Control System (EC-404) - Unit 3 Short Notes

UNIT 3: Control System Analysis and Design

1. System Modeling and Representation

1.1 Basic Control System Concepts

Open Loop vs Closed Loop Control Systems

[!TIP]

Exam Focus: Compare using accuracy, stability, complexity, and cost. Closed loop improves performance but may cause instability if not designed properly.

Feedback in Control Systems

• Significance: Reduces sensitivity to parameter variations, rejects disturbances, improves stability and accuracy.

• Performance Improvement:

  • Decreases steady-state error.

  • Increases bandwidth.

  • Improves transient response (e.g., reduces overshoot, settling time).

Poles and Zeros

• Pole: Root of denominator of transfer function \( G(s) \). System response component \( e^{p t} \).

• Zero: Root of numerator of \( G(s) \). Affects transient response shape.

• Physical Significance:

  • Poles in LHP (\( \text{Re}(s) < 0 \)): Stable, decaying response.

  • Poles on \( j\omega \)-axis: Marginally stable, sustained oscillations.

  • Poles in RHP (\( \text{Re}(s) > 0 \)): Unstable, growing response.

  • Zeros can cancel poles, but near-cancellation may cause undesirable transient.

1.2 Block Diagrams

Components:

• Block: Represents system component with transfer function.

• Summing Point: Adds/subtracts signals (\( \oplus, \ominus \)).

• Take-off Point: Splits signal for multiple paths.

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

• Feedback Path: Path from output to summing point.

Block Diagram Reduction Techniques

1. Series: \( G_1(s)G_2(s) \)

2. Parallel: \( G_1(s) \pm G_2(s) \)

3. Feedback: \( \frac{G(s)}{1 \pm G(s)H(s)} \) (negative feedback: \( 1 + G(s)H(s) \))

4. Moving Take-off Point: Across block, multiply/divide by block TF.

5. Moving Summing Point: Similar adjustment.

Transfer Function Determination: Reduce to single equivalent TF \( C(s)/R(s) \).

1.3 Signal Flow Graphs and Mason's Gain Formula

Signal Flow Graph Construction

• Nodes: System variables.

• Branches: Directed edges with gains (TFs).

• Loop: Closed path returning to same node.

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

Mason's Gain Formula

\[ T = \frac{C(s)}{R(s)} = \sum_{k=1}^{N} \frac{M_k \Delta_k}{\Delta} \]

where:

• \( M_k \): Gain of \( k \)-th forward path.

• \( \Delta = 1 - \sum L_i + \sum L_i L_j - \sum L_i L_j L_k + \cdots \)

  (\( L_i \): individual loop gains; \( L_i L_j \): product of non-touching loops).

• \( \Delta_k \): \( \Delta \) with loops touching \( k \)-th forward path removed.

Advantages over Block Diagram Reduction:

• Easier for complex interconnections.

• Systematic formula avoids iterative reduction.

• Directly gives TF without step-by-step simplification.

[!TIP]

Common Pitfall: Forgetting to exclude loops touching the forward path when computing \( \Delta_k \). Always list all loops, identify non-touching combinations.

1.4 Electrical Analogous Systems

Force-Voltage (F-V) Analogous (Torque-Voltage)

Force-Current (F-I) Analogous (Torque-Current)

[!TIP]

Exam Tip: F-V is direct analogous (impedance analogy). F-I is inverse analogous (mobility analogy). Remember: in F-I, inertia ↔ capacitance, damping ↔ conductance.

1.5 System Components: Actuators and Sensors

AC Servomotors

• Construction: Stator with two-phase windings (reference and control), rotor (squirrel-cage or solid).

• Working: AC voltage applied to control winding produces rotating magnetic field, inducing rotor current and torque. Speed proportional to control voltage.

• Assumptions for TF Derivation:

  1. Rotor inductance negligible.

  2. Constant field flux (reference winding).

  3. Small damping.

  4. No saturation.

• Transfer Function:

  \[ \frac{\Omega(s)}{V(s)} = \frac{K}{T_M s + 1}, \quad T_M = \frac{J}{B} \]

  where \( K \) = motor constant, \( T_M \) = mechanical time constant.

Stepper Motors

• Types:

  1. Variable Reluctance (VR): Toothed rotor, low cost, microstepping possible.

  2. Permanent Magnet (PM): Rotor with permanent magnets, high torque at low speed.

  3. Hybrid (HB): Combines VR and PM, high precision, used in CNC.

• Working: Each pulse rotates shaft by fixed step angle (\( \theta = \frac{360^\circ}{N_r \cdot N_s} \), where \( N_r \) = rotor teeth, \( N_s \) = stator teeth).

