Controllers and Compensators MCQ Quiz - Objective Question with Answer for Controllers and Compensators - Download Free PDF

Explanation:

Natural Frequency (ω_n)

Definition: Natural frequency (ω_n) is a fundamental property of a mechanical system that describes the rate at which the system oscillates in the absence of any driving or damping force. It is dependent on the system's stiffness and mass.

Mathematical Derivation:

For a control system, the natural frequency can be derived from the standard second-order differential equation that describes the system's motion. This equation is:

Where:

• J is the moment of inertia.
• k_p is the proportional gain or stiffness.
• θ is the angular displacement.

To find the natural frequency, we compare this equation to the standard form of the simple harmonic motion equation:

By comparing the coefficients, we get:

Therefore, the natural frequency is:

Correct Option Analysis:

The correct option is:

Option 2: ωn=kpJ" id="MathJax-Element-5-Frame" role="presentation" style="position: relative;" tabindex="0">ωn=kpJ−−√ωn=kpJ $\rm \omega_n=\sqrt{\frac{k_p}{J}}$

This option correctly represents the natural frequency for the given control system, as derived from the second-order differential equation of motion.

Additional Information

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

Option 1: ωn=1Jkp" id="MathJax-Element-6-Frame" role="presentation" style="position: relative;" tabindex="0">ωn=1Jkp−−−√ωn=1Jkp $\rm \omega_n=\sqrt{\frac{1}{Jk_p}}$

This option incorrectly suggests that the natural frequency is the square root of the reciprocal of the product of moment of inertia (J) and stiffness (k_p). The dimensions of this expression do not match those of frequency, making it an invalid option.

Option 3: ωn=Jkp" id="MathJax-Element-7-Frame" role="presentation" style="position: relative;" tabindex="0">ωn=Jkp−−−√ωn=Jkp $\rm \omega_n=\sqrt{Jk_p}$

This option suggests that the natural frequency is the square root of the product of moment of inertia (J) and stiffness (k_p). This expression is dimensionally incorrect for frequency and does not align with the standard form derived from the second-order differential equation.

Option 4: ωn=Jkp" id="MathJax-Element-8-Frame" role="presentation" style="position: relative;" tabindex="0">ωn=Jkp−−√ωn=Jkp $\rm \omega_n=\sqrt{\frac{J}{k_p}}$

This option suggests that the natural frequency is the square root of the ratio of moment of inertia (J) to stiffness (k_p). This is dimensionally incorrect for frequency and does not align with the standard form derived from the second-order differential equation.

Conclusion:

Understanding the derivation and the correct form of the natural frequency is crucial for analyzing the behavior of mechanical systems. The natural frequency is a key parameter that determines how a system responds to external disturbances. By correctly identifying the natural frequency using the derived formula, we can better design and control mechanical systems for optimal performance.

Concept

In control systems, particularly in closed-loop systems, the purpose of various controllers is to improve the performance of the system. The Proportional-Integral-Derivative (PID) controller is one of the most commonly used types. Each component of a PID controller (Proportional, Integral, and Derivative) serves a unique function to enhance the system's performance.

An integral controller (Ki) is designed specifically to address the steady-state error. Steady-state error is the difference between the desired and actual output of a system when the system has reached a steady state. The integral action of the controller accumulates the error over time and adjusts the system's control input to eliminate this error, thus reducing the steady-state error to zero.

Additional Information 

Let's look at the other options and understand why they are not correct:

1) To increase system bandwidth: Increasing system bandwidth is not the primary purpose of the integral controller. Bandwidth is more related to the speed of the system's response, which is generally influenced by the proportional (Kp) and derivative (Kd) components of the PID controller.

2) To adjust the damping ratio: The damping ratio is a measure of how oscillations in a system decay after a disturbance. This is primarily influenced by the proportional and derivative components of the PID controller, not the integral component.

3) To amplify the system gain: Amplifying system gain is related to the proportional controller (Kp), which adjusts the magnitude of the system's response to errors. The integral controller does not serve this purpose.

