Control System (EC-404) - Unit 5 Short Notes

1. FOUNDATIONAL CONCEPTS & SYSTEM REPRESENTATION

Block Diagram Representation

• Definition: Graphical representation of a system using blocks (transfer functions), summing points, and take-off points.

• Key Components:

  • Block: Represents a system or component with transfer function G(s).

  • Summing Point: Adds or subtracts signals (denoted by ⊕/⊖).

  • Take-off Point: Splits a signal to multiple paths.

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

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

• Open-loop vs. Closed-loop:

  | Feature | Open-loop | Closed-loop (Feedback) | | :--- | :--- | :--- | | Definition | No feedback; output does not affect input. | Output is fed back and compared with input. | | Accuracy | Low (sensitive to disturbances/parameter changes). | High (rejects disturbances, reduces sensitivity). | | Stability | Inherently stable. | Must be designed for stability. | | Complexity | Simple, inexpensive. | More complex, costly. | | Bandwidth | Typically higher. | Reduced due to feedback. |

• Reduction Rules: Series, parallel, feedback, and moving take-off/summing points (Shifting Rules).

Signal Flow Graph (SFG)

• Definition: Directed graph representing system equations. Nodes = variables, Branches = transfer functions.

• Terminology:

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

  • Loop: Path that starts and ends at same node, no other node repeated.

  • Non-touching Loops: Loops with no common nodes.

• Mason's Gain Formula (MGF):

$$T = \frac{C(s)}{R(s)} = \sum_{k=1}^{M} \frac{P_k \Delta_k}{\Delta}$$

Where:

*   `P_k` = gain of k<sup>th</sup> forward path.

*   `Δ` = 1 - (sum of all individual loop gains) + (sum of gains of all non-touching loop pairs) - ...

*   `Δ_k` = value of `Δ` for the part of graph **not touching** k<sup>th</sup> forward path.

> [!TIP] **Exam Focus:** MGF is **always** asked. Draw the SFG clearly, identify all loops and their touching/non-touching nature.

Analogous Systems

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

  | Mechanical (Translational) | Electrical | | :--- | :--- | | Force (F) | Voltage (V) | | Mass (M) | Inductance (L) | | Damping (B) | Resistance (R) | | Spring (K) | Capacitance (C) | | Displacement (x) | Charge (q) | | Velocity (v) | Current (i) |

• Force-Current (F-I) Analogy:

  | Mechanical (Translational) | Electrical | | :--- | :--- | | Force (F) | Current (I) | | Mass (M) | Capacitance (C) | | Damping (B) | Conductance (1/R) | | Spring (K) | Inverse Inductance (1/L) | | Displacement (x) | Flux (φ) |

• Rotational Systems: Replace F with Torque T, x with Angular Displacement θ, M with Moment of Inertia J, B with Viscous Friction B.

• Methods:

  • Direct Analogous: Directly map mechanical elements to electrical (F-V or F-I).

  • Inverse Analogous: Swap the roles of L and C (or R and 1/R) compared to direct analogy.

2. TIME DOMAIN ANALYSIS & SYSTEM RESPONSE

Standard Test Inputs

First-Order System

• Standard Form: G(s) = K / (τs + 1), where τ = time constant.

• Unit Step Response: c(t) = K(1 - e^{-t/τ}).

• Steady-State Error (SSE): e_ss = 1 - K (for unit step, if K is DC gain).

• Unit Ramp Response: c(t) = K( t - τ + τe^{-t/τ} ). SSE = τ/K (for ideal ramp t).

Second-Order System

• Standard Form:

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

*   `ω_n` = Undamped natural frequency (rad/s).

*   `ζ` (zeta) = Damping ratio (dimensionless).

• Step Response Cases:

  1. Underdamped (0 < ζ < 1): Oscillatory. Dominant poles: s = -ζω_n ± jω_n√(1-ζ²).

  2. Critically Damped (ζ = 1): Fastest non-oscillatory.

  3. Overdamped (ζ > 1): Slow, non-oscillatory.

• Performance Specifications (Underdamped):

  • Maximum Overshoot (M_p): M_p = e^{-ζπ/√(1-ζ²)}. \boxed{M_p = e^{\frac{-\zeta\pi}{\sqrt{1-\zeta^2}}}}

    Graph of M_p vs ζ is exponentially decaying.

