I. FUNDAMENTALS & SYSTEM REPRESENTATION
1.1 Basic Concepts & Classifications
-
Open-loop control system: Output does not affect input. No feedback. Simple, economical, but inaccurate with disturbances.
-
Closed-loop (feedback) control system: Output measured and fed back to input. Compensates for disturbances, reduces sensitivity, improves accuracy. May cause instability if not designed properly.
-
Components:
-
Plant: The system to be controlled.
-
Controller: Generates control signal.
-
Sensor/Transducer: Measures output.
-
Actuator: Drives the plant.
-
Reference Input: Desired value.
-
-
[!TIP] Feedback reduces sensitivity to parameter variations but introduces complexity. Always check stability in closed-loop design.
1.2 System Modeling & Analogies
-
Mechanical Translational: Mass (M), Spring (K), Damper (B). Equation: $$\displaystyle M\ddot{x} + B\dot{x} + Kx = F $$.
-
Mechanical Rotational: Inertia (J), Spring (K), Damper (B). Equation: $$\displaystyle J\ddot{\theta} + B\dot{\theta} + K\theta = T $$.
-
Electrical Analogies:
-
Force-Voltage (F-V) / Torque-Voltage: Analogous to series RLC circuit.
-
Force ↔ Voltage ($$\displaystyle F \leftrightarrow v $$)
-
Velocity ↔ Current ($$\displaystyle \dot{x} \leftrightarrow i $$)
-
Mass ↔ Inductance ($$\displaystyle M \leftrightarrow L $$)
-
Damper ↔ Resistance ($$\displaystyle B \leftrightarrow R $$)
-
Spring ↔ Inverse Capacitance ($$\displaystyle K \leftrightarrow 1/C $$)
-
-
Force-Current (F-I) / Torque-Current: Analogous to parallel RLC circuit.
-
Force ↔ Current ($$\displaystyle F \leftrightarrow i $$)
-
Velocity ↔ Voltage ($$\displaystyle \dot{x} \leftrightarrow v $$)
-
Mass ↔ Capacitance ($$\displaystyle M \leftrightarrow C $$)
-
Damper ↔ Conductance ($$\displaystyle B \leftrightarrow 1/R $$)
-
Spring ↔ Inverse Inductance ($$\displaystyle K \leftrightarrow 1/L $$)
-
-
-
Direct vs. Inverse Analogy: In direct analogy, analogous variables have same physical nature (e.g., force-voltage). In inverse, they are reciprocal (e.g., force-current).
1.3 Block Diagram & Signal Flow Graph (SFG)
-
Block Diagram Algebra:
-
Series: $$\displaystyle G_1 G_2 $$
-
Parallel: $$\displaystyle G_1 + G_2 $$
-
Feedback: $$\displaystyle \frac{G}{1 \pm GH} $$ (negative feedback uses $+$ in denominator).
-
-
Signal Flow Graph (SFG): Nodes (variables), branches (gains), direction.
-
Mason’s Gain Formula:
$$\boxed{T = \frac{\sum_{k=1}^{N} P_k \Delta_k}{\Delta}}$$
-
$$\displaystyle P_k $$: Gain of $$\displaystyle k^{th} $$ forward path.
-
$\Delta$: Determinant of the graph.
$$\Delta = 1 - \sum L_i + \sum L_i L_j - \sum L_i L_j L_k + \cdots$$
($$\displaystyle L_i $$: individual loop gains; $$\displaystyle L_i L_j $$: non-touching loop gains product).
-
$$\displaystyle \Delta_k $$: Cofactor of $\Delta$ after removing loops touching $$\displaystyle k^{th} $$ forward path.
-
Steps for Mason’s:
-
Identify all forward paths and their gains.
-
Identify all individual loops and their gains.
-
Compute $\Delta$: subtract sum of loop gains, add sum of products of two non-touching loops, etc.
-
For each forward path, compute $$\displaystyle \Delta_k $$ by removing loops touching that path.
-
Apply formula.
-
-
[!TIP] Mason’s formula avoids tedious block diagram reductions, especially with multiple loops and forward paths. Always check for non-touching loops.
1.4 Transfer Function (TF) Derivation
- Definition: Laplace transform of impulse response. Ratio of output to input under zero initial conditions.
$$\boxed{G(s) = \frac{C(s)}{R(s)}}$$
-
From Differential Equation: Take Laplace (assuming zero IC), solve for $C(s)/R(s)$.
-
From Block Diagram/SFG: Use block diagram algebra or Mason’s formula.
