UNIT 3: Control System Analysis & Design

Based on exhaustive analysis of RGPV past papers (JUN 2025, DEC 2024, JUN 2024, JUN 2023, NOV 2023, JUN 2022).

1.0 System Modeling & Representation

1.1 Analogous Systems

• Force-Voltage (F-V) Analogy: Force ↔ Voltage, Velocity ↔ Current, Displacement ↔ Charge, Friction ↔ Resistance, Mass ↔ Inductance, Spring ↔ Capacitance.

• Force-Current (F-I) Analogy: Force ↔ Current, Velocity ↔ Voltage, Displacement ↔ Flux, Friction ↔ Resistance, Mass ↔ Capacitance, Spring ↔ Inductance.

• Direct Analogous Method (Torque-Voltage): Rotational system elements mapped directly to electrical (J ↔ L, B ↔ R, K ↔ 1/C).

• Inverse Analogous Method (Torque-Current): Rotational system elements mapped inversely (J ↔ C, B ↔ R, K ↔ 1/L).

• Procedure: 1) Write system differential equations. 2) Replace mechanical variables with analogous electrical variables per chosen analogy. 3) Draw resulting circuit.

[!TIP] Exam questions often provide a mechanical network (mass, spring, damper) and ask for F-V or F-I analogy. Always write the torque/force equation first.

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

• SFG: Graphical representation of algebraic equations. Nodes = variables, Branches = gains, Direction = signal flow.

• Construction Rules from Block Diagram:

  1. Represent each variable as a node.

  2. Multiplyers become branch gains.

  3. Summing points become nodes with incoming branches.

  4. Take-off points are same node with multiple outgoing branches.

• Mason's Gain Formula:

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

Where:

*   $$\displaystyle P_k $$ = gain of k-th **forward path** (path from input to output node, no node repeated).

*   $\Delta$ = **Determinant** = $$\displaystyle 1 - \sum L_i + \sum L_i L_j - \sum L_i L_j L_k + ... $$

    *   $$\displaystyle L_i $$ = loop gain of each **individual loop**.

    *   $$\displaystyle L_i L_j $$ = product of gains of two **non-touching loops** (no common node).

*   $$\displaystyle \Delta_k $$ = **cofactor** of k-th forward path = value of $\Delta$ with all loops touching the k-th forward path removed.

• Advantages over Block Diagram:

  • Easier to visualize multiple forward paths & loops.

  • Systematic formula avoids trial-and-error reduction.

  • Clearly identifies non-touching loops.

[!TIP] Common Pitfall: Forgetting to remove all loops touching a forward path when calculating $$\displaystyle \Delta_k $$. Always list all loops first, then identify which touch each forward path.

1.3 Block Diagram Reduction

• Key Rules:

  1. Series: $$\displaystyle G_1 G_2 $$

  2. Parallel: $$\displaystyle G_1 + G_2 $$

  3. Feedback (Negative): $$\displaystyle \frac{G}{1+GH} $$

  4. Moving Summing Point: Can move across block if multiplied/divided by block gain.

  5. Moving Take-off Point: Can move across branch if multiplied/divided by branch gain.

• Procedure: Apply rules iteratively to reduce to single forward path and single feedback loop.

2.0 Time Domain Analysis & Performance Specifications

2.1 Standard Test Inputs (Laplace Domain)

2.2 First-Order System

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

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

  • Time constant τ: Time to reach 63.2% of final value.

  • Settling Time (2%): $$\displaystyle T_s \approx 4\tau $$.

• Steady-State Error:

  • Step input: $$\displaystyle e_{ss} = \frac{A}{1+K} $$ (if A=1, $$\displaystyle e_{ss} = \frac{1}{1+K} $$).

  • Ramp input: $$\displaystyle e_{ss} = \infty $$ (Type 0 system cannot track ramp).

2.3 Second-Order System (UNDERDAMPED - MOST IMPORTANT)

• Standard Form:

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

$\zeta$ = damping ratio, $$\displaystyle \omega_n $$ = natural frequency (rad/s).