4) To reduce the steady state error: This is the correct answer. The integral controller continuously sums the error over time and adjusts the control input to eliminate the steady-state error, ensuring the actual output matches the desired output.

When we talk about adding a pole to a transfer function in a control system, we refer to introducing an additional term in the denominator of the transfer function. This affects the frequency response and stability characteristics of the system.

Definitions:

• Lag Compensation:
  Lag compensation typically involves adding a pole and a zero such that the pole is closer to the origin than the zero (in the s-domain). This configuration slows down the system response and is commonly used to improve the steady-state performance (reduce steady-state error) while having a marginal impact on the transient response.
  Lead Compensation:
• Lead compensation, on the other hand, involves adding a zero and a pole where the zero is closer to the origin than the pole. This configuration speeds up the system response, enhances stability margins, and improves transient response by providing phase lead at higher frequencies.
  Effects of Adding a Pole:
• Adding a single pole to the system tends to reduce the overall system speed, increase lag, and potentially reduce the phase margin, which might adversely affect stability.
• Essentially, this means that the main result of adding a pole is akin to lag compensation, as it introduces a delay in the system's response to changes in input.

Proportion + Integral + Derivative:

The PID controller produces on output, which is the combination of outputs of proportional, integral, and derivative controllers.

This is defined in terms of differential equations as:

\(u\left( t \right) = {K_p}\;e\left( t \right) + {K_I}\smallint e\left( t \right) \cdot dt + {K_D} \cdot \frac{{de\left( t \right)}}{{dt}}\)

Applying Laplace transform, we get:

\(U\left( s \right) = \left( {{K_p} + \frac{{{K_I}}}{s} + {K_D}s} \right)E\left( s \right)\)

Transfer function will be:

\(\frac{{U\left( s \right)}}{{E\left( s \right)}} = {K_p} + \frac{{{K_I}}}{s} + {K_D} \cdot s\)

Integral control:

It is the control mode where the controller Output is proportional to the integral of the error with respect to time.

Integral controller output = k × integral of error with time, i.e.

\({G_C}\left( s \right) = \frac{{U\left( s \right)}}{{E\left( s \right)}} = \frac{{{K_I}}}{s}\)  

Proportional + Derivate:

The additive combination of proportional & Derivative control is known as P-D control.

Overall transfer function for a PD controller is given by:

\({G_C}\left( s \right) = \frac{{U\left( s \right)}}{{E\left( s \right)}} = {K_P} + s{K_D}\)

It is equivalent to a High-pass filter.

Proportion + Integral + Derivative (PID):

The PID controller produces on output, which is the combination of outputs of proportional, integral, and derivative controllers.

This is defined in terms of differential equations as:

\(u\left( t \right) = {K_p}\;e\left( t \right) + {K_I}\smallint e\left( t \right) \cdot dt + {K_D} \cdot \frac{{de\left( t \right)}}{{dt}}\)

Applying Laplace transform, we get:

\(U\left( s \right) = \left( {{K_p} + \frac{{{K_I}}}{s} + {K_D}s} \right)E\left( s \right)\)

Transfer function will be:

\(\frac{{U\left( s \right)}}{{E\left( s \right)}} = {K_p} + \frac{{{K_I}}}{s} + {K_D} \cdot s\)

Integral control:

It is the control mode where the controller Output is proportional to the integral of the error with respect to time.

Integral controller output = k × integral of error with time, i.e.

\({G_C}\left( s \right) = \frac{{U\left( s \right)}}{{E\left( s \right)}} = \frac{{{K_I}}}{s}\)  

Proportional + Derivate:

The additive combination of proportional & Derivative control is known as P-D control.

Overall transfer function for a PD controller is given by:

\({G_C}\left( s \right) = \frac{{U\left( s \right)}}{{E\left( s \right)}} = {K_P} + s{K_D}\)

It is equivalent to a High-pass filter.