-
A.C. Servomotor TF:
-
Assumptions: Linear torque-speed characteristic, negligible stator inductance, constant rotor resistance, small slip.
-
Torque $$\displaystyle T_e = K_t V_c $$, where $$\displaystyle V_c $$ is control voltage.
-
Mechanical equation: $$\displaystyle J\ddot{\theta} + B\dot{\theta} = K_t V_c $$.
-
Electrical time constant small → $$\displaystyle V_c \approx $$ input voltage.
-
Taking Laplace: $$\displaystyle J s^2 \Theta(s) + B s \Theta(s) = K_t V_c(s) $$.
-
$$\boxed{G(s) = \frac{\Theta(s)}{V_c(s)} = \frac{K_t}{J s^2 + B s} = \frac{K}{s(Ts+1)}}$$
where $$\displaystyle K = K_t/B $$, $$\displaystyle T = J/B $$ (mechanical time constant).
-
[!TIP] In A.C. servomotor, the rotor time constant is often neglected for simplicity, leading to a first-order TF.
II. TIME DOMAIN ANALYSIS & PERFORMANCE
2.1 Standard Test Input Signals
-
Step: $$\displaystyle u(t) = A $$, $$\displaystyle R(s) = A/s $$. Represents sudden command.
-
Ramp: $$\displaystyle u(t) = At $$, $$\displaystyle R(s) = A/s^2 $$. Represents linearly increasing demand.
-
Parabolic: $$\displaystyle u(t) = At^2/2 $$, $$\displaystyle R(s) = A/s^3 $$. Represents accelerating demand.
-
Impulse: $$\displaystyle u(t) = A\delta(t) $$, $$\displaystyle R(s) = A $$. Represents sudden shock.
-
[!TIP] Step is most common; steady-state error for ramp/parabolic indicates system type.
2.2 First-Order Systems
-
Standard TF: $$\displaystyle G(s) = \frac{K}{\tau s + 1} $$, $$\displaystyle \tau = \text{time constant} $$.
-
Unit Step Response: $$\displaystyle c(t) = K(1 - e^{-t/\tau}) $$.
-
Rise time $$\displaystyle t_r \approx 2.2\tau $$ (10% to 90%).
-
Settling time $$\displaystyle t_s $$ (2% criterion): $$\displaystyle t_s \approx 4\tau $$; (5%): $$\displaystyle t_s \approx 3\tau $$.
-
-
Unit Ramp Response: $$\displaystyle c(t) = K(t - \tau + \tau e^{-t/\tau}) $$.
-
Steady-state error $$\displaystyle e_{ss} = \tau $$ (for $$\displaystyle K=1 $$).
-
Velocity error constant $$\displaystyle K_v = \lim_{s\to 0} sG(s) = K/\tau $$, so $$\displaystyle e_{ss} = 1/K_v $$.
-
-
[!TIP] First-order systems never overshoot; response is exponential.
2.3 Second-Order Systems (Underdamped)
- Standard TF:
$$\boxed{G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}}$$
$\zeta$: damping ratio, $$\displaystyle \omega_n $$: natural frequency.
-
Unit Step Response: $$\displaystyle c(t) = 1 - \frac{1}{\sqrt{1-\zeta^2}} e^{-\zeta\omega_n t} \sin(\omega_d t + \phi) $$, where $$\displaystyle \omega_d = \omega_n\sqrt{1-\zeta^2} $$, $$\displaystyle \phi = \arccos(\zeta) $$.
-
Time Response Specifications:
- Maximum Overshoot ($$\displaystyle M_p $$): First peak above steady-state.
$$\boxed{M_p = e^{-\frac{\zeta\pi}{\sqrt{1-\zeta^2}}} \times 100\%}$$
> Derivation: At $$\displaystyle t_p = \pi/\omega_d $$, $$\displaystyle \sin(\omega_d t_p + \phi) = 1 $$, so $$\displaystyle c(t_p) = 1 + e^{-\zeta\pi/\sqrt{1-\zeta^2}} $$.
- Peak Time ($$\displaystyle t_p $$): Time to first peak.
$$\boxed{t_p = \frac{\pi}{\omega_d} = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}}}$$
- Rise Time ($$\displaystyle t_r $$): Time to first cross steady-state (0% to 100% for underdamped).
$$\boxed{t_r = \frac{\pi - \phi}{\omega_d} = \frac{\pi - \arccos(\zeta)}{\omega_n\sqrt{1-\zeta^2}}}$$
Approx: $$\displaystyle t_r \approx \frac{1.8}{\omega_n} $$ for $$\displaystyle \zeta=0.5 $$.