• Applications: Printers, plotters, robotics, 3D printers.

Tachogenerators

• Principle: Voltage output proportional to rotational speed (\( V_t = K_t \omega \)). DC or AC types.

• Use in Velocity Feedback: Output fed back to derivative controller to increase damping, improve transient response and stability.

2. Time Domain Analysis

2.1 Standard Test Signals

2.2 First Order System Response

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

• Unit Step Response:

  \[ y(t) = K\left(1 - e^{-t/T}\right) \]

  Time constant \( T \): time to reach 63.2% of final value.

• Unit Ramp Response:

  \[ y(t) = K\left(t - T\left(1 - e^{-t/T}\right)\right) \]

  Steady-state error \( e_{ss} = KT \).

• Steady-State Error Summary:

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

2.3 Second Order System Response

Standard Form:

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

where \( \omega_n \) = natural frequency, \( \zeta \) = damping ratio.

Step Response Cases:

• Underdamped (\( 0 < \zeta < 1 \)): Oscillatory, overshoot \( M_p \).

• Overdamped (\( \zeta > 1 \)): No overshoot, slower.

• Critically Damped (\( \zeta = 1 \)): Fastest without overshoot.

Time Domain Specifications:

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

   \[ M_p = e^{-\frac{\zeta\pi}{\sqrt{1-\zeta^2}}} \quad \text{(for step input)} \]

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

2. Peak Time (\( t_p \)):

   \[ t_p = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}} \]

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

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

   For \( \zeta < 0.69 \):

   \[ t_r \approx \frac{1.8}{\omega_n} \text{ to } \frac{1.9}{\omega_n} \quad \text{(approximate)} \]

   Exact: \( t_r = \frac{\pi - \theta}{\omega_d} \), where \( \theta = \tan^{-1}\left(\frac{\sqrt{1-\zeta^2}}{\zeta}\right) \), \( \omega_d = \omega_n\sqrt{1-\zeta^2} \).

4. Settling Time (\( t_s \)): Time to stay within 2% (or 5%) band.

   \[ t_s \approx \frac{4}{\zeta\omega_n} \quad (2\%), \quad t_s \approx \frac{3}{\zeta\omega_n} \quad (5\%) \]

5. Resonant Frequency (\( \omega_r \)) and Peak (\( M_r \)):

   For \( \zeta < \frac{1}{\sqrt{2}} \):

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

[!TIP]

Derivation Focus: \( M_p \) from \( y(t) \) peak value; \( t_p \) from \( \dot{y}(t)=0 \). Remember \( \omega_d = \omega_n\sqrt{1-\zeta^2} \). \( M_p \) decreases as \( \zeta \) increases.

2.4 Steady-State Error Analysis

Concept: Difference between desired output and actual output as \( t \to \infty \).
Static Error Coefficients:

• Position Error Constant (\( K_p \)):

  \[ K_p = \lim_{s \to 0} G(s)H(s) \]

• Velocity Error Constant (\( K_v \)):

  \[ K_v = \lim_{s \to 0} s G(s)H(s) \]

• Acceleration Error Constant (\( K_a \)):

  \[ K_a = \lim_{s \to 0} s^2 G(s)H(s) \]

Steady-State Error for Standard Inputs (Unity Feedback):

Generalized Error Coefficients:

For input \( r(t) = \frac{a_0 t^n}{n!} + \cdots + a_0 \):

\[ e_{ss} = \frac{a_0}{1 + E(0)}, \quad \text{where } E(s) = G(s)H(s) \text{ for type 0, etc.} \]

Limitations of Static Error Coefficient Method:

1. Only for stable systems (poles in LHP).

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

3. Cannot predict error for arbitrary inputs.

4. For unstable systems, \( e_{ss} \) may be infinite even if coefficients finite.

3. Stability Analysis

3.1 Stability Concepts

• BIBO Stability: Every bounded input yields bounded output.

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

• Marginal Stability: Poles on \( j\omega \)-axis (simple), response sustained oscillations.

