Control System (EX-405) - Unit 5 Short Notes

UNIT 5: CONTROL SYSTEMS - EXAM-FOCUSED SHORT NOTES

1. System Modeling & Representation

Signal Flow Graphs & Mason's Gain Formula

• Definition: A graphical representation of a system using nodes (variables) and branches (transfer functions).

  • Forward Path: Path from input node to output node, touching each node only once.

  • Loop: Path that starts and ends at the same node, without repeating any node.

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

• Mason's Gain Formula:

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

Where:

*   $$\displaystyle P_k $$ = Gain of the $$\displaystyle k^{th} $$ forward path.

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

*   $$\displaystyle \Delta_k $$ = Value of $\Delta$ for the part of the graph **not touching** the $$\displaystyle k^{th} $$ forward path.

• Step-by-Step Procedure:

  1. Identify all forward paths ($$\displaystyle P_k $$) and loops.

  2. Calculate $\Delta$.

  3. For each forward path, determine $$\displaystyle \Delta_k $$ by removing all loops touching that path and computing $\Delta$ for the remaining graph.

  4. Apply the formula.

• Advantages over Block Diagram:

  • Easier to apply for complex, interconnected systems.

  • Systematic, rule-based; less prone to error than successive reductions.

  • Easily identifies all forward paths and loops.

[!TIP] Exam Focus: Questions often involve systems with multiple forward paths and several touching/non-touching loops. Carefully list all loops and their combinations.

Electrical Analogies of Mechanical Systems

Two main analogies to convert mechanical systems (translational/rotational) into equivalent electrical networks.

• Derivation Principle: Based on mathematical similarity between governing differential equations.

  • Translational: $$\displaystyle M\ddot{x} + B\dot{x} + Kx = F $$

  • Rotational: $$\displaystyle J\ddot{\theta} + B\dot{\theta} + K\theta = T $$

[!TIP] Common Pitfall: Confusing which analogy maps mass to L vs. C. Remember: F-V is like a series RLC circuit (mass ~ inductor). F-I is like a parallel RLC circuit (mass ~ capacitor).

2. Time Domain Analysis & Standard Test Signals

Standard Test Input Signals

First Order System

• Transfer Function: $$\displaystyle G(s) = \frac{K}{\tau s + 1} $$, where $\tau$ = time constant.

• Unit Step Response: $$\displaystyle c(t) = K(1 - e^{-t/\tau}) $$.

  • At $$\displaystyle t = \tau $$, response reaches 63.2% of final value.

• Steady-State Error (ess):

  • Step input: $$\displaystyle e_{ss} = \frac{1}{1+K} $$ (for unit step).

  • Ramp input: $$\displaystyle e_{ss} = \tau $$ (non-zero, infinite velocity error constant).

Second Order System (Underdamped, $$\displaystyle \zeta < 1 $$)

• Standard Form:

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

Where $$\displaystyle \omega_n $$ = natural frequency, $\zeta$ = damping ratio.

• Unit Step Response Expression:

$$c(t) = 1 - \frac{e^{-\zeta\omega_n t}}{\sqrt{1-\zeta^2}} \sin(\omega_d t + \phi)$$

where $$\displaystyle \omega_d = \omega_n\sqrt{1-\zeta^2} $$ (damped frequency), $$\displaystyle \phi = \cos^{-1}(\zeta) $$.

• Key Performance Specifications (Derivations Crucial):

  • Maximum Overshoot ($$\displaystyle M_p $$):

$$M_p = e^{\frac{-\zeta\pi}{\sqrt{1-\zeta^2}}}$$

    > Graph of $$\displaystyle M_p $$ vs. $\zeta$ is exponential decay. $$\displaystyle M_p = 0\% $$ for $\zeta \ge 1$.

    \boxed{M_p = e^{\frac{-\zeta\pi}{\sqrt{1-\zeta^2}}}}

*   **Peak Time ($$\displaystyle t_p $$):**

$$t_p = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}} = \frac{\pi}{\omega_d}$$

    \boxed{t_p = \frac{\pi}{\omega_d}}

*   **Rise Time ($$\displaystyle t_r $$):** Time to go from 10% to 90% (or 0% to 100%) of final value.

