Control System (EC-404) - Unit 1 Short Notes

1.0 INTRODUCTION & FUNDAMENTAL CONCEPTS

1.1 Definition & Objectives of a Control System

• Definition: A control system is an interconnection of components that commands, directs, or regulates itself or another system to achieve a desired performance.

• Objectives:

  • Regulation: Maintain output at a set value despite disturbances.

  • Tracking: Make output follow a time-varying reference.

  • Stability: Ensure bounded output for bounded input (BIBO).

  • Performance: Fast response, minimal overshoot, zero steady-state error.

1.2 Classification of Control Systems

1.3 Significance and Effects of Feedback

• Advantages:

  • Reduces sensitivity to parameter variations and disturbances.

  • Improves stability (can stabilize unstable open-loop systems).

  • Enhances accuracy (reduces steady-state error).

  • Increases bandwidth (faster response).

  • Linearization effect: Feedback can linearize nonlinear systems around an operating point.

• Disadvantages:

  • Complexity & cost increase.

  • Risk of instability if not properly designed.

  • Reduction in gain (requires higher open-loop gain for same accuracy).

• Linearization Effect: In nonlinear systems, feedback around an operating point can make the closed-loop response approximately linear for small perturbations.

1.4 Control System Components & Terminology

• Plant: The physical system to be controlled.

• Controller/Compensator: Generates control signal based on error.

• Actuator: Amplifies controller signal to drive the plant.

• Sensor/Transducer: Measures output for feedback.

• Disturbance: Unwanted input affecting output.

• Reference Input (R): Desired command signal.

• Error (E): Difference between reference and feedback: $$\displaystyle E = R - H \cdot C $$.

• Effect of Location of Poles on Stability:

  • Left-Half Plane (LHP): $$\displaystyle \text{Re}(s) < 0 $$ → Stable (asymptotically stable).

  • Right-Half Plane (RHP): $$\displaystyle \text{Re}(s) > 0 $$ → Unstable.

  • Imaginary Axis: $$\displaystyle \text{Re}(s) = 0 $$ → Marginally stable (sustained oscillations).

  [!TIP] In Routh-Hurwitz, all first-column elements > 0 ensures all poles in LHP. Poles near imaginary axis → poorly damped (high overshoot).

2.0 SYSTEM MODELING AND REPRESENTATION

2.1 Block Diagram Representation

• Basic Elements:

  • Block: System component with transfer function (e.g., $G(s)$).

  • Summing Point: Algebraic sum of inputs (use + or −).

  • Take-off Point: Signal branching for measurement/feedback.

• Key Paths:

  • Forward Path: Path from input to output through blocks.

  • Feedback Path: Path from output back to summing point.

  • Open-Loop Transfer Function: $G(s)H(s)$ (product of forward and feedback).

  • Closed-Loop Transfer Function: $$\displaystyle T(s) = \frac{G(s)}{1 + G(s)H(s)} $$ (negative feedback).

• Block Diagram Reduction Rules:

  1. Combine cascaded blocks: $$\displaystyle G_1 G_2 $$.

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

  3. Feedback loop: $$\displaystyle \frac{G}{1 \pm GH} $$.

  4. Move take-off points forward/backward across summing points (adjust with blocks).

  5. Eliminate summing points by combining signals.

• Comparison:

  • Open Loop: Simple, no feedback, inaccurate, unstable to disturbances.

  • Closed Loop: Complex, feedback improves accuracy/stability, but may oscillate.

2.2 Signal Flow Graph (SFG)

• Terminology:

  • Node: Signal point (variable).

  • Branch: Directed edge with transmittance (gain).

  • Transmittance: Gain of branch (e.g., $G(s)$).

  • Input Node: Only outgoing branches (source).

  • Output Node: Only incoming branches (sink).

  • Mixed Node: Both incoming & outgoing.

• Mason's Gain Formula (MGF):

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

• $$\displaystyle P_k $$: Gain of $$\displaystyle k^{th} $$ forward path (input to output).

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

  • $$\displaystyle L_i $$: Loop gains (non-touching loops multiplied).

• $$\displaystyle \Delta_k $$: Co-factor of $$\displaystyle P_k $$ (remove loops touching $$\displaystyle P_k $$ from $\Delta$).