• Unit Step Response ($c(t)$ for $t \geq 0$):

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

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

• Key Specifications:

  1. Maximum Overshoot ($$\displaystyle M_p $$):

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

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

    *Graph: $$\displaystyle M_p $$ decreases as $\zeta$ increases. For $$\displaystyle \zeta=0.5 $$, $$\displaystyle M_p \approx 16.3\% $$; $$\displaystyle \zeta=0.707 $$, $$\displaystyle M_p \approx 4.3\% $$.*

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}}

3.  **Rise Time ($$\displaystyle t_r $$)** (0% to 100% for underdamped):

$$t_r \approx \frac{\pi - \phi}{\omega_d} = \frac{\pi - \cos^{-1}(\zeta)}{\omega_n\sqrt{1-\zeta^2}}$$

4.  **Settling Time ($$\displaystyle T_s $$)** (2% criterion):

$$T_s \approx \frac{4}{\zeta\omega_n}$$

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

5.  **Resonant Frequency ($$\displaystyle \omega_r $$) & Peak ($$\displaystyle M_r $$)**:

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

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

• Other Cases:

  • Critically Damped ($$\displaystyle \zeta=1 $$): Fastest response without overshoot. $$\displaystyle c(t) = 1 - (1+\omega_n t)e^{-\omega_n t} $$.

  • Overdamped ($$\displaystyle \zeta>1 $$): Slow, no overshoot. Two real poles.

[!TIP] Exam Focus: Deriving $$\displaystyle M_p $$ and $$\displaystyle t_p $$ is very common. Remember: $$\displaystyle M_p $$ occurs at $$\displaystyle t=t_p $$ where $$\displaystyle \omega_d t_p = \pi $$. Use $$\displaystyle c(t_p) = 1 + M_p $$.

2.4 Effect of Parameters

• $\zeta \uparrow$: $$\displaystyle M_p \downarrow $$, $$\displaystyle t_r \uparrow $$, $$\displaystyle T_s \downarrow $$, $$\displaystyle \omega_r $$ may not exist.

• $$\displaystyle \omega_n \uparrow $$: $$\displaystyle t_r \downarrow $$, $$\displaystyle t_p \downarrow $$, $$\displaystyle T_s \downarrow $$, response faster.

• Trade-off: Low $\zeta$ (fast, oscillatory) vs High $\zeta$ (slow, overdamped).

3.0 Steady-State Error Analysis

3.1 Concept

• Steady-State Error ($$\displaystyle e_{ss} $$): Difference between desired and actual output as $t \to \infty$.

$$e_{ss} = \lim_{t\to\infty} e(t) = \lim_{s\to0} sE(s)$$

For unity feedback, $$\displaystyle E(s) = \frac{R(s)}{1+G(s)} $$.

3.2 Static Error Coefficients (from Open-Loop TF $G(s)H(s)$)

3.3 Calculating $$\displaystyle e_{ss} $$ for Unity Feedback

[!TIP] Limitations: Only for stable open-loop systems and standard inputs (step, ramp, parabolic). Cannot predict transient behavior.

4.0 Stability Analysis in Time Domain

4.1 Stability Concepts

• BIBO Stability: Bounded Input → Bounded Output.

• Asymptotic Stability: All poles of closed-loop TF have negative real parts ($$\displaystyle Re(s) < 0 $$).

• Marginal Stability: Poles on imaginary axis (no RHP poles), sustained oscillations.

4.2 Routh-Hurwitz Criterion

• Characteristic Equation: $$\displaystyle a_n s^n + a_{n-1} s^{n-1} + ... + a_1 s + a_0 = 0 $$, $$\displaystyle a_n > 0 $$.

• Routh Array Construction:

  | $$\displaystyle s^n $$ | $$\displaystyle a_n $$ | $$\displaystyle a_{n-2} $$ | $$\displaystyle a_{n-4} $$ | ... | | $$\displaystyle s^{n-1} $$ | $$\displaystyle a_{n-1} $$ | $$\displaystyle a_{n-3} $$ | $$\displaystyle a_{n-5} $$ | ... | | $$\displaystyle s^{n-2} $$ | $$\displaystyle b_1 = \frac{a_{n-1}a_{n-2} - a_n a_{n-3}}{a_{n-1}} $$ | $$\displaystyle b_2 $$ | ... | | | $$\displaystyle s^{n-3} $$ | $$\displaystyle c_1 = \frac{b_1 a_{n-3} - a_{n-1} b_2}{b_1} $$ | $$\displaystyle c_2 $$ | ... | | | ... | ... | ... | ... | |

• Stability Condition: All elements of first column must be positive (no sign changes).

