Control System (EC-404) - Unit 2 Short Notes

1. Block Diagram & Signal Flow Graph Analysis

Mason’s Gain Formula

Used to find transfer function $$\displaystyle T = \frac{C(s)}{R(s)} $$ from a signal flow graph.

Formula:

$$ \boxed{T = \sum_{k=1}^{N} \frac{M_k \Delta_k}{\Delta}} $$

where:

• $$\displaystyle M_k $$ = gain of $$\displaystyle k^{th} $$ forward path

• $$\displaystyle \Delta = 1 - \sum L_i + \sum L_i L_j - \sum L_i L_j L_k + \cdots $$

  ($$\displaystyle L_i $$ = loop gains, $$\displaystyle L_i L_j $$ = non-touching loop products)

• $$\displaystyle \Delta_k $$ = $\Delta$ with loops touching $$\displaystyle k^{th} $$ forward path removed

Steps:

1. Identify all forward paths and their gains $$\displaystyle M_k $$.

2. Identify all individual loops (touching or not).

3. Compute $\Delta$ considering non-touching loop combinations.

4. For each forward path, compute $$\displaystyle \Delta_k $$ by removing loops touching it.

5. Apply formula.

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Common Pitfall: Forgetting to exclude all loops touching a forward path when computing $$\displaystyle \Delta_k $$.

Block Diagram Reduction

• Series blocks: multiply transfer functions.

• Parallel blocks: add transfer functions.

• Feedback: $$\displaystyle T = \frac{G}{1+GH} $$ (negative feedback).

• Take-off point: can move forward/backward with care.

Analogous Systems
Goal: Represent mechanical system by electrical analog for analysis.

Direct Analogy (Impedance Analogy):

• Preserve impedance analogy: $$\displaystyle F \leftrightarrow V $$, $$\displaystyle v \leftrightarrow I $$.

• Mechanical impedance $$\displaystyle Z_m = B + Ms + \frac{1}{Ks} $$ maps to electrical $$\displaystyle Z_e = R + Ls + \frac{1}{Cs} $$.

Inverse Analogy (Mobility Analogy):

• $$\displaystyle F \leftrightarrow I $$, $$\displaystyle v \leftrightarrow V $$.

• Mechanical mobility $$\displaystyle Y_m = \frac{1}{B} + \frac{s}{M} + \frac{1}{sK} $$ maps to electrical admittance $$\displaystyle Y_e = \frac{1}{R} + Cs + \frac{1}{Ls} $$.

2. Time Domain Analysis & System Response

Standard Test Signals

• Step: $$\displaystyle r(t) = A \cdot u(t) $$; $$\displaystyle R(s) = \frac{A}{s} $$

• Ramp: $$\displaystyle r(t) = A \cdot t \cdot u(t) $$; $$\displaystyle R(s) = \frac{A}{s^2} $$

• Parabolic: $$\displaystyle r(t) = \frac{A}{2} t^2 u(t) $$; $$\displaystyle R(s) = \frac{A}{s^3} $$

• Impulse: $$\displaystyle r(t) = A \delta(t) $$; $$\displaystyle R(s) = A $$

First-Order System
$$\displaystyle G(s) = \frac{K}{\tau s + 1} $$

Time Constant $\tau$: Time to reach 63.2% of final value for step input.

Second-Order System

Standard form:

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

• $\zeta$ = damping ratio

• $$\displaystyle \omega_n $$ = undamped natural frequency

Transient Specifications (Underdamped, $$\displaystyle \zeta < 1 $$):

• Peak Overshoot:

$$ \boxed{M_p = e^{-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}} \times 100\%} $$

• Peak Time:

$$ \boxed{T_p = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}}} $$

• Rise Time (0%→100%):

$$ \boxed{T_r = \frac{\pi - \beta}{\omega_d}}, \quad \beta = \tan^{-1}\left(\frac{\sqrt{1-\zeta^2}}{\zeta}\right), \quad \omega_d = \omega_n\sqrt{1-\zeta^2} $$

(Common approximation: $$\displaystyle T_r \approx \frac{1.8}{\omega_n} $$ for $\zeta \approx 0.5$)

• Settling Time (2% criterion):

$$ \boxed{T_s = \frac{4}{\zeta\omega_n}} $$

• Resonant Peak & Frequency (for $$\displaystyle \zeta < 0.707 $$):

$$ \omega_r = \omega_n\sqrt{1-2\zeta^2}, \quad M_r = \frac{1}{2\zeta\sqrt{1-\zeta^2}} $$

Effects of Controllers on Time Response

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Exam Focus: Derive $$\displaystyle M_p $$, $$\displaystyle T_p $$, $$\displaystyle T_r $$ from standard second-order step response $$\displaystyle c(t) = 1 - \frac{e^{-\zeta\omega_n t}}{\sqrt{1-\zeta^2}} \sin(\omega_d t + \phi) $$.

