIME DOMAIN ANALYSIS & PERFORMANCE

2.1 Standard Test Input Signals

2.2 First-Order System Response

• Transfer Function: G(s) = \frac{K}{τs + 1}

• Unit Step Response (R(s)=1/s):

$$c(t) = K \left(1 - e^{-t/τ}\right)$$

*   **Time Constant `τ`**: Time to reach 63.2% of final value.

*   **Rise Time `t_r`** (10% to 90%): `t_r ≈ 2.2τ`.

• Unit Ramp Response:

$$c(t) = K \left(t - τ + τ e^{-t/τ}\right)$$

*   **Steady-State Error**: `e_ss = lim_{t→∞} (r(t) - c(t)) = τ`.

2.3 Second-Order System Response

• Standard Form:

$$G(s) = \frac{\omega_n^2}{s^2 + 2ζω_n s + \omega_n^2}$$

Where `ω_n` = natural frequency, `ζ` = damping ratio.

• Underdamped Response (0 < ζ < 1):

$$c(t) = 1 - \frac{e^{-ζω_n t}}{\sqrt{1-ζ^2}} \sin(ω_d t + φ)$$

Where `ω_d = ω_n√(1-ζ²)` = damped frequency, `φ = arccos(ζ)`.

• Key Performance Specifications:

  • Maximum Overshoot M_p:

$$M_p = e^{\frac{-ζπ}{\sqrt{1-ζ^2}}} \times 100\%$$

    \boxed{M_p = \exp\left(-\frac{ζπ}{\sqrt{1-ζ^2}}\right)}

*   **Peak Time** `t_p`:

$$t_p = \frac{π}{ω_d} = \frac{π}{ω_n\sqrt{1-ζ^2}}$$

    \boxed{t_p = \frac{π}{ω_n\sqrt{1-ζ^2}}}

*   **Rise Time** `t_r` (0% to 100% for underdamped):

$$t_r ≈ \frac{π - φ}{ω_d}$$

*   **Settling Time** `t_s` (2% criterion):

$$t_s ≈ \frac{4}{ζω_n}$$

*   **Resonant Frequency** `ω_r` & **Resonance Peak** `M_r`:

$$ω_r = ω_n\sqrt{1-2ζ^2} \quad (for ζ < 1/\sqrt{2})$$

$$M_r = \frac{1}{2ζ\sqrt{1-ζ^2}}$$

• Critically Damped (ζ=1) & Overdamped (ζ>1):

  • No overshoot.

  • Slower response than underdamped with same ω_n.

  • t_r and t_s larger.

[!TIP] Derivation of M_p: Set dc/dt = 0 in c(t), solve for t = t_p, substitute back into c(t_p). M_p = c(t_p) - 1.

2.4 Performance Specifications & ζ-ω_n Relationship

• Trade-off: Higher ζ → Lower M_p, but longer t_s and t_r.

• Effect of ω_n: Higher ω_n → Faster response (smaller t_p, t_r, t_s), but may increase sensitivity.

• Graph: M_p vs. ζ is exponentially decaying. M_p=16% corresponds to ζ≈0.5; M_p=5% corresponds to ζ≈0.7.

3.0 STABILITY ANALYSIS (TIME DOMAIN)

3.1 Routh-Hurwitz Criterion

• Necessary & Sufficient Condition: All coefficients of characteristic polynomial a_n s^n + ... + a_0 = 0 must be positive AND no sign changes in the first column of Routh array.

• Routh Array Construction:

  | sⁿ | a_n | a_{n-2} | a_{n-4} | ... | | sⁿ⁻¹ | a_{n-1} | a_{n-3} | a_{n-5} | ... | | sⁿ⁻² | b_1 = (a_{n-1}a_{n-2} - a_n a_{n-3})/a_{n-1} | b_2 | ... | | | sⁿ⁻³ | c_1 = (b_1 a_{n-3} - a_{n-1} b_2)/b_1 | ... | | | | ... | ... | ... | | |

• Special Cases:

  1. Row of Zeros: Indicates symmetrical roots on jω axis (marginal stability). Form Auxiliary Equation from row above, differentiate, replace row with coefficients.

  2. First Element Zero: Replace ε → 0⁺ in that element and continue.

• Relative Stability: To check roots left of s = -σ line, substitute s = z - σ and apply Routh to polynomial in z.

[!TIP] Exam Pattern: "Find range of K for stability" → Form characteristic equation → Build Routh array → Condition: All first column elements > 0 → Solve inequalities for K.

