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

Transfer Function

Definition

Impulse response \(h(t)\) is the output when input is unit impulse \(\delta(t)\).

Stability from Impulse Response

System is BIBO stable if \(\int_0^{\infty} |h(t)| dt < \infty\)

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)

Standard Form

where:

• \(\zeta\): Damping ratio

• \(\omega_n\): Natural frequency (rad/s)

• \(\omega_d = \omega_n\sqrt{1-\zeta^2}\): Damped frequency

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

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

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

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: \(K_a = \lim_{s\to 0} s^2G(s)\)

Non-Unity Feedback Systems

For system with feedback \(H(s)\):

Error constants are calculated using \(G(s)H(s)\).

Disturbance Rejection

For disturbance \(D(s)\) at plant input:

Higher loop gain reduces disturbance effects.

Position Error Constant Formula

For \(G(s) = \dfrac{K \prod (s + z_i)}{s^N \prod (s + p_j)}\):

Procedure

For characteristic equation \(a_n s^n + a_{n-1} s^{n-1} + \dots + a_0 = 0\):

1. All coefficients must be positive (necessary condition)

2. Construct Routh array

3. Count sign changes in the first column

4. Number of sign changes = Number of RHP poles

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

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: \(\sigma_a = \dfrac{\sum \text{poles} - \sum \text{zeros}}{n-m}\)

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: \(K = \dfrac{\prod |s-p_j|}{\prod |s-z_i|}\)

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 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

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

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

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

Solution

Routh array:

For stability: All first column elements must be positive
\(K-1 > 0 \Rightarrow K > 1\) and \(4 > 0\) (always satisfied)
Also need \(K > 0\). Therefore: \(K > 1\). Answer: None exactly matches, but closest is A

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

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