3. Steady-State Error Analysis

Static Error Coefficients

Define system type = number of open-loop poles at origin.

Generalized Error Coefficients

For polynomial input $$\displaystyle r(t) = \frac{a_0}{m!} t^m + \cdots $$,

$$ \boxed{e_{ss} = \frac{a_0}{1 + E_m}}, \quad E_m = \lim_{s\to0} s^{m+1} G(s)H(s) $$

$$\displaystyle E_m $$ is generalized error coefficient.

Limitations of Static Error Coefficients

1. Only for stable systems (Routh necessary).

2. Only for polynomial inputs (step, ramp, parabola).

3. Does not predict error for arbitrary inputs.

4. Fails for systems with non-minimum phase zeros or unstable open-loop.

4. Stability Analysis

Routh-Hurwitz Criterion

Characteristic equation: $$\displaystyle a_n s^n + a_{n-1} s^{n-1} + \cdots + a_0 = 0 $$, $$\displaystyle a_n > 0 $$.

Necessary & Sufficient: All first-column elements > 0.

Special Cases:

1. Zero in first column: Replace with small $$\displaystyle \epsilon > 0 $$, continue. Check sign change in column.

2. Entire row zero: Form auxiliary equation from row above, differentiate, replace row. Indicates symmetrical roots (often imaginary axis).

Relative Stability (s = -σ line):

Shift $$\displaystyle s = z - \sigma $$. Apply Routh to $$\displaystyle P(z-\sigma)=0 $$. All first-column > 0 ⇒ all roots left of $$\displaystyle z = -\sigma $$ (i.e., $$\displaystyle Re(s) < -\sigma $$).

Root Locus Stability Assessment

• Closed-loop stable ⇔ all roots of $$\displaystyle 1 + G(s)H(s) = 0 $$ in LHP.

• From root locus: if for all $$\displaystyle K>0 $$, branches remain in LHP ⇒ stable for all $K$.

• Critical $K$: value where locus crosses imaginary axis (use Routh or angle condition).

Effect of Pole Location:

• LHP: stable (real part < 0).

• Imaginary axis: marginally stable (sustained oscillations).

• RHP: unstable (exponential growth).

5. Root Locus Technique

Construction Rules (Sketching)

1. Start/End Points: Start at open-loop poles ($$\displaystyle K=0 $$), end at open-loop zeros ($K \to \infty$). If $$\displaystyle n > m $$, $n-m$ branches go to ∞ along asymptotes.

2. Symmetry: About real axis.

3. Real-Axis Segments: Exists where number of real poles+zeros to right is odd.

4. Asymptotes:

   • Centroid: $$\displaystyle \sigma_a = \frac{\sum \text{poles} - \sum \text{zeros}}{n-m} $$

   • Angles: $$\displaystyle \theta_a = \frac{(2q+1)180^\circ}{n-m},\ q=0,1,\dots,n-m-1 $$

5. Breakaway/Break-in Points: On real axis, solve $$\displaystyle \frac{dK}{ds} = 0 $$ for $$\displaystyle K>0 $$.

6. Angle of Departure/Arrival:

   • From pole $p$: $$\displaystyle \angle D = 180^\circ - \sum \angle(\text{other poles to }p) + \sum \angle(\text{zeros to }p) $$

   • To zero $z$: $$\displaystyle \angle A = 180^\circ + \sum \angle(\text{poles to }z) - \sum \angle(\text{other zeros to }z) $$

7. Intersection with Imaginary Axis: Use Routh or angle condition at $$\displaystyle s=j\omega $$.

Breakaway/Break-in Points (Exact)

From $$\displaystyle K = -\frac{\prod |s - z_i|}{\prod |s - p_i|} $$, differentiate w.r.t. $s$:

$$ \frac{dK}{ds} = 0 \quad \Rightarrow \quad \sum \frac{1}{s - z_i} - \sum \frac{1}{s - p_i} = 0 $$

Solve for $s$ on real-axis segments.

