Introduction
Analysis of LTI Systems
Key Topics
-
Time response analysis (transient and steady-state)
-
Standard test signals and system responses
-
Steady-state error analysis
-
Routh-Hurwitz stability criterion
-
Root locus technique
-
Impulse and frequency response
-
Pole-zero analysis and system behavior
Objective
Analyze the behavior of Linear Time-Invariant (LTI) systems for GATE EE preparation.
System Representation
Transfer Function and System Representation
Transfer Function
Poles and Zeros
-
Poles: Values of \(s\) that make denominator zero
-
Zeros: Values of \(s\) that make numerator zero
-
Order: Highest power of \(s\) in denominator
Example
-
Zero at \(s = -2\)
-
Poles at \(s = -1, -3\)
-
Second-order system
Stability Condition
System is stable if and only if all poles lie in the left half of s-plane (negative real parts).
Impulse Response
Impulse Response Analysis
Definition
Impulse response \(h(t)\) is the output when input is unit impulse \(\delta(t)\).
Properties
-
Convolution: \(y(t) = x(t) * h(t)\)
-
Laplace: \(Y(s) = X(s) \cdot G(s)\)
-
System characterization
Example
For \(G(s) = \dfrac{1}{s+a}\):
Stability from Impulse Response
System is BIBO stable if \(\int_0^{\infty} |h(t)| dt < \infty\)
Time Response Analysis
Time Response Analysis
Components
-
Transient response: Initial dynamic behavior
-
Steady-state response: Long-term behavior
Standard Test Signals
-
Impulse: \(\delta(t)\) \(\leftrightarrow\) \(1\)
-
Step: \(u(t)\) \(\leftrightarrow\) \(\dfrac{1}{s}\)
-
Ramp: \(tu(t)\) \(\leftrightarrow\) \(\dfrac{1}{s^2}\)
-
Parabolic: \(\dfrac{t^2}{2}u(t)\) \(\leftrightarrow\) \(\dfrac{1}{s^3}\)
Example
First-order system:
First Order System Parameters
System: \(G(s) = \dfrac{K}{Ts + 1}\)
-
DC Gain: \(K\)
-
Time constant: \(T\) (time to reach 63.2% of final value)
-
Settling time: \(t_s = 4T\) (for 2% criterion)
Step Response
Ramp Response
Key Point
First-order systems never exhibit overshoot for step input.
Second Order Systems
Second Order System Analysis
Standard Form
-
\(\zeta\): Damping ratio
-
\(\omega_n\): Natural frequency (rad/s)
-
\(\omega_d = \omega_n\sqrt{1-\zeta^2}\): Damped frequency
Parameter | Formula |
---|---|
Peak overshoot (\(M_p\)) | \(e^{-\pi\zeta/\sqrt{1-\zeta^2}} \times 100\%\) |
Peak time (\(t_p\)) | \(\dfrac{\pi}{\omega_n\sqrt{1-\zeta^2}}\) |
Rise time (\(t_r\)) | \(\dfrac{\pi - \cos^{-1}\zeta}{\omega_n\sqrt{1-\zeta^2}}\) |
Settling time (\(t_s\)) | \(\dfrac{4}{\zeta\omega_n}\) (for 2% criterion) |
Damping Cases for Second Order Systems
Response Types Based on Damping Ratio
-
\(\zeta > 1\): Overdamped - No oscillations, slow response
-
\(\zeta = 1\): Critically damped - Fastest response without overshoot
-
\(0 < \zeta < 1\): Underdamped - Oscillatory response with overshoot
-
\(\zeta = 0\): Undamped - Sustained oscillations
Step Response Forms
Underdamped (\(0 < \zeta < 1\)):
Pole Locations
-
Real part: \(-\zeta\omega_n\)
-
Imaginary part: \(\pm\omega_d\)
Important Relations
-
\(\zeta = \cos\theta\) where \(\theta\) is angle from negative real axis
-
\(M_p\) decreases as \(\zeta\) increases
-
\(t_s\) is independent of \(\omega_d\)
Higher Order Systems
Dominant Pole Concept
For systems with multiple poles:
-
Dominant poles: Closest to imaginary axis
-
Determine the main characteristics of transient response
-
Other poles should be at least 5-10 times farther from imaginary axis
Example
Approximate as second-order system for initial analysis.
