Control Systems: Compensators and Controllers – Detailed GATE EE Notes

Introduction and Overview

Compensators and Controllers - Overview

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

Important for GATE

  • Compensator transfer functions and their effects

  • Design calculations for achieving desired specifications

  • Stability analysis with compensators

  • Time domain and frequency domain design approaches

PID Control Systems

PID Controller - Standard Form

Transfer Function

\[G_c(s) = K_p + \dfrac{K_i}{s} + K_d s = \dfrac{K_d s^2 + K_p s + K_i}{s}\]
  • \(K_p\): 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)

PID Controller Effects on System Response

Parameter Effects Table

Parameter Rise Time Overshoot Settling Time SS Error
\(K_p\) Decrease Increase Small change Decrease
\(K_i\) Decrease Increase Increase Eliminate
\(K_d\) Minor change Decrease Decrease No change

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

PID Tuning Methods

Ziegler-Nichols Method

Step 1: Find critical gain \(K_{cr}\) and period \(P_{cr}\)

  • Set \(K_i = K_d = 0\)

  • Increase \(K_p\) until system oscillates

  • Record \(K_{cr}\) and \(P_{cr}\)

Tuning Rules:

  • P: \(K_p = 0.5K_{cr}\)

  • PI: \(K_p = 0.45K_{cr}\), \(T_i = P_{cr}/1.2\)

  • PID: \(K_p = 0.6K_{cr}\), \(T_i = P_{cr}/2\), \(T_d = P_{cr}/8\)

Cohen-Coon Method

For first-order plus dead time (FOPDT):

\[G(s) = \dfrac{K e^{-\tau s}}{1 + T s}\]

Define: \(\tau/T\) ratio

PID Parameters:

\[\begin{aligned} K_p &= \dfrac{T}{K\tau}\left(1.35 + \dfrac{0.25\tau}{T}\right) \\ T_i &= \tau\left(2.5 - \dfrac{2\tau}{T}\right)/\left(1 - \dfrac{0.39\tau}{T}\right) \\ T_d &= \tau\left(0.37 - \dfrac{0.37\tau}{T}\right)/\left(1 - \dfrac{0.81\tau}{T}\right) \end{aligned}\]

Compensator Design

Lead Compensator

Transfer Function

\[G_c(s) = K_c \dfrac{1 + \alpha T s}{1 + T s} \quad \text{where } \alpha > 1\]
  • 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: \(\phi(\omega) = \tan^{-1}(\alpha T\omega) - \tan^{-1}(T\omega)\)

Design Steps

  1. Determine required phase lead: \(\phi_m = PM_{desired} - PM_{current} + 5^{\circ}\) to \(12^{\circ}\)

  2. Calculate \(\alpha = \dfrac{1+\sin\phi_m}{1-\sin\phi_m}\)

  3. Find \(\omega_m\) where gain is \(-10\log_{10}\sqrt{\alpha}\) dB

  4. Calculate \(T = \dfrac{1}{\omega_m\sqrt{\alpha}}\)

Lag Compensator

Transfer Function

\[G_c(s) = K_c \dfrac{1 + \beta T s}{1 + T s} \quad \text{where } \beta < 1\]
  • 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)

Design Steps

  1. Determine required steady-state error improvement

  2. Choose \(\beta = \dfrac{1}{K_{required}}\) where \(K_{required}\) is gain boost needed

  3. Place lag corner frequency at \(\omega = \dfrac{1}{T}\) (decade below \(\omega_{gc}\))

  4. Ensure minimal phase reduction at \(\omega_{gc}\)

Lead-Lag Compensator

Transfer Function

\[G_c(s) = K_c \dfrac{(1 + \alpha T_1 s)(1 + \beta T_2 s)}{(1 + T_1 s)(1 + T_2 s)}\]
where \(\alpha > 1\) (lead), \(\beta < 1\) (lag), \(T_1 < T_2\)

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

Design Techniques

Frequency Domain Design

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 Design

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

Design Procedure

  1. Determine desired pole locations (based on \(\zeta\), \(\omega_n\))

  2. Calculate angle deficiency at desired pole

  3. Place compensator zero/pole to satisfy angle criterion

  4. Calculate gain from magnitude criterion

State Feedback Control

State Feedback and Pole Placement

State Space Representation

\[\begin{aligned} \dot{x} &= Ax + Bu \\ y &= Cx + Du \end{aligned}\]
State feedback control law: \(u = -Kx + r\)

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

Pole Placement Design

Ackermann’s Formula

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

\[K = [0 \quad 0 \quad \ldots \quad 0 \quad 1] P_c^{-1} \alpha_c(A)\]
where \(\alpha_c(A) = A^n + a_1A^{n-1} + \ldots + a_nI\)

