Condition For Stability MCQ Quiz in తెలుగు - Objective Question with Answer for Condition For Stability - ముఫ్త్ [PDF] డౌన్‌లోడ్ కరెన్

Explanation:

Lyapunov's stability criterion is used to check the stability of a nonlinear control system.

Lyapunov's stability criterion: 

1 Lyapunov stability is named after Aleksandr Mikhailovich Lyapunov, a Russian mathematician who defended the thesis The General Problem of Stability of Motion regarding the asymptotic stability of nonlinear systems.

2 In simple terms, if the solutions that start out near an equilibrium point x_e stay near xe forever, then xe is Lyapunov stable. More strongly, if xe is Lyapunov stable and all solutions that start out near xe converge to xe, then xe is asymptotically stable.

Hence option (3) is the correct answer.

Important Points

Hough Transform:

Hough Transform is a popular technique to detect any shape if you can represent that shape in mathematical form. It can detect the shape even if it is broken or distorted a little bit.

Routh stability criterion and Root locus are used to check the stability of the linear control system.

Concept:

The output response of a system is equal to the sum of natural response and forced response.

Forced response: The response generated due to the pole of the input function is called the forced response.

Natural response: The response generated due to the pole of system function is called the natural response.

For linear systems, the stability is based on the natural response which leads to zero for a stable system, however, for an unstable system the natural response grows without bound and the forced response is overpowered by the natural response and the control is lost over the system. 

The response reaches a steady state after almost 5-time constants and then the difference in the input and output is termed as the steady-state error. 

The forced response remains and the natural response dies out for a stable system. 

Any system is called a stable system if the output of the system is bounded for any bounded input. If any of the bounded input yields unbounded output then the system is not stable. 

Characteristic equation:

1 + G(s) H(s) = 0

\( \Rightarrow 1 + K\left( {{s^2} - 2s + 2} \right)\frac{1}{{\left( {{s^2} + 2s + 4} \right)}} = 0\)

⇒ s² + 2s + 4 + Ks² – 2Ks + 2K = 0

⇒ (K + 1) s² + 2 (1 - K)s + (2K + 4) = 0

Since all co-efficient must be positive for stability in a second order polynomial,

-1 < K ; -∞ < K < 1 ; - 4 < 2 K < ∞

2K + 4 > 0; 2K > - 4; K > - 2

With all the conditions in hand we can write:

\( \Rightarrow -1 < K < 1\)

Concept: 

All the poles must have the negative real part.

Calculation:

Option 1:

(s² - 3s - 10) → (s + 2) (s - 5) Cascading with (s - 5), we get

⇒ \(\dfrac{1}{(s - 5)} (s + 2) ( s - 5)\)

⇒ (s + 2)

Option 2:

(s² + 2s - 35) → (s + 7) (s - 5)

Cascading with \(\dfrac{1}{(s - 5)}\), we get

⇒ \(\dfrac{1}{(s - 5)} (s + 7) ( s - 5)\)

⇒ s + 7

Option 3:

(s² + s - 6) → (s - 2) (s + 3)

does not give a stable system on cascading.

Option 4:

(s² - 2s + 12) → (s - 2)(s - 6)

does not give a stable system on cascading.

Explanation:

System Damping Analysis

The given characteristic equation of the closed-loop system is:

s² + 2s + 2 = 0

This is a second-order system, and its response is determined by analyzing its damping ratio (ζ) and natural frequency (ω_n).

Step 1: Standard Form of the Characteristic Equation

The standard form of a second-order system's characteristic equation is given by:

s² + 2ζω_ns + ω_n² = 0

Comparing the given equation (s² + 2s + 2 = 0) with the standard form:

• 2ζω_n = 2 → ζω_n = 1
• ω_n² = 2 → ω_n = √2

Step 2: Calculate the Damping Ratio (ζ)

From the first equation, ζ can be calculated as:

ζ = (1 / ω_n) = (1 / √2)

Therefore:

ζ = 0.707

Step 3: Determine the System's Damping Condition

The damping condition of a second-order system is determined by the value of the damping ratio (ζ):

• ζ > 1: Overdamped system
• ζ = 1: Critically damped system
• 0 < ζ < 1: Underdamped system
• ζ = 0: Undamped system

Since ζ = 0.707, which lies in the range 0 < ζ < 1, the system is underdamped.

Correct Option: Option 3 (Under damped)

Additional Information

To further analyze the other options, let's evaluate the damping conditions:

Option 1: Overdamped

In an overdamped system, the damping ratio (ζ) is greater than 1. This means that the system returns to its steady-state value without oscillating, but it does so slowly. Since ζ = 0.707 (less than 1) for the given characteristic equation, the system is not overdamped. Thus, this option is incorrect.

Option 2: Critically Damped

A critically damped system occurs when ζ = 1. In this case, the system returns to its steady-state value as quickly as possible without oscillating. However, for the given system, ζ = 0.707 (not equal to 1), so the system is not critically damped. Hence, this option is also incorrect.

Option 4: Undamped

An undamped system corresponds to ζ = 0. In this case, there is no damping force, and the system oscillates indefinitely at its natural frequency. Since ζ = 0.707 for the given system, it is not undamped. Therefore, this option is incorrect.

Option 5: Not Given in the Statement

This option does not apply to the given question as the characteristic equation directly provides enough information to determine the damping condition of the system.

Conclusion:

The damping ratio (ζ) is a critical parameter for determining the damping condition of a second-order system. Based on the given characteristic equation (s² + 2s + 2 = 0), we calculated ζ = 0.707, which classifies the system as underdamped. This means the system will exhibit oscillatory behavior with gradually decreasing amplitude over time.

Explanation:

sin ω_ot introduces poles on the jω axis. This will make the system marginally stable, and hence the stability will be ruined.

Also, sin ω_ot is also a periodic function, not letting to converge the impulse response since it exists from -∞ to ∞ time duration.

∴ The standard definition of stability precludes the sin ω₀t term in the impulse response.

Both Statement (I) and Statement (II) are individually true but Statement (II) is not the correct explanation of Statement (I)