• Number of RHP Roots = Number of sign changes in first column.

• Special Cases:

  1. Row of zeros: Indicate symmetrical roots on $j\omega$ axis. Form auxiliary equation from row above, differentiate, replace row.

  2. First element zero: Replace with small $$\displaystyle \epsilon > 0 $$, proceed, check sign as $\epsilon \to 0$.

[!TIP] Relative Stability: To check if roots lie left of line $$\displaystyle s = -\sigma $$, substitute $$\displaystyle s = z - \sigma $$ in CE, apply Routh to new polynomial in $z$.

4.3 Root Locus Technique

• Definition: Plot of closed-loop pole locations as gain $K$ varies from $0$ to $\infty$ for $$\displaystyle 1 + KG(s)H(s) = 0 $$.

• Construction Rules (for $$\displaystyle 0 < K < \infty $$):

  1. Branches: Number = number of open-loop poles ($n$).

  2. Start/End: Start at open-loop poles ($$\displaystyle K=0 $$), end at open-loop zeros ($$\displaystyle K=\infty $$) or $\infty$.

  3. Real Axis Segments: Exists if number of open-loop poles+zeros to the right is odd.

  4. Asymptotes (for $$\displaystyle n > m $$):

     • Number = $n - m$.

     • Angles: $$\displaystyle \theta = \frac{(2k+1)180^\circ}{n-m} $$, $$\displaystyle k=0,1,...,n-m-1 $$.

     • Centroid: $$\displaystyle \sigma = \frac{\sum \text{Re(poles)} - \sum \text{Re(zeros)}}{n-m} $$.

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

  6. Angle of Departure/Arrival (for complex poles/zeros):

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

     • Arrival at zero $$\displaystyle z_i $$: $$\displaystyle \angle \text{arrival} = 180^\circ + \sum \angle(\text{to poles}) - \sum \angle(\text{to other zeros}) $$.

  7. Imaginary Axis Crossing: Use Routh on CE or substitute $$\displaystyle s=j\omega $$ in $$\displaystyle 1+KG(j\omega)H(j\omega)=0 $$, separate real/imag parts.

• Stability from Root Locus: System stable if all closed-loop poles in LHP. Find $K$ range where locus lies entirely in LHP.

• Design Using Root Locus:

  1. For specified $\zeta$: Draw line from origin at angle $$\displaystyle \cos^{-1}(\zeta) $$ to negative real axis. Intersection with locus gives desired pole $$\displaystyle s = -\sigma \pm j\omega_d $$. Read corresponding $K$.

  2. For specified $$\displaystyle \omega_n $$: Draw circle of radius $$\displaystyle \omega_n $$ centered at origin. Intersection with locus gives pole, read $K$.

• Effects of Adding Poles/Zeros:

  • Adding Open-Loop Pole: Locus bends toward left half initially (more stable), but may bend right later. Asymptotes increase.

  • Adding Open-Loop Zero: Locus bends toward zero (more stable), reduces number of asymptotes, pulls branches to LHP.

[!TIP] Breakaway Calculation: Always verify $$\displaystyle K>0 $$ at candidate points. Use approximate sketch for complex loci.

5.0 Frequency Domain Analysis

5.1 Fundamentals

• Frequency Response: System response to sinusoidal input $$\displaystyle r(t)=A\sin\omega t $$. Output is $$\displaystyle c(t)=B\sin(\omega t + \phi) $$.

• Transfer Function at $$\displaystyle s=j\omega $$: $$\displaystyle G(j\omega) = |G(j\omega)| \angle G(j\omega) $$.

• Correlation: Low-frequency response ↔ steady-state, high-frequency ↔ transient sensitivity.