3.2 Root Locus Technique

• Definition: Plot of closed-loop pole locations as gain K varies from 0 → ∞.

• Construction Rules:

  1. Branches: Equal to number of open-loop poles n.

  2. Start/End Points: Start at OL poles (K=0), end at OL zeros (K→∞) or ∞.

  3. Asymptotes (if n > m):

     • Number = n - m

     • Angles: θ = \frac{(2q+1)180°}{n-m}, q = 0,1,...,n-m-1

     • Centroid: σ = \frac{\sum Re(poles) - \sum Re(zeros)}{n-m}

  4. Angle of Departure/Arrival:

     • From pole p_i: ∠Departure = 180° - Σ(angles to other poles) + Σ(angles to zeros)

     • To zero z_i: ∠Arrival = 180° - Σ(angles to other zeros) + Σ(angles to poles)

  5. Breakaway/Break-in Points:

     • Solve dK/ds = 0 for K(s) = \frac{Product(s - zeros)}{Product(s - poles)}.

     • Valid points: On real axis segment between poles/zeros.

  6. Imaginary Axis Crossing: Use Routh criterion on characteristic equation 1 + KG(s)H(s) = 0 or apply Angle Condition (∑∠G(jω)H(jω) = -180°).

• Stability Judgment: Closed-loop system unstable if any root locus branch lies in Right-Half Plane (RHP) for K > 0.

• Design Use: From root locus plot, read K for desired ζ (draw line from origin at angle cos⁻¹(ζ)).

[!TIP] Sketching Order: 1) Mark OL poles/zeros. 2) Draw asymptotes. 3) Plot on real axis. 4) Find breakaway points. 5) Determine departure angles. 6) Sketch symmetric curves.

4.0 FREQUENCY DOMAIN ANALYSIS & STABILITY

4.1 Bode Plot (Asymptotic & Exact)

• Magnitude Plot: 20 log₁₀ |G(jω)H(jω)| (dB) vs log ω.

• Phase Plot: ∠G(jω)H(jω) (degrees) vs log ω.

• Construction (Asymptotic):

  1. Write G(jω)H(jω) in standard form: K \frac{∏(1+jωT_z)}{∏(1+jωT_p)}.

  2. Magnitude:

     • Start at 20 log K (dB).

     • For each pole at origin: slope -20 dB/dec starting at ω=1.

     • For each zero at origin: slope +20 dB/dec.

     • For each simple pole at ω = 1/T_p: slope changes by -20 dB/dec at ω = 1/T_p.

     • For each simple zero at ω = 1/T_z: slope changes by +20 dB/dec.

  3. Phase:

     • Start: 0° (no zeros/poles at origin) or ±90°n (n poles/zeros at origin).

     • Each pole contributes 0° → -90° over 2 decades centered at ω = 1/T_p.

     • Each zero contributes 0° → +90° over 2 decades centered at ω = 1/T_z.

• Key Parameters:

  • Gain Crossover Frequency ω_gc: |G(jω)H(jω)| = 1 (0 dB).

  • Phase Crossover Frequency ω_pc: ∠G(jω)H(jω) = -180°.

  • Gain Margin (GM): GM = 1/|G(jω_pc)H(jω_pc)| (linear) or GM = -20 log|G(jω_pc)H(jω_pc)| (dB). Positive GM → Stable.

  • Phase Margin (PM): PM = 180° + ∠G(jω_gc)H(jω_gc). Positive PM → Stable.

• Stability Comment: For stable closed-loop, PM > 0 (typically > 30-50° for good transient response) and GM > 0 (typically > 6 dB).

[!TIP] Exact vs Asymptotic: Asymptotic is approximate. For accuracy, add correction factors at corner frequencies: ±3 dB for simple pole/zero, ±1 dB for second-order factors.

4.2 Polar Plot (Nyquist Open-Loop)

• Construction: Plot of G(jω)H(jω) (real vs imaginary) as ω varies 0 → ∞.

• Generalized Shapes:

  | System Type | Polar Plot Shape (ω: 0→∞) | | :--- | :--- | | Type 0 | Starts on +real axis, ends at origin (0,0). | | Type 1 | Starts at ∞ on -90° line, ends at finite point on +real axis. | | Type 2 | Starts at ∞ on -180° line, ends at origin. |

• Inverse Polar Plot: Plot of 1/[G(jω)H(jω)] (used in Nyquist criterion).

• Comparison with Bode: Bode separates magnitude/phase; Polar combines them in one plot. Polar shows encirclements directly.