• Steps to Apply MGF:

  1. Identify all forward paths & their gains $$\displaystyle P_k $$.

  2. Identify all loops & compute $\Delta$.

  3. For each $$\displaystyle P_k $$, compute $$\displaystyle \Delta_k $$ by removing loops touching that path.

  4. Compute $$\displaystyle T = \frac{\sum P_k \Delta_k}{\Delta} $$.

• Advantages over Block Diagram:

  • Easier for complex interconnections.

  • Visual identification of loops & paths.

  • Systematic application (no need for step-by-step reduction).

2.3 Transfer Function

• Definition: Ratio of Laplace transform of output to input under zero initial conditions:

$$G(s) = \frac{C(s)}{R(s)} \bigg|_{zero\;IC}$$

• Properties (for LTI systems):

  • Independent of input magnitude.

  • Poles: roots of denominator → system modes (stability).

  • Zeros: roots of numerator → affects transient response.

  • Causality: Proper ($\deg N \leq \deg D$).

  • Determined solely by system parameters.

• Derivation from Differential Equations:

  • Take Laplace transform (zero IC).

  • Solve algebraically for $C(s)/R(s)$.

• Transfer Function Decomposition:

  • Direct Decomposition: $$\displaystyle G(s) = G_1(s) G_2(s) ... $$ (cascaded).

  • Parallel Decomposition:

    • Standard: $$\displaystyle G(s) = G_1(s) + G_2(s) + \cdots $$ (blocks in parallel).

    • Non-Standard: Use partial fractions; each term has separate block.

  • Feedback Decomposition: $$\displaystyle G(s) = \frac{G_f}{1 - G_f H} $$ where $$\displaystyle G_f $$ is forward part.

2.4 Analogous Systems

• Force-Voltage (F-V) Analogy (Torque-Voltage for rotational):

  | Mechanical (Translational) | Electrical (F-V) | |-------------------------------|----------------------| | Force ($F$) | Voltage ($V$) | | Mass ($M$) | Inductance ($L$) | | Damping ($B$) | Resistance ($R$) | | Compliance ($1/K$) | Capacitance ($C$) | | Displacement ($x$) | Charge ($q$) | | Velocity ($\dot{x}$) | Current ($i$) |

• Force-Current (F-I) Analogy:

  | Mechanical | Electrical (F-I) | |----------------|----------------------| | Force ($F$) | Current ($I$) | | Mass ($M$) | Capacitance ($C$) | | Damping ($B$) | Conductance ($1/R$) | | Compliance | Inductance ($L$) | | Displacement | Flux ($\psi$) |

• Direct vs. Inverse:

  • Direct Analogy: Similar physical laws (e.g., F-V: $$\displaystyle F = M\ddot{x} + B\dot{x} + Kx $$ ↔ $$\displaystyle V = L\frac{di}{dt} + Ri + \frac{1}{C}\int i dt $$).

  • Inverse Analogy: Torque-Voltage (rotational F-V) or Torque-Current (rotational F-I).

• Electrical Analogous Networks: Replace mechanical elements with electrical equivalents per chosen analogy; write equations to get transfer function.

3.0 TIME DOMAIN ANALYSIS

3.1 Standard Test Input Signals

3.2 Time Response of First-Order Systems

• Standard Form: $$\displaystyle G(s) = \frac{K}{\tau s + 1} $$, $$\displaystyle \tau = \text{time constant} $$.

• Unit Step Response:

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

• At $$\displaystyle t = \tau $$, $$\displaystyle c(\tau) = 0.632K $$ (63.2% of final value).

• Unit Ramp Response:

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

• Steady-State Error:

  • Step: $$\displaystyle e_{ss} = 0 $$ (if $K$ finite? Actually for type 0: $$\displaystyle e_{ss} = 1/(1+K) $$; but first-order system typically type 0 if no pole at origin. Clarify: For $$\displaystyle G(s)=K/(τs+1) $$, step error $$\displaystyle e_{ss} = 1/(1+K) $$.)

  • Ramp: $$\displaystyle e_{ss} = \tau/K $$ (non-zero).

• Significance of $\tau$:

  • Time to reach 63.2% of final value for step.

  • Smaller $\tau$ → faster response.

  • Bandwidth $$\displaystyle \omega_b = 1/\tau $$ (rad/s).

3.3 Time Response of Second-Order Systems

• Standard Form:

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

• $$\displaystyle \omega_n $$: Undamped natural frequency (rad/s).

• $\zeta$: Damping ratio (dimensionless).

• Underdamped Case ($$\displaystyle 0 < \zeta < 1 $$):

  • Poles: $$\displaystyle s = -\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2} $$.

