Key Topics for GATE

• PID controller and its variants (P, PI, PD, PID)

• Lead, lag, and lead-lag compensators

• Compensation techniques (frequency domain, root locus)

• Controller design and tuning methods

• State feedback and pole placement

• Digital control and implementation

• Industrial applications and practical considerations

Transfer Function

• \(K_p\): Proportional gain (reduces rise time, steady-state error)

• \(K_i\): Integral gain (eliminates steady-state error for step inputs)

• \(K_d\): Derivative gain (reduces overshoot, improves stability)

Alternative Forms

• Ideal form: \(G_c(s) = K_p\left(1 + \dfrac{1}{T_i s} + T_d s\right)\) where \(T_i = \dfrac{K_p}{K_i}\), \(T_d = \dfrac{K_d}{K_p}\)

• Practical form: \(G_c(s) = K_p\left(1 + \dfrac{1}{T_i s} + \dfrac{T_d s}{1 + \dfrac{T_d s}{N}}\right)\) (with derivative filter)

Parameter Effects Table

PID Controller Variants

• P Controller: \(G_c(s) = K_p\) (simple, but has steady-state error)

• PI Controller: \(G_c(s) = K_p + \dfrac{K_i}{s}\) (most common in industry)

• PD Controller: \(G_c(s) = K_p + K_d s\) (improves stability, noise sensitive)

• I-PD Controller: \(G_c(s) = \dfrac{K_i}{s}\) in forward path, PD in feedback

Transfer Function

• Zero at \(s = -\dfrac{1}{\alpha T}\)

• Pole at \(s = -\dfrac{1}{T}\)

• Provides phase lead for \(\alpha > 1\)

Design Parameters

• Maximum phase lead: \(\phi_m = \sin^{-1}\left(\dfrac{\alpha-1}{\alpha+1}\right)\)

• Frequency of maximum phase lead: \(\omega_m = \dfrac{1}{T\sqrt{\alpha}}\)

• Gain at \(\omega_m\): \(|G_c(j\omega_m)| = K_c\sqrt{\alpha}\)

• Phase lead at any frequency: \(\phi(\omega) = \tan^{-1}(\alpha T\omega) - \tan^{-1}(T\omega)\)

Transfer Function

• Zero at \(s = -\dfrac{1}{\beta T}\)

• Pole at \(s = -\dfrac{1}{T}\)

• Provides attenuation at high frequencies

Design Parameters

• Maximum phase lag: \(\phi_m = \sin^{-1}\left(\dfrac{\beta-1}{\beta+1}\right)\) (negative)

• Frequency of maximum phase lag: \(\omega_m = \dfrac{1}{T\sqrt{\beta}}\)

• Gain at \(\omega_m\): \(|G_c(j\omega_m)| = K_c\sqrt{\beta}\)

• Attenuation factor: \(\dfrac{1}{\beta}\) (improves steady-state error)

Transfer Function

Design Approach

• Lead section: Improves transient response and stability

• Lag section: Improves steady-state accuracy

• Frequency separation: Ensure \(\dfrac{1}{T_1} >> \dfrac{1}{T_2}\) (typically 10:1 ratio)

Advantages

• Combines benefits of both lead and lag compensation

• Better overall system performance

• Flexibility in design parameters

• Reduced interaction between lead and lag sections

Bode Plot Method

• Gain Margin (GM): \(GM = \dfrac{1}{|G(j\omega_{pc})|}\) where \(\angle G(j\omega_{pc}) = -180^{\circ}\)

• Phase Margin (PM): \(PM = 180^{\circ} + \angle G(j\omega_{gc})\) where \(|G(j\omega_{gc})| = 1\)

• Typical specifications: GM \(\geq\) 6 dB, PM \(\geq 45^{\circ}\)

Design Steps

1. Draw Bode plot of uncompensated system

2. Determine current GM and PM

3. Choose compensator type based on requirements:

   • Lead: If PM needs improvement (transient response)

   • Lag: If steady-state error needs reduction

   • Lead-lag: If both transient and steady-state improvement needed

4. Calculate compensator parameters

5. Verify final specifications

Root Locus Method

• Objective: Place closed-loop poles at desired locations

• Angle criterion: \(\sum \angle \text{zeros} - \sum \angle \text{poles} = \pm 180^{\circ}(2k+1)\)

• Magnitude criterion: \(|G_c(s)G(s)| = 1\) at desired pole location

Compensator Effects on Root Locus

• Lead compensator: Pulls loci to the left (improves stability)

• Lag compensator: Minimal effect on transient response

• PD compensator: Adds zero, pulls loci left

• PI compensator: Adds pole at origin, may destabilize

State Space Representation

Closed-loop system: \(\dot{x} = (A - BK)x + Br\)