Design for Specified Damping Ratio $\zeta$

• Damping ratio line: line from origin at angle $$\displaystyle \cos^{-1}(\zeta) $$ with negative real axis.

• Intersection of root locus with this line gives desired closed-loop poles.

• Compute $K$ from magnitude condition: $$\displaystyle |G(s)H(s)| = 1 $$ at that $s$.

Effects of Adding Open-Loop Poles/Zeros

• Adding pole: Root locus shifts right, system becomes less stable, slower response.

• Adding zero: Attracts root locus, improves stability & speed (if placed appropriately).

Stability Regions & Critical K

• Stable region: all branches in LHP for given $K$ range.

• Critical $K$: value at which locus crosses $j\omega$-axis (marginal stability). Found via Routh or substituting $$\displaystyle s=j\omega $$ into characteristic equation and solving real/imag parts.

6. Frequency Response Analysis

Bode Plots

• Magnitude Plot: $$\displaystyle 20\log_{10}|G(j\omega)| $$ vs $\log\omega$ (dB vs log scale).

• Phase Plot: $\angle G(j\omega)$ vs $\log\omega$ (degrees vs log scale).

Asymptotic Construction:

1. Write $$\displaystyle G(s) = K \cdot \frac{\prod (1 + s/z_i)}{\prod (1 + s/p_i)} $$.

2. Magnitude:

   • Start at $20\log|K|$.

   • Each pole: slope $-20$ dB/dec starting at $$\displaystyle \omega = p_i $$.

   • Each zero: slope $+20$ dB/dec starting at $$\displaystyle \omega = z_i $$.

   • Corner frequencies: $$\displaystyle \omega = p_i $$ or $$\displaystyle z_i $$.

   • At $\omega \gg$ all corners, slope = $$\displaystyle 20(n_z - n_p) $$ dB/dec.

3. Phase:

   • Each pole: $$\displaystyle -90^\circ $$ starting ~1 decade before $$\displaystyle p_i $$, ending ~1 decade after.

   • Each zero: $$\displaystyle +90^\circ $$ similarly.

   • Sum contributions.

Key Frequencies:

• Gain crossover frequency $$\displaystyle \omega_{gc} $$: $$\displaystyle |G(j\omega_{gc})| = 1 $$ (0 dB).

• Phase crossover frequency $$\displaystyle \omega_{pc} $$: $$\displaystyle \angle G(j\omega_{pc}) = -180^\circ $$.

Stability Margins:

• Gain Margin (GM): Factor by which gain can increase before instability.

$$ \boxed{GM = \frac{1}{|G(j\omega_{pc})|}} \quad (\text{in linear}), \quad GM_{dB} = -20\log|G(j\omega_{pc})| $$

If $\angle G$ never reaches $$\displaystyle -180^\circ $$, GM = ∞.

• Phase Margin (PM): Additional phase lag required at $$\displaystyle \omega_{gc} $$ to reach $$\displaystyle -180^\circ $$.

$$ \boxed{PM = 180^\circ + \angle G(j\omega_{gc})} $$

Positive PM ⇒ stable; PM > 35°–60° desirable.

Polar Plots

Plot $G(j\omega)$ as $\omega$ varies from $0$ to $\infty$ in complex plane.

Inverse Polar Plot: Plot of $1/G(j\omega)$ (used in Nyquist).

Nyquist Criterion

• Plot $G(j\omega)H(j\omega)$ for $$\displaystyle \omega: 0^-\to\infty $$ and mirror for negative frequencies (closed contour).

• Stability: Closed-loop stable iff number of clockwise encirclements of $-1+ j0$ point, $N$, satisfies:

$$ \boxed{N = P - Z} $$

where $P$ = number of open-loop RHP poles, $Z$ = number of closed-loop RHP poles (should be 0 for stability).

• Gain Margin from Nyquist: Distance from plot to $-1$ point along real axis (if crossing negative real axis).

Log Magnitude vs. Phase Plot (Nichols Chart)

• Plot $20\log|G(j\omega)|$ vs $\angle G(j\omega)$ (both linear axes).

• Overlay constant $$\displaystyle M_p $$, $$\displaystyle T_s $$ curves for design.

• Directly read PM (phase at 0 dB) and GM (magnitude at $$\displaystyle -180^\circ $$).