  • Unit Step Response:

$$c(t) = 1 - \frac{e^{-\zeta\omega_n t}}{\sqrt{1-\zeta^2}} \sin\left(\omega_n\sqrt{1-\zeta^2} t + \phi\right), \quad \phi = \cos^{-1}(\zeta)$$

• Transient Performance Specifications:

  • Delay Time ($$\displaystyle T_d $$): Time to reach 50% of final value.

$$T_d \approx \frac{1 + 0.7\zeta}{\omega_n} \quad (\text{approx})$$

• Rise Time ($$\displaystyle T_r $$): Time from 10% to 90% (underdamped).

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

• Peak Time ($$\displaystyle T_p $$): Time to first peak.

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

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

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

> [!TIP] $$\displaystyle M_p $$ depends **only** on $\zeta$. For $$\displaystyle M_p = 16\% $$, $\zeta \approx 0.5$.

• Settling Time ($$\displaystyle T_s $$): Time to stay within ±2% (or 5%) band.

$$T_s = \frac{4}{\zeta\omega_n} \quad (2\%) \quad \text{or} \quad T_s = \frac{3}{\zeta\omega_n} \quad (5\%)$$

• Effect of $\zeta$ and $$\displaystyle \omega_n $$:

  • $\zeta \uparrow$: Less overshoot, longer $$\displaystyle T_r $$, $$\displaystyle T_p $$, $$\displaystyle T_s $$ (slower).

  • $$\displaystyle \omega_n \uparrow $$: Faster response (all times $\downarrow$), same $$\displaystyle M_p $$ if $\zeta$ fixed.

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

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

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

3.4 Steady-State Error Analysis

• Definition: $$\displaystyle e_{ss} = \lim_{t\to\infty} e(t) $$ (difference between desired and actual output).

• Unity Feedback System:

$$E(s) = \frac{R(s)}{1 + G(s)} \quad \Rightarrow \quad e_{ss} = \lim_{s\to 0} sE(s) = \lim_{s\to 0} \frac{sR(s)}{1 + G(s)}$$

• Static Error Coefficients (for type 0,1,2 systems):

  | System Type | Position Error $$\displaystyle K_p $$ | Velocity Error $$\displaystyle K_v $$ | Acceleration Error $$\displaystyle K_a $$ | |-----------------|-------------------------|--------------------------|-----------------------------| | Type 0 | $$\displaystyle K_p = \lim_{s\to0} G(s) $$ | $\infty$ | $\infty$ | | Type 1 | $\infty$ | $$\displaystyle K_v = \lim_{s\to0} sG(s) $$ | $\infty$ | | Type 2 | $\infty$ | $\infty$ | $$\displaystyle K_a = \lim_{s\to0} s^2G(s) $$ |

• Steady-State Error for Standard Inputs:

  | Input | Type 0 | Type 1 | Type 2 | |-------------|------------------|------------------|------------------| | Step | $$\displaystyle \frac{1}{1+K_p} $$ | $0$ | $0$ | | Ramp | $\infty$ | $$\displaystyle \frac{1}{K_v} $$ | $0$ | | Parabolic| $\infty$ | $\infty$ | $$\displaystyle \frac{1}{K_a} $$ |

• Generalized Error Coefficients:

$$e_{ss} = \frac{1}{1 + K_p} \quad \text{(step)}$$

For ramp/parabolic, use $$\displaystyle K_v $$, $$\displaystyle K_a $$.

• Limitations of Static Error Coefficient Method:

  • Only applicable to stable unity feedback systems.

  • Only for standard inputs (step, ramp, parabolic).

  • Fails for non-unity feedback or unstable open-loop.

  • Does not give error for transient period.

• Error Series (Routh's Error Coefficient):

  • For polynomial input $$\displaystyle r(t) = a_0 + a_1 t + a_2 t^2 + \cdots $$,

  • $$\displaystyle e_{ss} = \frac{a_0}{1+K_p} + \frac{a_1}{K_v} + \frac{a_2}{K_a} + \cdots $$ (if all finite).

  [!TIP] If any $$\displaystyle K_p, K_v, K_a $$ is infinite, corresponding term is zero.

4.0 STABILITY ANALYSIS IN TIME DOMAIN (ROUTH-HURWITZ)

4.1 Concept of Stability

• BIBO Stability: Bounded input → bounded output.

• Asymptotic Stability: All poles in LHP; response decays to zero.

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

• Instability: At least one pole in RHP; response grows unbounded.