Controllability

• Controllability matrix: \(P_c = [B \quad AB \quad A^2B \quad \ldots \quad A^{n-1}B]\)

• System is completely controllable if \(\text{rank}(P_c) = n\)

• Necessary condition for arbitrary pole placement

Observability

• Observability matrix: \(P_o = [C^T \quad A^TC^T \quad (A^2)^TC^T \quad \ldots \quad (A^{n-1})^TC^T]^T\)

• System is completely observable if \(\text{rank}(P_o) = n\)

• Required for state estimation

Ackermann’s Formula

For desired characteristic polynomial \(\alpha_c(s) = s^n + a_1s^{n-1} + \ldots + a_n\):

Bass-Gura Formula

If system is in controllable canonical form:

Discrete PID Algorithm

Z-Transform Representation

Position form: \(G_c(z) = K_p + \dfrac{K_i T_s z}{z-1} + K_d \dfrac{z-1}{T_s z}\)

Discretization Methods

• Backward difference: \(s = \dfrac{z-1}{T_s z}\)

• Tustin (Bilinear): \(s = \dfrac{2}{T_s}\dfrac{z-1}{z+1}\)

• Forward difference: \(s = \dfrac{z-1}{T_s}\)

Sampling Considerations

• Sampling frequency: \(f_s \geq 10 \times \text{bandwidth}\) (rule of thumb)

• Nyquist criterion: \(f_s > 2f_{max}\) to avoid aliasing

• Shannon’s theorem: Perfect reconstruction requires \(f_s \geq 2f_{max}\)

Practical Implementation Issues

• Integrator windup: Limit integral term accumulation

• Derivative kick: Use derivative of output instead of error

• Quantization effects: Finite word length in digital systems

• Computational delay: Effect of processing time

Cascade Control

• Structure: Inner loop (secondary) + Outer loop (primary)

• Advantages: Better disturbance rejection, improved stability

• Design rule: Inner loop bandwidth >> Outer loop bandwidth

• Applications: Temperature control, flow control systems

Feedforward Control

• Principle: Anticipate disturbances before they affect output

• Transfer function: \(G_{ff}(s) = -\dfrac{G_d(s)}{G_p(s)}\)

• Limitation: Requires measurable disturbances

• Combination: Usually combined with feedback control

Ratio Control

• Maintains constant ratio between two flow rates

• Wild flow (uncontrolled) and controlled flow

• Common in chemical process industries

Adaptive Control

• Self-tuning regulators (STR): Online parameter estimation

• Model reference adaptive control (MRAC): Follow reference model

• Gain scheduling: Pre-computed gains for different operating points

• Applications: Aircraft control, process control with varying parameters

Robust Control

• \(\mathrm{H}_\infty\) control: Minimize worst-case performance

• \(\mu\)-synthesis: Structured uncertainty handling

• Sliding mode control: Insensitive to parameter variations

• Loop shaping: Frequency domain robust design

Model Predictive Control (MPC)

• Optimize control actions over prediction horizon

• Handle constraints explicitly

• Widely used in process industries

Solution

Given: \(T = 0.01\), \(\alpha T = 0.1\), therefore \(\alpha = 10\)

(a) Maximum phase lead:

(b) Frequency of maximum phase lead:

Solution

Required phase lead = \(50^{\circ} - 17^{\circ} + 5^{\circ} = 38^{\circ}\) (adding \(5^{\circ}\) safety margin)

Calculate \(\alpha\):

Find current gain crossover frequency from \(|G(j\omega_{gc})| = 1\):

New gain crossover should be at \(\omega_m\):

Therefore: \(T = \dfrac{1}{\omega_m\sqrt{\alpha}} = \dfrac{1}{4.59 \times 2.05} = 0.106\) s

Solution

For unit step input \(e(t) = u(t)\), the Laplace transform is \(E(s) = \dfrac{1}{s}\)

Controller output:

Taking inverse Laplace transform:

Where:

• \(2u(t)\): Proportional term

• \(tu(t)\): Integral term

• \(0.5\delta(t)\): Derivative term (impulse)

Solution

Given: \(A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}\), \(B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\)

Check controllability: \(P_c = [B \quad AB] = \begin{bmatrix} 0 & 1 \\ 1 & -3 \end{bmatrix}\)

\(\det(P_c) = -1 \neq 0\), so system is controllable.

