Control System (EX-405) - Unit 4 Short Notes

UNIT 4: CONTROL SYSTEM - SHORT NOTES

1.0 SYSTEM MODELING & REPRESENTATION

1.1 Mason's Gain Formula

Definition: A method to determine the overall transfer function of a system represented by a signal flow graph without simplification. Formula:

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

where:

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

• $\Delta$ = $$\displaystyle 1 - \sum $$ (individual loop gains) $$\displaystyle + \sum $$ (gain products of all non-touching loop pairs) $-$ ...

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

Step-by-Step Procedure:

1. Identify all forward paths from input to output and their gains ($$\displaystyle P_k $$).

2. Identify all individual loops and compute their gains.

3. Identify all possible combinations of non-touching loops (loops that share no common node).

4. Compute $$\displaystyle \Delta = 1 - (\text{sum of all loop gains}) + (\text{sum of gain products of all non-touching loop pairs}) - (\text{sum of triple products}) + ... $$

5. For each forward path $$\displaystyle P_k $$, form $$\displaystyle \Delta_k $$ by removing all loops touching that forward path and computing the $\Delta$ for the remaining sub-graph.

6. Apply the formula.

Advantages over Block Diagram Reduction:

• Systematic, less prone to error for complex systems.

• Can be applied directly to a signal flow graph.

• Handles multiple forward paths and loops efficiently.

[!TIP] Common Pitfall: Forgetting to consider higher-order non-touching loop combinations (triples, etc.) in $\Delta$.

1.2 Electrical Analogies of Mechanical Systems

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

Key: F-V & T-V are Direct Analogies (effort ↔ voltage, flow ↔ current). F-I & T-I are Inverse Analogies (effort ↔ current, flow ↔ voltage).

1.3 Transfer Function Derivation

From Differential Equation:

Take Laplace transform (assuming zero initial conditions). Solve for $C(s)/R(s)$.

Example: $$\displaystyle \frac{d^2 y}{dt^2} + 6\frac{dy}{dt} + 8y = 16e^{-t} $$

$$\displaystyle s^2 Y(s) + 6s Y(s) + 8 Y(s) = 16 \frac{1}{s+1} $$

$$\displaystyle \boxed{T.F. = \frac{Y(s)}{R(s)} = \frac{16}{(s+1)(s^2+6s+8)}} $$

From A.C. Servomotor (Two-Phase):
Assumptions:

1. Constant field current (linear magnetization).

2. Rotor time constant $$\displaystyle L_r/R_r \approx 0 $$ (negligible).

3. Small angular displacement.

4. No saturation.

Working: Control voltage $$\displaystyle V_c $$ applied to control winding produces flux $$\displaystyle \phi_c \propto V_c $$. Torque $$\displaystyle T \propto \phi_c \times \phi_f \propto V_c $$. This torque accelerates the rotor (inertia $J$, damping $B$). Equation: $$\displaystyle J\frac{d^2\theta}{dt^2} + B\frac{d\theta}{dt} = K_t V_c $$
Laplace: $$\displaystyle J s^2 \Theta(s) + B s \Theta(s) = K_t V_c(s) $$
Back-EMF: $$\displaystyle V_b = K_b \frac{d\theta}{dt} \xrightarrow{\mathcal{L}} K_b s \Theta(s) $$
Control Voltage: $$\displaystyle V_c(s) = V_r(s) - V_b(s) = V_r(s) - K_b s \Theta(s) $$

Substitute and solve:

$$ (Js^2 + Bs)\Theta(s) = K_t [V_r(s) - K_b s \Theta(s)] $$

$$ \boxed{\frac{\Theta(s)}{V_r(s)} = \frac{K_t}{Js^2 + (B + K_t K_b)s}} $$

From Block Diagrams/Signal Flow Graphs: Use reduction rules (series, parallel, feedback) or Mason's Gain Formula.

2.0 TIME DOMAIN ANALYSIS & PERFORMANCE SPECIFICATIONS

2.1 Standard Test Input Signals

2.2 First Order System Response

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

Unit Step Response ($$\displaystyle R(s)=1/s $$):

$$ C(s) = \frac{K}{s(1+sT)} = \frac{K}{s} - \frac{K}{s+1/T} $$

$$ c(t) = K(1 - e^{-t/T}), \quad t \ge 0 $$

• Time Constant ($\tau$): Time to reach $63.2\%$ of final value.