Approximation Guidelines
-
Neglect poles that are 5+ times farther than dominant poles
-
Use dominant pole approximation for quick analysis
-
Verify with complete system analysis if needed
Steady-State Error
Steady-State Error Analysis
Error Signal and Final Value Theorem
For unity feedback system: \(E(s) = \dfrac{R(s)}{1 + G(s)}\)
Steady-state error: \(e_{ss} = \lim_{t \to \infty} e(t) = \lim_{s \to 0} sE(s)\)
System Type and Error Constants
System type is determined by the number of integrators (\(1/s\)) in \(G(s)\):
-
Position constant: \(K_p = \lim_{s\to 0} G(s)\)
-
Velocity constant: \(K_v = \lim_{s\to 0} sG(s)\)
-
Acceleration constant: \(K_a = \lim_{s\to 0} s^2G(s)\)
Input | Type 0 | Type 1 | Type 2 |
---|---|---|---|
Step (\(\dfrac{1}{s}\)) | \(\dfrac{1}{1+K_p}\) | 0 | 0 |
Ramp (\(\dfrac{1}{s^2}\)) | \(\infty\) | \(\dfrac{1}{K_v}\) | 0 |
Parabolic (\(\dfrac{1}{s^3}\)) | \(\infty\) | \(\infty\) | \(\dfrac{1}{K_a}\) |
Generalized Error Analysis
Non-Unity Feedback Systems
For system with feedback \(H(s)\):
Disturbance Rejection
For disturbance \(D(s)\) at plant input:
Position Error Constant Formula
For \(G(s) = \dfrac{K \prod (s + z_i)}{s^N \prod (s + p_j)}\):
Key Insight
-
Increasing system type reduces steady-state error
-
Higher gain improves accuracy but may affect stability
-
Trade-off between accuracy and stability
Stability Analysis
Routh-Hurwitz Criterion
Procedure
For characteristic equation \(a_n s^n + a_{n-1} s^{n-1} + \dots + a_0 = 0\):
-
All coefficients must be positive (necessary condition)
-
Construct Routh array
-
Count sign changes in the first column
-
Number of sign changes = Number of RHP poles
Example
For \(s^3 + 4s^2 + 6s + 4 = 0\):
Special Cases
-
Zero in first column: Replace with small \(\epsilon\)
-
Row of zeros: Use auxiliary equation
-
Marginal stability: Poles on imaginary axis
Advanced Stability Concepts
Marginal Stability
System has poles on imaginary axis. From row of zeros:
-
Form auxiliary equation from previous row
-
Differentiate to get next row
-
Marginal frequency from auxiliary equation
Example
For \(s^4 + 2s^3 + 3s^2 + 2s + 2 = 0\):
System is marginally stable with oscillation at \(\omega = 1\) rad/s.
Note
When row of zeros occurs, replace that row with derivative of auxiliary equation.
Root Locus
Root Locus Technique
Basic Concept
Root locus shows the path of closed-loop poles as gain \(K\) varies from 0 to \(\infty\).
For unity feedback: \(1 + KG(s)H(s) = 0\)
Construction Rules
-
Locus starts at open-loop poles (\(K=0\))
-
Locus ends at zeros or infinity (\(K=\infty\))
-
Real-axis segments: Exist where total #poles + #zeros to the right is odd
-
Asymptotes: \(n-m\) branches to infinity
-
Asymptote angles: \(\dfrac{(2q+1)\pi}{n-m}\), \(q = 0,1,2,\ldots,(n-m-1)\)
-
Centroid: \(\sigma_a = \dfrac{\sum \text{poles} - \sum \text{zeros}}{n-m}\)
Root Locus - Advanced Rules
Breakaway/Break-in Points
Solve \(\dfrac{dK}{ds} = 0\) where \(K = -\dfrac{1}{G(s)H(s)}\)
Alternative: \(\sum_{i=1}^n \dfrac{1}{s-p_i} = \sum_{j=1}^m \dfrac{1}{s-z_j}\)
Angle and Magnitude Conditions
For any point \(s\) on root locus:
-
Angle condition: \(\sum \angle(s-z_i) - \sum \angle(s-p_j) = (2k+1)\pi\)
-
Magnitude condition: \(K = \dfrac{\prod |s-p_j|}{\prod |s-z_i|}\)
Example
For \(G(s) = \dfrac{K}{s(s+4)}\), breakaway point:
Root Locus - Additional Rules
Departure/Arrival Angles
For complex poles/zeros:
-
Departure angle from complex pole:
\[\theta_d = 180^{\circ} - \sum \angle(\text{pole to other poles}) + \sum \angle(\text{pole to zeros})\] -
Arrival angle at complex zero:
\[\theta_a = 180^{\circ} + \sum \angle(\text{zero to poles}) - \sum \angle(\text{zero to other zeros})\]
Intersection with Imaginary Axis
-
Use Routh-Hurwitz criterion
-
Find value of \(K\) that makes system marginally stable
-
Solve auxiliary equation for intersection points
Pole-Zero Analysis
Effect of Poles and Zeros on Response