7. Compensation Techniques

Lead Compensation

• Purpose: Increase phase margin, improve transient response (reduce overshoot, settling time).

• Network:

$$ G_c(s) = \frac{1 + \tau s}{1 + \alpha \tau s}, \quad 0 < \alpha < 1, \quad \tau = RC $$

Circuit: Series $R$-$C$ with parallel $R$-$C$? Actually, typical lead: $$\displaystyle R_1 $$, $$\displaystyle C_1 $$ in series, $$\displaystyle R_2 $$ parallel with $$\displaystyle C_1 $$? Standard: $$\displaystyle G_c(s) = K_c \frac{1 + T s}{1 + \alpha T s} $$, $$\displaystyle \alpha < 1 $$.

• Maximum Phase Lead:

$$ \phi_m = \sin^{-1}\left(\frac{1-\alpha}{1+\alpha}\right), \quad \omega_m = \frac{1}{T\sqrt{\alpha}} $$

• Design (Bode):

  1. Determine required PM boost = desired PM – existing PM + safety (5°–12°).

  2. Find $$\displaystyle \alpha = \frac{1-\sin\phi_m}{1+\sin\phi_m} $$.

  3. Place $$\displaystyle \omega_m $$ at gain crossover of uncompensated system (or adjust $$\displaystyle K_c $$ to set $$\displaystyle \omega_{gc} $$).

  4. $$\displaystyle T = 1/(\omega_m\sqrt{\alpha}) $$.

  5. $$\displaystyle K_c = 1/|G(j\omega_m)| $$ to set magnitude 0 dB at $$\displaystyle \omega_{gc} $$.

Lag Compensation

• Purpose: Improve steady-state error (increase $$\displaystyle K_v $$ or $$\displaystyle K_p $$) without significantly affecting transient.

• Network:

$$ G_c(s) = \frac{1 + \tau s}{1 + \beta \tau s}, \quad \beta > 1 $$

• Effect: Low-frequency gain increase by factor $\beta$, high-frequency attenuation $-20\log\beta$ dB/dec.

• Design (Bode):

  1. Determine required $$\displaystyle K_v $$ or $$\displaystyle K_p $$ from $$\displaystyle e_{ss} $$ spec.

  2. Choose $$\displaystyle \beta = \frac{\text{required }K_v}{\text{existing }K_v} $$.

  3. Place corner frequencies $$\displaystyle \omega_1 = 1/\tau $$, $$\displaystyle \omega_2 = \beta/\tau $$ such that $$\displaystyle \omega_1 $$ is a decade below $$\displaystyle \omega_{gc} $$ of uncompensated, $$\displaystyle \omega_2 $$ near $$\displaystyle \omega_{gc} $$ to minimize phase lag.

  4. Adjust $$\displaystyle K_c $$ if needed.

Lag-Lead Compensation

• Purpose: Simultaneously improve transient (PM) and steady-state ($$\displaystyle K_v $$).

• Network: Cascade of lead and lag:

$$ G_c(s) = K_c \frac{(1+T_1 s)(1+T_2 s)}{(1+\alpha T_1 s)(1+\frac{1}{\beta} T_2 s)}, \quad \alpha < 1, \beta > 1 $$

• Design:

  1. Use lead to meet PM spec first (temporary).

  2. Then use lag to boost low-frequency gain without disturbing PM much (place lag corners appropriately).

  3. Adjust $$\displaystyle K_c $$ to finalize.

Compensator Design Using Root Locus

• For damping ratio $\zeta$: find intersection of root locus with $\zeta$-line, compute $K$.

• Add compensator pole/zero to shift locus toward desired region.

• Lead: zero closer to origin than pole → attracts locus left.

• Lag: pole closer to origin than zero → pushes locus right but mostly at low $K$.

8. Controllers

P, I, D, PID Transfer Functions

Tuning Methods (Ziegler-Nichols implied)

• P: Set $$\displaystyle K_p $$ to achieve desired response (trial).

• PI: Add $$\displaystyle K_i $$ to eliminate offset, watch for oscillations.

• PD: Add $$\displaystyle K_d $$ to improve stability.

• PID:

  1. Set $I$ and $D$ to zero, increase $P$ until sustained oscillations (critical $$\displaystyle K_u $$, period $$\displaystyle P_u $$).

