Control Systems MCQ Quiz - Objective Question with Answer for Control Systems - Download Free PDF

Explanation:

Bode Amplitude Plot for a First-Order Lowpass System

Definition: A first-order lowpass system is a type of linear system that allows low-frequency signals to pass through while attenuating high-frequency signals. The system is characterized by a single pole in its transfer function, and the Bode plot is used to graphically represent the frequency response of the system.

The Bode amplitude plot typically consists of two regions:

• A flat region at low frequencies (where the system behaves as a constant gain).
• A sloped region at high frequencies (where the system attenuates the signal).

Working Principle: The transfer function of a first-order lowpass system can be expressed mathematically as:

H(s) = K / (1 + s/ω_c)

Where:

• K is the DC gain of the system.
• s is the Laplace variable.
• ω_c is the cutoff angular frequency of the system.

The frequency response of this transfer function can be evaluated by substituting s = jω, where ω is the frequency in radians per second. The magnitude of the frequency response is given by:

|H(jω)| = K / √(1 + (ω/ω_c)²)

Bode Amplitude Plot Analysis:

• Low Frequencies: When ω c, the term (ω/ω_c)² becomes negligible, and the magnitude of the frequency response approximates to |H(jω)| ≈ K. On the Bode plot, this appears as a horizontal line parallel to the frequency axis with a constant gain in decibels.
• High Frequencies: When ω >> ω_c, the term (ω/ω_c)² dominates, and the magnitude of the frequency response approximates to |H(jω)| ≈ K / (ω/ω_c). In decibels, this corresponds to a slope of -20 dB/decade, indicating that the amplitude decreases by 20 dB for every tenfold increase in frequency.

Correct Option Analysis:

The correct option is:

Option 2: Line with slope -20 dB/decade.

This option correctly describes the behavior of the Bode amplitude plot of a first-order lowpass system. At frequencies much greater than the cutoff frequency, the amplitude decreases at a rate of -20 dB/decade, which is characteristic of a first-order system.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: Line with slope -40 dB/decade.

This description is incorrect. A slope of -40 dB/decade is characteristic of a second-order system, not a first-order system. In a second-order system, the amplitude decreases at a faster rate (two poles contribute to the attenuation), whereas in a first-order system, the slope is limited to -20 dB/decade.

Option 3: Line with slope +20 dB/decade.

This option is incorrect because it represents a system where the amplitude increases with frequency, which is not the behavior of a first-order lowpass system. A first-order lowpass system attenuates high-frequency signals rather than amplifying them.

Option 4: Straight line parallel to the frequency axis.

This option is partially correct but incomplete. While the amplitude plot of a first-order lowpass system is indeed a straight line parallel to the frequency axis at low frequencies, this description fails to account for the sloped region at higher frequencies where the amplitude decreases at a rate of -20 dB/decade.

Option 5: None of the above.

This option is incorrect because Option 2 accurately describes the behavior of the Bode amplitude plot for a first-order lowpass system.

Conclusion:

Understanding the frequency response and Bode plot characteristics of a first-order lowpass system is crucial for analyzing its behavior in various applications. The amplitude plot consists of a flat region at low frequencies and a sloped region with a slope of -20 dB/decade at high frequencies. This characteristic distinguishes first-order systems from higher-order systems, which exhibit steeper slopes in their amplitude plots.

Explanation:

Signal Flow Graph

Definition: A Signal Flow Graph (SFG) is a topological representation of a set of linear algebraic equations or differential equations. It is a graphical method used to represent the relationships between variables in a system. The graph consists of nodes (representing system variables) and directed branches (representing causal relationships and transfer functions) that connect the nodes. SFG is widely used in control systems to analyze and simplify the representation of complex systems.

Working Principle:

In an SFG, each variable in the system is represented as a node, and the relationships between these variables are represented as directed branches. The direction of the branch indicates the cause-and-effect relationship, while the weight of the branch represents the transfer function or gain between the variables. By using graphical techniques such as Mason's Gain Formula, the overall transfer function of the system can be determined efficiently.

Key Features of Signal Flow Graphs:

• Nodes represent system variables (e.g., input, output, and intermediate variables).
• Directed branches represent causal relationships between variables, with weights indicating the transfer function or gain.
• It is applicable only to linear time-invariant systems.
• SFGs are particularly useful for analyzing feedback systems and determining overall system behavior.

Applications:

• Analyzing control systems, particularly feedback systems.
• Simplifying and solving complex systems of linear equations.
• Deriving the transfer function of a system.
• Modeling and analyzing electrical circuits, mechanical systems, and other dynamic systems.