5.2 Bode Plot (Magnitude & Phase)

• Construction Procedure (Asymptotic):

  1. Write $G(j\omega)$ in standard form: $$\displaystyle G(j\omega) = K \cdot \frac{\prod (1+j\omega T_i)}{\prod (1+j\omega T_j)} \cdot (j\omega)^\pm N $$.

  2. Magnitude Plot (dB vs log$\omega$):

     • K: $20\log|K|$ dB horizontal line.

     • $j\omega$ (pole at origin): Slope $+20$ dB/dec starting at $$\displaystyle \omega=1 $$.

     • $1/(j\omega)$ (zero at origin): Slope $-20$ dB/dec.

     • $1+j\omega T$ (first-order zero): $0$ dB until $$\displaystyle \omega=1/T $$, then $+20$ dB/dec. Corner at $$\displaystyle \omega_c = 1/T $$.

     • $1/(1+j\omega T)$ (first-order pole): $0$ dB until $$\displaystyle \omega=1/T $$, then $-20$ dB/dec.

     • Quadratic ($$\displaystyle \omega_n^2/(s^2+2\zeta\omega_n s+\omega_n^2) $$): If $$\displaystyle \zeta < 0.707 $$, resonant peak. For $\zeta \geq 0.707$, treat as two first-order poles at $$\displaystyle \omega_n $$.

  3. Phase Plot (degrees vs log$\omega$):

     • K: $$\displaystyle 0^\circ $$ or $$\displaystyle 180^\circ $$ depending on sign.

     • $j\omega$: $$\displaystyle +90^\circ $$ constant.

     • $1/(j\omega)$: $$\displaystyle -90^\circ $$ constant.

     • $1+j\omega T$: $$\displaystyle 0^\circ $$ at low $\omega$, $$\displaystyle +90^\circ $$ at high $\omega$, transition over 2 decades centered at $$\displaystyle \omega_c=1/T $$. Approximate: $$\displaystyle 0^\circ $$ at $$\displaystyle \omega=0.1\omega_c $$, $$\displaystyle +90^\circ $$ at $$\displaystyle \omega=10\omega_c $$.

     • $1/(1+j\omega T)$: $$\displaystyle 0^\circ $$ to $$\displaystyle -90^\circ $$ similarly.

     • Quadratic: Phase from $$\displaystyle 0^\circ $$ to $$\displaystyle -180^\circ $$, transition centered at $$\displaystyle \omega_n $$.

  4. Combine all factors.

• Key Metrics from Bode Plot:

  • Gain Crossover Frequency ($$\displaystyle \omega_{gc} $$): Where $$\displaystyle |G(j\omega)| = 1 $$ (0 dB).

  • Phase Crossover Frequency ($$\displaystyle \omega_{pc} $$): Where $$\displaystyle \angle G(j\omega) = -180^\circ $$.

  • Gain Margin (GM): $$\displaystyle GM = \frac{1}{|G(j\omega_{pc})|} $$ (absolute) or $$\displaystyle 20\log\frac{1}{|G(j\omega_{pc})|} $$ (dB). Positive GM means stable.

  • Phase Margin (PM): $$\displaystyle PM = 180^\circ + \angle G(j\omega_{gc}) $$. Positive PM means stable.

• Stability Comment:

  • GM > 0 dB (or >1) and PM > 0° → Closed-loop stable.

  • GM = 0 dB, PM = 0° → Marginally stable (sustained oscillations).

  • Larger GM/PM → Greater relative stability.

[!TIP] Example: For $$\displaystyle G(s)=\frac{242(s+5)}{s(s+1)(s^2+5s+121)} $$:

1. Factor: $$\displaystyle 242 \cdot \frac{5(1+s/5)}{s(1+s)(121(1+s/5)^2+...)} \approx \frac{10(1+s/5)}{s(1+s)(1+0.1s)^2} $$.

2. Magnitude: Start $$\displaystyle 20\log10=20 $$ dB, -20 dB/dec from pole at origin, -40 dB/dec from two poles at ~10 rad/s, +20 dB/dec from zero at 5 rad/s.