- Settling Time ($$\displaystyle t_s $$): Time to stay within ±2% (or 5%) of steady-state.
$$\boxed{t_s(2\%) \approx \frac{4}{\zeta\omega_n}, \quad t_s(5\%) \approx \frac{3}{\zeta\omega_n}}$$
- Resonant Frequency ($$\displaystyle \omega_r $$) & Peak ($$\displaystyle M_r $$): For $$\displaystyle \zeta < 1/\sqrt{2} $$.
$$\boxed{\omega_r = \omega_n\sqrt{1-2\zeta^2}}$$
$$\boxed{M_r = \frac{1}{2\zeta\sqrt{1-\zeta^2}}}$$
-
Effect of $\zeta$ and $$\displaystyle \omega_n $$:
-
$\zeta \uparrow$: $$\displaystyle M_p \downarrow $$, $$\displaystyle t_r \uparrow $$, $$\displaystyle t_s \downarrow $$, $$\displaystyle \omega_r \downarrow $$.
-
$$\displaystyle \omega_n \uparrow $$: Response faster ($$\displaystyle t_r, t_p, t_s \downarrow $$), but overshoot unchanged.
-
-
[!TIP] Memorize overshoot formula. For $$\displaystyle \zeta=0.5 $$, $$\displaystyle M_p \approx 16.3\% $$; $$\displaystyle \zeta=0.707 $$, $$\displaystyle M_p \approx 4.3\% $$.
2.4 Steady-State Error Analysis
-
Definition: $$\displaystyle e_{ss} = \lim_{t\to\infty} e(t) = \lim_{s\to 0} sE(s) $$, where $$\displaystyle E(s) = R(s)/(1+G(s)H(s)) $$.
-
System Type: Number of open-loop poles at origin in $G(s)H(s)$.
-
Static Error Coefficients:
- Position error constant ($$\displaystyle K_p $$): For step input.
$$\boxed{K_p = \lim_{s\to 0} G(s)H(s)}$$
- Velocity error constant ($$\displaystyle K_v $$): For ramp input.
$$\boxed{K_v = \lim_{s\to 0} sG(s)H(s)}$$
- Acceleration error constant ($$\displaystyle K_a $$): For parabolic input.
$$\boxed{K_a = \lim_{s\to 0} s^2 G(s)H(s)}$$
-
Steady-State Error for Unity Feedback:
| System Type | Step Input $A$ | Ramp Input $At$ | Parabolic Input $$\displaystyle At^2/2 $$ | |-------------|----------------|------------------|--------------------------| | 0 | $$\displaystyle \frac{A}{1+K_p} $$ | $\infty$ | $\infty$ | | 1 | 0 | $$\displaystyle \frac{A}{K_v} $$ | $\infty$ | | 2 | 0 | 0 | $$\displaystyle \frac{A}{K_a} $$ |
-
Limitations of Static Error Coefficients:
-
Applicable only for stable closed-loop systems.
-
Valid only for standard inputs (step, ramp, parabola).
-
Cannot predict transient error behavior.
-
-
Generalized Error Coefficients: From Taylor series of $1/(1+G(s)H(s))$ around $$\displaystyle s=0 $$. Used for arbitrary inputs.
III. STABILITY ANALYSIS (CLASSICAL)
3.1 Concept of Stability
-
BIBO Stability: Every bounded input yields bounded output.
-
Internal Stability: All poles of closed-loop transfer function have negative real parts (LHP).
-
Relative Stability: How far poles are from imaginary axis. Measured by:
-
Gain Margin (GM): Factor by which gain can be multiplied before instability.
-
Phase Margin (PM): Additional phase lag required to reach instability.
-
-
[!TIP] PM > 0 and GM > 0 (in dB: GM > 0 dB, PM > 0°) imply stability. Larger margins mean more stable.
3.2 Routh-Hurwitz (R-H) Criterion
-
Routh Array Formation:
For $$\displaystyle a_n s^n + a_{n-1} s^{n-1} + \cdots + a_0 = 0 $$.
-
First row: $$\displaystyle a_n, a_{n-2}, a_{n-4}, \dots $$
-
Second row: $$\displaystyle a_{n-1}, a_{n-3}, a_{n-5}, \dots $$
-
Subsequent rows: $$\displaystyle b_i = \frac{a_{n-1} \cdot (\text{row above element}) - a_n \cdot (\text{row above next})}{a_{n-1}} $$, etc.
