Introduction and Overview
Compensators and Controllers - Overview
Key Topics for GATE
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PID controller and its variants (P, PI, PD, PID)
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Lead, lag, and lead-lag compensators
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Compensation techniques (frequency domain, root locus)
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Controller design and tuning methods
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State feedback and pole placement
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Digital control and implementation
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Industrial applications and practical considerations
Important for GATE
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Compensator transfer functions and their effects
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Design calculations for achieving desired specifications
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Stability analysis with compensators
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Time domain and frequency domain design approaches
PID Control Systems
PID Controller - Standard Form
Transfer Function
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\(K_p\): Proportional gain (reduces rise time, steady-state error)
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\(K_i\): Integral gain (eliminates steady-state error for step inputs)
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\(K_d\): Derivative gain (reduces overshoot, improves stability)
Alternative Forms
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Ideal form: \(G_c(s) = K_p\left(1 + \dfrac{1}{T_i s} + T_d s\right)\) where \(T_i = \dfrac{K_p}{K_i}\), \(T_d = \dfrac{K_d}{K_p}\)
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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
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P Controller: \(G_c(s) = K_p\) (simple, but has steady-state error)
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PI Controller: \(G_c(s) = K_p + \dfrac{K_i}{s}\) (most common in industry)
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PD Controller: \(G_c(s) = K_p + K_d s\) (improves stability, noise sensitive)
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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 \(P_{cr}\)
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Set \(K_i = K_d = 0\)
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Increase \(K_p\) until system oscillates
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Record \(K_{cr}\) and \(P_{cr}\)
Tuning Rules:
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P: \(K_p = 0.5K_{cr}\)
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PI: \(K_p = 0.45K_{cr}\), \(T_i = P_{cr}/1.2\)
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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):
Define: \(\tau/T\) ratio
PID Parameters:
Compensator Design
Lead Compensator
Transfer Function
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Zero at \(s = -\dfrac{1}{\alpha T}\)
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Pole at \(s = -\dfrac{1}{T}\)
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Provides phase lead for \(\alpha > 1\)
Design Parameters
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Maximum phase lead: \(\phi_m = \sin^{-1}\left(\dfrac{\alpha-1}{\alpha+1}\right)\)
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Frequency of maximum phase lead: \(\omega_m = \dfrac{1}{T\sqrt{\alpha}}\)
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Gain at \(\omega_m\): \(|G_c(j\omega_m)| = K_c\sqrt{\alpha}\)
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Phase lead at any frequency: \(\phi(\omega) = \tan^{-1}(\alpha T\omega) - \tan^{-1}(T\omega)\)
Design Steps
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Determine required phase lead: \(\phi_m = PM_{desired} - PM_{current} + 5^{\circ}\) to \(12^{\circ}\)
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Calculate \(\alpha = \dfrac{1+\sin\phi_m}{1-\sin\phi_m}\)
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Find \(\omega_m\) where gain is \(-10\log_{10}\sqrt{\alpha}\) dB
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Calculate \(T = \dfrac{1}{\omega_m\sqrt{\alpha}}\)
Lag Compensator
Transfer Function
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Zero at \(s = -\dfrac{1}{\beta T}\)
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Pole at \(s = -\dfrac{1}{T}\)
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Provides attenuation at high frequencies
Design Parameters
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Maximum phase lag: \(\phi_m = \sin^{-1}\left(\dfrac{\beta-1}{\beta+1}\right)\) (negative)
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Frequency of maximum phase lag: \(\omega_m = \dfrac{1}{T\sqrt{\beta}}\)
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Gain at \(\omega_m\): \(|G_c(j\omega_m)| = K_c\sqrt{\beta}\)
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Attenuation factor: \(\dfrac{1}{\beta}\) (improves steady-state error)
Design Steps
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Determine required steady-state error improvement
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Choose \(\beta = \dfrac{1}{K_{required}}\) where \(K_{required}\) is gain boost needed
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Place lag corner frequency at \(\omega = \dfrac{1}{T}\) (decade below \(\omega_{gc}\))
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Ensure minimal phase reduction at \(\omega_{gc}\)
Lead-Lag Compensator
Transfer Function
Design Approach
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Lead section: Improves transient response and stability
