Control System (EX-405) - Unit 2 Short Notes

UNIT 2: Control System - Analysis and Design

1. System Modeling and Representation

Mason's Gain Formula

Definition: A technique to find the overall transfer function of a system represented by a signal flow graph (SFG) without reduction.

Formula:

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

where:

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

• $\Delta$ = $1 -$ (sum of all individual loop gains) $+$ (sum of gains of all possible two non-touching loops) $-$ (sum of gains of all possible three non-touching loops) $+ \dots$

• $$\displaystyle \Delta_k $$ = value of $\Delta$ for that part of the graph which does not touch the $$\displaystyle k^{th} $$ forward path.

Components:

• Forward Paths: Paths from input to output node.

• Loops: Closed paths starting and ending at same node.

• Non-touching Loops: Loops that share no common nodes.

[!TIP] Exam Tip: Mason's formula is powerful for complex SFGs with many loops. Always identify all forward paths and all loops (touching and non-touching) before applying.

Advantages over Block Diagram Reduction:

• Systematic and less prone to error for complex interconnections.

• Can be applied directly to system equations without drawing block diagrams.

• Handles multiple feedback loops and feedforward paths efficiently.

Electrical Analogous Systems

Purpose: Represent mechanical/rotational systems as equivalent electrical circuits for analysis.

Two Main Analogies:

Methods:

• Direct Analogy: Replace mechanical elements with their direct F-V or F-I counterparts.

• Inverse Analogy: Replace using the opposite analogy table (e.g., mass as capacitance in F-I).

[!TIP] Common Pitfall: Confusing which analogy maps velocity to current vs. voltage. Remember: F-V: $v \propto I$ (like current through an inductor); F-I: $v \propto V$ (like voltage across a capacitor).

Block Diagram Reduction

Basic Rules:

1. Series: $$\displaystyle G_1(s) \cdot G_2(s) $$

2. Parallel: $$\displaystyle G_1(s) + G_2(s) $$

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

4. Moving Summing Point: Can shift across blocks by multiplying/dividing by the block transfer function.

5. Moving Take-off Point: Similar rule as summing point.

Procedure: Apply rules iteratively to simplify to a single forward path and single feedback loop.

Signal Flow Graphs (SFG)

Construction from System Equations:

1. Represent each variable as a node.

2. Draw directed branches between nodes as per the equations (branch gain = coefficient).

3. Identify input (source) and output (sink) nodes.

Application of Mason's Formula: (See above section).

2. Time Domain Analysis

Standard Test Signals

First-Order System Response

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

Unit Step Response:

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

• Steady-state value: $$\displaystyle c(\infty) = K $$

• Time constant $\tau$: Time to reach 63.2% of final value.

Unit Ramp Response:

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

• Steady-state error: $$\displaystyle e_{ss} = \tau $$ (for unit ramp, $$\displaystyle A=1 $$).

Second-Order System Response

Standard Form (Unity Feedback):

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

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

Underdamped Response ($$\displaystyle 0 < \zeta < 1 $$):

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

$$ t_r \approx \frac{\pi - \beta}{\omega_d}, \quad \text{where } \omega_d = \omega_n\sqrt{1-\zeta^2}, \quad \beta = \cos^{-1}(\zeta) $$

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

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

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

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

Key Relationship: $$\displaystyle \ln(M_p) = \frac{-\zeta\pi}{\sqrt{1-\zeta^2}} $$. Plot of $$\displaystyle M_p $$ vs. $\zeta$ is exponential decay.

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

$$ t_s \approx \frac{4}{\zeta\omega_n} \quad (2\% \text{ criterion}) $$

Resonant Frequency ($$\displaystyle \omega_r $$) & Resonance Peak ($$\displaystyle M_r $$): (For $$\displaystyle \zeta < 0.707 $$)

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

Steady-State Error Analysis

General Error: $$\displaystyle E(s) = R(s) - C(s) $$.