  • Peak Time (T_p): Time to first peak. T_p = π / (ω_n√(1-ζ²)). \boxed{T_p = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}}}

  • Rise Time (T_r): Time to go from 10% to 90% (or 0% to 100%) of final value.

    For 0 < ζ < 0.9: T_r ≈ (π - φ)/ω_d, where φ = tan⁻¹(√(1-ζ²)/ζ), ω_d = ω_n√(1-ζ²).

  • Settling Time (T_s): Time to stay within a band (usually 2% or 5%).

    T_s ≈ 4/(ζω_n) (2% criterion). \boxed{T_s \approx \frac{4}{\zeta\omega_n} \ (2%)}

  • Resonant Frequency (ω_r) & Peak (M_r):

    ω_r = ω_n√(1 - 2ζ²) (exists only for ζ < 1/√2 ≈ 0.707).

    M_r = 1 / (2ζ√(1-ζ²)). \boxed{M_r = \frac{1}{2\zeta\sqrt{1-\zeta^2}}}

System Type & Error Constants

• Type of System: Number of pure integrators (1/s factors) in open-loop G(s)H(s) at origin.

  • Type 0: No pole at origin.

  • Type 1: One pole at origin.

  • Type 2: Two poles at origin.

• Static Error Coefficients (for Unity Feedback):

  • Position Error Constant (K_p): K_p = lim_{s→0} G(s)H(s). SSE to step = 1/(1+K_p).

  • Velocity Error Constant (K_v): K_v = lim_{s→0} s·G(s)H(s). SSE to ramp = 1/K_v.

  • Acceleration Error Constant (K_a): K_a = lim_{s→0} s²·G(s)H(s). SSE to parabolic = 1/K_a.

• Generalized Error Coefficients & Series:

  E(s) = R(s) - C(s) = \frac{R(s)}{1 + G(s)H(s)}

  For R(s) = A/s^m, e_ss = \frac{A}{m!} \left[ \frac{d^m}{ds^m} \left( \frac{1}{1+G(s)H(s)} \right) \right]_{s=0}.

• Limitations of Static Error Coefficients:

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

  • Cannot predict SSE for arbitrary inputs.

  • Do not give information about transient response.

  • Assume system is stable.

3. STABILITY ANALYSIS (Routh-Hurwitz & Root Locus)

Routh-Hurwitz Criterion

• Concept: Algebraic test for stability of linear time-invariant systems by examining the characteristic polynomial a_n s^n + ... + a_0 = 0.

• Routh Array Construction:

  
  s^n     a_n     a_{n-2}   a_{n-4} ...
  
  s^{n-1} a_{n-1} a_{n-3}   a_{n-5} ...
  
  s^{n-2} b_1    b_2       b_3 ...
  
  s^{n-3} c_1    c_2       ...
  
  ...
  
  s^0      f_1
  
  

  Where b_1 = (a_{n-1}a_{n-2} - a_n a_{n-3}) / a_{n-1}, etc.

• Stability Condition: All elements of the first column must be positive for all roots to have negative real parts (LHP).

• Special Cases:

  1. Row of Zeros: Indicate symmetrical roots about origin (e.g., pure imaginary, or pairs ±a ± bj). Form Auxiliary Equation from the row above, differentiate, replace the zero row.

  2. First Element Zero: Replace ε → 0⁺ in that element and reconstruct array.

• Finding K for Stability: Treat K as a variable in first column. Apply sign change conditions.

• Marginal Stability: First column has zero but no sign changes → roots on jω axis. Frequency found from auxiliary equation.

Root Locus Technique

• Definition: Graphical plot of closed-loop pole locations as system gain K varies from 0 to ∞.

• Construction Rules (for G(s)H(s) = K·P(s)/Q(s)):

  1. Branches: Equal to max(deg(P), deg(Q)).

  2. Start/End Points: Start at open-loop poles (K=0), end at open-loop zeros (K=∞). If zeros < poles, (n-m) branches go to ∞ (asymptotes).