• Steady-State Value: $$\displaystyle c(\infty) = K $$.

Unit Ramp Response ($$\displaystyle R(s)=1/s^2 $$):

$$ C(s) = \frac{K}{s^2(1+sT)} = \frac{K}{s^2} - \frac{KT}{s} + \frac{KT}{s+1/T} $$

$$ c(t) = K(t - T + Te^{-t/T}) $$

• Steady-State Error: $$\displaystyle e_{ss} = \lim_{t\to\infty} [r(t) - c(t)] = T = \tau $$.

[!TIP] For a first-order system, $$\displaystyle e_{ss} $$ for ramp input equals the time constant $\tau$.

2.3 Second Order System Characteristics

Standard Form (Unity Feedback):

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

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

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

Step Response:

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

Key Specifications (Derived):

1. Peak Time ($$\displaystyle t_p $$): Time of first maximum overshoot.

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

2. Maximum Overshoot ($$\displaystyle M_p $$): $$\displaystyle (c_{max} - 1) \times 100\% $$.

$$ \boxed{M_p = e^{\frac{-\pi\zeta}{\sqrt{1-\zeta^2}}} \quad (\text{decimal}), \quad M_p\% = 100 \cdot e^{\frac{-\pi\zeta}{\sqrt{1-\zeta^2}}}} $$

Graph: $$\displaystyle M_p $$ decreases as $\zeta$ increases.

3. Rise Time ($$\displaystyle t_r $$): Time to go from $0.1$ to $0.9$ (or $0$ to $1$) of final value.

   Approx.:

$$ t_r \approx \frac{\pi - \phi}{\omega_n\sqrt{1-\zeta^2}}, \quad \phi = \tan^{-1}\frac{\sqrt{1-\zeta^2}}{\zeta} $$

4. Settling Time ($$\displaystyle t_s $$): Time to stay within a tolerance band (e.g., $\pm2\%$).

$$ \boxed{t_s \approx \frac{4}{\zeta\omega_n} \quad (\text{for } 2\%)}, \quad t_s \approx \frac{3}{\zeta\omega_n} \quad (\text{for } 5\%) $$

Critically Damped ($$\displaystyle \zeta = 1 $$) & Overdamped ($$\displaystyle \zeta > 1 $$)

• Critically Damped: Fastest response without overshoot.

$$ c(t) = 1 - (1 + \omega_n t)e^{-\omega_n t} $$

• Overdamped: Slower, no overshoot. Two real poles.

2.4 Resonant Frequency & Peak

Applies only to underdamped systems ($$\displaystyle 0 < \zeta < 0.707 $$).

• Resonant Frequency ($$\displaystyle \omega_r $$): Frequency at which magnitude $|M(j\omega)|$ peaks.

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

• Resonance Peak ($$\displaystyle M_r $$): Magnitude at $$\displaystyle \omega_r $$.

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

[!NOTE] $$\displaystyle M_r $$ exists only if $$\displaystyle \zeta < 1/\sqrt{2} \approx 0.707 $$.

3.0 STEADY-STATE ERROR ANALYSIS

3.1 Static Error Coefficients

For a unity feedback system with open-loop transfer function $G(s)$:

• Position Error Coefficient ($$\displaystyle K_p $$): For step input.

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

• Velocity Error Coefficient ($$\displaystyle K_v $$): For ramp input.

$$ K_v = \lim_{s\to 0} s G(s) $$

• Acceleration Error Coefficient ($$\displaystyle K_a $$): For parabolic input.

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

3.2 Steady-State Error ($$\displaystyle e_{ss} $$) Calculation

General Formula:

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

Type of System (Number of poles at origin in $G(s)$):

3.3 Limitations of Static Error Coefficient Method

1. Only for stable systems (unstable systems have infinite $$\displaystyle e_{ss} $$).

2. Only for standard inputs (step, ramp, parabolic). Not valid for arbitrary inputs.

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

4. Assumes unity feedback or can be extended with care.

5. Only gives steady-state error, not the error signal shape.

4.0 STABILITY ANALYSIS IN TIME DOMAIN

4.1 Routh-Hurwitz Stability Criterion

Necessary & Sufficient Condition: All coefficients of characteristic polynomial must be positive AND all elements of the first column of the Routh array must be positive.