Lag compensator:

Transfer function:

If it is in the form of

 \(\frac{{1 + aTs}}{{1 + Ts}}\), then a < 1

If it is in the form of

 \(\frac{{s + a}}{{s + b}}\), 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:

 \({\omega _m} = \frac{1}{{T\sqrt a }}\)

Maximum phase lag:

 \({\phi _m} = {\sin ^{ - 1}}\left( {\frac{{a - 1}}{{a + 1}}} \right)\)

ϕm is negative

Lead compensator:

Transfer function:

If it is in the form of

 \(\frac{{1 + aTs}}{{1 + Ts}}\), then a > 1

If it is in the form of

 \(\frac{{s + a}}{{s + b}}\), then a < b

Pole zero plot:

The zero is nearer to the origin.

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

Maximum phase lead frequency:

 \({\omega _m} = \frac{1}{{T\sqrt a }}\)

Maximum phase lead:

 \({\phi _m} = {\sin ^{ - 1}}\left( {\frac{{a - 1}}{{a + 1}}} \right)\)

ϕm is positive

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 - 

\({G_C}\left( s \right) = \frac{{U\left( s \right)}}{{E\left( s \right)}} = \frac{{{K_p}}}{s(s+1)}\)

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

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. the transient response is improved
• Improves gain margin, phase margin, and resonant peak
• Increases the input noise
• Improves the stability

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

Concept:

In general, the lead and lag compensator is represented by the below transfer function

\(\frac{{{V_0}\left( s \right)}}{{{V_i}\left( s \right)}} = k\frac{{s + a}}{{s + b}}\)

If a > b then that is lag compensator because pole comes first.

If a < b then that is the lead compensator since zero comes first.

Analysis:

Lead compensator:

1) When sinusoidal input applied to this it produces sinusoidal output with the phase lead input.

2) It speeds up the Transient response and increases the margin for stability.

A circuit diagram is as shown:

Response is:

\(\frac{{{V_0}\left( s \right)}}{{{V_i}\left( s \right)}} = \frac{{{R_2}\left( {1 + s{C_1}{R_1}} \right)}}{{{R_1} + {R_2} + s{C_1}{R_1}{R_2}}}\)

\(\frac{{{V_0}\left( s \right)}}{{{V_i}\left( s \right)}} = \frac{{1 + s\tau }}{{1 + \alpha s\tau }}\)

Lead constant \(\alpha = \frac{{{R_2}}}{{{R_1} + {R_2}}}\) < 1

Important Points

Concept:

The standard T/F of the compensator is 

\(\frac{{1 + aTs}}{{1 + Ts}}\)

Maximum phase lead

\( {ϕ _m} = {\sin ^{ - 1}}\left( {\frac{{a - 1}}{{a + 1}}} \right)\\ \)

Maximum phase lead frequency, 

\({\omega _m} = \frac{1}{{T\sqrt a }}\)

Calculation:

The given transfer function is,

\(T/F = 4(\frac{{1 + 0.15s}}{{1 + 0.05s}})\)

By comparing both transfer functions,

aT = 0.15

T = 0.05

a = 3

Maximum phase lead

\(\begin{array}{l} {ϕ _m} = {\sin ^{ - 1}}\left( {\frac{{a - 1}}{{a + 1}}} \right)\\ = {\sin ^{ - 1}}\left( {\frac{{3 - 1}}{{3 + 1}}} \right)\\ = {\sin ^{ - 1}}\left( {\frac{1}{{2}}} \right) \end{array}\)

= sin-1 (0.5)

ϕ_m = 30° 

Lead compensator:

Transfer function:

If it is in the form of \(\frac{{1 + aTs}}{{1 + Ts}}\), then a > 1

If it is in the form of \(\frac{{s + a}}{{s + b}}\), then a < b

Maximum phase lead frequency: \({\omega _m} = \frac{1}{{T\sqrt a }}\)

Maximum phase lead: \({\phi _m} = {\sin ^{ - 1}}\left( {\frac{{a - 1}}{{a + 1}}} \right)\)

ϕ_m is positive

Pole zero plot:

The zero is nearer to the origin.