3. Find $$\displaystyle \omega_{gc} $$ (0 dB crossing), read phase there → PM.

5.3 Polar Plot (Nyquist without encirclements)

• Construction: Plot $G(j\omega)$ as $\omega$ goes $$\displaystyle 0^+ \to \infty $$. Start at $$\displaystyle \omega=0 $$ (low-freq limit), end at $$\displaystyle \omega=\infty $$ (high-freq limit).

• Typical Shapes:

  • Type 0 (no pole at origin): Starts on +real axis, ends at origin (if proper), may go through all quadrants.

  • Type 1 (one pole at origin): Starts at $\infty$ on $$\displaystyle -90^\circ $$ line, ends at finite point on real axis.

  • Type 2 (two poles at origin): Starts at $\infty$ on $$\displaystyle -180^\circ $$ line, ends at origin.

• Effect of Adding Poles:

  • Pole at origin: Adds $$\displaystyle -90^\circ $$ to starting angle.

  • Pole at $$\displaystyle s=-1/T_i $$ (real, LHP): Plot shifts rightward (increases real part at mid-freq).

  • Pole at $$\displaystyle s=-1/T_r $$ (complex): Adds phase lag, plot dips downward.

5.4 Nyquist Stability Criterion

• Contour: Encloses entire RHP (imaginary axis from $-j\infty$ to $+j\infty$, large semicircle in LHP).

• Definitions:

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

  • $N$ = Number of clockwise encirclements of $(-1, j0)$ point by Nyquist plot of $G(s)H(s)$.

  • $Z$ = Number of closed-loop RHP poles (poles of $$\displaystyle 1+G(s)H(s)=0 $$ in RHP).

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

• Closed-Loop Stability Condition: $$\displaystyle Z = 0 $$ (no RHP poles) → $$\displaystyle N = P $$.

• Procedure:

  1. Determine $P$ from open-loop TF.

  2. Sketch Nyquist plot of $G(s)H(s)$ for $$\displaystyle \omega: 0^+ \to \infty $$, then mirror for $$\displaystyle \omega: -\infty \to 0^- $$ (if needed).

  3. Count clockwise encirclements $N$ of $(-1, j0)$.

  4. Apply $$\displaystyle Z = N + P $$. If $$\displaystyle Z=0 $$, closed-loop stable.

• Relative Stability: Distance of Nyquist plot from $(-1, j0)$ indicates gain/phase margins. Plot crossing real axis left of $-1$ indicates instability for some $K$.

• Example: For $$\displaystyle G(s)H(s)=\frac{K}{s(s+2)(s+10)} $$:

  • $$\displaystyle P=0 $$ (all LHP poles).

  • Nyquist starts at $\infty$ on $$\displaystyle -90^\circ $$ line, ends at origin.

  • For stability, plot must not encircle $(-1, j0)$. Find $$\displaystyle K_{max} $$ where plot just touches $(-1, j0)$ (using $$\displaystyle \angle G(j\omega)=-180^\circ $$ → $$\displaystyle \omega_{pc} $$, then $$\displaystyle |G(j\omega_{pc})|=1/K_{max} $$).

6.0 Controller Design & Compensation

6.1 PID Controllers

• Transfer Function: $$\displaystyle G_c(s) = K_p + \frac{K_i}{s} + K_d s $$.

• Effects:

  • P (Proportional): Increases $$\displaystyle K_p $$, reduces $$\displaystyle e_{ss} $$ for Type ≥0, but may reduce PM.

  • I (Integral): Increases system type by 1 → eliminates $$\displaystyle e_{ss} $$ for step/ramp/parabolic. Adds pole at origin → reduces PM, slows response.

  • D (Derivative): Adds zero → increases PM, reduces $$\displaystyle T_s $$, improves stability. Sensitive to noise.

• Advantages/Disadvantages:

  | Controller | Advantages | Disadvantages | | :--- | :--- | :--- | | P | Simple, reduces $$\displaystyle e_{ss} $$ (Type 0) | Finite $$\displaystyle e_{ss} $$, stability issues | | I | Zero $$\displaystyle e_{ss} $$ for Type 0 | Slow response, instability | | D | Improves transient, stability | Noise amplification | | PID | Good steady-state & transient | Tuning complex, noise sensitive |

6.2 Compensation Techniques

• Need: Improve stability (PM), steady-state accuracy ($$\displaystyle K_v $$), or speed ($$\displaystyle \omega_n $$).