Construction of Routh Array:

For $$\displaystyle a_n s^n + a_{n-1} s^{n-1} + ... + a_1 s + a_0 = 0 $$:

Special Cases:

1. Zero in First Column: Replace zero with a small $$\displaystyle \epsilon > 0 $$, complete array, then let $\epsilon \to 0$. Sign change indicates instability.

2. Entire Row Zero: Indicates symmetrical roots about origin. Form auxiliary equation from the row above, differentiate, replace the zero row with coefficients of derivative, and continue.

Number of RHP Roots: Equal to number of sign changes in the first column.

4.2 Root Locus Technique

Definition: Plot of closed-loop pole locations as a system parameter (usually gain $K$) varies from $0$ to $\infty$.

Sketching Rules (Complete):

1. Starting Points: Open-loop poles ($$\displaystyle K=0 $$).

2. Ending Points: Open-loop zeros (finite) + ($n-m$) zeros at infinity.

3. Real-Axis Segments: Exists to the left of an odd number of real poles/zeros.

4. Asymptotes: ($n-m$) asymptotes with angles $$\displaystyle \theta_A = \frac{(2q+1)180^\circ}{n-m} $$, $$\displaystyle q=0,1,...,n-m-1 $$. Centroid:

$$ \sigma_A = \frac{\sum \text{Re}(p_i) - \sum \text{Re}(z_i)}{n-m} $$

5. Breakaway/Break-in Points: On real axis, solve $$\displaystyle \frac{dK}{ds} = 0 $$ from $$\displaystyle K = -\frac{\text{Denominator of } G(s)H(s)}{\text{Numerator of } G(s)H(s)} $$.

6. Angle of Departure/Arrival: From complex pole $$\displaystyle p_i $$:

$$ \angle \text{departure} = 180^\circ - \sum \angle(\text{from other poles}) + \sum \angle(\text{to zeros}) $$

7. Intersection with Imaginary Axis: Use Routh-Hurwitz on characteristic equation $$\displaystyle 1 + KG(s)H(s) = 0 $$ to find marginal $K$ and $\omega$.

8. Symmetry: About real axis.

Procedure for Finding $K$ for Specified $\zeta$:

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

2. Intersection of this line with root locus gives desired pole location $$\displaystyle s = -\sigma \pm j\omega $$.

3. Compute $K$ using magnitude condition: $$\displaystyle K = \frac{1}{|G(s)H(s)|} $$ at that $s$.

Stability Comment: System stable if all closed-loop poles are in the Left Half Plane (LHP). As $K$ increases, poles may cross into RHP → instability.

Effects of Adding Open-Loop Poles/Zeros:

4.3 Relative Stability & Damping Ratio

• Relative Stability: Measure of how far poles are from imaginary axis (or $j\omega$ axis). Indicated by damping ratio $\zeta$ or phase margin.

• From root locus, for a given $K$, read pole location $$\displaystyle s = \sigma \pm j\omega_d $$, then $$\displaystyle \zeta = \cos(\angle s) $$ from negative real axis.

• Higher $\zeta$ → better relative stability (less oscillatory).

5.0 FREQUENCY RESPONSE ANALYSIS

5.1 Bode Plot (Magnitude & Phase)

Construction Procedure:

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

2. Magnitude Plot: Express in dB: $20\log|G(j\omega)H(j\omega)|$.

   • Corner Frequencies: $$\displaystyle \omega = 1/T $$ for each first-order factor.

   • Asymptotic Approximation:

     • Constant $K$: $20\log K$ dB.

     • Integral factor $1/(j\omega)$: Slope $-20$ dB/dec, starting at $$\displaystyle \omega=1 $$.

     • Derivative factor $j\omega$: Slope $+20$ dB/dec, starting at $$\displaystyle \omega=1 $$.

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

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

     • Quadratic factor: Similar with $40$ dB/dec change.

   • Exact Plot: Add corrections at corner frequencies ($\pm 3$ dB for first-order).