-
-
Stability Condition: All elements of first column > 0.
-
Number of RHP Roots: Equal to number of sign changes in first column.
-
Special Cases:
-
Row of zeros: Indicate symmetrical roots on imaginary axis or repeated roots. Form auxiliary equation from row above, differentiate, replace row of zeros with coefficients.
-
First element zero: Replace with small $$\displaystyle \epsilon > 0 $$, continue, then take limit $\epsilon \to 0$.
-
-
Application: Find range of $K$ for stability by ensuring all first column elements positive.
-
[!TIP] If first column has a zero, immediately suspect imaginary axis roots. Use auxiliary equation to find frequency of oscillation.
Root Locus (RL) Method
-
Definition: Plot of closed-loop pole locations as gain $K$ varies from $0$ to $\infty$.
-
Construction Rules (180° System):
-
Start at open-loop poles ($$\displaystyle K=0 $$), end at open-loop zeros ($$\displaystyle K=\infty $$). If zeros < poles, $P-Z$ branches go to $\infty$ along asymptotes.
-
Asymptotes: Angles $$\displaystyle \theta = \frac{(2q+1)180^\circ}{P-Z} $$, $$\displaystyle q=0,1,\dots,P-Z-1 $$. Centroid $$\displaystyle \sigma = \frac{\sum \text{poles} - \sum \text{zeros}}{P-Z} $$.
-
Breakaway/Break-in Points: On real axis, solve $$\displaystyle \frac{dK}{ds}=0 $$ from $$\displaystyle K = -\frac{1}{G(s)H(s)} $$.
-
Angle of Arrival/Departure: At complex poles/zeros, $$\displaystyle \sum \text{angles to other poles/zeros} = \pm 180^\circ $$.
-
Imaginary Axis Crossing: Use Routh or substitute $$\displaystyle s=j\omega $$ in characteristic equation.
-
Symmetry: About real axis.
-
-
Sketching Steps:
-
Mark open-loop poles/zeros.
-
Determine asymptotes and centroid.
-
Identify real-axis segments (to left of odd number of poles/zeros).
-
Find breakaway points (if any).
-
Compute angles at complex poles/zeros.
-
Sketch smooth curves following rules.
-
-
Commenting on Stability: If any branch lies in RHP for some $K$, system unstable for that $K$.
-
Design for Specified $\zeta$: Draw line from origin at angle $$\displaystyle \cos^{-1}(\zeta) $$, intersection with root locus gives desired pole location; read corresponding $K$.
-
Effects of Adding Poles/Zeros:
-
Adding pole: Root locus bends toward left initially but may go to RHP for high $K$; reduces stability.
-
Adding zero: Attracts root locus, improves stability and transient response.
-
-
[!TIP] Breakaway points are found by solving $$\displaystyle dK/ds=0 $$, but verify they lie on real-axis segments between poles.
3.3 Frequency Domain Stability Measures
-
Gain Margin (GM): $$\displaystyle GM = \frac{1}{|G(j\omega_{pc})|} $$, where $$\displaystyle \omega_{pc} $$ is phase crossover frequency (phase = -180°). In dB: $$\displaystyle GM_{dB} = -20\log_{10}|G(j\omega_{pc})| $$.
-
Phase Margin (PM): $$\displaystyle PM = 180^\circ + \angle G(j\omega_{gc}) $$, where $$\displaystyle \omega_{gc} $$ is gain crossover frequency (magnitude = 1). In radians: $$\displaystyle PM = \pi + \arg(G(j\omega_{gc})) $$.
-
Relationship: Larger GM/PM → more stable, lower overshoot. Approx: $$\displaystyle \zeta \approx \frac{PM}{100} $$ for PM between 0° and 70°.
-
Bode Plot Construction:
-
Magnitude Plot: $|G(j\omega)|$ in dB: $$\displaystyle 20\log_{10}|G(j\omega)| $$. Asymptotic: straight lines with slopes $0, \pm20, \pm40,...$ dB/decade. Corner frequencies at poles/zeros.
-
Phase Plot: $\angle G(j\omega)$ in degrees. Asymptotic: start at $$\displaystyle 0^\circ $$ (no poles at origin), $$\displaystyle -90^\circ $$ per pole at origin, etc. Actual phase may need correction at corners.
-
Steps:
-
Write $G(s)$ in standard form: $$\displaystyle G(s) = K \frac{\prod (1+sT_z)}{\prod (1+sT_p)} $$.
