Automation & Control: Stability of Linear Systems

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Questions and Answers

What is the characteristic equation of the unity negative feedback control system represented by the open-loop transfer function K(s + 2) / ((s + 1)(s + 4))?

(s + 2) = -K(s + 1)(s + 4)

1 + K(s + 2) / ((s + 1)(s + 4)) = 0

K(s + 1)(s + 4) + (s + 2) = 0

K(s + 2) + (s + 1)(s + 4) = 0 (correct)

What indicates that the system has infinite gain margin based on the root locus?

When the root locus does not cross the jω axis. (correct)

When the system has a stable feedback controller.

When the gain factor K is set to zero.

When the root locus crosses the jω axis.

In which plane can the closed loop poles of the system exist?

Only in the first quadrant.

Anywhere on the complex plane. (correct)

Only in the imaginary axis.

Only on the real axis.

What does the term 'gain margin' represent in the context of root locus?

The factor by which the gain factor K can be increased before instability occurs. Signup and view all the answers

For a gain factor K = 2, how can the closed-loop poles be determined?

By substituting K = 2 in the characteristic equation. Signup and view all the answers

What happens to the closed-loop pole locations as the gain k is varied?

They move continuously on the s-plane as k changes. Signup and view all the answers

What is the effect on system response when the poles become complex?

The transient response becomes underdamped. Signup and view all the answers

What is the significance of the break point in a root locus plot?

It's where poles become complex. Signup and view all the answers

How does increasing gain k affect the percentage overshoot?

It increases the percentage overshoot. Signup and view all the answers

What remains unchanged with increasing gain during the transient response?

The real part of the complex poles. Signup and view all the answers

What effect does the real part of complex poles have on settling time?

It does not affect the settling time. Signup and view all the answers

What is the resulting expression after simplifying the closed-loop transfer function?

$\frac{k}{(s + 4)(s + 20) + k}$ Signup and view all the answers

What is the effect of the system response when the gain is set to k = 20?

The poles will change location leading to different transient responses. Signup and view all the answers

What represents the zeros of a linear system's transfer function?

The roots of the numerator Signup and view all the answers

Which MATLAB command is used to find the roots of a polynomial?

roots(P) Signup and view all the answers

How are poles represented in a pzmap generated by MATLAB?

As crosses Signup and view all the answers

What characterizes a stable linear time-invariant control system?

The output eventually returns to its equilibrium state Signup and view all the answers

What does a marginally stable linear system mean?

The output continues oscillating forever without settling Signup and view all the answers

In the context of stability, what does it mean if a system is unstable?

The output diverges without bounds from its equilibrium state Signup and view all the answers

In the given first-order differential equation context, which component is a constant factor?

'a' Signup and view all the answers

What is the time response y(t) of a system when a Laplace transform is applied with impulse input?

y(t) = e^{-at} Signup and view all the answers

What is the condition for a system to be classified as stable?

All closed-loop poles are in the left half of the s-plane. Signup and view all the answers

What occurs when at least one coefficient of the characteristic equation is negative?

The system is unstable. Signup and view all the answers

In the given example, which characteristic equation represents a stable system?

q(s) = s^3 + 3s^2 + 2s + 3 Signup and view all the answers

What is the role of a compensator in a control system?

To modify the output signal based on an input error signal. Signup and view all the answers

If a system has all coefficients of the characteristic equation positive, what can be said about the system?

The stability of the system cannot be determined. Signup and view all the answers

Which of the following statements about the closed-loop transfer function is true?

It is represented as kG / (1 + kGH). Signup and view all the answers

What type of stability is indicated by y(∞) = 1 in a control system?

Neutral (marginally stable) Signup and view all the answers

For which case is y(∞) equal to ∞ and what does it indicate?

a < 0; indicates instability. Signup and view all the answers

Study Notes

Automation & Control - Stability of Linear Feedback Systems

Stability Concept: Fundamental to control systems. An unstable closed-loop system is impractical. Many systems (like aircraft) are inherently open-loop unstable. Control design requires predicting system behavior from component knowledge.
Stability Criteria: A stable system returns to equilibrium after an initial condition. Marginally stable systems oscillate indefinitely. Unstable systems diverge from equilibrium.

Automation & Control - Mathematical Description

First-Order Differential Equation: dy(t)/dt + ay(t) = r(t)
Initial Condition: y(0) = 0; a is a constant.
Impulse Input (Laplace Transform): Y(s) = 1/(s + a)
Time Response: y(t) = e^-at
Stability Cases:
- a > 0: Stable
- a = 0: Neutral (marginally stable)
- a < 0: Unstable

Automation & Control - Graphical Representation

S-Plane: Systems are stable if all closed-loop poles have negative real parts (lie in the left-half of the s-plane).
Real and Imaginary Representation: Poles and zeros are plotted on the s-plane's real and imaginary axes (in complex number form: σ + jω).

Automation & Control - Root Locus Plotting

System Closed-Loop Transfer Function: T(s)= [kG]/(1 + kGH)
Characteristic Equation: 1 + kGH = 0
Coefficient Condition for Stability: A stable system has positive coefficients for the closed-loop characteristic equation's terms.
Root Locus Plots: Show closed-loop pole location changes as gain 'k' varies. Used to assess system stability and performance as gain changes. Poles on the imaginary axis (jω axis) indicate instability. Gain factors lead to adjustments of root position.

Automation & Control - Gain Margin

Definition: Gain margin is the factor by which the gain can increase before instability.
Root Locus Determination: Gain margin is found where the root locus crosses the imaginary axis (jω axis) of the s-plane—the value(k) of gain at the crossover point is the "value of K at crossover." Then gain margin = [desirable K value)/ (calculated K value)
Infinite Gain Margin: No crossing of imaginary axis implies infinite gain margin.

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Description

This quiz focuses on the stability of linear feedback systems in automation and control. It covers key concepts like stability criteria, mathematical descriptions using differential equations, and graphical representations in the S-plane. Assess your understanding of these fundamental principles essential for effective control system design.