    For underdamped: $$\displaystyle t_r \approx \frac{\pi - \phi}{\omega_d} $$.

*   **Settling Time ($$\displaystyle t_s $$):** Time to stay within a band (usually ±2% or ±5%).

$$t_s \approx \frac{4}{\zeta\omega_n} \quad (\text{2\% band}) \quad \text{or} \quad \frac{3}{\zeta\omega_n} \quad (\text{5\% band})$$

    \boxed{t_s \approx \frac{4}{\zeta\omega_n}}

• Resonant Frequency ($$\displaystyle \omega_r $$) & Peak ($$\displaystyle M_r $$):

$$\omega_r = \omega_n\sqrt{1-2\zeta^2} \quad (\text{for } \zeta < \frac{1}{\sqrt{2}})$$

$$M_r = \frac{1}{2\zeta\sqrt{1-\zeta^2}}$$

> Exists only for $$\displaystyle \zeta < 0.707 $$.

[!TIP] Exam Focus: Deriving $$\displaystyle M_p $$ and $$\displaystyle t_p $$ from the step response expression is a repeatedly asked 7-mark question. Practice the steps: find $$\displaystyle dc(t)/dt=0 $$, solve for $$\displaystyle t_p $$, substitute back for $$\displaystyle c(t_p) $$.

3. Steady-State Error Analysis

Static Error Coefficients

For open-loop TF $$\displaystyle G(s)H(s) = \frac{K \prod (s+z_i)}{s^N \prod (s+p_i)} $$ (N = system type).

Steady-State Error ($$\displaystyle e_{ss} $$) for Standard Inputs

Generalized Error Coefficients

For input $$\displaystyle r(t) = \frac{a_0 t^n}{n!} + ... + a_0 $$ (polynomial):

$$e_{ss} = \frac{a_0}{E_n(0)} \quad \text{where} \quad E(s) = 1 + G(s)H(s)$$

$$\displaystyle E_n(0) $$ is the $$\displaystyle n^{th} $$ derivative of $E(s)$ evaluated at $$\displaystyle s=0 $$.

Limitations of Static Error Coefficients

1. Only applicable for stable open-loop systems (or systems with poles in LHP).

2. Only valid for polynomial inputs (step, ramp, parabolic).

3. Does not provide transient response information (overshoot, settling time).

4. Fails for unstable open-loop systems (ess may be finite even if CL unstable).

[!TIP] Common Pitfall: Forgetting that $$\displaystyle K_p, K_v, K_a $$ are defined for the open-loop transfer function $G(s)H(s)$. Also, note that Type 0 system has finite ess for step, infinite for ramp/parabolic.

4. Stability Analysis in Time Domain

Routh-Hurwitz Stability Criterion

• Concept: Determines number of closed-loop poles in Right-Half Plane (RHP) without solving characteristic equation.

• Condition for Stability: All elements of the first column of Routh array must be positive (for stable polynomial with positive coefficients).

• Formation of Routh Array:

  For $$\displaystyle a_n s^n + a_{n-1} s^{n-1} + ... + a_0 = 0 $$.

  | $$\displaystyle s^n $$ | $$\displaystyle a_n $$ | $$\displaystyle a_{n-2} $$ | $$\displaystyle a_{n-4} $$ | ... | | $$\displaystyle s^{n-1} $$ | $$\displaystyle a_{n-1} $$ | $$\displaystyle a_{n-3} $$ | $$\displaystyle a_{n-5} $$ | ... | | $$\displaystyle s^{n-2} $$ | $$\displaystyle b_1 = \frac{a_{n-1}a_{n-2} - a_n a_{n-3}}{a_{n-1}} $$ | $$\displaystyle b_2 $$ | ... | | | $$\displaystyle s^{n-3} $$ | $$\displaystyle c_1 = \frac{b_1 a_{n-3} - a_{n-1} b_2}{b_1} $$ | ... | | | | ... | ... | ... | | |

• Special Cases:

  1. First element zero: Replace with a small positive $\epsilon$ and continue. Sign change indicates instability.

  2. Entire row zero: Indicates symmetrical roots about origin (purely imaginary or RHP-LHP pairs). Form auxiliary equation from the row above, differentiate, and replace the zero row with coefficients of the derivative.