Routh-Hurwitz Stability Criterion

• Routh Array Construction:

  For \( a_n s^n + a_{n-1}s^{n-1} + \cdots + a_0 = 0 \):

  \[ \begin{array}{c|cc} s^n & a_n & a_{n-2} & \cdots \\ s^{n-1} & a_{n-1} & a_{n-3} & \cdots \\ s^{n-2} & b_1 & b_2 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ s^0 & \cdots & & \\ \end{array} \]

  where \( b_1 = \frac{a_{n-1}a_{n-2} - a_n a_{n-3}}{a_{n-1}} \), etc.

• Special Cases:

  1. Row of Zeros: Form auxiliary equation from row above, differentiate, replace zero row.

  2. First Element Zero: Replace with small \( \epsilon > 0 \), continue, check sign as \( \epsilon \to 0 \).

• Stability Condition: All elements of first column positive → no sign changes → stable.

• Shifted Axis (\( s = -\sigma \)): Substitute \( s = s' - \sigma \) to check stability relative to line \( \text{Re}(s) = -\sigma \).

[!TIP]

Exam Alert: Routh array errors common in special cases. For row of zeros, auxiliary equation is key. For \( \epsilon \) method, track sign changes carefully.

3.2 Root Locus Technique

Definition: Plot of closed-loop poles as gain \( K \) varies from 0 to \( \infty \).

Properties:

1. Symmetry about real axis.

2. Real-axis segments: where number of open-loop poles/zeros to right is odd.

3. Asymptotes:

   • Number: \( p - z \) (poles minus zeros).

   • Angles: \( \frac{(2q+1)180^\circ}{p-z} \), \( q=0,1,\dots,p-z-1 \).

   • Centroid: \( \sigma = \frac{\sum \text{poles} - \sum \text{zeros}}{p - z} \).

4. Breakaway/Break-in points: On real axis, solve \( \frac{dK}{ds} = 0 \) (from \( K = -\frac{1}{G(s)H(s)} \)).

5. Intersection with \( j\omega \)-axis: Use Routh or angle criterion.

6. Angles of departure/arrival: From complex poles/zeros.

Construction Steps:

1. Mark open-loop poles/zeros.

2. Determine real-axis segments.

3. Draw asymptotes.

4. Find breakaway/break-in points (real-axis only).

5. Find \( j\omega \)-axis crossing (Routh on characteristic equation).

6. Sketch locus, add curvatures.

Stability Analysis:

• If locus lies entirely in LHP → stable for all \( K \).

• If crosses \( j\omega \)-axis → marginal stability at crossing \( K \).

• If enters RHP → unstable for \( K \) beyond crossing.

Finding \( K \) for Specified \( \zeta \) or \( \omega_n \):

• Draw \( \zeta \) line from origin (angle \( \cos^{-1}\zeta \)).

• Intersection with locus gives desired pole \( s \).

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

Marginal Stability: Frequency of sustained oscillations = imaginary part of \( j\omega \)-axis crossing.

[!TIP]

Breakaway Points: Must lie on real-axis segments between poles. Solve \( \frac{dK}{ds}=0 \), check if \( K>0 \) and point between poles/zeros. Common mistake: ignoring sign of \( K \).

3.3 Effects of Open-Loop Poles and Zeros on Root Locus

• Adding Poles: Locus bends leftward (more negative real part), tends to destabilize (slower response, more overshoot).

• Adding Zeros: Locus bends rightward (towards zero), tends to stabilize (faster response, less overshoot). Zeros attract locus.

4. Frequency Domain Analysis

4.1 Bode Plots

Construction Procedure:

• Magnitude Plot: \( 20\log|G(j\omega)| \) vs \( \log\omega \) (dB).

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

• Rules for Factors:

  | Factor | Magnitude (dB) | Phase (degrees) | |--------------------------|----------------------------------------|----------------------------------| | Gain \( K \) | \( 20\log K \) (constant shift) | 0 | | Pole at origin (\( 1/s \)) | \(-20 \text{ dB/dec}\) starting at \( \omega=1 \) | \(-90^\circ\) constant | | Zero at origin (\( s \)) | \(+20 \text{ dB/dec}\) | \(+90^\circ\) | | Real pole \( 1/(Ts+1) \) | \(-20 \text{ dB/dec}\) after \( \omega=1/T \), \(-3\text{dB}\) at \( \omega=1/T \) | \(-90^\circ\) after \( \omega=1/T \) | | Real zero \( (Ts+1) \) | \(+20 \text{ dB/dec}\) after \( \omega=1/T \) | \(+90^\circ\) after \( \omega=1/T \) | | Complex poles/zeros | \(\pm40 \text{ dB/dec}\) at high \( \omega \) | \(\pm180^\circ\) at high \( \omega \) |

Key Frequencies:

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

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

Gain Margin (GM) and Phase Margin (PM):

\[ \text{GM} = \frac{1}{|G(j\omega_{pc})|} \quad (\text{in linear}), \quad \text{GM}_{\text{dB}} = -20\log|G(j\omega_{pc})| \]

\[ \text{PM} = 180^\circ + \angle G(j\omega_{gc}) \quad (\text{positive for stability}) \]

Relative Stability Assessment:

• PM > 0 and GM > 1 (0 dB) → stable.