4.2 Routh-Hurwitz (R-H) Stability Criterion

• Necessary Condition: All coefficients of characteristic polynomial $$\displaystyle a_n s^n + \cdots + a_0 = 0 $$ must be positive.

• Sufficient Condition: All elements of first column of Routh array positive.

• Formation of Routh Array:

  • Row 1: $$\displaystyle a_n, a_{n-2}, a_{n-4}, \ldots $$

  • Row 2: $$\displaystyle a_{n-1}, a_{n-3}, a_{n-5}, \ldots $$

  • Row $i$: $$\displaystyle b_1 = \frac{a_{n-1}a_{n-2} - a_n a_{n-3}}{a_{n-1}}, \ldots $$ etc.

• Special Cases:

  1. First element zero: Replace with small $$\displaystyle \epsilon > 0 $$, complete array, then let $\epsilon \to 0$. Check sign changes.

  2. Entire row zero: Indicates symmetrical roots (e.g., $$\displaystyle s = \pm j\omega $$). Form auxiliary equation from row above, differentiate, replace zero row.

• Application to Find Range of K:

  • Write characteristic equation with parameter $K$.

  • Construct Routh array; impose all first-column elements $$\displaystyle > 0 $$.

  • Solve inequalities for $K$.

• Relative Stability: Roots to left of line $$\displaystyle s = -\sigma $$.

  • Substitute $$\displaystyle s = z - \sigma $$ (shift origin) and apply R-H to polynomial in $z$.

4.3 Root Locus (RL) Technique

• Definition: Plot of closed-loop pole locations as gain $K$ varies from $0$ to $\infty$.

• Properties & Construction Rules:

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

  2. Number of Branches: Equal to number of poles $n$.

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

  4. Asymptotes:

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

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

  5. Breakaway/Break-in Points:

     • On real axis, solve $$\displaystyle \frac{dK}{ds} = 0 $$ (from $$\displaystyle K = -\frac{\prod (s+z_i)}{\prod (s-p_i)} $$).

     • Valid if point lies on real-axis segment.

  6. Angle of Departure/Arrival:

     • From complex pole: $$\displaystyle \phi_d = 180^\circ - \sum \text{angles to other poles} + \sum \text{angles to zeros} $$.

     • To complex zero: $$\displaystyle \phi_a = 180^\circ + \sum \text{angles to poles} - \sum \text{angles to other zeros} $$.

  7. Intersection with Imaginary Axis: Use Routh array (set first-column element = 0) or substitute $$\displaystyle s=j\omega $$ into characteristic equation.

• Step-by-Step Procedure:

  1. Locate open-loop poles/zeros.

  2. Determine real-axis segments.

  3. Find asymptotes & centroid.

  4. Compute breakaway points.

  5. Find angles of departure/arrival for complex poles/zeros.

  6. Locate intersection with $j\omega$-axis.

  7. Sketch smoothly, respecting symmetry about real axis.

• Stability Analysis: If any branch lies in RHP → unstable for that $K$. Stable if all branches in LHP.

• Determining $K$ for Specified $\zeta$ or $$\displaystyle \omega_n $$:

  • Draw line from origin at angle $$\displaystyle \theta = \cos^{-1}(\zeta) $$ (for $\zeta$-line) or constant $$\displaystyle \omega_n $$ circle.

  • Intersection with root locus gives desired pole location.

  • Compute $$\displaystyle K = \frac{1}{|G(s)H(s)|} $$ at that $s$.

• Effect of Adding Open-Loop Poles/Zeros:

  • Adding pole: Root locus shifts right → tends to destabilize.

  • Adding zero: Root locus shifts left → tends to stabilize (attracts branches).

• Design Example: For desired $$\displaystyle M_p $$ → find $\zeta$ from $$\displaystyle M_p $$ formula. For desired $$\displaystyle T_s $$ → $$\displaystyle \zeta\omega_n \geq 4/T_s $$. Use RL to find $K$ giving pole with that $\zeta$ and $$\displaystyle \omega_n $$.

5.0 FREQUENCY DOMAIN ANALYSIS

5.1 Frequency Response Concepts

• Definition: Steady-state response of system to sinusoidal input $$\displaystyle r(t) = A\sin(\omega t) $$.

• Output is sinusoid of same frequency $\omega$ but different amplitude & phase: $$\displaystyle c(t) = B\sin(\omega t + \phi) $$.

• Correlation with Time Response: Frequency response (Bode, Nyquist) reveals stability margins, resonant peaks, bandwidth, which relate to time-domain specs (overshoot, settling time).