Desired characteristic polynomial: \((s+1-j)(s+1+j) = s^2 + 2s + 2\)

Current characteristic polynomial: \(\det(sI-A) = s^2 + 3s + 2\)

Using Bass-Gura formula: \(K = [a_2 - \alpha_2 \quad a_1 - \alpha_1] = [2-2 \quad 3-2] = [0 \quad 1]\)

Therefore: \(u = -[0 \quad 1]x = -x_2\)

Controller Types

• PID: Most versatile, \(G_c(s) = K_p + \dfrac{K_i}{s} + K_d s\)

• Lead: Improves transient response, \(\phi_m = \sin^{-1}\dfrac{\alpha-1}{\alpha+1}\)

• Lag: Improves steady-state accuracy, provides attenuation

• Lead-lag: Combines advantages of both

Design Methods

• Frequency domain: Based on Bode plots, GM/PM specifications

• Root locus: Pole placement based on desired transient response

• State feedback: Arbitrary pole placement using Ackermann’s formula

• Digital implementation: Z-transform, sampling considerations

Stability Margins

• Phase Margin: \(PM = 180^{\circ} + \angle G(j\omega_{gc})\) where \(|G(j\omega_{gc})| = 1\)

• Gain Margin: \(GM = \dfrac{1}{|G(j\omega_{pc})|}\) where \(\angle G(j\omega_{pc}) = -180^{\circ}\)

• Typical values: GM \(\geq\) 6 dB, PM \(\geq~45^{\circ}\)

Lead Compensator Design

• Maximum phase lead: \(\phi_m = \sin^{-1}\left(\dfrac{\alpha-1}{\alpha+1}\right)\)

• Frequency of max phase: \(\omega_m = \dfrac{1}{T\sqrt{\alpha}}\)

• Alpha calculation: \(\alpha = \dfrac{1+\sin\phi_m}{1-\sin\phi_m}\)

• Gain at \(\omega_m\): \(|G_c(j\omega_m)| = K_c\sqrt{\alpha}\)

State Feedback

• Controllability: \(\text{rank}([B \quad AB \quad A^2B \quad \ldots \quad A^{n-1}B]) = n\)

• Ackermann’s formula: \(K = [0 \quad 0 \quad \ldots \quad 1] P_c^{-1} \alpha_c(A)\)

• Closed-loop poles: Eigenvalues of \((A-BK)\)

Problem-Solving Tips

• Always check units and dimensions

• Verify stability margins after compensation

• Use approximations for quick calculations (e.g., \(\sin^{-1}(0.6) \approx 37^{\circ}\))

• Remember standard values: \(\sqrt{2} \approx 1.414\), \(\sqrt{3} \approx 1.732\), \(\sqrt{10} \approx 3.16\)

• Practice frequency domain and time domain correlations

When to Use Which Controller

• P Controller: Simple systems, some steady-state error acceptable

• PI Controller: Most industrial applications, eliminates steady-state error

• PID Controller: Complex systems requiring precise control

• Lead Compensator: Improve transient response, increase bandwidth

• Lag Compensator: Improve steady-state accuracy, reduce bandwidth

• Lead-Lag: When both transient and steady-state improvement needed

Design Trade-offs

• Stability vs. Performance

• Bandwidth vs. Noise rejection

• Settling time vs. Overshoot

• Steady-state accuracy vs. System complexity

• Robustness vs. Optimality

Common Mathematical Values

• \(\pi \approx 3.14159\), \(e \approx 2.718\)

• \(\sqrt{2} \approx 1.414\), \(\sqrt{3} \approx 1.732\), \(\sqrt{5} \approx 2.236\), \(\sqrt{10} \approx 3.162\)

• \(\ln(2) \approx 0.693\), \(\ln(10) \approx 2.303\)

• \(\sin(30^{\circ}) = 0.5\), \(\sin(45^{\circ}) = 0.707\), \(\sin(60^{\circ}) = 0.866\)

• \(\cos(30^{\circ}) = 0.866\), \(\cos(45^{\circ}) = 0.707\), \(\cos(60^{\circ}) = 0.5\)

Useful Approximations for GATE

• \(\sin^{-1}(0.6) \approx 37^{\circ}\), \(\sin^{-1}(0.707) = 45^{\circ}\), \(\sin^{-1}(0.8) \approx 53^{\circ}\)

• \(\tan^{-1}(1) = 45^{\circ}\), \(\tan^{-1}(0.5) \approx 26.6^{\circ}\), \(\tan^{-1}(2) \approx 63.4^{\circ}\)

• For small angles: \(\sin\theta \approx \theta\) (in radians), \(\cos\theta \approx 1\)

• \(20\log_{10}(2) \approx 6\) dB, \(20\log_{10}(10) = 20\) dB

Common Mistakes to Avoid

• Confusing lead and lag compensator effects

• Incorrect angle calculations in root locus

• Sign errors in state feedback design

• Units mismatch in frequency domain calculations

• Forgetting to check controllability/observability

Next Topics to Study

• State Space Analysis (Canonical forms, Solution of state equations)

• Nonlinear Control Systems (Describing functions, Phase plane analysis)

• Optimal Control (LQR, LQG, Kalman filter)

• Digital Signal Processing (Z-transform, Digital filters)