• Larger PM/GM → more stable, less oscillatory.

• \( \omega_{gc} \) ≈ bandwidth: higher → faster response.

4.2 Polar Plots

Construction Procedure:

• Plot \( |G(j\omega)| \angle G(j\omega) \) for \( \omega = 0 \to \infty \).

• Start at \( \omega=0 \): magnitude \( |G(0)| \), phase \( \angle G(0) \).

• End at \( \omega=\infty \): magnitude 0 (if proper), phase \( -90^\circ \times (\text{number of poles} - \text{zeros}) \).

• Sketch curve, noting intersections with real/imaginary axes.

Polar Plots for Type 0, 1, 2 Systems:

Effect of Adding Poles/Zeros:

• Pole at origin: Rotates plot -90° (adds -90° phase).

• Pole at \( s = -1/T \): Bends plot toward -90° (more negative phase).

• Zero at origin: Rotates +90°.

• Zero at \( s = -1/T \): Bends toward 0° (less negative phase).

Inverse Polar Plot: Plot of \( 1/G(j\omega) \). Useful when \( G(s) \) has pole at origin (infinite start).

• Difference from Bode: Polar shows magnitude and phase simultaneously; Bode separates them.

4.3 Nyquist Stability Criterion

Nyquist Plot Construction:

• Map contour \( s \): RHP semicircle + \( j\omega \)-axis (indent around origin if poles on \( j\omega \)).

• Plot \( G(s)H(s) \) for \( s \) on contour.

Nyquist Stability Condition:

\[ N = P - Z \]

• \( N \): Net encirclements of \(-1 + j0\) point clockwise.

• \( P \): Poles of \( G(s)H(s) \) in RHP.

• \( Z \): Zeros of \( 1 + G(s)H(s) \) in RHP (closed-loop unstable poles).

• Closed-loop stable iff \( Z = 0 \Rightarrow N = P \).

Stability Analysis for Poles on \( j\omega \)-axis:

• Indent contour around pole(s) with small semicircle.

• \( N \) includes encirclements from indentation.

• Modified: \( N = P - Z \), but \( P \) includes poles on \( j\omega \)-axis? Actually, \( P \) = poles in open RHP only; poles on \( j\omega \) are not in RHP, but indentation accounts for them in \( N \).

• Rule: If \( G(s)H(s) \) has poles on \( j\omega \), Nyquist plot goes to infinity near those frequencies; indentation adds half encirclements.

Gain Margin from Nyquist Plot:

• Distance from Nyquist plot to \(-1\) point along real axis: \( \text{GM} = |G(j\omega_{pc})|^{-1} \).

• If plot crosses real axis left of \(-1\), GM > 1 (stable); right of \(-1\), GM < 1 (unstable).

4.4 Log Magnitude vs Phase Plot (Nichols Chart)

• Construction: Plot \( 20\log|G(j\omega)| \) (dB) vs \( \angle G(j\omega) \) (degrees) for all \( \omega \).

• Nichols Chart: Overlay contours of constant \( M_p \), \( \omega_r \), \( t_s \), etc.

• Use: Direct reading of stability margins and transient specs from plot; useful for design.

5. Design and Compensation

5.1 Controllers

Tuning Methods:

• Ziegler-Nichols: Ultimate gain \( K_u \) and period \( P_u \) from sustained oscillations.

• Cohen-Coon: For processes with large dead time.

5.2 Compensation Techniques

Phase Lead Compensation

• Network: RC series with parallel C? Actually, lead: \( R_1, C_1 \) series, \( R_2 \) parallel? Standard:

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

  \( \tau = R_1C \), \( \alpha\tau = (R_1||R_2)C \).

• Effect on Bode: Increases gain at high freq, adds positive phase (max \( \phi_m = \sin^{-1}\frac{1-\alpha}{1+\alpha} \)).

• Effect on Root Locus: Attracts locus rightward, increases damping.