• Lead Compensation ($$\displaystyle \alpha > 1 $$):

$$G_c(s) = \frac{1+\alpha T s}{1+ T s}$$

*   **Zero at $$\displaystyle \omega_z = 1/(\alpha T) $$**, **pole at $$\displaystyle \omega_p = 1/T $$** ($$\displaystyle \omega_z < \omega_p $$).

*   **Maximum Phase Lead**: $$\displaystyle \phi_m = \sin^{-1}\left(\frac{\alpha-1}{\alpha+1}\right) $$ at $$\displaystyle \omega_m = \frac{1}{T\sqrt{\alpha}} $$.

*   **Design (Bode)**:

    1.  Find $$\displaystyle \omega_{gc} $$ needed for desired $$\displaystyle \omega_n $$ (from specs).

    2.  Required PM = Desired PM + safety (5°-12°) - Current PM.

    3.  From $$\displaystyle \phi_m $$, find $$\displaystyle \alpha = \frac{1+\sin\phi_m}{1-\sin\phi_m} $$.

    4.  Place $$\displaystyle \omega_m = \omega_{gc,new} $$.

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

    6.  Adjust $K$ to meet $$\displaystyle |G_c G|_{db}=0 $$ at new $$\displaystyle \omega_{gc} $$.

*   **Effects**: ↑ PM, ↑ $$\displaystyle \omega_{gc} $$ (faster), ↑ bandwidth.

• Lag Compensation ($$\displaystyle \alpha > 1 $$):

$$G_c(s) = \frac{1+ T s}{1+ \alpha T s}$$

*   **Pole at $$\displaystyle \omega_p = 1/(\alpha T) $$**, **zero at $$\displaystyle \omega_z = 1/T $$** ($$\displaystyle \omega_z < \omega_p $$).

*   **Purpose**: Improve $$\displaystyle K_v $$ or $$\displaystyle K_p $$ without affecting high-frequency response much.

*   **Design (Bode)**:

    1.  Find $$\displaystyle K_{v,new} $$ from spec → required low-frequency magnitude ↑.

    2.  Choose $$\displaystyle \omega_z $$ ~ one decade below $$\displaystyle \omega_{gc,old} $$.

    3.  From magnitude difference at $$\displaystyle \omega_z $$, find $\alpha$.

    4.  Place pole at $$\displaystyle \omega_p = \omega_z / \alpha $$.

    5.  $K$ adjusted so $$\displaystyle |G_c G|_{db}=0 $$ at same $$\displaystyle \omega_{gc} $$ (approx).

*   **Effects**: ↑ $$\displaystyle K_v $$/$$\displaystyle K_p $$, ↓ bandwidth, may ↓ PM slightly.

• Lag-Lead Compensation:

$$G_c(s) = \frac{(1+T_1 s)(1+\alpha_2 T_2 s)}{(1+\alpha_1 T_1 s)(1+ T_2 s)} \quad (\alpha_1>1, \alpha_2>1)$$

*   Combines lead (phase boost) and lag (low-freq gain).

*   **Design Steps**:

    1.  Meet $$\displaystyle K_v $$ spec using lag part (place zero/pole low).

    2.  Meet PM spec using lead part (place zero/pole near $$\displaystyle \omega_{gc} $$).

    3.  Cascade or combine, adjust $K$.

6.3 PD Controller Design (Root Locus)

• $$\displaystyle G_c(s) = K_p(1 + T_d s) $$ adds a zero at $$\displaystyle s = -1/T_d $$.

• Procedure: For desired $\zeta$ or $$\displaystyle T_s $$, find required closed-loop pole location from second-order specs. On root locus, find $K$ and $$\displaystyle T_d $$ such that zero attracts locus to that point.

• Critically Damped ($$\displaystyle \zeta=1 $$): Place zero on real axis to the left of desired pole to pull locus left.