3. Phase Plot: Sum of phases of all factors.

   • Approximate: $$\displaystyle -45^\circ $$ to $$\displaystyle +45^\circ $$ transition centered at $$\displaystyle \omega=1/T $$ for first-order factors.

   • Quadratic: $$\displaystyle -90^\circ $$ to $$\displaystyle 0^\circ $$ transition over 2 decades.

Key Frequencies:

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

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

Stability Margins:

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

$$ GM = \frac{1}{|G(j\omega_{pc})H(j\omega_{pc})|}, \quad \text{in dB: } GM_{dB} = -|G(j\omega_{pc})H(j\omega_{pc})|_{dB} $$

• Phase Margin (PM): Additional phase lag required at $$\displaystyle \omega_{gc} $$ to reach $$\displaystyle -180^\circ $$.

$$ PM = 180^\circ + \angle G(j\omega_{gc})H(j\omega_{gc}) $$

Stability Condition (for minimum phase systems):

• Closed-loop stable if: $$\displaystyle PM > 0^\circ $$ and $$\displaystyle GM > 1 $$ (or $$\displaystyle GM_{dB} > 0 $$ dB).

• Larger PM/GM → better relative stability.

5.2 Polar Plot (Nyquist Plot for Open-Loop)

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

Standard Shapes:

Construction Steps:

1. Find $G(j0)H(j0)$ (low freq) and $$\displaystyle \lim_{\omega\to\infty} G(j\omega)H(j\omega) $$ (high freq).

2. Find intersections with real/imag axes (set imaginary/real part to zero).

3. Determine direction of plot (increasing $\omega$).

4. Sketch smooth curve.

5.3 Nyquist Stability Criterion

Concept: Relates closed-loop stability to open-loop frequency response encirclements of $(-1, j0)$.

Definitions:

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

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

• $Z$ = Number of closed-loop poles of $1+G(s)H(s)$ in RHP (i.e., unstable closed-loop poles).

Fundamental Relation:

$$ \boxed{N = P - Z} $$

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

• If $$\displaystyle P=0 $$ (open-loop stable), Nyquist must not encircle $(-1, j0)$.

• If $$\displaystyle P>0 $$, Nyquist must encircle $(-1, j0)$ clockwise $P$ times.

Application:

1. Sketch Nyquist plot for $G(s)H(s)$.

2. Count $P$ (poles of OLTF in RHP).

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

4. Compute $$\displaystyle Z = P - N $$.

5. Closed-loop stable iff $$\displaystyle Z=0 $$.

Range of $K$ for Stability: For $$\displaystyle G(s)H(s) = K \cdot G_0(s)H_0(s) $$, find $K$ such that Nyquist of $$\displaystyle G_0(s)H_0(s) $$ scaled by $K$ just touches $-1$. Use $$\displaystyle |KG_0(j\omega)H_0(j\omega)| = 1 $$ and $$\displaystyle \angle = -180^\circ $$ (or Routh).

5.4 Gain Margin & Phase Margin (Revisit)

• From Bode: $GM$ at $$\displaystyle \omega_{pc} $$, $PM$ at $$\displaystyle \omega_{gc} $$.

• From Nyquist: $GM$ = reciprocal of distance from $(-1,0)$ to plot crossing real axis (if it does). $PM$ = phase at gain crossover.

• Interpretation: Larger margins → system can tolerate more gain/phase variations before instability → better relative stability.

6.0 COMPENSATION & CONTROLLER DESIGN

6.1 Need & Types of Compensation

Need: Improve stability, steady-state accuracy, transient response (rise time, overshoot) of an existing system.

6.2 PID Controller

$$ G_c(s) = K_p + \frac{K_i}{s} + K_d s = K_p \left(1 + \frac{1}{T_i s} + T_d s\right) $$

6.3 Compensator Design using Frequency Response (Lead Example)

Given: Desired $$\displaystyle K_v $$ and Phase Margin $$\displaystyle PM_{des} $$ for a Type 1 system. Steps:

1. Determine $$\displaystyle \omega_{gc} $$ from $$\displaystyle K_v $$:

   $$\displaystyle K_v = \lim_{s\to 0} s G(s) = |G(j\omega)| \cdot \omega $$ at low $\omega$.

   For Type 1, $$\displaystyle K_v = K / \text{(product of time constants)} $$.