Correct Option Analysis:

The correct option is:

Option 3: Topological representation of set of differential equations.

This option correctly describes the Signal Flow Graph. It is a topological representation of a set of differential equations or linear algebraic equations, showcasing the relationships between system variables in a graphical form. The SFG provides an intuitive and systematic way to analyze and compute the transfer function of a system, making it a valuable tool in control system analysis.

Important Information

To further understand the analysis, let’s evaluate the other options:

Option 1: Polar plot

A Polar Plot is a graphical representation of a complex function, typically used in control systems and signal processing to represent the frequency response of a system. It plots the magnitude and phase of a system's transfer function in polar coordinates. While polar plots are useful in frequency domain analysis, they are not related to the Signal Flow Graph, which is a topological representation of equations in the time domain.

Option 2: Bode plot

A Bode Plot is another frequency domain analysis tool that represents the magnitude and phase of a system's transfer function as separate plots on a logarithmic scale. It is widely used for stability analysis and design in control systems. However, like the polar plot, it is not related to the Signal Flow Graph, which focuses on the relationships between variables in a system using a topological approach.

Option 4: Truth table

A Truth Table is a tabular representation of the logical relationships between inputs and outputs in a digital system. It is commonly used in digital electronics and Boolean algebra to analyze and design logic circuits. While truth tables are essential in digital system design, they have no connection to Signal Flow Graphs, which are used for analyzing linear systems.

Option 5: Not provided

No specific information is given for this option. However, it is clear that none of the options other than Option 3 correctly describe the Signal Flow Graph.

Conclusion:

The Signal Flow Graph is a powerful tool for representing and analyzing linear systems. It uses a topological approach to model the relationships between system variables and provides an efficient way to compute the transfer function. By understanding its principles and applications, one can effectively analyze complex systems and gain insights into their behavior. The correct answer, Option 3, accurately reflects the nature and purpose of the Signal Flow Graph, while the other options pertain to unrelated concepts in control systems, signal processing, or digital electronics.

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Explanation:

Phase Lead and Lag Compensators

The transfer functions of phase lead and lag compensators are given as:

• Phase lead compensator: (s + z<sub>1</sub>) / (s + p<sub>1</sub>)
• Phase lag compensator: (s + z<sub>2</sub>) / (s + p<sub>2</sub>)

To determine the correct option, we analyze the conditions for these compensators:

Phase Lead Compensator:

• In a phase lead compensator, the purpose is to advance the phase of the system by introducing a zero z<sub>1</sub> and a pole p<sub>1</sub>.
• The zero z<sub>1</sub> is designed to be larger than the pole p<sub>1</sub> (z<sub>1</sub> > p<sub>1</sub>).
• This arrangement ensures that the phase contribution is positive in the frequency range of interest, which improves the transient response of the system.

Phase Lag Compensator:

• In a phase lag compensator, the purpose is to delay the phase of the system by introducing a zero z<sub>2</sub> and a pole p<sub>2</sub>.
• The pole p<sub>2</sub> is designed to be larger than the zero z<sub>2</sub> (z<sub>2</sub> 2).
• This arrangement ensures that the phase contribution is negative in the frequency range of interest, which improves the steady-state accuracy of the system.

Correct Option Analysis:

The correct option is:

Option 1: z<sub>1</sub> > p<sub>1</sub> and z<sub>2</sub> 2

This option satisfies the conditions for both the phase lead and lag compensators. In the phase lead compensator, the zero is greater than the pole (z<sub>1</sub> > p<sub>1</sub>), and in the phase lag compensator, the zero is smaller than the pole (z<sub>2</sub> 2). These arrangements ensure proper phase advancement and delay for their respective compensators.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: z<sub>1</sub> > p<sub>1</sub> and z<sub>2</sub> > p<sub>2</sub>

This option is incorrect as it assumes that the zero is greater than the pole for both the lead and lag compensators. While this is true for the phase lead compensator (z<sub>1</sub> > p<sub>1</sub>), it contradicts the condition for the phase lag compensator, where the pole must be greater than the zero (z<sub>2</sub> 2).

Option 3: z<sub>1</sub> 1 and z<sub>2</sub> 2

This option is incorrect as it assumes that the zero is smaller than the pole for both the lead and lag compensators. While this is true for the phase lag compensator (z<sub>2</sub> 2), it contradicts the condition for the phase lead compensator, where the zero must be greater than the pole (z<sub>1</sub> > p<sub>1</sub>).