7.0 State-Space Analysis

7.1 State Variable Representation

• State Equation: $$\displaystyle \dot{x}(t) = A x(t) + B u(t) $$

• Output Equation: $$\displaystyle y(t) = C x(t) + D u(t) $$

• State Vector $x(t)$: Minimal set of variables determining future state.

• Canonical Forms (from TF $$\displaystyle G(s)=\frac{Y(s)}{U(s)} $$):

  • Controllable Canonical: $$\displaystyle A = \begin{bmatrix} 0 & 1 & ... & 0 \\ 0 & 0 & ... & 0 \\ ... & ... & ... & ... \\ -a_0 & -a_1 & ... & -a_{n-1} \end{bmatrix} $$, $$\displaystyle B = \begin{bmatrix} 0 \\ 0 \\ ... \\ 1 \end{bmatrix} $$, $$\displaystyle C = [b_0 - a_0 d, ..., b_{n-1} - a_{n-1} d] $$.

  • Observable Canonical: $A$ transpose of controllable, $$\displaystyle C = [1, 0, ..., 0] $$, $B$ accordingly.

7.2 Eigenvalues & Eigenvectors

• Eigenvalues $$\displaystyle \lambda_i $$: Roots of $$\displaystyle \det(sI - A) = 0 $$. Same as poles of system.

• Eigenvectors $$\displaystyle v_i $$: Non-zero vectors satisfying $$\displaystyle (A - \lambda_i I)v_i = 0 $$.

• Modal Matrix $V$: Matrix of eigenvectors as columns. $$\displaystyle V = [v_1, v_2, ..., v_n] $$.

• Diagonalization: If $V$ non-singular, $$\displaystyle V^{-1} A V = \Lambda = \text{diag}(\lambda_1, \lambda_2, ..., \lambda_n) $$.

• Significance: Solution $$\displaystyle x(t) = V e^{\Lambda t} V^{-1} x(0) + ... $$. Each eigenvector mode decays/grows at rate $$\displaystyle e^{\lambda_i t} $$.

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

• Definition: Matrix function satisfying $$\displaystyle \frac{d}{dt}\phi(t) = A\phi(t) $$, $$\displaystyle \phi(0)=I $$.

• Properties:

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

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

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

  4. $$\displaystyle \phi(t_1)\phi^{-1}(t_2) = \phi(t_1-t_2) $$

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

• Significance: Gives zero-input response: $$\displaystyle x_{zi}(t) = \phi(t)x(0) $$.

• Computation Methods:

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

  2. Cayley-Hamilton: If $A$ has distinct eigenvalues, $$\displaystyle \phi(t) = \sum_{i=1}^{n} e^{\lambda_i t} \frac{V_i}{(\lambda_i - \lambda_j)...} $$ where $$\displaystyle V_i = \frac{(v_i w_i^T)}{w_i^T v_i} $$ (using left eigenvectors $$\displaystyle w_i $$).

7.4 Solution of State Equations

• General Solution:

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

• Zero-Input Response (ZIR): $$\displaystyle x_{zi}(t) = \phi(t)x(0) $$ (due to initial state only).

• Zero-State Response (ZSR): $$\displaystyle x_{zs}(t) = \int_0^t \phi(t-\tau) B u(\tau) d\tau $$ (due to input only, $$\displaystyle x(0)=0 $$).

8.0 Components & Special Topics (Short Notes)

8.1 AC Servomotor

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

• Working: Control winding voltage $$\displaystyle V_c $$ (proportional to error) creates rotating magnetic field. Torque $$\displaystyle \propto V_c \sin\theta $$ (where $\theta$ = rotor angle). At small $\theta$, $\sin\theta \approx \theta$ → torque $\propto$ error → linearization.

• Assumptions for TF Derivation:

  1. Linear magnetization (no saturation).

  2. Negligible rotor dynamics (inertia $$\displaystyle J_r \approx 0 $$).

  3. Constant field current in reference winding.

  4. Small operating range ($\theta$ small).

• Transfer Function (Speed Output):

$$G(s) = \frac{\omega(s)}{V_c(s)} = \frac{K}{ (T_m s + 1)(T_e s + 1) }$$

where $K$ = gain, $$\displaystyle T_m $$ = mechanical time constant ($J/B$), $$\displaystyle T_e $$ = electrical time constant ($$\displaystyle L_r/R_r $$).