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Lag section: Improves steady-state accuracy
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Frequency separation: Ensure \(\dfrac{1}{T_1} >> \dfrac{1}{T_2}\) (typically 10:1 ratio)
Advantages
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Combines benefits of both lead and lag compensation
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Better overall system performance
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Flexibility in design parameters
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Reduced interaction between lead and lag sections
Design Techniques
Frequency Domain Design
Bode Plot Method
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Gain Margin (GM): \(GM = \dfrac{1}{|G(j\omega_{pc})|}\) where \(\angle G(j\omega_{pc}) = -180^{\circ}\)
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Phase Margin (PM): \(PM = 180^{\circ} + \angle G(j\omega_{gc})\) where \(|G(j\omega_{gc})| = 1\)
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Typical specifications: GM \(\geq\) 6 dB, PM \(\geq 45^{\circ}\)
Design Steps
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Draw Bode plot of uncompensated system
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Determine current GM and PM
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Choose compensator type based on requirements:
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Lead: If PM needs improvement (transient response)
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Lag: If steady-state error needs reduction
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Lead-lag: If both transient and steady-state improvement needed
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Calculate compensator parameters
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Verify final specifications
Root Locus Design
Root Locus Method
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Objective: Place closed-loop poles at desired locations
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Angle criterion: \(\sum \angle \text{zeros} - \sum \angle \text{poles} = \pm 180^{\circ}(2k+1)\)
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Magnitude criterion: \(|G_c(s)G(s)| = 1\) at desired pole location
Compensator Effects on Root Locus
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Lead compensator: Pulls loci to the left (improves stability)
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Lag compensator: Minimal effect on transient response
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PD compensator: Adds zero, pulls loci left
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PI compensator: Adds pole at origin, may destabilize
Design Procedure
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Determine desired pole locations (based on \(\zeta\), \(\omega_n\))
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Calculate angle deficiency at desired pole
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Place compensator zero/pole to satisfy angle criterion
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Calculate gain from magnitude criterion
State Feedback Control
State Feedback and Pole Placement
State Space Representation
Closed-loop system: \(\dot{x} = (A - BK)x + Br\)
Controllability
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Controllability matrix: \(P_c = [B \quad AB \quad A^2B \quad \ldots \quad A^{n-1}B]\)
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System is completely controllable if \(\text{rank}(P_c) = n\)
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Necessary condition for arbitrary pole placement
Observability
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Observability matrix: \(P_o = [C^T \quad A^TC^T \quad (A^2)^TC^T \quad \ldots \quad (A^{n-1})^TC^T]^T\)
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System is completely observable if \(\text{rank}(P_o) = n\)
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Required for state estimation
Pole Placement Design
Ackermann’s Formula
For desired characteristic polynomial \(\alpha_c(s) = s^n + a_1s^{n-1} + \ldots + a_n\):
Bass-Gura Formula
If system is in controllable canonical form:
Design Steps
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Check controllability of \((A,B)\)
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Specify desired pole locations
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Form desired characteristic polynomial
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Calculate feedback gain matrix \(K\) using Ackermann’s formula
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Verify closed-loop pole locations
Digital Control Implementation
Digital PID Controller
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
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Backward difference: \(s = \dfrac{z-1}{T_s z}\)
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Tustin (Bilinear): \(s = \dfrac{2}{T_s}\dfrac{z-1}{z+1}\)
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Forward difference: \(s = \dfrac{z-1}{T_s}\)
Digital Implementation Issues
Sampling Considerations
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Sampling frequency: \(f_s \geq 10 \times \text{bandwidth}\) (rule of thumb)
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Nyquist criterion: \(f_s > 2f_{max}\) to avoid aliasing
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Shannon’s theorem: Perfect reconstruction requires \(f_s \geq 2f_{max}\)
Practical Implementation Issues
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Integrator windup: Limit integral term accumulation
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Derivative kick: Use derivative of output instead of error