  3. Sign changes in first column: Number of sign changes = number of RHP poles.

Finding Range of K for Stability & Oscillation Frequency

• Apply Routh to characteristic equation $$\displaystyle 1 + G(s)H(s) = 0 $$ (contains parameter K).

• Condition: All first column elements > 0 → gives inequalities in K.

• Marginal stability (oscillations): Occurs when a row becomes zero. The frequency of sustained oscillations $\omega$ is found by solving the auxiliary equation $$\displaystyle A(s) = 0 $$ for $$\displaystyle s = j\omega $$.

[!TIP] Exam Focus: Questions like "find relation between K and T for stability" or "find marginal K and frequency of oscillations" are very common. Master the entire row zero case and auxiliary equation method.

5. Root Locus Technique

Definition & Basic Rules

• Root Locus: Plot of closed-loop pole locations as gain $K$ varies from $0$ to $\infty$ for the system $$\displaystyle T(s) = \frac{K G(s)}{1 + K G(s)} $$.

• Construction Rules Summary:

  1. Branches: Equal to number of open-loop poles (or zeros, whichever is greater).

  2. Start/End Points: Start at OL poles ($$\displaystyle K=0 $$), end at OL zeros ($K \to \infty$). If zeros < poles, $p-z$ branches go to $\infty$ along asymptotes.

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

  4. Asymptotes: For branches going to $\infty$.

     • Centroid: $$\displaystyle \sigma_a = \frac{\sum \text{Re}(p_i) - \sum \text{Re}(z_i)}{p - z} $$

     • Angles: $$\displaystyle \theta_a = \frac{(2k+1)180^\circ}{p-z}, \quad k = 0, \pm1, \pm2... $$

  5. Breakaway/Break-in Points: On real axis segments. Solve $$\displaystyle \frac{dK}{ds} = 0 $$ where $$\displaystyle K = -\frac{1}{G(s)H(s)} $$.

  6. Angle of Departure/Arrival: For complex poles/zeros.

     • Departure from pole $$\displaystyle p_i $$: $$\displaystyle \angle \text{Departure} = 180^\circ - \sum \text{angles to other poles} + \sum \text{angles to zeros} $$.

  7. Intersection with Imaginary Axis: Use Routh-Hurwitz on characteristic equation or angle condition ($$\displaystyle \angle G(j\omega)H(j\omega) = \pm180^\circ $$).

Sketching Complete Root Locus (Example: $$\displaystyle G(s)H(s) = \frac{K}{s(s+2)(s^2+6s+25)} $$)

1. OL poles: $0, -2, -3\pm j4$. No OL zeros. 4 branches.

2. Real axis: Segment between $0$ and $-2$.

3. Asymptotes: $$\displaystyle p-z=4 $$, centroid at $$\displaystyle \sigma_a = \frac{(0-2-3-3) - 0}{4} = -2 $$. Angles: $$\displaystyle 45^\circ, 135^\circ, -45^\circ, -135^\circ $$.

4. Breakaway point on real axis between 0 and -2 (solve $$\displaystyle dK/ds=0 $$).

5. Departure angles from complex poles $-3\pm j4$.

6. Imaginary axis crossing: Use Routh on $$\displaystyle s^4 + 11s^3 + 43s^2 + 50s + K = 0 $$. Find $K$ for marginal stability.

Design Applications

• Finding K for specified $\zeta$: Draw a line from origin at angle $$\displaystyle \theta = \cos^{-1}(\zeta) $$. Intersection with RL gives desired pole location and corresponding K.

• PD Controller: Adds a zero at $$\displaystyle s = -1/T_d $$. RL attracts branches to the left, improving damping. Can achieve critical damping by placing zero appropriately.

• Effect of Adding Poles/Zeros:

  • Adding OL zero: Attracts RL branches, generally improves stability and transient response.

  • Adding OL pole: Repels RL branches, tends to destabilize, slows response.

[!TIP] Exam Focus: "Sketch complete root locus with approximate breakaway points" and "Comment on stability" are extremely frequent. Always state the range of K for stability based on your sketch.

6. Frequency Domain Analysis (Bode, Nyquist, Polar)

Bode Plot Construction

• Procedure:

  1. Write $G(j\omega)H(j\omega)$ in standard form: $$\displaystyle K \cdot \frac{\prod (1+j\omega/z_i)}{\prod (1+j\omega/p_i)} $$.