   Choose $$\displaystyle \omega_{gc} $$ such that $$\displaystyle |G(j\omega_{gc})H(j\omega_{gc})| \cdot \omega_{gc} = K_v $$.

2. Find required phase lead:

   $$\displaystyle \phi_m = PM_{des} - \angle G(j\omega_{gc})H(j\omega_{gc}) - 5^\circ $$ (safety margin).

3. Calculate $\alpha$:

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

4. Determine $T$:

   $$\displaystyle \omega_{max} = \frac{1}{T\sqrt{\alpha}} = \omega_{gc} $$ (place compensator zero at $$\displaystyle \omega_{gc} $$ for maximum phase lead).

   $$\displaystyle \Rightarrow T = \frac{1}{\omega_{gc}\sqrt{\alpha}} $$.

5. Form compensator: $$\displaystyle G_c(s) = \frac{1+sT}{1+s\alpha T} $$.

6. Verify: Replot Bode with $$\displaystyle G_c(s)G(s)H(s) $$, check $PM$ and $$\displaystyle K_v $$.

Lag Design: To improve $$\displaystyle K_v $$ while maintaining existing $PM$. Place zero near $$\displaystyle \omega_{gc} $$ and pole at $\beta$ times lower frequency ($$\displaystyle \beta > 1 $$).

6.4 PD Controller Tuning

$$\displaystyle G_c(s) = K_p(1 + T_d s) $$.

• Adds a zero at $$\displaystyle s = -1/T_d $$.

• Effect on Root Locus: Attracts root locus to left → increases damping, reduces $$\displaystyle t_s $$.

• Design for Critical Damping:

  Closed-loop poles must be real and equal. For a given plant $G(s)$, characteristic equation: $$\displaystyle 1 + K_p(1+T_d s)G(s) = 0 $$.

  Choose $$\displaystyle T_d $$ so that the zero cancels a plant pole or is placed appropriately to make the locus break on real axis at a double point.

  Settling Time ($$\displaystyle t_s $$): For critically damped second-order system, $$\displaystyle t_s \approx 4/\sigma $$ where $\sigma$ is real pole location.

7.0 SYSTEM COMPONENTS & ACTUATORS

7.1 Servomotors

A.C. Servomotor (Two-Phase Induction Motor):

• Construction: Stator has two windings: main (excited by constant AC) and control (excited by variable AC from amplifier). Rotor: squirrel-cage.

• Working: Control voltage $$\displaystyle V_c $$ produces flux $$\displaystyle \phi_c \propto V_c $$. Interaction with main flux $$\displaystyle \phi_m $$ produces torque $$\displaystyle T \propto \phi_m \phi_c \sin\delta \propto V_c $$. Direction controlled by phase of $$\displaystyle V_c $$.

• Torque-Speed Characteristics: Nearly linear for small displacements. Good for position control.

• Advantages: Rugged, low maintenance, no commutation.

• Disadvantages: Non-linear at high speeds, limited torque, requires two power supplies.

D.C. Servomotor:

• Construction: Armature-controlled (field constant). Permanent magnet or separately excited.

• Transfer Function: $$\displaystyle \frac{\Theta(s)}{V_a(s)} = \frac{K_t}{(L_a s + R_a)(J s + B) + K_t K_b} \approx \frac{K}{(Js+B)(\tau_a s + 1)} $$ (if $$\displaystyle L_a $$ small).

• Advantages: Linear, good speed control, high starting torque.

• Disadvantages: Commutator maintenance, brush wear.

7.2 Stepper Motors

• Principle: Rotates in discrete steps (e.g., $$\displaystyle 1.8^\circ $$/step) by sequentially energizing stator windings.

• Open-Loop Control: No feedback needed; number of pulses = angular displacement.

• Advantages: No cumulative error, simple drive, good for positioning.

• Disadvantages: Resonance at high speeds, low torque at high speeds.

• Applications: Printers, plotters, CNC machines.

7.3 Tacho-Generators (Techo-Generators)

• Principle: DC generator with permanent magnet field. Output voltage $$\displaystyle V_o \propto $$ rotational speed $\omega$.

$$ V_o(s) = K_t \cdot \Omega(s) $$

• Use: Velocity feedback in control systems (e.g., in servo systems to improve damping).