5.2 Bode Plot (Asymptotic & Exact)

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

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

• Construction Rules:

  • Pole/Zero at Origin:

    • Zero: slope +20 dB/dec starting at $$\displaystyle \omega=1 $$, phase +90°.

    • Pole: slope −20 dB/dec, phase −90°.

  • Simple Non-Origin Pole/Zero:

    • Zero at $$\displaystyle s = -1/T $$: magnitude corner at $$\displaystyle \omega = 1/T $$, slope +20 dB/dec after; phase from $$\displaystyle 0.1\omega_c $$ to $$\displaystyle 10\omega_c $$ (0°→90°).

    • Pole at $$\displaystyle s = -1/T $$: corner at $$\displaystyle \omega = 1/T $$, slope −20 dB/dec; phase 0°→−90°.

  • Complex Poles/Zeros:

    • Approximate as two real poles/zeros if $\zeta$ small? Actually, for underdamped pair, resonant peak appears. Asymptotic: two corners at $$\displaystyle \omega_n $$ with −40 dB/dec slope.

  • System Type:

    • Type 0: Low-freq magnitude constant ($20\log K$).

    • Type 1: Low-freq slope −20 dB/dec.

    • Type 2: Low-freq slope −40 dB/dec.

  • High-Frequency Asymptote: Slope = −20 × (number of poles) dB/dec.

• Steps to Draw Bode:

  1. Write $G(j\omega)$ in factored form.

  2. Identify $K$, poles/zeros at origin, finite poles/zeros.

  3. Draw low-frequency asymptote.

  4. Add slope changes at each corner frequency.

  5. Apply corrections at 1/√2 and √2 times corner for ±3 dB.

  6. For phase, add contributions: zero: +90° over 2 decades; pole: −90°.

• Key Frequencies:

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

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

• Stability Margins:

  • Gain Margin (GM): $$\displaystyle GM = \frac{1}{|G(j\omega_{pc})|} $$ (absolute) or $$\displaystyle 20\log_{10}(GM) $$ (dB). $$\displaystyle GM > 1 $$ (0 dB) → stable.

  • Phase Margin (PM): $$\displaystyle PM = 180^\circ + \angle G(j\omega_{gc}) $$. $$\displaystyle PM > 0^\circ $$ → stable.

• Relative Stability: Larger GM & PM → more stable (greater damping). $PM \approx \zeta$ (roughly: $\zeta \approx PM/100$ for small PM).

• Log Magnitude vs. Phase Plot (Nichols Chart): Plot of $20\log|G|$ vs $\angle G$ (not vs $\omega$). Used for direct reading of GM/PM and closed-loop response.

5.3 Polar Plot (Nyquist without encirclements)

• Definition: Plot of $\text{Re}[G(j\omega)]$ vs $\text{Im}[G(j\omega)]$ as $\omega$ from $0 \to \infty$.

• Construction:

  • Start at $$\displaystyle \omega=0 $$: $G(0)$ (real, finite or infinite).

  • As $\omega \to \infty$: approaches origin (if proper).

  • For Type 0: starts at $K$, ends at origin.

  • For Type 1: starts at $\infty$ on negative real axis, ends at origin.

  • For Type 2: starts at $\infty$ on negative imaginary axis? Actually, Type 2: $$\displaystyle G(s) \approx K/s^2 $$ → $$\displaystyle G(j\omega) \approx -K/\omega^2 $$ (real negative), so starts at $-\infty$ on negative real axis.

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

• Comparison with Bode: Bode separates magnitude & phase; Polar combines both in one plot but less precise for reading margins.

5.4 Nyquist Stability Criterion

• Contour Mapping: Map open-loop contour $N$ encirclements of $(-1, j0)$ → closed-loop poles in RHP: $$\displaystyle Z = N + P $$.

  • $P$: Open-loop poles in RHP.

  • $N$: Clockwise encirclements of $(-1, j0)$ by Nyquist plot of $G(s)H(s)$.

  • $Z$: Closed-loop poles in RHP (instability if $$\displaystyle Z > 0 $$).

• Nyquist Contour: Encloses entire RHP; includes imaginary axis and large semicircle.

• Application:

  1. Determine $P$ (count open-loop RHP poles).

  2. Sketch Nyquist plot of $G(j\omega)H(j\omega)$ for $$\displaystyle \omega: 0^-\to 0^+\to \infty $$ (include large semicircle if needed).