4.3 Nyquist Stability Criterion

• Nyquist Contour: Semi-circle in RHP (s = σ + jω, σ from -∞ to +∞, ω from +∞ to -∞).

• Mapping Principle: Contour mapped by F(s) = 1 + G(s)H(s).

• Encirclement N: Number of clockwise encirclements of -1 + j0 point by Nyquist plot of G(s)H(s).

• Poles in RHP P: Number of open-loop unstable poles.

• Closed-loop Unstable Poles Z:

  \boxed{Z = P - N}

  • Closed-loop stable iff Z = 0 (no poles in RHP).

• Application:

  1. Plot Nyquist of G(s)H(s).

  2. Count P (poles of G(s)H(s) in RHP).

  3. Count N (clockwise encirclements of -1 by Nyquist plot).

  4. Compute Z. If Z=0, closed-loop stable.

• Gain Margin from Nyquist: GM = 1/|G(jω_pc)H(jω_pc)| where ω_pc is frequency where plot crosses negative real axis. Distance from crossing point to -1.

[!TIP] Key: N is clockwise positive. For P=0 (stable open-loop), N must be 0 for stability (no encirclement of -1). For P>0, N must equal P for stability.

4.4 Gain Margin & Phase Margin

• Gain Margin (GM): Factor by which gain can be multiplied before system becomes unstable. GM = 1/|G(jω_pc)H(jω_pc)|. High GM → Less sensitive to gain changes.

• Phase Margin (PM): Additional phase lag required to reach -180° at ω_gc. PM = 180° + ∠G(jω_gc)H(jω_gc). High PM → Less oscillatory, more damped response.

• Relationship: Both measure relative stability (how close to instability). Empirical: PM ≈ ζ (for low ζ). PM > 30° usually desired.

5.0 STEADY-STATE ERROR ANALYSIS

5.1 Static Error Coefficients

• Position Error Constant K_p:

$$K_p = \lim_{s→0} G(s)H(s)$$

• Velocity Error Constant K_v:

$$K_v = \lim_{s→0} s G(s)H(s)$$

• Acceleration Error Constant K_a:

$$K_a = \lim_{s→0} s^2 G(s)H(s)$$

• Derivation: From E(s) = \frac{R(s)}{1 + G(s)H(s)}, apply Final Value Theorem (FVT) for standard inputs.

5.2 Steady-State Error e_ss

• Type of System: Number of pure integrators (poles at origin) in G(s) (open-loop).

  | System Type | Input | e_ss (Unit Feedback) | | :--- | :--- | :--- | | Type 0 | Step A | \frac{A}{1+K_p} | | | Ramp At | ∞ | | | Parabolic At²/2 | ∞ | | Type 1 | Step | 0 | | | Ramp | \frac{A}{K_v} | | | Parabolic | ∞ | | Type 2 | Step | 0 | | | Ramp | 0 | | | Parabolic | \frac{A}{K_a} |

• Generalized Error Coefficients:

$$e_{ss} = \frac{1}{1 + K_p} \text{ (step)}; \quad e_{ss} = \frac{1}{K_v} \text{ (ramp)}; \quad e_{ss} = \frac{1}{K_a} \text{ (parabolic)}$$

5.3 Limitations of Static Error Coefficients

1. Only valid for stable Type 0 & Type 1 systems.

2. Does not predict e_ss for unstable Type ≥1 systems (FVT not applicable).

3. Inaccurate for inputs other than standard step/ramp/parabolic.

4. Does not give transient information.

[!TIP] Remember: K_p, K_v, K_a depend only on open-loop TF G(s)H(s), not on closed-loop stability. Always check stability first (Routh/Root Locus).

6.0 CONTROLLERS & COMPENSATION DESIGN

6.1 Basic Controllers

6.2 Need for Compensation

• Improve transient response (increase PM, ζ, reduce t_s).

• Improve steady-state accuracy (increase K_v, K_a).

• Reduce sensitivity to parameter variations.

• Achieve controllability/observability in state-space.

6.3 Lead Compensation

• Phase Lead Network:

$$G_c(s) = \frac{1 + sT}{1 + αsT}, \quad 0 < α < 1$$

*   Maximum phase lead: `φ_m = sin⁻¹\left(\frac{1-α}{1+α}\right)` at `ω_m = 1/(T√α)`.

*   `α` determines `φ_m`; `T` sets `ω_m`.

• Effect:

  • Increases Phase Margin (adds positive phase).