  2. Identify corner frequencies ($$\displaystyle \omega = |p_i|, |z_i| $$).

  3. Magnitude Plot: Start at $$\displaystyle 20\log_{10}|K| $$ (dB). Apply ±20 dB/dec slope change at each pole/zero.

  4. Phase Plot: Start at $$\displaystyle 0^\circ $$ (if no poles/zeros at origin) or $$\displaystyle -90^\circ \times N $$. Add $$\displaystyle -45^\circ $$ to $$\displaystyle -90^\circ $$ transition around each pole/zero.

• Type 0,1,2 Systems:

  • Type 0: Low-freq mag slope = 0 dB/dec. Phase starts near $$\displaystyle 0^\circ $$.

  • Type 1: Low-freq mag slope = -20 dB/dec. Phase starts near $$\displaystyle -90^\circ $$.

  • Type 2: Low-freq mag slope = -40 dB/dec. Phase starts near $$\displaystyle -180^\circ $$.

• Key Frequencies:

  • Gain Crossover ($$\displaystyle \omega_{gc} $$): Where $$\displaystyle |G(j\omega)H(j\omega)| = 1 $$ (0 dB). $$\displaystyle PM = 180^\circ + \angle G(j\omega_{gc})H(j\omega_{gc}) $$.

  • Phase Crossover ($$\displaystyle \omega_{pc} $$): Where $$\displaystyle \angle G(j\omega)H(j\omega) = -180^\circ $$. $$\displaystyle GM = -20\log_{10}|G(j\omega_{pc})H(j\omega_{pc})| $$ (in dB).

• Stability Criteria (Unity Feedback):

  • Stable: $$\displaystyle PM > 0^\circ $$ (and $$\displaystyle GM > 0 $$ dB).

  • Marginally Stable: $$\displaystyle PM = 0^\circ $$ (or $$\displaystyle GM = 0 $$ dB).

  • Desirable: $$\displaystyle PM \approx 30^\circ - 60^\circ $$ for good transient response.

[!TIP] Common Pitfall: Phase at $$\displaystyle \omega_{gc} $$ must be calculated accurately. Remember phase contribution from a factor $$\displaystyle (1+j\omega/\omega_c) $$ is $$\displaystyle \tan^{-1}(\omega/\omega_c) $$. For $$\displaystyle \omega \ll \omega_c $$, phase ≈ $$\displaystyle 0^\circ $$; for $$\displaystyle \omega \gg \omega_c $$, phase ≈ $$\displaystyle 90^\circ $$.

Polar Plot (Nyquist without encirclements)

• Definition: Plot of $|G(j\omega)H(j\omega)| \angle G(j\omega)H(j\omega)$ as $\omega$ varies from $0$ to $\infty$.

• Construction for Type 0,1,2:

  • Type 0: Starts at $$\displaystyle K \angle 0^\circ $$, ends at $$\displaystyle 0 \angle -90^\circ \times (\text{# poles}) $$.

  • Type 1: Starts at $$\displaystyle \infty \angle -90^\circ $$, ends at $$\displaystyle 0 \angle -90^\circ \times (\text{# poles} - 1) $$.

  • Type 2: Starts at $$\displaystyle \infty \angle -180^\circ $$, ends at $$\displaystyle 0 \angle -90^\circ \times (\text{# poles} - 2) $$.

• Effect of Adding Poles:

  • Pole at origin: Plot starts at $\infty$ with angle $$\displaystyle -90^\circ \times N $$.

  • Pole at $$\displaystyle s = -1/T_i $$: Introduces a "dip" or loop, shifting the plot.

Nyquist Stability Criterion

• Contour Mapping: Map the Nyquist contour (encircling RHP) through $G(s)H(s)$.

• Nyquist Equation: $$\displaystyle N = P - Z $$

  • $N$ = Net clockwise encirclements of -1 point by Nyquist plot.

  • $P$ = Number of open-loop RHP poles (poles of $G(s)H(s)$).

  • $Z$ = Number of closed-loop RHP poles (poles of $1+G(s)H(s)$).

• Stability Condition: For closed-loop stability, $$\displaystyle Z = 0 $$. Therefore, $$\displaystyle N = P $$.