-
Find $$\displaystyle \omega_{gc} $$ and $$\displaystyle \omega_{pc} $$ from intersection of 0 dB and -180° lines.
-
Draw asymptotes, then sketch actual considering phase/magnitude corrections.
-
-
Determine GM/PM from Bode:
-
$$\displaystyle \omega_{pc} $$: where phase = -180°, read magnitude $|G|$, $$\displaystyle GM = 1/|G| $$.
-
$$\displaystyle \omega_{gc} $$: where magnitude = 0 dB, read phase $\phi$, $$\displaystyle PM = 180 + \phi $$.
-
-
-
Polar Plot (Nyquist for open-loop without encirclements):
-
Plot of $G(j\omega)$ as $\omega$ from $0$ to $\infty$.
-
Type 0: Starts at real positive, ends at origin.
-
Type 1: Starts at infinity on negative imaginary axis, ends at finite real point.
-
Type 2: Starts at infinity on positive real axis, ends at origin.
-
Effect of Poles:
-
Pole at origin ($$\displaystyle s=0 $$): Adds $$\displaystyle -90^\circ $$ to phase, plot starts on negative imaginary axis.
-
Pole at $$\displaystyle s=-1/T $$: Adds $$\displaystyle -90^\circ $$ phase shift, bends plot downward.
-
-
-
Nyquist Stability Criterion:
-
Contour: Large semicircle in RHP enclosing entire RHP.
-
Nyquist Plot: Locus of $G(s)H(s)$ as $s$ traverses contour.
-
$$\displaystyle N = Z - P $$: Number of clockwise encirclements of (-1, j0) point.
-
$P$: Number of open-loop RHP poles.
-
$Z$: Number of closed-loop RHP poles (for stability, $$\displaystyle Z=0 $$).
-
-
Stability Condition: $$\displaystyle N = P $$ (i.e., Nyquist plot must encircle (-1, j0) exactly $P$ times clockwise).
-
Range of $K$: For $$\displaystyle G(s)H(s) = K \cdot G_0(s) $$, plot $$\displaystyle G_0(j\omega) $$, find where it crosses real axis at (-1, j0); $$\displaystyle K_{max} $$ from magnitude at that point.
-
-
[!TIP] For unity feedback, if open-loop stable ($$\displaystyle P=0 $$), then closed-loop stable iff Nyquist plot does NOT encircle (-1, j0).
IV. COMPENSATION & CONTROLLER DESIGN
4.1 Need for Compensation
-
Improve transient response (reduce overshoot, settling time).
-
Improve steady-state accuracy (reduce error).
-
Increase stability margins (GM, PM).
-
Meet conflicting specifications (e.g., high $$\displaystyle K_v $$ vs. high PM).
4.2 Compensation Techniques
-
Lag Compensator:
- TF:
$$\boxed{G_c(s) = \frac{1+sT}{1+s\alpha T}, \quad \alpha > 1}$$
Zero at $$\displaystyle z = -1/T $$, pole at $$\displaystyle p = -1/(\alpha T) $$ (pole closer to origin).
-
Effect on Bode: Increases low-frequency gain (improves $$\displaystyle K_v $$, $$\displaystyle K_a $$), little effect on high-frequency phase. Adds -20 dB/dec slope after zero, then +20 dB/dec after pole (net 0 dB/dec at high freq).
-
Design: Place zero a decade below new $$\displaystyle \omega_c $$, pole at $\alpha/10$ of zero frequency, $\alpha$ from required $$\displaystyle K_v $$ ratio.
-
Lead Compensator:
- TF:
$$\boxed{G_c(s) = \frac{1+sT}{1+sT/\beta}, \quad \beta > 1}$$
Zero at $$\displaystyle z = -1/T $$, pole at $$\displaystyle p = -\beta/T $$ (zero closer to origin).
-
Effect on Bode: Adds positive phase (maximum $$\displaystyle \phi_m = \sin^{-1}\frac{\beta-1}{\beta+1} $$), increases bandwidth, improves PM.
-
Design Steps:
-
Determine uncompensated $K$ from steady-state spec (e.g., $$\displaystyle K_v $$).
-
Find $$\displaystyle \omega_c $$ where $$\displaystyle |G(j\omega_c)| = 1/K $$ (or desired gain).
-
Required PM at $$\displaystyle \omega_c $$: $$\displaystyle PM_{req} = PM_{desired} + 5^\circ $$ (safety).
-
Compute $\beta$ from $$\displaystyle \sin(PM_{req}) = \frac{\beta-1}{\beta+1} $$.