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

  4. Compute $$\displaystyle Z = N + P $$. Stable if $$\displaystyle Z = 0 $$.

• Gain/Phase Margin from Nyquist:

  • GM: Distance from plot to $(-1, j0)$ along real axis (if crossing negative real axis).

  • PM: Angle between negative real axis and plot at $$\displaystyle |G(j\omega)H(j\omega)|=1 $$ intersection.

• Relative Stability: Larger GM/PM → more robust.

6.0 COMPENSATION TECHNIQUES

6.1 Need for Compensation

• Improve transient response (overshoot, settling time).

• Improve steady-state accuracy (reduce error).

• Enhance stability (increase PM/GM).

• Meet conflicting specifications (e.g., fast response vs. low overshoot).

6.2 Types of Compensators

• Lead Compensator (Phase Lead):

  • Circuit: RC network with $$\displaystyle R_1, R_2, C $$.

  • Transfer Function: $$\displaystyle G_c(s) = \frac{1 + sT}{1 + \alpha sT}, \quad \alpha < 1 $$.

    • Zero at $$\displaystyle s = -1/T $$, pole at $$\displaystyle s = -1/(\alpha T) $$ (zero to right of pole).

  • Effect on Bode:

    • Magnitude: Increases around $$\displaystyle \omega = 1/\sqrt{\alpha}T $$.

    • Phase: Adds positive phase (lead) up to $$\displaystyle \phi_{\max} = \sin^{-1}\left(\frac{1-\alpha}{1+\alpha}\right) $$.

  • Design Procedure (Bode-based):

    1. Compute required PM boost: $$\displaystyle \phi_m = \text{desired PM} - \text{current PM} + 5^\circ $$ (safety).

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

    3. Find $$\displaystyle \omega_c $$ (new gain crossover) from magnitude condition: $$\displaystyle |G(j\omega_c)| \cdot |G_c(j\omega_c)| = 1 $$.

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

    5. Verify PM & GM.

• Lag Compensator (Phase Lag):

  • Circuit: Similar but $$\displaystyle R_1, R_2 $$ swapped; $$\displaystyle \beta > 1 $$.

  • Transfer Function: $$\displaystyle G_c(s) = \frac{1 + sT}{1 + \beta sT}, \quad \beta > 1 $$.

    • Pole at $$\displaystyle s = -1/T $$, zero at $$\displaystyle s = -1/(\beta T) $$ (pole to right of zero).

  • Effect on Bode:

    • Low-frequency magnitude boost (improves $$\displaystyle K_v $$).

    • Small phase lag (negative) near $$\displaystyle \omega = 1/T $$.

  • Design Procedure:

    1. Determine required $$\displaystyle K_v $$ from steady-state spec.

    2. Choose $$\displaystyle \beta = \frac{K_{v,\text{new}}}{K_{v,\text{old}}} $$.

    3. Place corner $$\displaystyle \omega_c = 1/T $$ at least a decade below $$\displaystyle \omega_{gc} $$ (to minimize phase effect).

    4. Adjust $K$ to meet magnitude at $$\displaystyle \omega_{gc} $$.

• Lag-Lead Compensator:

  • Combined: $$\displaystyle G_c(s) = \frac{(1+sT_1)(1+sT_2)}{(1+\alpha sT_1)(1+\beta sT_2)} $$ with $$\displaystyle \alpha<1 $$, $$\displaystyle \beta>1 $$.

  • When Employed: Both transient (PM) and steady-state ($$\displaystyle K_v $$) specs required.

  • Design Steps:

    1. Design lead for PM (as above).

    2. Design lag for $$\displaystyle K_v $$ (place lag corner far below $$\displaystyle \omega_{gc} $$).

    3. Cascade; verify.

6.3 Compensator Design Examples

• Lead Design Example: Given $G(s)$, desired PM = 50°, current PM = 20°. $$\displaystyle \phi_m = 50-20+5=35° $$. $$\displaystyle \alpha = (1-\sin35°)/(1+\sin35°) ≈ 0.18 $$. Find $$\displaystyle \omega_c $$ from $$\displaystyle |G(j\omega_c)| \cdot \frac{1}{\sqrt{\alpha}} = 1 $$ (since lead max gain $1/\sqrt{\alpha}$). Then $$\displaystyle T = 1/(\omega_c\sqrt{\alpha}) $$.