• Procedure:

  1. Plot Nyquist for $$\displaystyle \omega: 0^+ \to \infty $$ and $$\displaystyle \omega: \infty \to 0^+ $$ (mirror image if real).

  2. Count $P$ from open-loop TF.

  3. Count $N$ (encirclements of -1).

  4. Apply $$\displaystyle Z = P - N $$. If $$\displaystyle Z=0 $$, system stable.

• Gain Margin from Nyquist: The factor by which gain can be multiplied before plot passes through -1. If Nyquist crosses real axis at $$\displaystyle x < 0 $$, $$\displaystyle GM = 1/|x| $$.

[!TIP] Exam Focus: Sketching Nyquist for $$\displaystyle G(s)H(s) = \frac{K}{s(s+2)(s+10)} $$ and finding K for stability is a classic question. Remember: $$\displaystyle P=1 $$ (pole at origin). For stability, Nyquist must encircle -1 point once clockwise ($$\displaystyle N=-1 $$) so that $$\displaystyle Z = 1 - (-1) = 0 $$.

7. Compensation Techniques

Lead Compensator Design (Frequency Domain)

Given $$\displaystyle K_v $$ and $$\displaystyle PM_{spec} $$:

1. Determine required velocity constant: $$\displaystyle K_v = \lim_{s\to0} s G_c(s)G(s) \ge \text{spec} $$.

2. Find uncompensated $$\displaystyle PM_{un} $$ from Bode of $G(s)$.

3. Phase boost needed: $$\displaystyle \phi_m = PM_{spec} - PM_{un} + 5^\circ $$ (safety).

4. Calculate $$\displaystyle \alpha = \frac{1 - \sin\phi_m}{1 + \sin\phi_m} $$.

5. Find $$\displaystyle \omega_{max} $$ (frequency where $|G(j\omega)|$ has magnitude $$\displaystyle 20\log_{10}(1/\sqrt{\alpha}) $$). This is new $$\displaystyle \omega_{gc} $$.

6. $$\displaystyle T = 1/(\omega_{max} \sqrt{\alpha}) $$.

7. $K$ adjusted to satisfy $$\displaystyle K_v $$ requirement.

Lag Compensator Design

1. Determine required $$\displaystyle K_v $$ or $$\displaystyle K_a $$.

2. Find $K$ from uncompensated system to meet $$\displaystyle K_v $$ spec: $$\displaystyle K_{new} = \beta K_{old} $$.

3. Choose $\beta$ such that $$\displaystyle \beta = K_{new}/K_{old} $$.

4. Place lag zero at $$\displaystyle \omega_{gc} $$ of uncompensated system (or 1/10th of new $$\displaystyle \omega_{gc} $$) to minimize phase effect.

5. $$\displaystyle T_i = 10/\omega_{gc} $$ (pole 10x left of zero).

[!TIP] Design Tip: For lead, $$\displaystyle \omega_{gc} $$ shifts right (faster response). For lag, $$\displaystyle \omega_{gc} $$ shifts left (slightly slower). Lag-lead balances this.

8. State Space Analysis (Modern Control)

State Model

• State Variables: Minimum set of variables $$\displaystyle (x_1, x_2, ..., x_n) $$ that completely describe system dynamics.

• State Equations:

$$\dot{x} = Ax + Bu$$

$$y = Cx + Du$$

Where $x$ = state vector, $u$ = input, $y$ = output.

• Choice of State Variables: Often chosen as capacitor voltages and inductor currents (electrical) or positions and velocities (mechanical).

State Transition Matrix $$\displaystyle \phi(t) = e^{At} $$

• Definition: Matrix function that maps initial state to future state: $$\displaystyle x(t) = \phi(t)x(0) $$ (for $$\displaystyle u=0 $$).

• Properties:

  1. $$\displaystyle \phi(0) = I $$ (identity).

  2. $$\displaystyle \phi(t_2)\phi(t_1) = \phi(t_1 + t_2) $$.

  3. $$\displaystyle \phi^{-1}(t) = \phi(-t) $$.

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

  5. $$\displaystyle \phi(t) = \mathcal{L}^{-1}\left[(sI - A)^{-1}\right] $$.