  2. Use Z-N table: $$\displaystyle K_p = 0.6K_u $$, $$\displaystyle K_i = 1.2K_u/P_u $$, $$\displaystyle K_d = 0.075K_u P_u $$ for PID.

9. State Space Analysis

State Variables & State Equations

• State $\mathbf{x}(t)$: minimal set of variables determining future state.

• State-space model:

$$ \dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}u, \quad y = \mathbf{C}\mathbf{x} + \mathbf{D}u $$

• Canonical Forms:

  • Controllable canonical: $A$ in companion form from transfer function denominator.

  • Observable canonical: $$\displaystyle A^T $$ in controllable canonical form.

Controllability & Observability

• Controllability: Can state be driven to origin in finite time?

$$ \boxed{\text{Rank } \mathcal{C} = \text{Rank } [\mathbf{B}\ \mathbf{AB}\ \cdots\ \mathbf{A}^{n-1}\mathbf{B}] = n} $$

• Observability: Can state be estimated from output?

$$ \boxed{\text{Rank } \mathcal{O} = \text{Rank } [\mathbf{C}^T\ (\mathbf{CA})^T\ \cdots\ (\mathbf{CA}^{n-1})^T] = n} $$

State Transition Matrix $\Phi(t)$

• Defined: $$\displaystyle \Phi(t) = e^{\mathbf{A}t} $$.

• Properties:

  1. $$\displaystyle \Phi(0) = \mathbf{I} $$.

  2. $$\displaystyle \Phi^{-1}(t) = \Phi(-t) $$.

  3. $$\displaystyle \Phi(t_1)\Phi(t_2) = \Phi(t_1+t_2) $$.

  4. $$\displaystyle \frac{d}{dt}\Phi(t) = \mathbf{A}\Phi(t) = \Phi(t)\mathbf{A} $$.

• Computation:

  • If $\mathbf{A}$ diagonalizable: $$\displaystyle \mathbf{A} = \mathbf{V}\Lambda\mathbf{V}^{-1} $$, then $$\displaystyle \Phi(t) = \mathbf{V}e^{\Lambda t}\mathbf{V}^{-1} $$, where $$\displaystyle e^{\Lambda t} = \text{diag}(e^{\lambda_i t}) $$.

  • If not: use Jordan form or inverse Laplace: $$\displaystyle \Phi(t) = \mathcal{L}^{-1}\{(s\mathbf{I}-\mathbf{A})^{-1}\} $$.

Eigenvalues & Eigenvectors

• Eigenvalues $$\displaystyle \lambda_i $$: roots of $$\displaystyle \det(s\mathbf{I}-\mathbf{A})=0 $$ → system poles.

• Eigenvectors $$\displaystyle \mathbf{v}_i $$: solve $$\displaystyle (\mathbf{A}-\lambda_i\mathbf{I})\mathbf{v}_i = 0 $$.

• Modal decomposition: $$\displaystyle \mathbf{x}(t) = \sum c_i \mathbf{v}_i e^{\lambda_i t} $$. Stability iff all $$\displaystyle \text{Re}(\lambda_i) < 0 $$.

Solution of State Equations

• Homogeneous: $$\displaystyle \mathbf{x}(t) = \Phi(t)\mathbf{x}(0) $$.

• Forced: $$\displaystyle \mathbf{x}(t) = \Phi(t)\mathbf{x}(0) + \int_0^t \Phi(t-\tau)\mathbf{B}u(\tau)d\tau $$.

• For zero initial state: $$\displaystyle \mathbf{X}(s) = (s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}U(s) $$.

10. Special Systems & Components

AC Servomotor

• Construction: Two-phase induction motor with reference winding (voltage $$\displaystyle V_r $$) and control winding (voltage $$\displaystyle V_c $$).

• Assumptions:

  1. Constant field flux (linear magnetization).

  2. Rotor inductance and resistance referred to stator constant.

  3. Small operating range → linear torque-speed.

• Torque: $$\displaystyle T \propto I_c $$ (control current).

• Transfer Function:

  Derive from torque equation $$\displaystyle J\ddot{\theta} + B\dot{\theta} = K_t I_c $$, and voltage equation $$\displaystyle V_c = R_c I_c + L_c \frac{dI_c}{dt} + K_b \dot{\theta} $$ (back EMF).