  • Increases bandwidth and speed (reduces t_s).

  • Increases K_v slightly (if zero near origin).

• Design Procedure (Bode):

  1. From specs, determine required PM_new = PM_required + safety (5-12°).

  2. Find φ_max = PM_new - PM_current (from uncompensated Bode at ω_gc).

  3. Determine α = (1 - sin φ_max)/(1 + sin φ_max).

  4. Find ω_m (where uncompensated phase = PM_new - φ_max - 90°? Actually: ω_gc should be at ω_m after compensation. Adjust K so that |G_c(jω_m)G(jω_m)H(jω_m)| = 1).

  5. Calculate T = 1/(ω_m √α).

  6. Final K chosen to meet K_v or gain crossover.

6.4 Lag Compensation

• Phase Lag Network:

$$G_c(s) = \frac{1 + sT}{1 + βsT}, \quad β > 1$$

*   Maximum phase lag: `φ_m ≈ - (1 - α)/(1 + α)`? Actually for lag: `α = 1/β`, `φ_m = sin⁻¹\left(\frac{1-α}{1+α}\right)` but negative.

*   `β` determines attenuation (and thus `K_v` improvement).

• Effect:

  • Increases system type → Reduces e_ss (improves K_v).

  • Slightly reduces PM (adds negative phase at ω_gc).

  • Lowers bandwidth.

• Design Procedure (Bode):

  1. Determine K from K_v requirement: K_v = lim_{s→0} s G(s) = K (for Type 1).

  2. Draw Bode with this K. Check PM. If PM insufficient, design lead first, then add lag to adjust K_v without affecting PM much (place lag pole/zero near origin, ω_gc >> 1/T).

6.5 Lag-Lead Compensation

• Combined Network: G_c(s) = G_lead(s) G_lag(s).

• When Used: Need simultaneous improvement in both transient (PM) and steady-state (K_v).

• Design Steps:

  1. Design lead to meet PM spec (as in 6.3).

  2. Check K_v from resulting K. If insufficient, design lag to increase K without disturbing PM (place lag corner freq << ω_gc).

  3. Alternatively, design simultaneously using Nichols or Bode.

6.6 Design Examples (Pattern)

• Given: G(s), specs: K_v = ?, PM ≥ ?°.

• Lead Design:

  1. Find K from K_v (if Type 1: K_v = K).

  2. Plot Bode with K. Find PM_current.

  3. φ_needed = PM_req + 5° - PM_current.

  4. α = (1 - sin φ_needed)/(1 + sin φ_needed).

  5. Find ω_gc_desired where uncompensated phase = -180° + PM_req + φ_needed.

  6. T = 1/(ω_gc_desired √α).

  7. Verify PM at new ω_gc.

• Lag Design (if needed after lead):

  1. β = K_v_required / K_v_current.

  2. Place ω_z = ω_gc/10, ω_p = ω_z/β (so that |G_c| ≈ 1 at ω_gc).

7.0 STATE SPACE ANALYSIS

7.1 State Space Representation

• State Variables: Minimum set x₁(t), x₂(t), ..., xₙ(t) that completely describes system dynamics.

• State Vector: x(t) = [x₁ x₂ ... xₙ]ᵀ.

• State Equations:

$$\dot{x}(t) = A x(t) + B u(t)$$

$$y(t) = C x(t) + D u(t)$$

Where `A` (n×n), `B` (n×r), `C` (m×n), `D` (m×r).

• Modeling from Transfer Function:

  • Controllable Canonical Form: Companion matrix A.

  • Observable Canonical Form: Transpose of controllable form Aᵀ.

• Transfer Function Decomposition:

  • Direct Decomposition: G(s) = C(sI - A)⁻¹B + D.

  • Parallel Decomposition: G(s) = Σ_{i=1}^n \frac{r_i}{s - p_i} (residues r_i from partial fraction of (sI-A)⁻¹B).

7.2 Controllability & Observability

• Controllability: Ability to transfer state from any initial to any final in finite time with suitable input.

  • Kalman's Rank Condition:

    \boxed{\text{Rank } \mathcal{C} = n}

    Where Controllability Matrix:

$$\mathcal{C} = [B \quad AB \quad A^2B \quad ... \quad A^{n-1}B]$$

• Observability: Ability to determine initial state from output history.

  • Kalman's Rank Condition:

    \boxed{\text{Rank } \mathcal{O} = n}

    Where Observability Matrix:

$$\mathcal{O} = \begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^{n-1} \end{bmatrix}$$

• Duality: (A,B) controllable ↔ (Aᵀ,Cᵀ) observable.