• Solution of State Equations:

$$x(t) = \phi(t)x(0) + \int_0^t \phi(t-\tau) Bu(\tau) d\tau$$

Eigenvalues & Eigenvectors

• Eigenvalues ($$\displaystyle \lambda_i $$): Roots of $$\displaystyle \det(sI - A) = 0 $$. They are the system poles. Determine stability, natural response modes.

• Eigenvectors ($$\displaystyle v_i $$): Solve $$\displaystyle (A - \lambda_i I)v_i = 0 $$. Define modal directions.

• Diagonalization: If $A$ has distinct eigenvalues, $$\displaystyle A = V \Lambda V^{-1} $$, where $$\displaystyle V = [v_1, v_2, ...] $$ (modal matrix), $$\displaystyle \Lambda = \text{diag}(\lambda_1, \lambda_2, ...) $$.

  Then $$\displaystyle e^{At} = V e^{\Lambda t} V^{-1} $$.

• For Repeated Eigenvalues: Use Jordan form.

[!TIP] Exam Focus: Computing eigenvalues/eigenvectors for a 2x2 or 3x3 matrix $A$ is common. For $$\displaystyle A = \begin{bmatrix}0 & 1\\-2 & -3\end{bmatrix} $$: $$\displaystyle \det(sI-A)=s^2+3s+2=0 \Rightarrow \lambda_1=-1, \lambda_2=-2 $$. Eigenvectors: for $$\displaystyle \lambda=-1 $$, $$\displaystyle v_1 = [1, -1]^T $$; for $$\displaystyle \lambda=-2 $$, $$\displaystyle v_2 = [1, -2]^T $$.

9. Actuators & Components

AC Servomotor

• Construction: Two-phase induction motor. Stator has two windings (reference & control) 90° apart. Rotor is squirrel-cage.

• Assumptions for TF Derivation:

  1. Constant field flux (control voltage controls torque directly).

  2. Linear torque-speed relationship.

  3. Negligible rotor time constant ($$\displaystyle T_r \approx 0 $$).

  4. No slip.

• Transfer Function:

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

Where $K$ = motor constant, $$\displaystyle T_M $$ = mechanical time constant ($J/B$).

Stepper Motor

• Working: Digital motor. Rotor moves in discrete steps (e.g., 1.8°/step) when stator windings are energized in sequence.

• Advantages: Open-loop position control, no feedback needed, precise positioning, holds position at rest.

• Disadvantages: Resonance at high speeds, torque drops rapidly with speed, needs driver circuit.

Tacho-Generator

• Principle: Generates voltage $$\displaystyle V_t \propto \omega $$ (shaft speed). Used as a speed feedback sensor in control systems (e.g., in speed control loops).

10. Miscellaneous & Fundamental Concepts

Feedback in Control Systems

• Significance:

  1. Error Reduction: Reduces sensitivity to parameter variations and disturbances.

  2. Stability Trade-off: Can destabilize a stable open-loop system if gain is too high.

  3. Improves Bandwidth & Response: Can speed up or slow down response.

  4. Reduces Steady-State Error: Increases system type.

Open Loop vs. Closed Loop

Poles & Zeros

• Poles: Roots of denominator of $G(s)$. Determine natural response (stability, speed).

  • LHP pole: decaying mode.

  • RHP pole: unstable, growing mode.

  • Imag axis pole: marginally stable (oscillation).

• Zeros: Roots of numerator. Affect transient response shape (undershoot, direction) and controllability/observability.

Transfer Function Derivation

• General Steps:

  1. Write differential equations for the system (using Kirchhoff's laws, Newton's laws).

  2. Take Laplace transform (assume zero ICs).

  3. Express output/input ratio to get $$\displaystyle G(s) = C(s)/R(s) $$.

• Example (Mechanical): $$\displaystyle M\ddot{x} + B\dot{x} + Kx = F \Rightarrow G(s) = X(s)/F(s) = \frac{1}{Ms^2 + Bs + K} $$.

[!TIP] Final Reminder: In exams, always sketch diagrams (root locus, Bode, Nyquist) clearly. Label axes, critical points (breakaway, crossover frequencies, margins). For derivations, state assumptions clearly. For design problems, verify final specs (e.g., check $$\displaystyle M_p $$ from obtained $\zeta$).