  Neglect $$\displaystyle L_c $$ (small):

$$ \frac{\theta(s)}{V_c(s)} = \frac{K_t / R_c}{s\left(\frac{J}{B}s + 1\right)\left(\frac{R_c}{K_b} s + 1\right)} \approx \frac{K}{s(Ts+1)} $$

where $$\displaystyle T = J/B $$ (mechanical time constant).

DC Servomotor (Brief)

• Armature control: $$\displaystyle T = K_t I_a $$, $$\displaystyle V_a = L_a\frac{dI_a}{dt} + R_a I_a + K_b \omega $$.

• Transfer function: $$\displaystyle \frac{\omega(s)}{V_a(s)} = \frac{K_t}{(J s + B)(L_a s + R_a) + K_t K_b} $$.

Stepper Motors

• Construction: Stator with multiple phases, rotor (permanent magnet or variable reluctance).

• Working: Each pulse to a phase causes rotor to move fixed step angle (e.g., 1.8°/step for 200 steps/rev).

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

• Advantages: No feedback needed, precise positioning, holds position at rest.

• Disadvantages: Resonance at high speeds, torque drops with speed.

Tacho-Generators

• Principle: DC generator or AC tachometer producing voltage proportional to speed ($$\displaystyle V_{tg} = K_t \omega $$).

• Use: Provide speed feedback in control systems (e.g., in PID controllers for derivative action).

Linearization Effect of Feedback

• Feedback reduces sensitivity to parameter variations and linearizes nonlinearities within the loop.

• For a nonlinear element $$\displaystyle y = f(x) $$, closed-loop transfer approximates linear if feedback is high.

Relative Stability Concepts

• Gain Margin (GM): Multiplicative factor allowed before instability.

• Phase Margin (PM): Additional phase lag allowed before instability.

• Phase Crossover: Frequency where phase = $$\displaystyle -180^\circ $$.

• Gain Crossover: Frequency where magnitude = 1 (0 dB).

• Higher GM/PM ⇒ more stable, less oscillatory.

11. Miscellaneous & Short Note Topics

Block Diagram vs. Closed-Loop Comparison

Transfer Function Decomposition

• Direct Decomposition: Express $G(s)$ as sum of simpler transfer functions (e.g., partial fractions). Useful for state-space realization.

• Example: $$\displaystyle G(s) = \frac{1}{s(s+1)(s+2)} = \frac{1/2}{s} + \frac{-1}{s+1} + \frac{1/2}{s+2} $$.

Polar Plot for Different System Types

• Type 0: Ends at origin along $-\text{Im}$ axis if $$\displaystyle n_p > n_z $$.

• Type 1: Starts at $\infty$ along $$\displaystyle +90^\circ $$, ends on $-\text{Re}$ axis.

• Type 2: Starts at $\infty$ along $$\displaystyle +180^\circ $$, ends at origin along $-\text{Im}$.

Nyquist vs. Bode Plot Comparison

Effect of Open-Loop Poles/Zeros on Root Locus

• Poles: Attract locus; more poles → more branches to ∞, centroid shifts right.

• Zeros: Attract locus; adding zeros pulls branches left (improves stability).

• Asymptote angles depend on $n-m$.

Log Magnitude vs. Phase Plot (Nichols)

• Plot $20\log|G|$ vs $\angle G$ (linear axes).

• Constant $$\displaystyle M_p $$, $$\displaystyle T_s $$ curves superimposed → direct readout of time-domain specs from frequency response.

• Useful for design with multiple constraints.

State Space Representation Benefits

1. Handles MIMO systems naturally.

2. Incorporates initial conditions directly.

3. Easily extends to nonlinear systems (via state-space).

4. Unified framework for controllability/observability.

5. Digital computation friendly.

[!TIP]

Exam Focus:

• Mason’s formula: always compute $\Delta$ carefully, include all non-touching loop combinations.

• Root locus breakaway: solve $$\displaystyle \frac{dK}{ds}=0 $$, verify $$\displaystyle K>0 $$ and on real-axis segment.

• Bode phase margin: always at $$\displaystyle \omega_{gc} $$, not $$\displaystyle \omega_{pc} $$.

• Routh: if first column zero, use $\epsilon$ and check sign change.

• Steady-state error: identify system type correctly from $G(s)H(s)$ (poles at origin).

• Lead compensator: $$\displaystyle \alpha < 1 $$, phase lead positive; lag: $$\displaystyle \beta > 1 $$, phase lag negative.