• Advantages: Smooth operation, low inertia, good speed control.

• Disadvantages: Limited torque, maintenance (brushes in some types), nonlinear at large angles.

8.2 Stepper Motor

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

• Types: Variable reluctance, permanent magnet, hybrid.

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

8.3 Tacho-Generator

• Principle: DC generator or AC tachometer. Output voltage $$\displaystyle V_o \propto $$ rotational speed $\omega$.

• Use: Rate feedback in servosystems to damp oscillations, improve stability.

8.4 Relative Stability

• Concept: How far system poles are from RHP/jω-axis. Measures "stability margin".

• Measures:

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

  • Phase Margin (PM): Additional phase lag that can be tolerated.

  • Sensitivity: $$\displaystyle \left| \frac{1}{1+G(s)H(s)} \right| $$ – lower near crossover = less sensitive to parameter changes.

8.5 Gain Margin & Phase Margin

• From Bode:

  • GM: At $$\displaystyle \omega_{pc} $$ (phase = -180°), $$\displaystyle GM = 1/|G(j\omega_{pc})| $$.

  • PM: At $$\displaystyle \omega_{gc} $$ (mag = 1), $$\displaystyle PM = 180^\circ + \angle G(j\omega_{gc}) $$.

• From Nyquist: Distance from $(-1,j0)$ to plot crossing real axis (GM) and to plot at $$\displaystyle \omega_{gc} $$ (PM).

• Rule of Thumb: PM ≈ 40°-60° for good transient response.

9.0 Miscellaneous & Problem-Solving

9.1 Determining System Type

• Count number of poles at origin in open-loop TF $G(s)H(s)$.

• Type 0: 0 poles at s=0.

• Type 1: 1 pole at s=0.

• Type 2: 2 poles at s=0, etc.

9.2 Design Problems Summary

• Find K from $$\displaystyle M_p $$ & $$\displaystyle T_p $$:

$$M_p = e^{-\frac{\zeta\pi}{\sqrt{1-\zeta^2}}} \Rightarrow \text{solve for } \zeta$$

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

Then from standard form, find $K$ (for $$\displaystyle G(s)=\frac{K}{s(s+a)(s+b)} $$, $$\displaystyle \omega_n^2 = K/(ab) $$, $$\displaystyle 2\zeta\omega_n = a+b $$).

• Find K and T from $$\displaystyle M_p $$ & $$\displaystyle \omega_r $$:

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

Solve for $\zeta$, $$\displaystyle \omega_n $$, then match coefficients.

• Lead Compensator Design (Bode):

  1. Find $$\displaystyle \omega_{gc} $$ from $$\displaystyle \omega_n $$ spec.

  2. Required PM = Desired PM + safety - Current PM.

  3. $$\displaystyle \phi_m = \text{Required PM} $$.

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

  5. $$\displaystyle \omega_m = \omega_{gc,new} $$.

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

  7. Adjust $K$ so $$\displaystyle |G_c G| = 0 $$ dB at $$\displaystyle \omega_{gc,new} $$.

9.3 Stability from Different Plots

9.4 Routh-Hurwitz Special Cases

• Missing Term: Indicates at least one root on imaginary axis → unstable or marginal.

• First Element Zero: Replace with $$\displaystyle \epsilon \to 0^+ $$, continue. Sign of $\epsilon$ row determines sign change.

• Row of Zeros: Use auxiliary equation $$\displaystyle A(s)=0 $$ (from row above), differentiate, replace zero row with coefficients of $dA/ds$.

9.5 Finding Number of RHP/LHP Roots

• Routh: Number of sign changes = number of RHP roots.

• Argument Principle (Nyquist): $$\displaystyle N = Z - P $$. If $P$ known, $$\displaystyle Z = N + P $$ gives RHP closed-loop poles.

\boxed{\text{END OF UNIT 3 NOTES}}
Always practice sketch-based questions (Root Locus, Bode, Nyquist) from past papers. Derive key formulas ($$\displaystyle M_p $$, $$\displaystyle t_p $$, $$\displaystyle e_{ss} $$) at least once.