Pole Effects
-
Real poles: Exponential terms \(e^{-\sigma t}\)
-
Complex poles: Oscillatory terms \(e^{-\sigma t}\cos(\omega t + \phi)\)
-
Dominant poles: Closest to imaginary axis (slowest modes)
-
Distance from origin: Related to natural frequency
Zero Effects
-
LHP zeros: Reduce overshoot, speed up response
-
RHP zeros: Cause undershoot, non-minimum phase behavior
-
Near pole-zero cancellation: Reduces effect of both
-
Zeros at origin: Derivative action
Design Guidelines
-
Place poles in LHP for stability
-
Use zeros to shape transient response
-
Avoid pole-zero cancellation in RHP
-
Consider sensitivity to parameter variations
Performance Specifications
Time Domain Performance Specifications
Standard Specifications
-
Rise time (\(t_r\)): Time to go from 10% to 90% of final value
-
Peak time (\(t_p\)): Time to reach maximum overshoot
-
Settling time (\(t_s\)): Time to stay within 2% or 5% of final value
-
Maximum overshoot (\(M_p\)): Maximum deviation from final value
Relationships for Second-Order Systems
-
Faster rise time \(\Rightarrow\) Higher \(\omega_n\)
-
Lower overshoot \(\Rightarrow\) Higher \(\zeta\)
-
Faster settling \(\Rightarrow\) Higher \(\zeta\omega_n\)
-
Trade-off between speed and overshoot
Typical Design Requirements
-
\(M_p < 20\%\) \(\Rightarrow\) \(\zeta > 0.45\)
-
\(t_s < 4\) sec \(\Rightarrow\) \(\zeta\omega_n > 1\)
GATE Questions
GATE Practice Questions
Example 1
A unity feedback system has \(G(s) = \dfrac{K}{s(s+3)}\). The value of \(K\) for damping ratio \(\zeta = 0.5\) is:
-
1
-
3
-
9
-
27
Solution
Characteristic equation: \(s^2 + 3s + K = 0\)
Standard form: \(s^2 + 2\zeta\omega_n s + \omega_n^2 = 0\)
Compare: \(\omega_n^2 = K\), \(2\zeta\omega_n = 3\)
For \(\zeta = 0.5\): \(\omega_n = \dfrac{3}{2 \times 0.5} = 3\)
Thus, \(K = \omega_n^2 = 9\). Answer: C
GATE Practice Questions (Contd.)
Example 2
The steady-state error of a unity feedback system with \(G(s) = \dfrac{10}{s(s+2)}\) to a unit ramp input is:
-
0
-
0.2
-
0.5
-
\(\infty\)
Solution
System type = 1 (one integrator)
Velocity constant: \(K_v = \lim_{s\to 0} s \cdot \dfrac{10}{s(s+2)} = \dfrac{10}{2} = 5\)
Steady-state error for ramp: \(e_{ss} = \dfrac{1}{K_v} = \dfrac{1}{5} = 0.2\)
Answer: B
GATE Practice Questions (Contd.)
Example 3
For the characteristic equation \(s^3 + 4s^2 + Ks + 4 = 0\), the range of \(K\) for stability is:
-
\(K > 0\)
-
\(0 < K < 4\)
-
\(0 < K < 16\)
-
\(K > 4\)
Solution
Routh array:
\(K-1 > 0 \Rightarrow K > 1\) and \(4 > 0\) (always satisfied)
Also need \(K > 0\). Therefore: \(K > 1\). Answer: None exactly matches, but closest is A
GATE Practice Questions (Contd.)
Example 4
The peak overshoot of a second-order system with \(\zeta = 0.6\) is approximately:
-
9.5%
-
15.2%
-
25.4%
-
30.8%
Solution
Peak overshoot: \(M_p = e^{-\pi\zeta/\sqrt{1 -\zeta^2}} \times 100\%\)
For \(\zeta = 0.6\): \(M_p = e^{-\pi \times 0.6/\sqrt{1-0.6^2}} \times 100\%\) \(M_p = e^{-\pi \times 0.6/\sqrt{0.64}} \times 100\%\) \(M_p = e^{-\pi \times 0.6/0.8} \times 100\%\) \(M_p = e^{-2.356} \times 100\% = 0.095 \times 100\% = 9.5\%\)
Answer: A
Summary
Summary
Key Takeaways
-
Transfer function analysis provides complete system characterization
-
Pole locations determine stability and transient response nature
-
Time domain specifications help in controller design
-
Steady-state error analysis ensures accuracy requirements
-
Routh-Hurwitz criterion is essential for stability analysis
-
Root locus technique shows parameter sensitivity
GATE Preparation Tips
-
Practice standard forms and their responses
-
Master Routh-Hurwitz for all special cases
-
Understand pole-zero effects on system behavior
-
Focus on steady-state error calculations
-
Learn root locus construction rules
Important Formulas to Remember
-
Second-order parameters: \(M_p\), \(t_p\), \(t_s\), \(t_r\)
-
Error constants: \(K_p\), \(K_v\), \(K_a\)
-
Stability conditions and Routh array