Phase Lag Compensation

• Network:

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

• Effect: Increases low-frequency gain (improves \( K_v, K_a \)), reduces bandwidth, adds negative phase (small).

• Use: When steady-state error specification is stringent but transient is acceptable.

Lag-Lead Compensation

• Network: Cascade of lag and lead networks.

• Design: Lead for PM/transient, lag for steady-state.

• Applications: Systems requiring both fast response and low steady-state error.

5.3 Compensator Design

Design for Phase Margin (Bode Approach):

1. Given \( G(s) \), compute uncompensated PM.

2. Required additional phase \( \phi_m = \text{PM}_{\text{req}} - \text{PM}_{\text{uncomp}} + 5^\circ \sim 12^\circ \) (safety).

3. Choose \( \alpha = \frac{1 - \sin\phi_m}{1 + \sin\phi_m} \).

4. Find \( \omega_m \) where \( |G(j\omega)| = \frac{1}{\alpha} \) (0 dB crossing after compensation).

5. \( \tau = 1/(\omega_m\sqrt{\alpha}) \).

6. \( K \) adjusted to meet gain crossover.

Design for Steady-State Error (Type Improvement):

• Increase system type by adding integrator (I or lag).

• Lag: choose \( \beta \) to achieve required \( K_v \) without affecting PM much.

Design for Combined Specifications:

• Example: Given \( G(s) = \frac{K}{s(Ts+1)} \), design lead for PM=50°, \( K_v=10 \).

  • \( K_v = \lim_{s\to0} s \cdot G_c(s)G(s) = K \cdot \text{dc gain of } G_c = 10 \).

  • Lead adds phase at \( \omega_m \), adjust \( K \) and \( \alpha, \tau \).

Example: PD Controller for Critical Damping

• PD: \( G_c(s) = K_p + K_d s = K_p(1 + T_d s) \).

• For second-order system \( G(s)=\frac{\omega_n^2}{s(s+2\zeta\omega_n)} \), PD adds zero at \( s = -1/T_d \).

• Critical damping: \( \zeta = 1 \). Place zero to cancel slow pole or adjust locus.

[!TIP]

Design Steps: Always sketch uncompensated Bode, compute margins, determine required phase lead, choose \( \alpha \), find \( \omega_m \), compute \( \tau \), verify. For lag, place \( \omega_c \) a decade below crossover to minimize phase effect.

6. State Space Analysis

6.1 State Space Representation

• State Variables: Minimal set \( x_1, x_2, \dots, x_n \) to describe future.

• State Equations:

  \[ \dot{x} = A x + B u, \quad y = C x + D u \]

  where \( x \in \mathbb{R}^n \), \( u \in \mathbb{R}^m \), \( y \in \mathbb{R}^p \).

• Transfer Function:

  \[ G(s) = C(sI - A)^{-1}B + D \]

Transfer Function Decomposition (Direct Decomposition):

From TF \( G(s) = \frac{b_m s^m + \cdots + b_0}{s^n + a_{n-1}s^{n-1} + \cdots + a_0} \) (proper, \( n \ge m \)):

• Controllable Canonical Form:

  \[ A = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ -a_0 & -a_1 & -a_2 & \cdots & -a_{n-1} \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{bmatrix}, \quad C = \begin{bmatrix} b_0 - a_0 d & b_1 - a_1 d & \cdots & b_{n-1} - a_{n-1}d \end{bmatrix} \]

  with \( d = 0 \) if \( m < n \), \( d = b_n \) if \( m=n \).

6.2 State Transition Matrix (\( e^{At} \))

• Definition: \( \Phi(t) = e^{At} \), solution of \( \dot{\Phi} = A\Phi \), \( \Phi(0)=I \).

• Properties:

  1. \( \Phi(0) = I \)

  2. \( \Phi(t_1)\Phi(t_2) = \Phi(t_1+t_2) \)

  3. \( \Phi^{-1}(t) = \Phi(-t) \)

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

  5. \( \Phi(t) = \mathcal{L}^{-1}\left\{ (sI - A)^{-1} \right\} \)

Solution of State Equations:

\[ x(t) = \Phi(t)x(0) + \int_0^t \Phi(t-\tau) B u(\tau) d\tau \]

Computation Methods:

1. Laplace: \( \Phi(t) = \mathcal{L}^{-1}\{(sI - A)^{-1}\} \).

2. Cayley-Hamilton: Use characteristic polynomial.

3. Eigenvalues/Eigenvectors: If \( A = V \Lambda V^{-1} \), then \( \Phi(t) = V e^{\Lambda t} V^{-1} \).

6.3 Controllability and Observability

Controllability: Ability to drive state from any \( x(0) \) to origin in finite time.