-
$$\displaystyle T = 1/(\omega_c \sqrt{\beta}) $$.
-
Verify new $$\displaystyle \omega_c $$ and PM.
-
-
Lag-Lead Compensator:
-
TF: $$\displaystyle G_c(s) = \frac{(1+sT_1)(1+sT_2)}{(1+s\alpha T_1)(1+sT_2/\beta)} $$, with $$\displaystyle \alpha>1 $$, $$\displaystyle \beta>1 $$.
-
When Used: When both SSE improvement (lag) and transient response (lead) needed.
-
Design: First design lead for PM, then lag to adjust low-frequency gain without disturbing PM much.
-
-
[!TIP] Lead compensator improves PM but reduces gain at low freq; lag improves low-freq gain but adds phase lag near crossover. Lag-lead combines both.
4.3 PID Controllers
-
Proportional (P): $$\displaystyle u(t) = K_p e(t) $$.
- Effects: Reduces rise time, decreases steady-state error, increases overshoot, may destabilize.
-
Integral (I): $$\displaystyle u(t) = K_i \int e(t) dt $$.
- Effects: Eliminates steady-state error for step/ramp, increases overshoot and settling time, may cause instability.
-
Derivative (D): $$\displaystyle u(t) = K_d \frac{de(t)}{dt} $$.
- Effects: Reduces overshoot and settling time, improves stability, amplifies noise.
-
PID: $$\displaystyle u(t) = K_p e(t) + K_i \int e(t) dt + K_d \frac{de(t)}{dt} $$.
-
Combines benefits: good transient response and zero SSE.
-
Disadvantages: Three parameters to tune, derivative amplifies noise.
-
-
[!TIP] In practice, PI is common for SSE elimination with moderate transient; PD for improved stability; PID for high precision.
V. STATE-SPACE ANALYSIS (MODERN METHODS)
5.1 State Variable Representation
-
State: Minimum set of variables $$\displaystyle (x_1, x_2, \dots, x_n) $$ such that knowledge at $$\displaystyle t_0 $$ and input $u(t)$ determines future state and output.
-
State Equations:
$$\boxed{\dot{x} = Ax + Bu}$$
$$\boxed{y = Cx + Du}$$
$x$: state vector ($n \times 1$), $u$: input vector ($m \times 1$), $y$: output vector ($p \times 1$).
$A$: system matrix ($n \times n$), $B$: input matrix ($n \times m$), $C$: output matrix ($p \times n$), $D$: feedthrough matrix ($p \times m$).
-
From Transfer Function:
- Controllable Canonical Form: For $$\displaystyle G(s) = \frac{b_0 s^m + b_1 s^{m-1} + \cdots + b_m}{s^n + a_1 s^{n-1} + \cdots + a_n} $$.
$$A = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ -a_n & -a_{n-1} & -a_{n-2} & \cdots & -a_1 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{bmatrix}, \quad C = \begin{bmatrix} b_m - a_1 b_0 & b_{m-1} - a_2 b_0 & \cdots & b_0 \end{bmatrix} \text{ (if } n=m\text{)}$$
- Observable Canonical Form: Transpose of controllable form (swap $$\displaystyle A \leftrightarrow A^T $$, $$\displaystyle B \leftrightarrow C^T $$).
5.2 State Transition Matrix ($\Phi(t)$)
- Definition:
$$\boxed{\Phi(t) = e^{At}}$$
-
Properties:
-
$$\displaystyle \Phi(0) = I $$.
-
$$\displaystyle \Phi(t_1)\Phi(t_2) = \Phi(t_1+t_2) $$.
-
$$\displaystyle \Phi^{-1}(t) = \Phi(-t) $$.
-
$$\displaystyle \frac{d}{dt}\Phi(t) = A\Phi(t) = \Phi(t)A $$.
-
$\Phi(t)$ satisfies $$\displaystyle \ddot{\Phi} = A\Phi $$.
-
-
Significance: Solution of state equations:
$$\boxed{x(t) = \Phi(t)x(0) + \int_0^t \Phi(t-\tau)Bu(\tau) d\tau}$$
First term: zero-input response; second: zero-state response.
-
Computation:
-
Laplace Inverse: $$\displaystyle \Phi(t) = \mathcal{L}^{-1}\{(sI - A)^{-1}\} $$.
-
Cayley-Hamilton: Use characteristic equation to express $$\displaystyle e^{At} $$ as polynomial in $A$.
-
5.3 Eigenvalues & Eigenvectors
-
Eigenvalues ($\lambda$): Roots of $$\displaystyle \det(A - \lambda I) = 0 $$.