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Quantization effects: Finite word length in digital systems
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Computational delay: Effect of processing time
Anti-windup Techniques
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Conditional integration (stop integration when saturated)
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Back-calculation (reduce integral term when saturated)
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Tracking mode (reset integral when switching modes)
Advanced Control Concepts
Advanced Control Strategies
Cascade Control
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Structure: Inner loop (secondary) + Outer loop (primary)
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Advantages: Better disturbance rejection, improved stability
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Design rule: Inner loop bandwidth >> Outer loop bandwidth
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Applications: Temperature control, flow control systems
Feedforward Control
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Principle: Anticipate disturbances before they affect output
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Transfer function: \(G_{ff}(s) = -\dfrac{G_d(s)}{G_p(s)}\)
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Limitation: Requires measurable disturbances
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Combination: Usually combined with feedback control
Ratio Control
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Maintains constant ratio between two flow rates
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Wild flow (uncontrolled) and controlled flow
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Common in chemical process industries
Adaptive and Robust Control
Adaptive Control
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Self-tuning regulators (STR): Online parameter estimation
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Model reference adaptive control (MRAC): Follow reference model
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Gain scheduling: Pre-computed gains for different operating points
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Applications: Aircraft control, process control with varying parameters
Robust Control
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\(\mathrm{H}_\infty\) control: Minimize worst-case performance
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\(\mu\)-synthesis: Structured uncertainty handling
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Sliding mode control: Insensitive to parameter variations
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Loop shaping: Frequency domain robust design
Model Predictive Control (MPC)
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Optimize control actions over prediction horizon
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Handle constraints explicitly
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Widely used in process industries
GATE Problem Solving
GATE Practice Problems - Lead Compensator
Problem 1
A lead compensator has transfer function:
Solution
Given: \(T = 0.01\), \(\alpha T = 0.1\), therefore \(\alpha = 10\)
(a) Maximum phase lead:
(b) Frequency of maximum phase lead:
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 \(\alpha\):
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
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:
Taking inverse Laplace transform:
Where:
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\(2u(t)\): Proportional term
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\(tu(t)\): Integral term
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\(0.5\delta(t)\): 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
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PID: Most versatile, \(G_c(s) = K_p + \dfrac{K_i}{s} + K_d s\)
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Lead: Improves transient response, \(\phi_m = \sin^{-1}\dfrac{\alpha-1}{\alpha+1}\)
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Lag: Improves steady-state accuracy, provides attenuation
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Lead-lag: Combines advantages of both
Design Methods
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Frequency domain: Based on Bode plots, GM/PM specifications
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Root locus: Pole placement based on desired transient response
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State feedback: Arbitrary pole placement using Ackermann’s formula
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Digital implementation: Z-transform, sampling considerations
Important Formulas for GATE
Stability Margins
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Phase Margin: \(PM = 180^{\circ} + \angle G(j\omega_{gc})\) where \(|G(j\omega_{gc})| = 1\)
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Gain Margin: \(GM = \dfrac{1}{|G(j\omega_{pc})|}\) where \(\angle G(j\omega_{pc}) = -180^{\circ}\)
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Typical values: GM \(\geq\) 6 dB, PM \(\geq~45^{\circ}\)
Lead Compensator Design
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Maximum phase lead: \(\phi_m = \sin^{-1}\left(\dfrac{\alpha-1}{\alpha+1}\right)\)
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Frequency of max phase: \(\omega_m = \dfrac{1}{T\sqrt{\alpha}}\)
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Alpha calculation: \(\alpha = \dfrac{1+\sin\phi_m}{1-\sin\phi_m}\)
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Gain at \(\omega_m\): \(|G_c(j\omega_m)| = K_c\sqrt{\alpha}\)
State Feedback