Option 4: z<sub>1</sub> 1 and z<sub>2</sub> > p<sub>2</sub>

This option is incorrect as it assumes that the zero is smaller than the pole for the lead compensator and larger for the lag compensator. This contradicts the correct conditions for both compensators. In the phase lead compensator, the zero must be greater than the pole (z<sub>1</sub> > p<sub>1</sub>), and in the phase lag compensator, the pole must be greater than the zero (z<sub>2</sub> 2).

Conclusion:

Understanding the conditions for phase lead and lag compensators is essential for ensuring proper system performance. A phase lead compensator requires the zero to be greater than the pole, while a phase lag compensator requires the pole to be greater than the zero. The correct option, Option 1, satisfies these conditions, making it the appropriate choice.

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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 Underdamped system
• ζ = 0: Undamped system

Since ζ = 0.707, which lies in the range 0 , 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:

Control System: Movement of Summing Point in Block Diagrams

Definition: In control systems, a summing point is a node where different signals are algebraically summed. These summing points are crucial for defining the relationships between input and output signals. The movement of a summing point in a block diagram impacts the mathematical representation of the control system and its feedback path.

Working Principle: When a summing point is moved across a block in a control system, it alters the feedback path and the overall system behavior. The movement of the summing point follows specific rules to ensure that the system's mathematical representation remains consistent. These rules are based on the nature of the block and the feedback configuration.

Correct Option Analysis:

The correct option is:

Option 1: Multiplication of the G(s) in the feedback path.

Explanation: When a summing point is moved to the right side of a block (with transfer function G(s)) in a block diagram, the feedback path gets multiplied by the transfer function G(s). This happens because the signal passing through the summing point now encounters the block G(s) before reaching its destination. Mathematically, if the original feedback path was H(s), it becomes H(s) × G(s) after the movement of the summing point.

Example: Suppose we have a system with the following configuration:

• The summing point is initially on the left of the block G(s).
• The feedback path is represented by H(s).

After moving the summing point to the right side of the block G(s), the feedback path changes to H(s) × G(s). This rule ensures that the system's mathematical representation remains consistent while modifying the block diagram.

Importance: This concept is important in control system analysis and design because it helps engineers simplify or rearrange block diagrams without altering the system's behavior. Understanding these transformations is crucial for tasks such as deriving transfer functions and analyzing system stability.

Additional Information:

To further understand the analysis, let’s evaluate the other options:

Option 2: Multiplication of the 1/G(s) in the feedback path.

This option is incorrect. Moving the summing point to the right side of the block G(s) does not lead to multiplication by 1/G(s). Instead, the feedback path is multiplied by G(s), as explained in the correct option analysis. Multiplication by 1/G(s) would occur under a different transformation, such as moving a takeoff point across a block.

Option 3: Addition of gain block.

This option is incorrect. Moving a summing point does not involve the addition of a gain block to the system. Instead, it modifies the feedback path by altering the mathematical relationship between the signals, as described in the correct option analysis.

Option 4: Subtraction of gain block.

This option is also incorrect. Moving a summing point does not involve the subtraction of a gain block. The movement affects the feedback path by introducing a multiplication factor, not by adding or subtracting blocks.

Conclusion:

Understanding the rules for moving summing points and blocks in control system diagrams is essential for accurate analysis and design. In this case, moving a summing point to the right side of a block results in the multiplication of the feedback path by the block's transfer function (G(s)). This transformation preserves the system's behavior while allowing for simplifications or modifications in the block diagram representation.

Concept:

Time constant 

Calculation:

Taking Laplace transform, we get

40 s X(s) + 2X(s) = 12(s)

Pole will be at -1/20. 

Time constant 

• The root locus is the strongest tool for determining stability and the transient response of the system as it gives the exact pole-zero location and also their effect on the response
• A Bode plot is a useful tool that shows the gain and phase response of a given LTI system for different frequencies
• The Nyquist plot in addition to providing absolute stability also gives information on the relative stability of stable systems and degree of instability of the unstable system
• Routh-Hurwitz criterion is used to find the range of the gain for stability and gives information regarding the location of poles

Concept:

A transfer function is defined as the ratio of Laplace transform of the output to the Laplace transform of the input by assuming initial conditions are zero.

TF = L[output]/L[input]

For unit impulse input i.e. r(t) = δ(t)

⇒ R(s) = δ(s) = 1

Now transfer function = C(s)

Therefore, the transfer function is also known as the impulse response of the system.