-
Same as poles of transfer function.
-
Stability: All $$\displaystyle \text{Re}(\lambda) < 0 $$ for asymptotic stability.
-
-
Eigenvectors ($v$): Non-zero vectors satisfying $$\displaystyle (A - \lambda I)v = 0 $$.
-
Modal Matrix ($M$): Matrix of eigenvectors as columns. If eigenvalues distinct, $M$ invertible and $$\displaystyle A = M \Lambda M^{-1} $$, where $$\displaystyle \Lambda = \text{diag}(\lambda_1, \lambda_2, \dots) $$.
-
Jordan Form: For repeated eigenvalues, generalized eigenvectors needed.
-
[!TIP] Eigenvalues determine stability; eigenvectors determine mode shapes. For 2x2, solve $$\displaystyle \lambda^2 - (a_{11}+a_{22})\lambda + \det(A) = 0 $$.
5.4 Solution of State Equations
-
Homogeneous Solution: $$\displaystyle x_h(t) = e^{At}x(0) = \Phi(t)x(0) $$.
-
Forced Solution (Zero-State): $$\displaystyle x_{zs}(t) = \int_0^t \Phi(t-\tau)Bu(\tau) d\tau $$.
-
Complete Response: $$\displaystyle x(t) = x_h(t) + x_{zs}(t) $$.
-
For $$\displaystyle u(t)=0 $$, $$\displaystyle x(t) = e^{At}x(0) $$.
-
[!TIP] For time-invariant systems, always use state transition matrix. For scalar systems, $$\displaystyle e^{At} $$ reduces to $$\displaystyle e^{\lambda t} $$ for each eigenvalue.
VI. SYSTEM COMPONENTS & ACTUATORS
6.1 Servomotors
-
A.C. Servomotor (Two-Phase Induction Motor):
-
Construction: Stator with two windings (reference and control) 90° apart. Rotor: squirrel-cage or drag-cup.
-
Principle: Control voltage on control winding produces torque proportional to voltage (for small slip). Reference winding fed constant voltage.
-
TF Derivation:
-
Electrical time constant negligible → control voltage $$\displaystyle V_c $$ appears across rotor.
-
Torque $$\displaystyle T_e = K_t V_c $$.
-
Mechanical: $$\displaystyle J\ddot{\theta} + B\dot{\theta} = K_t V_c $$.
-
Laplace: $$\displaystyle (Js^2 + Bs)\Theta(s) = K_t V_c(s) $$.
-
-
$$\boxed{G(s) = \frac{\Theta(s)}{V_c(s)} = \frac{K_t}{Js^2 + Bs} = \frac{K}{s(Ts+1)}}$$
where $$\displaystyle K = K_t/B $$, $$\displaystyle T = J/B $$.
-
Advantages: No commutation, low maintenance, smooth operation.
-
Disadvantages: Low efficiency, nonlinear at high speeds, limited torque.
-
D.C. Servomotor: Brushed motor, better control linearity, but requires maintenance (brushes).
6.2 Stepper Motors
-
Principle: Digital motor; each pulse rotates shaft by fixed step angle $$\displaystyle \theta_s = 360^\circ / N $$, where $N$ = number of steps per revolution.
- Rotating magnetic field causes rotor to step.
-
Types:
-
Variable Reluctance (VR): Toothed rotor, low cost, high torque at low speed.
-
Permanent Magnet (PM): Rotor with permanent magnets, detent torque, medium torque.
-
Hybrid (HB): Combines VR and PM, high precision, high torque.
-
-
Advantages: Open-loop control, precise positioning, holds position without feedback.
-
Disadvantages: Resonance at certain speeds, torque drops at high speed, needs driver circuit.
-
[!TIP] Stepper motors are used in printers, CNC machines where precise open-loop positioning is needed.
6.3 Tacho-Generators
-
Principle: D.C. or A.C. generator whose output voltage $$\displaystyle V_o $$ is proportional to rotational speed $\omega$: $$\displaystyle V_o = K_t \omega $$.
-
Use: Feedback element for velocity measurement in speed control systems.
-
Advantages: Simple, rugged.
-
Disadvantages: Contact wear (in D.C.), limited frequency response.
VII. DESIGN APPLICATION & PROBLEM-SOLVING
7.1 Parameter Determination from Specifications
-
For second-order system $$\displaystyle G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} $$:
-
Given $$\displaystyle M_p $$: $$\displaystyle \zeta = \frac{-\ln(M_p)}{\sqrt{\pi^2 + \ln^2(M_p)}} $$.