• Advantages: Simple, reliable.

• Disadvantages: Mechanical wear, limited frequency response.

8.0 STATE-SPACE ANALYSIS (Advanced Topics)

8.1 State-Space Representation

State Variables: Minimal set of variables $$\displaystyle x_1, x_2, ..., x_n $$ that completely describe system dynamics. State Equation:

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

Output Equation:

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

where $\mathbf{x}$ = state vector, $u$ = input, $y$ = output.

• Controllability: Ability to transfer state from any initial to any final in finite time via input.

• Observability: Ability to determine initial state from output history.

8.2 Eigenvalues & Eigenvectors

Definition: For matrix $\mathbf{A}$, scalar $\lambda$ and non-zero vector $\mathbf{v}$ are eigenvalue and eigenvector if:

$$ \mathbf{A} \mathbf{v} = \lambda \mathbf{v} $$

Significance:

• Eigenvalues ($\lambda$): Are the system poles (roots of $$\displaystyle \det(s\mathbf{I} - \mathbf{A}) = 0 $$). Determine stability ($$\displaystyle \text{Re}(\lambda) < 0 $$ for stable).

• Eigenvectors ($\mathbf{v}$): Define mode shapes (direction of state evolution corresponding to each pole).

Computation:

1. Solve characteristic equation: $$\displaystyle \det(\mathbf{A} - \lambda \mathbf{I}) = 0 $$ for $$\displaystyle \lambda_i $$.

2. For each $$\displaystyle \lambda_i $$, solve $$\displaystyle (\mathbf{A} - \lambda_i \mathbf{I}) \mathbf{v}_i = \mathbf{0} $$ for $$\displaystyle \mathbf{v}_i $$ (non-trivial solution).

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

• Char. eq.: $$\displaystyle \lambda^2 + 3\lambda + 2 = 0 \Rightarrow \lambda_1 = -1, \lambda_2 = -2 $$.

• For $$\displaystyle \lambda_1=-1 $$: $$\displaystyle (\mathbf{A}+\mathbf{I})\mathbf{v}_1=0 \Rightarrow \begin{bmatrix} 1 & 1 \\ -2 & -2 \end{bmatrix}\mathbf{v}_1=0 \Rightarrow \mathbf{v}_1 = \begin{bmatrix} 1 \\ -1 \end{bmatrix} $$.

• For $$\displaystyle \lambda_2=-2 $$: $$\displaystyle (\mathbf{A}+2\mathbf{I})\mathbf{v}_2=0 \Rightarrow \begin{bmatrix} 2 & 1 \\ -2 & -1 \end{bmatrix}\mathbf{v}_2=0 \Rightarrow \mathbf{v}_2 = \begin{bmatrix} 1 \\ -2 \end{bmatrix} $$.

8.3 State Transition Matrix ($\boldsymbol{\Phi}(t)$)

Definition:

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

Solution of State Equation (Zero Input):

$$ \mathbf{x}(t) = \boldsymbol{\Phi}(t) \mathbf{x}(0) $$

Solution with Input:

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

Properties:

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

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

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

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

5. $\boldsymbol{\Phi}(t)$ is the fundamental matrix; its columns are independent solutions.

Significance: Predicts unforced state evolution, fundamental to controllability/observability tests and system response.

Computation Methods:

• Laplace: $$\displaystyle \boldsymbol{\Phi}(t) = \mathcal{L}^{-1}\left[ (s\mathbf{I} - \mathbf{A})^{-1} \right] $$.

• Cayley-Hamilton: For low-order systems.

• Diagonalization: If $\mathbf{A}$ has distinct eigenvalues, $$\displaystyle \boldsymbol{\Phi}(t) = \mathbf{V} e^{\boldsymbol{\Lambda}t} \mathbf{V}^{-1} $$, where $\mathbf{V}$ = eigenvector matrix, $\boldsymbol{\Lambda}$ = diag$$\displaystyle (\lambda_i) $$.

END OF UNIT 4 NOTES
Always cross-reference derivations with past paper patterns (e.g., Jun 2025: Mason's, AC Servo TF, Mp-ζ, PD tuning, steady-state error, Routh, root locus, Bode, Nyquist, polar, ωr/Mr, lag-lead).