  3. Asymptotes: Angles θ_a = (2q+1)180°/(n-m), q=0,1,...,(n-m-1).

     Centroid σ = (Σ real poles - Σ real zeros) / (n-m).

  4. Angle of Departure/Arrival: From complex pole/zero.

     θ_depart = 180° - Σ(angles to other poles) + Σ(angles to zeros).

  5. Breakaway/Break-in Points: On real axis, where multiple branches meet. Solve dK/ds = 0 for K(s) = -Q(s)/P(s).

  6. Intersection with jω axis: Use Routh or substitute s=jω into characteristic equation.

• Finding K for Specified ζ: Draw ζ-line (line from origin at angle cos⁻¹(ζ)). Intersection of root locus with this line gives desired pole location s = -ζω_n ± jω_n√(1-ζ²). Compute K from magnitude condition |G(s)H(s)| = 1.

• Effect of Adding Poles/Zeros:

  • Adding Open-loop Pole: Root locus shifts rightward (towards RHP), system becomes less stable.

  • Adding Open-loop Zero: Root locus shifts leftward (towards LHP), system becomes more stable, attracts branches.

4. FREQUENCY RESPONSE ANALYSIS (Bode, Nyquist, Polar)

Bode Plot (Magnitude & Phase)

• Construction Procedure (Asymptotic):

  1. Write G(jω)H(jω) in standard form: K·(jω)^N · Π(1+jω/z_i) / Π(1+jω/p_i).

  2. Magnitude (dB): 20 log|G(jω)H(jω)|. Corner frequencies at ω = z_i or p_i.

     • 1/s (integrator): -20 dB/dec slope, 0 dB at ω=1.

     • 1/(1+jωT) (first-order lag): 0 dB/dec to -20 dB/dec at ω=1/T.

     • (1+jωT) (first-order lead): +20 dB/dec to 0 dB/dec at ω=1/T.

  3. Phase (degrees): Sum of phases of each factor.

     • 1/s: -90° constant.

     • 1/(1+jωT): 0° to -90°, approx -45° at ω=1/T.

     • (1+jωT): 0° to +90°, approx +45° at ω=1/T.

  4. Add magnitude/phase contributions.

• Key Frequencies:

  • Gain Crossover Frequency (ω_gc): |G(jω)H(jω)| = 1 (0 dB). ω where magnitude plot crosses 0 dB.

  • Phase Crossover Frequency (ω_pc): ∠G(jω)H(jω) = -180°. ω where phase plot crosses -180°.

• Stability Margins (from Bode):

  • Gain Margin (GM): Factor by which gain can be increased before instability. GM = 1 / |G(jω_pc)H(jω_pc)| (in linear) or GM (dB) = -|G(jω_pc)H(jω_pc)|_{dB}.

  • Phase Margin (PM): Additional phase lag required to reach -180° at ω_gc. PM = 180° + ∠G(jω_gc)H(jω_gc).

  [!TIP] Relative Stability: PM > 0 and GM > 1 (or > 0 dB) imply closed-loop stability. Larger PM/GM → more stable, but slower response.

Polar Plot (Nyquist Plot)

• Definition: Plot of G(jω)H(jω) (complex plane) as ω varies from 0 to ∞.

• Construction for Type 0,1,2 Systems:

  • Type 0: Starts at (K, 0°), ends at (0, -90°·(n-m)).

  • Type 1: Starts at (∞, 0°) (from high mag at low ω), ends at (0, -90°·(n-m)).

  • Type 2: Starts at (∞, 0°), ends at (0, -180°·(n-m)) (or +180°).

• Inverse Polar Plot: Plot of 1/[G(jω)H(jω)]. Useful for design.

• Comparison with Bode: Polar shows magnitude & phase simultaneously; Bode separates them on log scale.

Nyquist Stability Criterion

• Concept: Relates open-loop stability (P = # of RHP poles of G(s)H(s)) to closed-loop stability via encirclements N of (-1, j0) point by Nyquist plot.

