GATE EE

Control Systems: Compensators and Controllers – Detailed GATE EE Notes

Lecture Notes

SEC 01

Introduction and Overview

1Compensators and Controllers - Overview
1Key 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

1Important 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

SEC 02

PID Control Systems

1PID Controller - Standard Form
1Transfer Function
\[G_c(s) = K_p + \dfrac{K_i}{s} + K_d s = \dfrac{K_d s^2 + K_p s + K_i}{s}\]
1Alternative Forms
1PID Controller Effects on System Response
1Parameter 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
1PID Controller Variants
1PID Tuning Methods
1Ziegler-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: \(K_p = 0.6K_{cr}\), \(T_i = P_{cr}/2\), \(T_d = P_{cr}/8\)

1Cohen-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:

\[\begin{aligned} K_p &= \dfrac{T}{K\tau}\left(1.35 + \dfrac{0.25\tau}{T}\right) \\ T_i &= \tau\left(2.5 - \dfrac{2\tau}{T}\right)/\left(1 - \dfrac{0.39\tau}{T}\right) \\ T_d &= \tau\left(0.37 - \dfrac{0.37\tau}{T}\right)/\left(1 - \dfrac{0.81\tau}{T}\right) \end{aligned}\]
SEC 03

Compensator Design

1Lead Compensator
1Transfer Function
\[G_c(s) = K_c \dfrac{1 + \alpha T s}{1 + T s} \quad \text{where } \alpha > 1\]
1Design Parameters
1Design 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 \(T = \dfrac{1}{\omega_m\sqrt{\alpha}}\)

1Lag Compensator
1Transfer Function
\[G_c(s) = K_c \dfrac{1 + \beta T s}{1 + T s} \quad \text{where } \beta < 1\]
1Design Parameters
1Design 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 \(\omega_{gc}\)

1Lead-Lag Compensator
1Transfer 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), \(T_1 < T_2\)
1Design Approach
1Advantages
SEC 04

Design Techniques

1Frequency Domain Design
1Bode 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 \(\geq 45^{\circ}\)

1Design 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

1Root Locus Design
1Root Locus Method
1Compensator Effects on Root Locus
1Design 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

SEC 05

State Feedback Control

1State Feedback and Pole Placement
1State Space Representation
\[\begin{aligned} \dot{x} &= Ax + Bu \\ y &= Cx + Du \end{aligned}\]
State feedback control law: \(u = -Kx + r\)

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

1Controllability
1Observability
1Pole Placement Design
1Ackermann’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 \(\alpha_c(A) = A^n + a_1A^{n-1} + \ldots + a_nI\)
1Bass-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
1Design 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

SEC 06

Digital Control Implementation

1Digital PID Controller
1Discrete 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
1Z-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: \(G_c(z) = K_p + \dfrac{K_i T_s z}{z-1} + K_d \dfrac{z-1}{T_s z}\)

1Discretization Methods
1Digital Implementation Issues
1Sampling Considerations
1Practical Implementation Issues
1Anti-windup Techniques
SEC 07

Advanced Control Concepts

1Advanced Control Strategies
1Cascade 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

1Feedforward Control
1Ratio Control
1Adaptive and Robust Control
1Adaptive Control
1Robust Control
1Model Predictive Control (MPC)
SEC 08

GATE Problem Solving

1GATE Practice Problems - Lead Compensator
1Problem 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
1Solution

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:

\[\omega_m = \dfrac{1}{T\sqrt{\alpha}} = \dfrac{1}{0.01\sqrt{10}} = \dfrac{1}{0.01 \times 3.16} = 31.6 \text{ rad/s}\]
1GATE Practice Problems - Phase Margin
1Problem 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 \(50^{\circ}\).

1Solution

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

1GATE Practice Problems - PID Response
1Problem 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.

1Solution

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:

1GATE Practice Problems - State Feedback
1Problem 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 \(s = -1 \pm j\).

1Solution

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

SEC 09

Summary and Key Formulas

1Summary - Key Points
1Controller 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

1Design Methods
1Important Formulas for GATE
1Stability Margins
1Lead Compensator Design
1State Feedback
1Common GATE Question Types
1Frequently Asked Topics
1Problem-Solving Tips
1Quick Reference - Controller Selection
1When to Use Which Controller
1Design Trade-offs
1Numerical Values and Approximations
1Common Mathematical Values
1Useful Approximations for GATE
1Final Tips for GATE Success
1Exam Strategy
1Common Mistakes to Avoid
1Next Topics to Study