Transfer function = L[IR]

IR = L-1 [TF]

Calculation:

Given the differential equation is,

By applying the Laplace transform,

s² Y(s) + 4 Y(s) = 6 R(s)

Poles are the roots of the denominator in the transfer function.

⇒ s² + 4 = 0

⇒ s = ±2j

Lag compensator:

Transfer function:

If it is in the form of

 , then a

If it is in the form of

 , then a > b

Pole zero plot:

The pole is nearer to the origin.

Filter: It is a low pass filter (LPF)

Maximum phase lag frequency:

Maximum phase lag:

ϕm is negative

Lead compensator:

Transfer function:

If it is in the form of

 , then a > 1

If it is in the form of

 , then a

Pole zero plot:

The zero is nearer to the origin.

Filter: It is a high pass filter (HPF).

Maximum phase lead frequency:

Maximum phase lead:

ϕm is positive

Concept:

KP = position error constant = 

Kv = velocity error constant = 

Ka = acceleration error constant = 

Steady-state error for different inputs is given by

From the above table, it is clear that for type – 1 system, a system shows zero steady-state error for step-input.

Concept:

Closed-loop transfer function = 

For unity negative feedback system Open-loop transfer function (G(s) H(s)) can be found by subtracting the numerator term from the denominator term 

Application:

Open-loop transfer Function

For DC gain s = 0

∴ open-loop gain 

Concept:

Minimum phase system: It is a system in which poles and zeros will not lie on the right side of the s-plane.  In particular, zeros will not lie on the right side of the s-plane.

For a minimum phase system,

Where P & Z are finite no. of poles and zeros of G(s)H(s)

Non-minimum phase system: It is a system in which some of the poles and zeros may lie on the right side of the s-plane. In particular, zeros lie on the right side of the s-plane.

Stable system: A system is said to be stable if all the poles lie on the left side of the s-plane.

Application:

As one zero lies in the right side of the s-plane, it is a non-minimum phase transfer function.

As there no poles on the right side of the s-plane, it is a stable system.

Explanation:

The controller is a device that is used to alter or maintain the transient state & steady-state region performance parameter as per our requirement.

Proportional Controller- 

The standard Proportional Controller as shown:

In space-form - 

In time-domain form - 

p(t) = Kp e(t) + po

Where,

po = controller output with zero error  

Kp = proportional gain constant.

Some effects of the proportional controller are as follows:

• The P-controller can stabilize a first-order system, can give a near-zero error, and improves the settling time by increasing the bandwidth.
• It also helps in reducing the steady-state error which makes the system more stable.
• The slow response of an over-damped system can be made faster with the help of the proportional controller. Hence option (2) is the correct answer.

​Important Points

Effects of Proportional Integral (PI) controllers:

• Increases the type of the system by one
• Rise time and settling time increases and Bandwidth decreases
• The speed of response decreased i.e. transient response becomes slower
• Decreases the steady-state error and steady-state response is improved
• Decreases the stability

Effects of Proportional Derivative (PD) controllers:

• Decreases the type of the system by one
• Reduces the rise time and settling time
• Rise time and settling time decreases and Bandwidth increases
• The speed of response is increased i.e. transient response is improved
• Improves gain margin, phase margin, and resonant peak
• Increases the input noise
• Improves the stability

The transfer function of a control system is defined as the ratio of the Laplace transform of the output variable to Laplace transform of the input variable assuming all initial conditions to be zero.

It is also defined as the Laplace transform of the impulse response.

If the input is represented by R(s) and the output is represented by C(s), then the transfer function will be:

Routh-Hurwitz criterion:

• Using the Routh-Hurwitz method, the stability information can be obtained without the need to solve the closed-loop system poles. This can be achieved by determining the number of poles that are in the left-half or right-half plane and on the imaginary axis.
• This involves checking the roots of the characteristic polynomial of a linear system to determine its stability.
• It is used to determine the absolute stability of a system.

Other methods of determining stability include:

Root locus:

• This method gives the position of the roots of the characteristic equation as the gain K is varied.
• With Root locus (unlike the case with Routh-Hurwitz criterion), we can do both analysis (i.e., for each gain value we know where the closed-loop poles are) and design (i.e., on the curve we can search for a gain value that results in the desired closed-loop poles).

Nyquist plot:

• This method is mainly used for assessing the stability of a system with feedback.
• While Nyquist is a graphical technique, it only provides a limited amount of intuition for why a system is stable or unstable, or how to modify an unstable system to be stable.