Static Error Coefficients (for Unity Feedback):

Definitions:

• $$\displaystyle K_p = \lim_{s \to 0} G(s) $$

• $$\displaystyle K_v = \lim_{s \to 0} sG(s) $$

• $$\displaystyle K_a = \lim_{s \to 0} s^2 G(s) $$

Generalized Error Coefficients:

$$ e_{ss} = \frac{1}{1 + K_p} \quad (\text{step}), \quad e_{ss} = \frac{1}{K_v} \quad (\text{ramp}), \quad e_{ss} = \frac{1}{K_a} \quad (\text{parabolic}) $$

[!TIP] Limitations: Static coefficients only apply to Type 0, 1, 2 systems and stable open-loop $G(s)$. For unstable systems or inputs not exactly polynomial (e.g., sinusoid), use final value theorem directly.

Controller Characteristics

Proportional (P) Controller: $$\displaystyle G_c(s) = K_p $$

• Effect: Increases system gain, reduces steady-state error, but can decrease relative stability (increase overshoot).

• Disadvantage: Cannot eliminate steady-state error for ramp/parabolic inputs.

Integral (I) Controller: $$\displaystyle G_c(s) = \frac{K_i}{s} $$

• Effect: Increases system type by 1, eliminates steady-state error for step/ramp/parabolic (depending on new type). Slows response, can cause instability.

• Disadvantage: Lowers phase margin, increases settling time.

Derivative (D) Controller: $$\displaystyle G_c(s) = K_d s $$

• Effect: Improves transient response (reduces overshoot, settling time), increases damping. Predicts error, adds phase lead.

• Disadvantage: Amplifies noise, not physically realizable (requires pure differentiator).

PID Controller: $$\displaystyle G_c(s) = K_p + \frac{K_i}{s} + K_d s $$

• Advantages: Combines benefits—good steady-state accuracy (I) and improved transient response (D).

• Disadvantages: Parameter tuning complex, D amplifies noise, I can cause initial windup.

PD Controller Design for Critical Damping:

For a plant $$\displaystyle G(s) = \frac{K}{s(s+a)} $$, with PD controller $$\displaystyle G_c(s) = K_p(1 + T_d s) $$.

• Closed-loop char. eq.: $$\displaystyle s^2 + (a + K_p K T_d)s + K_p K = 0 $$

• Critical damping condition: $$\displaystyle \zeta = 1 \Rightarrow (a + K_p K T_d)^2 = 4 K_p K $$

• Solve for $$\displaystyle T_d $$ given $$\displaystyle K_p, K, a $$.

• Settling time (2%): $$\displaystyle t_s = \frac{4}{\omega_n} = \frac{4}{\sqrt{K_p K}} $$ (since $$\displaystyle \zeta=1 $$, $$\displaystyle \omega_n = \sqrt{K_p K} $$).

3. Stability Analysis

Routh-Hurwitz Criterion

Condition: System stable iff all elements of first column of Routh array have same sign (no sign changes).

Procedure:

1. Write characteristic equation: $$\displaystyle a_n s^n + a_{n-1} s^{n-1} + \dots + a_0 = 0 $$.

2. Construct Routh array.

3. Count sign changes in first column → number of roots in RHS.

Special Cases:

1. Zero in first column: Replace zero with small $$\displaystyle \epsilon > 0 $$, continue. Sign change count depends on sign of element above/below.

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

3. First element zero, others non-zero: Replace zero with $\epsilon$ or multiply equation by factor to avoid.

[!TIP] Exam Tip: For "entire row zero," always form the auxiliary equation $$\displaystyle A(s)=0 $$ from the row above, find its roots (which are symmetric), then replace the zero row with coefficients of $dA(s)/ds$.

Root Locus Technique

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

Construction Rules:

1. Number of branches: = number of open-loop poles.

2. Symmetry: About real axis.

3. Real-axis segments: Exists to left of an odd number of real-axis poles/zeros.

4. Asymptotes: For $$\displaystyle n > m $$ branches going to infinity.

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

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

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 points lie on root locus.

6. Angles of departure/arrival:

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

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

7. Intersection with imaginary axis: Use Routh criterion on char. eq. $$\displaystyle 1 + KG(s)H(s)=0 $$, find $K$ where sign change occurs → gives $\omega$.

Determining $K$ for specified $\zeta$:

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

• Intersection of this line with root locus gives desired pole location $$\displaystyle s = -\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2} $$.

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

[!TIP] Common Pitfall: Breakaway points must satisfy both $$\displaystyle \frac{dK}{ds}=0 $$ and lie on real-axis root locus segment. Always verify.