• Controllability Matrix:

  \[ Q_c = \left[ B \quad AB \quad A^2B \quad \cdots \quad A^{n-1}B \right] \]

• Test: System controllable iff \( \text{rank}(Q_c) = n \).

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

• Observability Matrix:

  \[ Q_o = \begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^{n-1} \end{bmatrix} \]

• Test: System observable iff \( \text{rank}(Q_o) = n \).

Duality: \( (A,B) \) controllable \( \Leftrightarrow \) \( (A^T, C^T) \) observable.

6.4 Eigenvalues and Eigenvectors

• Eigenvalues (\( \lambda \)): Roots of \( \det(sI - A) = 0 \). Determine stability (all \( \text{Re}(\lambda) < 0 \) stable).

• Eigenvectors (\( v \)): Non-zero vectors satisfying \( (A - \lambda I)v = 0 \).

• Computation: For each \( \lambda \), solve \( (A - \lambda I)v = 0 \).

• Modal Decomposition:

  If \( A \) has distinct eigenvalues, \( A = V \Lambda V^{-1} \), where \( V = [v_1, v_2, \dots, v_n] \), \( \Lambda = \text{diag}(\lambda_1, \lambda_2, \dots, \lambda_n) \).

  Then \( \Phi(t) = V e^{\Lambda t} V^{-1} \), and state response \( x(t) = \sum c_i v_i e^{\lambda_i t} \).

Relationship with System Dynamics:

• Each eigenvalue-eigenvector pair corresponds to a mode.

• Real eigenvalue: exponential mode.

• Complex conjugate pair: damped oscillatory mode.

• Stability requires all eigenvalues in LHP.

[!TIP]

Exam Alert: For \( 2\times2 \) matrix, eigenvalues from \( \lambda^2 - (a_{11}+a_{22})\lambda + (a_{11}a_{22}-a_{12}a_{21}) = 0 \). Eigenvectors: solve \( (A-\lambda I)v=0 \), set one component arbitrarily.

7. Advanced Concepts and Effects

7.1 Effect of Feedback

• Linearization: Feedback linearizes nonlinear systems around operating point (describing function approximation).

• Improvements:

  • Stability: Can stabilize unstable open-loop systems.

  • Bandwidth: Increases (higher \( \omega_{gc} \)).

  • Disturbance Rejection: Reduces effect of disturbances at input/output.

  • Parameter Sensitivity: Reduces sensitivity to plant parameter variations (if loop gain high).

  • Steady-State Accuracy: Improves with integral action.

7.2 Location of Poles and System Stability

• LHP Poles: Stable, decaying exponentials. Distance from origin → speed (larger \( |\sigma| \) faster decay).

• RHP Poles: Unstable, growing exponentials.

• \( j\omega \)-Axis Poles: Marginally stable, pure sinusoids.

• Complex Poles: Damped oscillations; damping ratio \( \zeta \) from angle with real axis: \( \zeta = \cos\theta \), where \( \theta = \angle(s) \).

• Real Poles: Non-oscillatory; dominant pole (closest to \( j\omega \)-axis) dictates response speed.

7.3 Relative Stability Measures

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

• Phase Margin (PM): Additional phase lag allowable before instability. Larger PM → less oscillatory.

• Gain Crossover Frequency (\( \omega_{gc} \)): Bandwidth indicator; higher \( \omega_{gc} \) → faster response but more noise sensitivity.

• Phase Crossover Frequency (\( \omega_{pc} \)): Related to gain margin; lower \( \omega_{pc} \) often better.

• Interpretation:

  • PM ≈ \( \zeta \times 100\% \) for second-order systems.

  • GM > 6 dB (≈1.995) and PM > 30° typically acceptable.

  • \( \omega_{gc} \) ≈ \( \omega_n \) for second-order.

[!TIP]

Relative Stability vs Absolute: Routh gives absolute (all poles LHP). Bode/Nyquist give relative (how close to instability). PM/GM quantify robustness.

End of Unit 3 Notes
Focus on derivations (Mason, overshoot, steady-state error, Routh, root locus rules), design procedures (compensators), and interpretation of frequency plots. Practice past paper problems for exam readiness.