Bass-Gura Formula

If system is in controllable canonical form:

\[K = [a_n - \alpha_n \quad a_{n-1} - \alpha_{n-1} \quad \ldots \quad a_1 - \alpha_1]\]
where \(\{a_i\}\) are original system coefficients, \(\{\alpha_i\}\) are desired coefficients

Design Steps

  1. Check controllability of \((A,B)\)

  2. Specify desired pole locations

  3. Form desired characteristic polynomial

  4. Calculate feedback gain matrix \(K\) using Ackermann’s formula

  5. Verify closed-loop pole locations

Digital Control Implementation

Digital PID Controller

Discrete PID Algorithm

\[u(k) = K_p e(k) + K_i T_s \sum_{i=0}^{k} e(i) + K_d \dfrac{e(k)-e(k-1)}{T_s}\]
where \(T_s\) is sampling time, \(k\) is sample number

Z-Transform Representation

\[G_c(z) = K_p + \dfrac{K_i T_s}{1-z^{-1}} + K_d \dfrac{1-z^{-1}}{T_s}\]

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

Digital Implementation Issues

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

Anti-windup Techniques

  • Conditional integration (stop integration when saturated)

  • Back-calculation (reduce integral term when saturated)

  • Tracking mode (reset integral when switching modes)

Advanced Control Concepts

Advanced Control Strategies

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 and Robust Control

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

GATE Problem Solving

GATE Practice Problems - Lead Compensator

Problem 1

A lead compensator has transfer function:

\[G_c(s) = \dfrac{1 + 0.1s}{1 + 0.01s}\]
Find: (a) Maximum phase lead, (b) Frequency of maximum phase lead

Solution

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

(a) Maximum phase lead:

\[\phi_m = \sin^{-1}\left(\dfrac{\alpha-1}{\alpha+1}\right) = \sin^{-1}\left(\dfrac{10-1}{10+1}\right) = \sin^{-1}\left(\dfrac{9}{11}\right) = 54.74^{\circ}\]

(b) Frequency of maximum phase lead:

\[\omega_m = \dfrac{1}{T\sqrt{\alpha}} = \dfrac{1}{0.01\sqrt{10}} = \dfrac{1}{0.01 \times 3.16} = 31.6 \text{ rad/s}\]

GATE Practice Problems - Phase Margin

Problem 2

A system with \(G(s) = \dfrac{10}{s(s+2)}\) has phase margin of \(17^{\circ}\) at gain crossover frequency. Design a lead compensator to achieve phase margin of \(50^{\circ}\).

Solution

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

Calculate \(\alpha\):

\[\alpha = \dfrac{1+\sin 38^{\circ}}{1-\sin 38^{\circ}} = \dfrac{1+0.616}{1-0.616} = \dfrac{1.616}{0.384} = 4.21\]

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

\[\left|\dfrac{10}{j\omega_{gc}(j\omega_{gc}+2)}\right| = 1 \Rightarrow \omega_{gc} = 2.24 \text{ rad/s}\]

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

\[\omega_m = \omega_{gc}\sqrt{\alpha} = 2.24\sqrt{4.21} = 4.59 \text{ rad/s}\]

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

GATE Practice Problems - PID Response

Problem 3

A PID controller \(G_c(s) = 2 + \dfrac{1}{s} + 0.5s\) is subjected to unit step input. Find the controller output in time domain.

Solution

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

Controller output:

\[U(s) = G_c(s)E(s) = \left(2 + \dfrac{1}{s} + 0.5s\right) \cdot \dfrac{1}{s} = \dfrac{2}{s} + \dfrac{1}{s^2} + 0.5\]

Taking inverse Laplace transform:

\[u(t) = 2u(t) + tu(t) + 0.5\delta(t)\]

Where:

  • \(2u(t)\): Proportional term

  • \(tu(t)\): Integral term

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

GATE Practice Problems - State Feedback

Problem 4

For the system \(\dot{x} = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}x + \begin{bmatrix} 0 \\ 1 \end{bmatrix}u\), design state feedback to place poles at \(s = -1 \pm j\).

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

Summary and Key Formulas

Summary - Key Points

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

Important Formulas for GATE

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 \((A-BK)\)

Common GATE Question Types

Frequently Asked Topics

  • PID tuning: Ziegler-Nichols method, parameter effects

  • Lead compensator: Maximum phase lead frequency calculation

  • Lag compensator: Steady-state error improvement

  • Phase/Gain margins: Stability analysis with compensators

  • Root locus: Effect of compensators on pole locations

  • State feedback: Controllability check, pole placement

  • Digital control: Discretization, sampling effects

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

Quick Reference - Controller Selection

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

Numerical Values and Approximations

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

Final Tips for GATE Success

Exam Strategy

  • Time management: Allocate 2-3 minutes per mark

  • Practice: Solve previous years’ questions repeatedly

  • Conceptual clarity: Understand physical significance

  • Formula sheet: Memorize key formulas and their applications

  • Numerical skills: Practice mental calculations

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)