• Stability Condition: N = P → Closed-loop stable. N ≠ P → Unstable.

  • N = # clockwise encirclements of (-1, j0).

  • P = # of open-loop poles in RHP.

• Procedure:

  1. Sketch Nyquist contour (entire RHP + jω axis).

  2. Map through G(s)H(s) to get Nyquist plot.

  3. Count N (clockwise positive).

  4. Determine P from open-loop TF.

  5. Apply Z = P - N (closed-loop RHP poles). Z=0 → stable.

• Gain Margin from Nyquist: GM = distance from (-1, j0) to plot along negative real axis, if plot crosses it.

5. COMPENSATION & CONTROLLER DESIGN

Need for Compensation

To modify system dynamics to meet both transient (stability, overshoot, settling time) and steady-state (SSE) specifications, which are often conflicting.

Compensation Techniques

Lead Compensation Design (Bode Method)

1. From G(s), find PM_old. PM_required → ΔPM = PM_req - PM_old + 5° (safety).

2. φ_m = ΔPM (max phase from lead network). α = (1 - sin φ_m)/(1 + sin φ_m).

3. ω_m (new ω_gc) = frequency where |G(jω)H(jω)| = -20 log √α dB (i.e., magnitude before adding lead = -20 log √α).

4. T = 1/(ω_m √α).

5. Verify PM and ω_gc with G_c(s)G(s).

6. Adjust K if needed to set ω_gc exactly.

Lag Compensation Design (Bode Method)

1. From G(s), find K_v or K_p. Compute required K_v_req from SSE spec.

2. β = K_v_req / K_v_old (gain increase needed at low ω).

3. Place lag pole at ω_p = ω_gc/10 (to not affect PM much). Zero at ω_z = ω_p/β.

4. Verify K_v and check PM (should be similar to uncompensated at ω_gc).

PID Controller

• Structure: G_c(s) = K_p + K_i/s + K_d s = K_p(1 + 1/(T_i s) + T_d s).

• Effects:

  • P (Proportional): Increases K_p → reduces SSE for Type ≥1, but reduces PM (can destabilize).

  • I (Integral): Increases system type → eliminates SSE for step/ramp/parabolic. Adds -90° phase lag → reduces stability.

  • D (Derivative): Adds phase lead → improves PM, reduces overshoot, improves stability. Sensitive to noise.

• Composite (PI, PD, PID): Balance between steady-state accuracy and transient response.

6. ADVANCED TOPICS & APPLICATIONS

A.C. Servomotor

• Assumptions (Two-phase, constant voltage):

  1. Two stator windings 90° apart, one fed by reference voltage V_ref, other by control voltage V_c.

  2. Rotor inertia & damping negligible.

  3. Magnetic linearity (torque T pto I_r).

  4. No saturation, no induced EMF in control winding.

• Transfer Function Derivation:

  T(s) = K_t I_r(s), J d²θ/dt² = T(s), V_c(s) = (R_c + sL_c) I_r(s) + K_b s θ(s) (back EMF).

  Combining:

$$\frac{\Theta(s)}{V_c(s)} = \frac{K_t}{(R_c + sL_c)(Js^2) + K_t K_b s} \approx \frac{K}{s(T_1 s + 1)}$$

after approximations (L_c small, neglect Js² term in denominator).

Stepper Motor (Short Note)

• Working: Converts digital pulses into angular displacement. Rotor moves in discrete steps (θ_step = 360°/(steps/rev)).

• Applications: Printers, plotters, CNC machines, robotics where precise positioning needed.

• Advantages: Open-loop control, no feedback sensor needed, high holding torque.

• Disadvantages: Resonance at high speeds, loses steps under overload.

State Space Analysis

• State Space Representation:

  ẋ(t) = A x(t) + B u(t) (State Equation)

  y(t) = C x(t) + D u(t) (Output Equation)

  Where x = state vector, u = input, y = output.

• State Transition Matrix (Φ(t)):

  Φ(t) = e^{At}. Solution: x(t) = Φ(t)x(0) + ∫₀ᵗ Φ(t-τ)Bu(τ)dτ.

• Properties of Φ(t):

  1. Φ(0) = I (Identity).