7.3 State Transition Matrix Φ(t)

• Definition: Φ(t) = e^{At}.

• Solution of State Equation (zero-input + zero-state):

$$x(t) = Φ(t)x(0) + \int_0^t Φ(t-τ) B u(τ) dτ$$

• Properties:

  1. Φ(0) = I

  2. Φ(t₁)Φ(t₂) = Φ(t₁ + t₂)

  3. Φ(-t) = [Φ(t)]⁻¹

  4. dΦ(t)/dt = A Φ(t) = Φ(t) A

  5. Φ(t) = \mathcal{L}^{-1}[(sI - A)^{-1}]

• Calculation:

  • Laplace Inverse: Φ(t) = \mathcal{L}^{-1}[(sI - A)^{-1}].

  • Cayley-Hamilton: For 2×2/3×3, find e^{At} = α₀(t)I + α₁(t)A + α₂(t)A² using eigenvalues.

7.4 Eigenvalues & Eigenvectors

• Definitions: For matrix A, scalar λ and vector v satisfy Av = λv.

• Significance:

  • Eigenvalues λ_i = closed-loop poles (for A matrix of state-space).

  • Determine stability (Re(λ) < 0 for all i).

  • Determine natural response modes (e^{λ_i t}).

• Calculation: Solve det(A - λI) = 0 for λ. For each λ, solve (A - λI)v = 0 for v.

• Modal Matrix V: Matrix of eigenvectors [v₁ v₂ ... vₙ].

• Diagonalization: If V invertible, A = V Λ V⁻¹, where Λ = diag(λ₁, λ₂, ..., λₙ).

• Relation to Φ(t):

$$e^{At} = V e^{Λt} V^{-1} = V \begin{bmatrix} e^{λ₁t} & & \\ & \ddots & \\ & & e^{λₙt} \end{bmatrix} V^{-1}$$

[!TIP] Controllability/Observability Check: For A in controllable canonical form, always controllable. For observable canonical form, always observable.

8.0 ADVANCED TOPICS & SHORT NOTES

8.1 Linearization of Nonlinear Systems

• Concept: Approximate nonlinear system ẋ = f(x,u) around an operating point (x₀, u₀) using Taylor series (ignore higher-order terms).

$$\delta\dot{x} = \frac{∂f}{∂x}\bigg|_{(x₀,u₀)} \delta x + \frac{∂f}{∂u}\bigg|_{(x₀,u₀)} \delta u$$

Where `δx = x - x₀`, `δu = u - u₀`.

• Effect of Feedback: Feedback linearization uses feedback to cancel nonlinearities, making overall system linear.

8.2 Effect of Pole Location on Stability & Response

• Real Poles (σ < 0):

  • Left of origin → stable.

  • More negative (σ large magnitude) → faster response.

  • Dominant pole (closest to origin) dictates response speed.

• Complex Poles (σ ± jω_d):

  • σ determines decay rate (stability margin).

  • ω_d determines oscillation frequency.

  • ζ = -σ/√(σ²+ω_d²) determines damping.

• Pole-Zero Cancellation: Cancelling a pole with a zero can remove a mode but risks instability if cancelled pole is unstable or lightly damped.

8.3 Log Magnitude vs. Phase Plot

• Plot of 20 log|G(jω)H(jω)| vs ∠G(jω)H(jω) (both in dB/degrees).

• Relation to Nichols Chart: Nichols chart is a standardized version showing constant M_p, ω_r contours.

• Use: Directly read gain/phase margins from intersection with 0 dB and -180° lines. Assess stability margins visually.

8.4 Relative Stability

• Beyond Absolute Stability: Measures how close system is to instability.

• Quantification:

  • Gain Margin (GM): Factor of safety against gain increase.

  • Phase Margin (PM): Factor of safety against phase lag.

  • Delay Margin: Maximum time delay before instability.

• Higher GM/PM → More Robustly Stable.

8.5 Comparative Analysis of Plots

[!TIP] Quick Recall: Bode = dB vs log ω (separate); Polar = Re vs Im (single curve); Nyquist = mapping of entire contour (includes infinite semicircle).

END OF UNIT 4 NOTES
Aligned with RGPV past papers (JUN 2025 - JUN 2022). Focus on derivations (overshoot, servomotor), rule applications (Routh, Root Locus, Mason), and design (Bode-based compensator).