-
Given $$\displaystyle t_p $$: $$\displaystyle \omega_n = \frac{\pi}{t_p \sqrt{1-\zeta^2}} $$.
-
Given $$\displaystyle \omega_r $$: $$\displaystyle \omega_r = \omega_n\sqrt{1-2\zeta^2} $$ → $$\displaystyle \omega_n = \frac{\omega_r}{\sqrt{1-2\zeta^2}} $$ (valid for $$\displaystyle \zeta < 1/\sqrt{2} $$).
-
Then $K$ from standard form (if unity feedback, closed-loop TF = $G/(1+G)$).
-
-
Example: Given $$\displaystyle M_p=26\% $$, $$\displaystyle \omega_r=8 $$ rad/s.
-
$$\displaystyle \zeta = 0.4 $$ (from $$\displaystyle M_p $$ table or formula).
-
$$\displaystyle \omega_n = 8 / \sqrt{1-2(0.4)^2} = 8 / \sqrt{0.68} \approx 9.7 $$ rad/s.
-
$$\displaystyle M_r = 1/(2\zeta\sqrt{1-\zeta^2}) \approx 1.37 $$.
-
7.2 Compensator Design from Frequency Domain Specs
-
Lead Compensator Design for $$\displaystyle K_v $$ and PM:
-
From $$\displaystyle K_v $$: $$\displaystyle K_v = \lim_{s\to 0} sG(s) = K/T $$ (for type 1 with integrator). Determine required $K$.
-
Plot uncompensated Bode with $K$ from step 1. Find $$\displaystyle \omega_c $$ where $$\displaystyle |G(j\omega_c)| = 1 $$ (0 dB). Note current PM.
-
Required PM at $$\displaystyle \omega_c $$: $$\displaystyle PM_{req} = PM_{desired} + 5^\circ $$ (safety).
-
Compute $\beta$: $$\displaystyle \sin(PM_{req}) = \frac{\beta-1}{\beta+1} $$ → $$\displaystyle \beta = \frac{1+\sin(PM_{req})}{1-\sin(PM_{req})} $$.
-
$$\displaystyle T = 1/(\omega_c \sqrt{\beta}) $$.
-
New compensator: $$\displaystyle G_c(s) = \frac{1+sT}{1+sT/\beta} $$.
-
Verify new Bode plot: new $$\displaystyle \omega_c \approx $$ old $$\displaystyle \omega_c $$, new PM = $$\displaystyle PM_{desired} $$.
-
-
Lag Compensator for $$\displaystyle K_v $$:
-
After lead design, if $$\displaystyle K_v $$ still low, add lag.
-
Place zero at $$\displaystyle \omega_z = \omega_c/10 $$, pole at $$\displaystyle \omega_p = \omega_z/\alpha $$, where $$\displaystyle \alpha = K_{v,req}/K_{v,uncomp} $$.
-
Ensure lag pole is a decade below new $$\displaystyle \omega_c $$ to avoid affecting PM.
-
7.3 Stability Analysis from Different Plots
-
Root Locus: If any branch in RHP for $$\displaystyle K>0 $$, unstable for that $K$. Comment on relative stability by how far poles are from imaginary axis.
-
Bode Plot:
-
Stable if $$\displaystyle PM > 0^\circ $$ (and $$\displaystyle GM > 0 $$ dB).
-
$$\displaystyle PM = 0^\circ $$ → marginally stable (sustained oscillations at $$\displaystyle \omega_{pc} $$).
-
$$\displaystyle PM < 0^\circ $$ → unstable.
-
-
Nyquist Plot:
-
Count $P$ (open-loop RHP poles).
-
Count $N$ (clockwise encirclements of -1 point).
-
Closed-loop stable if $$\displaystyle N = P $$.
-
If $N \neq P$, number of closed-loop RHP poles $$\displaystyle Z = N + P $$.
-
-
Frequency of Sustained Oscillations: When system is marginally stable.
-
From Routh: Row of zeros → solve auxiliary equation → $$\displaystyle \omega = \sqrt{\text{constant}} $$.
-
From Bode: $$\displaystyle \omega_{pc} $$ when phase = -180°.
-
From Root Locus: Imaginary axis crossing points.
-
[!TIP] In exams, always state stability condition clearly: e.g., "From Nyquist, $$\displaystyle P=0 $$, $$\displaystyle N=0 $$, so closed-loop stable." For marginal stability, give oscillation frequency.