  2. Φ(-t) = Φ⁻¹(t).

  3. Φ(t₁ + t₂) = Φ(t₁)Φ(t₂).

  4. dΦ(t)/dt = A Φ(t) = Φ(t) A.

• Controllability (Kalman's Rank Condition):

  System is controllable if rank[ B AB A²B ... A^{n-1}B ] = n (n = order).

• Eigenvalues & Eigenvectors:

  • Eigenvalues of A = closed-loop poles (if u=0).

  • Eigenvectors define mode shapes.

  • Diagonalization: A = VΛV⁻¹, where Λ = diag(λ_i), V = eigenvector matrix.

  • Solution: x(t) = V e^{Λt} V⁻¹ x(0) + ... (modal decomposition).

7. DESIGN PROBLEMS & INTEGRATED APPLICATIONS

Designing Compensators for Given Specs

• Typical Specs: PM ≥ X°, K_v ≥ Y sec⁻¹, SSE ≤ Z%.

• Procedure:

  1. Determine system type from G(s).

  2. Check if K_v (or K_p, K_a) meets SSE spec. If not, need lag or PI to boost low-freq gain.

  3. Check PM from Bode of G(s). If PM insufficient, need lead or PD to add phase.

  4. If both needed, use lag-lead or cascade compensators.

  5. Design lead first (sets ω_gc), then design lag (placed at ω_gc/10).

Finding K and Parameters from Performance Specs

• Given M_p and ω_r (or T_p):

  1. From M_p, find ζ: ζ = √[ ln²(M_p) / (π² + ln²(M_p)) ].

  2. From ω_r = ω_n√(1-2ζ²), find ω_n.

  3. For standard 2nd order G(s) = ω_n²/(s²+2ζω_n s+ω_n²), compare with given G(s) to find K and other time constants.

• Example (PD Controller for Critical Damping):

  Given G(s) = K/(s(s+1)) with PD G_c(s) = K_p(1+T_d s). Closed-loop T(s) = K_p(K)(1+T_d s) / [s²(s+1) + K_p K (1+T_d s)].

  For critical damping (repeated real pole), set discriminant = 0 of characteristic polynomial s³ + s² + K_p K T_d s + K_p K = 0. Solve for T_d in terms of K_p K. Then T_s = 4/(σ) where σ is the repeated pole.

Marginal Stability Analysis

• Using Routh: Find K where first column has a zero (but no sign change). Use auxiliary equation to find ω of oscillation.

• Using Root Locus: Find K where locus crosses jω axis (imaginary axis). ω from angle condition.

• Using Nyquist: Find K where Nyquist plot passes through (-1, j0). K_marginal = 1/|G(jω)H(jω)| at that ω.

Commenting on Stability from Plots

• Root Locus: If all branches for K>0 lie in LHP → stable for all K. If branches cross into RHP → unstable for K beyond crossover.

• Bode: PM > 0 and GM > 1 → stable. PM < 0 or GM < 1 → unstable.

• Nyquist: N = P → stable. N ≠ P → unstable. Count encirclements of (-1, j0).

[!CAUTION] Common Pitfalls:

1. Mason's Gain Formula: Forgetting Δ_k excludes only the forward path in question, not all touching loops.

2. Routh Array: Arithmetic errors in computing elements. Zero row mishandling.

3. Root Locus: Misapplying angle condition (sum of angles to poles minus sum to zeros = odd multiple of 180°). Breakaway point calculation: dK/ds = 0 on real axis segments between poles/zeros.

4. Bode Plot: Incorrect corner frequencies, slope changes. Remember: 1+sT adds +20 dB/dec starting at ω=1/T.

5. Nyquist: Forgetting to map the entire contour (including large semicircle in RHP). Counting encirclements direction correctly (clockwise positive).

6. Error Coefficients: Using wrong coefficient for input type. Remember: Type 0 → finite K_p only; Type 1 → finite K_v only; Type 2 → finite K_a only.

7. Compensator Design: Placing lag zero/pole too close to ω_gc (degrades PM). Not adding safety margin (5-12°) to ΔPM.