Relative Stability

• Gain Margin (GM): Factor by which gain can be increased before instability. $$\displaystyle GM = \frac{1}{|G(j\omega_{pc})|} $$ (in linear), $$\displaystyle 20\log_{10}(GM) $$ in dB. Positive GM means stable.

• Phase Margin (PM): Additional phase lag required to reach $$\displaystyle -180^\circ $$ at gain crossover frequency $$\displaystyle \omega_{gc} $$. $$\displaystyle PM = 180^\circ + \angle G(j\omega_{gc}) $$. Positive PM means stable.

• Relationship: Larger GM/PM → more stable (more damping, less oscillatory). For good transient response, typically $$\displaystyle PM \approx 30^\circ - 60^\circ $$.

4. Frequency Response Analysis

Bode Plot Construction

Rules for Asymptotic Approximation:

Procedure:

1. Write $G(j\omega)$ in standard form: $$\displaystyle K \cdot \frac{\prod (1 + j\omega T_z)}{\prod (1 + j\omega T_p)} $$.

2. Mark corner frequencies ($1/T$) on log scale.

3. Sketch magnitude: start at $20\log|K|$, apply slopes at each corner.

4. Sketch phase: sum phases of each factor, use approximate straight-line segments.

5. Corrections: For better accuracy, add $\pm 3$ dB at each corner for first-order factors; for quadratic factors, peak near $$\displaystyle \omega_r $$ if $$\displaystyle \zeta < 0.707 $$.

Key Frequencies from Bode:

• Gain crossover frequency ($$\displaystyle \omega_{gc} $$): Where $$\displaystyle |G(j\omega)| = 1 $$ (0 dB). Read phase at $$\displaystyle \omega_{gc} $$ → PM = 180° + phase.

• Phase crossover frequency ($$\displaystyle \omega_{pc} $$): Where $$\displaystyle \angle G(j\omega) = -180° $$. Read magnitude at $$\displaystyle \omega_{pc} $$ → GM = 1/|G(j\omega_{pc})|.

Stability Assessment (Unity Feedback):

• Stable if $$\displaystyle PM > 0^\circ $$ (or $$\displaystyle GM > 1 $$).

• Marginally stable if $$\displaystyle PM = 0^\circ $$ (or $$\displaystyle GM = 1 $$).

• Unstable if $$\displaystyle PM < 0^\circ $$ (or $$\displaystyle GM < 1 $$).

Polar Plot (Nyquist for Open-Loop)

Construction:

1. Vary $\omega$ from $0$ to $\infty$.

2. Plot $Re[G(j\omega)]$ vs. $Im[G(j\omega)]$.

3. For $\omega \to \infty$, point tends to origin.

4. For $\omega \to 0$:

   • Type 0: $$\displaystyle G(0) = K $$ (finite real).

   • Type 1: $$\displaystyle G(j\omega) \approx \frac{K}{j\omega} \to -j\infty $$ (downward).

   • Type 2: $$\displaystyle G(j\omega) \approx \frac{K}{(j\omega)^2} \to -\infty $$ (leftward).

Effect of Adding Poles:

• Pole at origin: Plot starts at infinity (down for Type 1, left for Type 2).

• Pole at $$\displaystyle s = -1/T_i $$: Low-frequency asymptote rotates by $$\displaystyle -90^\circ $$.

• Pole at $$\displaystyle s = -1/T_r $$: High-frequency asymptote rotates by $$\displaystyle -90^\circ $$.

Nyquist Stability Criterion

General Formula: $$\displaystyle N = P - Z $$

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

• $P$ = number of open-loop poles in RHS.

• $Z$ = number of closed-loop poles in RHS (for stability, $$\displaystyle Z=0 $$).

For Unity Feedback ($$\displaystyle H(s)=1 $$):

• If $$\displaystyle P=0 $$ (open-loop stable), system stable iff $$\displaystyle N=0 $$ (no encirclement of $-1$).

• If $P \neq 0$, need $$\displaystyle N = P $$ for closed-loop stability.

Gain Margin from Nyquist: Distance from $(-1, j0)$ to the plot along real axis at phase crossover.

[!TIP] Exam Tip: For $$\displaystyle G(s)H(s) = \frac{K}{s(s+1)(s+2)} $$, $$\displaystyle P=0 $$. Sketch Nyquist (starts at $-\infty$, goes clockwise, ends at origin). Find $K$ such that plot just touches $-1$ point → that $K$ is critical gain.