• Lag Design Example: Given $G(s)$, max steady-state error for ramp $$\displaystyle e_{ss} \leq 0.02 $$ → $$\displaystyle K_v \geq 50 $$. If current $$\displaystyle K_v = 10 $$, need $$\displaystyle \beta = 5 $$. Place $$\displaystyle \omega_c = 1/T $$ at $$\displaystyle \omega_{gc}/10 $$.

7.0 CONTROLLERS

• Composite Controllers: Advanced structures (e.g., lead-lag, state feedback) for multi-objective design.

8.0 STATE SPACE ANALYSIS (MODERN CONTROL THEORY)

8.1 State Space Representation

• State Variables: Minimum set $$\displaystyle \mathbf{x}(t) = [x_1, x_2, \ldots, x_n]^T $$ that completely describes system dynamics.

• State Equations:

$$\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}u$$

$$y = \mathbf{C}\mathbf{x} + D u$$

• $\mathbf{A}$: System matrix ($n \times n$).

• $\mathbf{B}$: Input matrix ($n \times r$).

• $\mathbf{C}$: Output matrix ($m \times n$).

• $D$: Feedthrough matrix ($m \times r$).

• Transfer Function:

$$G(s) = \mathbf{C}(s\mathbf{I} - \mathbf{A})^{-1}\mathbf{B} + D$$

• Poles = eigenvalues of $\mathbf{A}$.

• Block Diagram: Integrators for $$\displaystyle \dot{x}_i = \text{combination of states} + \text{input} $$.

8.2 Solution of State Equations

• State Transition Matrix $\Phi(t)$:

$$\Phi(t) = e^{\mathbf{A}t}$$

• Properties:

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

  2. $$\displaystyle \Phi^{-1}(t) = e^{-\mathbf{A}t} = \Phi(-t) $$ (if $\mathbf{A}$ nonsingular).

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

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

• Significance: Propagates initial state: $$\displaystyle \mathbf{x}(t) = \Phi(t)\mathbf{x}(0) $$ (zero input).

• General Solution:

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

• First term: zero-input response.

• Second term: zero-state response.

8.3 Controllability & Observability

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

  • Kalman's Test: Controllability matrix $$\displaystyle \mathbf{Q}_c = [\mathbf{B} \; \mathbf{A}\mathbf{B} \; \mathbf{A}^2\mathbf{B} \; \cdots \; \mathbf{A}^{n-1}\mathbf{B}] $$.

    • System controllable iff $$\displaystyle \text{rank}(\mathbf{Q}_c) = n $$.

• Observability: Ability to determine initial state from output over finite time.

  • Observability matrix $$\displaystyle \mathbf{Q}_o = [\mathbf{C}^T \; (\mathbf{CA})^T \; \cdots \; (\mathbf{CA}^{n-1})^T]^T $$.

    • System observable iff $$\displaystyle \text{rank}(\mathbf{Q}_o) = n $$.

• Example: For $$\displaystyle \dot{x} = Ax + Bu $$, with $$\displaystyle A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}, B = \begin{bmatrix} 0 \\ 1 \end{bmatrix} $$:

  $$\displaystyle \mathbf{Q}_c = [B \; AB] = \begin{bmatrix} 0 & 1 \\ 1 & -3 \end{bmatrix} $$, $$\displaystyle \text{rank}=2 $$ → controllable.

8.4 Eigenvalues & Eigenvectors

• Definition: For matrix $\mathbf{A}$, $\lambda$ is eigenvalue if $$\displaystyle \det(\mathbf{A} - \lambda\mathbf{I}) = 0 $$; corresponding eigenvector $\mathbf{v} \neq 0$ satisfies $$\displaystyle (\mathbf{A} - \lambda\mathbf{I})\mathbf{v} = 0 $$.

• Significance:

  • Eigenvalues = poles of transfer function → determine stability & natural response modes.

  • Eigenvectors define mode shapes (direction in state space).

• Computation:

  • Solve characteristic equation $$\displaystyle \det(s\mathbf{I} - \mathbf{A}) = 0 $$ for eigenvalues $$\displaystyle s_i $$.

  • For each $$\displaystyle \lambda_i $$, solve $$\displaystyle (\mathbf{A} - \lambda_i\mathbf{I})\mathbf{v}_i = 0 $$ for eigenvector $$\displaystyle \mathbf{v}_i $$.

• Example: $$\displaystyle A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix} $$:

  • $$\displaystyle \det(s\mathbf{I}-A) = s(s+3) + 2 = s^2+3s+2=0 $$ → $$\displaystyle \lambda_1=-1, \lambda_2=-2 $$.