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Controllability: \(\text{rank}([B \quad AB \quad A^2B \quad \ldots \quad A^{n-1}B]) = n\)
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Ackermann’s formula: \(K = [0 \quad 0 \quad \ldots \quad 1] P_c^{-1} \alpha_c(A)\)
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Closed-loop poles: Eigenvalues of \((A-BK)\)
Common GATE Question Types
Frequently Asked Topics
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PID tuning: Ziegler-Nichols method, parameter effects
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Lead compensator: Maximum phase lead frequency calculation
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Lag compensator: Steady-state error improvement
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Phase/Gain margins: Stability analysis with compensators
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Root locus: Effect of compensators on pole locations
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State feedback: Controllability check, pole placement
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Digital control: Discretization, sampling effects
Problem-Solving Tips
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Always check units and dimensions
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Verify stability margins after compensation
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Use approximations for quick calculations (e.g., \(\sin^{-1}(0.6) \approx 37^{\circ}\))
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Remember standard values: \(\sqrt{2} \approx 1.414\), \(\sqrt{3} \approx 1.732\), \(\sqrt{10} \approx 3.16\)
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Practice frequency domain and time domain correlations
Quick Reference - Controller Selection
When to Use Which Controller
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P Controller: Simple systems, some steady-state error acceptable
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PI Controller: Most industrial applications, eliminates steady-state error
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PID Controller: Complex systems requiring precise control
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Lead Compensator: Improve transient response, increase bandwidth
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Lag Compensator: Improve steady-state accuracy, reduce bandwidth
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Lead-Lag: When both transient and steady-state improvement needed
Design Trade-offs
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Stability vs. Performance
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Bandwidth vs. Noise rejection
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Settling time vs. Overshoot
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Steady-state accuracy vs. System complexity
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Robustness vs. Optimality
Numerical Values and Approximations
Common Mathematical Values
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\(\pi \approx 3.14159\), \(e \approx 2.718\)
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\(\sqrt{2} \approx 1.414\), \(\sqrt{3} \approx 1.732\), \(\sqrt{5} \approx 2.236\), \(\sqrt{10} \approx 3.162\)
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\(\ln(2) \approx 0.693\), \(\ln(10) \approx 2.303\)
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\(\sin(30^{\circ}) = 0.5\), \(\sin(45^{\circ}) = 0.707\), \(\sin(60^{\circ}) = 0.866\)
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\(\cos(30^{\circ}) = 0.866\), \(\cos(45^{\circ}) = 0.707\), \(\cos(60^{\circ}) = 0.5\)
Useful Approximations for GATE
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\(\sin^{-1}(0.6) \approx 37^{\circ}\), \(\sin^{-1}(0.707) = 45^{\circ}\), \(\sin^{-1}(0.8) \approx 53^{\circ}\)
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\(\tan^{-1}(1) = 45^{\circ}\), \(\tan^{-1}(0.5) \approx 26.6^{\circ}\), \(\tan^{-1}(2) \approx 63.4^{\circ}\)
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For small angles: \(\sin\theta \approx \theta\) (in radians), \(\cos\theta \approx 1\)
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\(20\log_{10}(2) \approx 6\) dB, \(20\log_{10}(10) = 20\) dB
Final Tips for GATE Success
Exam Strategy
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Time management: Allocate 2-3 minutes per mark
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Practice: Solve previous years’ questions repeatedly
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Conceptual clarity: Understand physical significance
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Formula sheet: Memorize key formulas and their applications
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Numerical skills: Practice mental calculations
Common Mistakes to Avoid
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Confusing lead and lag compensator effects
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Incorrect angle calculations in root locus
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Sign errors in state feedback design
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Units mismatch in frequency domain calculations
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Forgetting to check controllability/observability
Next Topics to Study
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State Space Analysis (Canonical forms, Solution of state equations)
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Nonlinear Control Systems (Describing functions, Phase plane analysis)
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Optimal Control (LQR, LQG, Kalman filter)
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Digital Signal Processing (Z-transform, Digital filters)