5. Controller Design and Compensation

Need for Compensation

• Improve relative stability (increase PM/GM).

• Improve steady-state accuracy (increase error constants).

• Improve transient response (reduce overshoot, settling time).

• Often trade-offs exist (e.g., increasing gain improves SSE but reduces stability).

Lead Compensation

Transfer Function: $$\displaystyle G_c(s) = K_c \frac{1 + T s}{1 + \alpha T s} $$, where $$\displaystyle \alpha < 1 $$.

• Zero at $$\displaystyle z = -1/T $$, pole at $$\displaystyle p = -1/(\alpha T) $$ (pole left of zero).

• Effect: Adds positive phase (phase lead) in frequency range $$\displaystyle \frac{1}{\sqrt{\alpha}T} < \omega < \frac{1}{\alpha T} $$. Increases PM, improves transient response.

• Design Steps (for PM specification):

  1. From given $G(s)$, find $$\displaystyle \omega_{gc}' $$ (uncompensated gain crossover).

  2. Determine required PM (add $$\displaystyle 5^\circ-12^\circ $$ safety).

  3. Find $$\displaystyle \phi_m = \text{required PM} - \angle G(j\omega_{gc}') $$.

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

  5. Find $$\displaystyle \omega_{max} $$ (frequency of max phase lead) = $$\displaystyle \frac{1}{T\sqrt{\alpha}} $$. Often choose $$\displaystyle \omega_{max} \approx \omega_{gc}' $$.

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

  7. Determine $$\displaystyle K_c $$ so that $$\displaystyle |G_c(j\omega_{max})G(j\omega_{max})| = 1 $$.

Lag Compensation

Transfer Function: $$\displaystyle G_c(s) = K_c \frac{1 + T s}{1 + \beta T s} $$, where $$\displaystyle \beta > 1 $$.

• Zero at $$\displaystyle z = -1/T $$, pole at $$\displaystyle p = -1/(\beta T) $$ (pole left of zero, very close to origin).

• Effect: Increases low-frequency gain (improves SSE) with minimal effect on high-frequency phase (transient response almost unchanged).

• Design Steps (for $$\displaystyle K_v $$ or $$\displaystyle K_p $$):

  1. Determine required error constant $$\displaystyle K_{v,\text{req}} $$ or $$\displaystyle K_{p,\text{req}} $$.

  2. From uncompensated $G(s)$, find existing $$\displaystyle K_v $$ or $$\displaystyle K_p $$.

  3. Required gain increase: $$\displaystyle \beta \approx \frac{K_{v,\text{req}}}{K_{v,\text{uncomp}}} $$.

  4. Choose $T$ so that pole-zero pair is near origin (e.g., $$\displaystyle T = 10 \times $$ time constant of dominant poles) to avoid affecting transient response.

  5. Adjust $$\displaystyle K_c $$ to meet both gain and error constant specs.

Lag-Lead Compensation

Combined: $$\displaystyle G_c(s) = K_c \frac{(1 + T_1 s)(1 + T_2 s)}{(1 + \alpha T_1 s)(1 + \beta T_2 s)} $$, with $$\displaystyle \alpha < 1 $$ (lead part), $$\displaystyle \beta > 1 $$ (lag part).

• When employed: Need simultaneous improvement in transient (PM) and steady-state (error constant).

• Design Steps:

  1. Design lead compensator first to meet PM requirement (as above).

  2. Check if steady-state error spec is met. If not, add lag network to boost low-frequency gain without disturbing PM significantly.

  3. Choose lag pole-zero very close to origin (e.g., at $$\displaystyle \omega = 0.1 \times $$ crossover freq).

6. State-Space Analysis

State Variables and State Equations

State: Minimal set of variables $$\displaystyle x_1, x_2, \dots, x_n $$ that completely determine future system behavior. State-Space Representation:

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

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

where $\mathbf{x}$ = state vector, $u$ = input, $y$ = output, $\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D}$ = system matrices.

Eigenvalues and Eigenvectors

Definition: For matrix $\mathbf{A}$, eigenvalue $\lambda$ and eigenvector $\mathbf{v}$ satisfy $$\displaystyle \mathbf{A} \mathbf{v} = \lambda \mathbf{v} $$. Significance:

• Eigenvalues = roots of characteristic equation $$\displaystyle \det(s\mathbf{I} - \mathbf{A}) = 0 $$ → determine stability (all $$\displaystyle \text{Re}(\lambda) < 0 $$ for stability) and natural response modes.