  • For $$\displaystyle \lambda_1=-1 $$: $$\displaystyle (A+I)v=0 $$ → $$\displaystyle \begin{bmatrix} 1 & 1 \\ -2 & -2 \end{bmatrix}v=0 $$ → $$\displaystyle v_1 = \begin{bmatrix} 1 \\ -1 \end{bmatrix} $$.

  • For $$\displaystyle \lambda_2=-2 $$: $$\displaystyle (A+2I)v=0 $$ → $$\displaystyle \begin{bmatrix} 2 & 1 \\ -2 & -1 \end{bmatrix}v=0 $$ → $$\displaystyle v_2 = \begin{bmatrix} 1 \\ -2 \end{bmatrix} $$.

9.0 ELECTRICAL MACHINES & DEVICES IN CONTROL SYSTEMS

9.1 AC Servomotor

• Construction & Working:

  • Two-phase induction motor: stator has two windings (control & reference) 90° apart.

  • Rotor: squirrel-cage or drag-cup.

  • Control winding fed by voltage $$\displaystyle V_c $$ (amplified error), reference winding by fixed voltage $$\displaystyle V_r $$.

  • Air-gap flux rotates; rotor develops torque proportional to $$\displaystyle V_c $$ (for small signals).

• Assumptions for Transfer Function:

  1. Constant air-gap flux (linear magnetic circuit).

  2. Small operating range (linearization).

  3. Inertia & friction lumped.

  4. Neglect stator dynamics (high bandwidth).

• Transfer Function Derivation (simplified):

  • Torque $$\displaystyle T_e \propto V_c $$ → $$\displaystyle T_e = K_t V_c $$.

  • Mechanical equation: $$\displaystyle J\ddot{\theta} + B\dot{\theta} = T_e $$.

  • Electrical: Control winding inductance/resistance → first-order lag.

  • Final form: $$\displaystyle \frac{\Theta(s)}{V_c(s)} = \frac{K}{(sT_1+1)(sT_2+1)} $$ or $$\displaystyle \frac{K}{s(T_ms+1)(T_Ls+1)} $$ depending on model.

• Advantages: Smooth operation, high torque at low speeds, simple construction.

• Disadvantages: Nonlinear (saturation), limited torque-speed curve, requires two-phase supply.

9.2 DC Servomotor (Brief)

• Armature-controlled: $$\displaystyle V_a = L_a \frac{di_a}{dt} + R_a i_a + e_b $$, $$\displaystyle e_b = K_b \omega $$, $$\displaystyle T = K_t i_a $$.

• Transfer function: $$\displaystyle \frac{\Theta(s)}{V_a(s)} = \frac{K_t}{(J s + B)(L_a s + R_a) + K_t K_b} $$ (often reduced to $$\displaystyle \frac{K}{s(Ts+1)} $$ if $$\displaystyle L_a $$ small).

9.3 Stepper Motors

• Principle: Digital input pulses → discrete angular steps (open-loop).

• Types:

  • Variable Reluctance (VR): Toothed rotor, stator windings energized sequentially.

  • Permanent Magnet (PM): Rotor with permanent magnets; detent torque.

  • Hybrid: Combines VR & PM; high resolution, high torque.

• Advantages: Precise positioning, no feedback needed, holds position.

• Applications: Printers, CNC machines, robotics.

9.4 Tacho-Generators

• Principle: Voltage output proportional to rotational speed: $$\displaystyle V_t = K_t \omega $$.

• Use as Speed Feedback Sensor: Connected to motor shaft; provides feedback for speed control loops (e.g., in PI controllers for zero steady-state speed error).

[!IMPORTANT] Exam Focus Areas from Past Papers:

• Mason's Gain Formula: Always compute $\Delta$ and $$\displaystyle \Delta_k $$ correctly; identify non-touching loops.

• Root Locus: Sketching rules (asymptotes, breakaway, imaginary axis crossing) frequently asked.

• Bode/Nyquist: Drawing plots and finding GM/PM; stability determination.

• Steady-State Error: Static coefficients table and error calculations for type 0,1,2.

• Compensator Design: Lead for PM, lag for $$\displaystyle K_v $$; design steps.

• AC Servomotor: Transfer function derivation with assumptions.

• Routh-Hurwitz: Special cases (zero row, first element zero); range of $K$.

• State Space: Controllability test, state transition matrix properties.

• Short Notes: Linearization effect, pole location effect, Nichols chart, servomotors, tacho-generators.