• Eigenvectors determine the direction (mode shape) of each natural response component.

• Used in modal decomposition, controllability/observability analysis.

State Transition Matrix $\Phi(t)$

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

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

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

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

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

5. $\Phi(t)$ satisfies $$\displaystyle \dot{\Phi}(t) = \mathbf{A}\Phi(t) $$, $$\displaystyle \Phi(0)=\mathbf{I} $$.

Solution of State Equations (zero-input + zero-state):

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

Significance: $\Phi(t)$ propagates initial state forward in time; the convolution integral gives forced response.

7. System Components and Special Topics

Servomotors

AC Servomotor (Two-Phase Induction Motor):

• Construction: Stator has two windings (control and reference) $$\displaystyle 90^\circ $$ apart. Rotor: squirrel-cage or drag-cup.

• Working: Control winding voltage magnitude/phase controls torque. Reference winding gets constant voltage. Rotating magnetic field induces eddy currents in rotor, producing torque.

• Transfer Function Derivation (Assumptions):

  1. Negligible rotor inductance.

  2. Constant field flux (linear magnetization).

  3. Small torque-speed nonlinearity.

$$ \frac{\Theta(s)}{E_c(s)} = \frac{K_m}{(T_m s + 1)(T_e s + 1)} \approx \frac{K_m}{T_m s + 1} \quad (\text{if } T_e \ll T_m) $$

where $$\displaystyle K_m $$ = motor gain, $$\displaystyle T_m $$ = mechanical time constant, $$\displaystyle T_e $$ = electrical time constant.

Advantages: Smooth speed control, high torque at low speeds, no brushes. Disadvantages: Requires two-phase supply (or capacitor for phase shift), less efficient than DC servo.

Stepper Motors (Short Note)

• Operation: Digital motor; each input pulse rotates shaft by fixed step angle (e.g., $$\displaystyle 1.8^\circ $$).

• Types: Variable reluctance, permanent magnet, hybrid.

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

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

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

Tacho-Generators (Short Note)

• Principle: DC generator whose output voltage proportional to shaft speed: $$\displaystyle V_t = K_t \omega $$.

• Use: Speed feedback in control systems (e.g., in servos for damping).

• Advantages: Simple, reliable, linear over range.

• Disadvantages: Mechanical wear, brushes, limited frequency response.

Feedback in Control Systems

Significance:

1. Reduces Sensitivity: System performance less sensitive to parameter variations.

2. Improves Stability: Can stabilize unstable open-loop systems.

3. Rejects Disturbances: Reduces effect of external disturbances.

4. Improves Accuracy: Reduces steady-state error.

5. Widens Bandwidth: Increases frequency range of operation.

How Feedback Improves Performance:

• Disturbance Rejection: Disturbance entering after feedback point is attenuated by $1/(1+GH)$.

• Parameter Variation: Sensitivity $$\displaystyle S = \frac{1}{1+GH} $$ → large $GH$ reduces sensitivity.

• Stability: Properly designed feedback can move closed-loop poles to LHS.

Poles and Zeros

• Poles: Roots of denominator of transfer function $G(s)$. Determine natural response modes and stability (RHP poles → unstable).

• Zeros: Roots of numerator. Affect transient response shape (e.g., non-minimum phase zeros cause undershoot).

• Importance: Location in s-plane dictates response characteristics (damping, frequency, speed). Used in root locus, Bode, etc.

8. Short Note Topics (Integrated Above)

• Nyquist Plot: See Section 4.

• Polar Plot for Type 0, 1, 2: Type 0 ends at finite real; Type 1 goes to $-\jmath\infty$; Type 2 goes to $-\infty$.

• Gain and Phase Margin: See Section 3.

• Relative Stability: Quantified by GM/PM; larger margins → more stable.

• Compensation Techniques: Lead, Lag, Lag-Lead (Section 5).

• Eigenvalues and Eigenvectors: See Section 6.

• Servomotors: See Section 7.

• Stepper Motors: See Section 7.

• Tacho-Generators: See Section 7.

• P, I, D Controllers: See Section 2.