Filter: It is a high pass filter (HPF)

Effect on the system:

• Rise time and settling time decreases and Bandwidth increases
• The transient response becomes faster
• The steady-state response is not affected
• Improves the stability

Lag compensator:

Transfer function:

If it is in the form of \(\frac{{1 + aTs}}{{1 + Ts}}\), then a < 1

If it is in the form of \(\frac{{s + a}}{{s + b}}\), then a > b

Maximum phase lag frequency: \({\omega _m} = \frac{1}{{T\sqrt a }}\)

Maximum phase lag: \({\phi _m} = {\sin ^{ - 1}}\left( {\frac{{a - 1}}{{a + 1}}} \right)\)

ϕ_m is negative

Pole zero plot:

The pole is nearer to the origin.

Filter: It is a low pass filter (LPF)

Effect on the system:

• Rise time and settling time increases and Bandwidth decreases
• The transient response becomes slower
• The steady-state response is improved
• Stability decreases

Analysis:

Lag Compensator:

\(\frac{{{V_0}\left( s \right)}}{{{V_1}\left( s \right)}} = \frac{{1 + ST}}{{1 + \beta ST}}\)

Where,

\(\beta = \frac{{{R_1} + {R_2}}}{{{R_2}}},\;\beta > 1\)

In Lag Compensator, the steady-state error is reduced So, steady-state Response is increased.

In lag compensator, Transient response decreases.

Lead Compensator:

\(\frac{{{V_0}\left( s \right)}}{{{V_i}\left( s \right)}} = \frac{{\alpha \left( {1 + ST} \right)}}{{1 + \alpha ST}}\)

\(\alpha = \frac{{{R_2}}}{{{R_1} + {R_2}}}\;;\alpha < 1\)

In lead comp. steady-state error increased So, the steady-state response is decreased.

In lead comp. Transient response Improves.

Lag-lead Compensator:

Lag lead Compensator is a combined form of a lead compensator and lag comp compensator.

\(\frac{{{V_0}\left( s \right)}}{{{V_i}\left( s \right)}} = \frac{{\left( {S + {Z_1}} \right)\left( {S + {Z_2}} \right)}}{{\left( {S + {P_1}} \right)\left( {S + {P_2}} \right)}}\)

\(\beta = \underbrace {\frac{{{Z_1}}}{{{P_1}}} > 1}_{Lag}, ~ \alpha = \underbrace {\frac{{{Z_2}}}{{{P_2}}} < 1}_{Lead}\)

Lag compensator:

Transfer function:

If it is in the form of \(\frac{{1 + aTs}}{{1 + Ts}}\), then a < 1

If it is in the form of \(\frac{{s + a}}{{s + b}}\), then a > b

Maximum phase lag frequency:

\({\omega _m} = \frac{1}{{T\sqrt a }}\)

Maximum phase lag::

\({\phi _m} = {\sin ^{ - 1}}\left( {\frac{{a - 1}}{{a + 1}}} \right)\)

ϕm is negative

Pole zero plot:

The pole is nearer to the origin.

Filter: It is a low pass filter (LPF)

Effect on the system:

• Rise time and settling time increases and Bandwidth decreases
• The transient response becomes slower
• The steady-state response is improved
• Stability decreases

Lead compensator:

Transfer function:

If it is in the form of \(\frac{{1 + aTs}}{{1 + Ts}}\), then a > 1

If it is in the form of \(\frac{{s + a}}{{s + b}}\), then a < b

Maximum phase lead frequency:

\({\omega _m} = \frac{1}{{T\sqrt a }}\)

Maximum phase lead:

\({\phi _m} = {\sin ^{ - 1}}\left( {\frac{{a - 1}}{{a + 1}}} \right)\)

ϕm is positive

Pole zero plot:

The zero is nearer to the origin.

Filter: It is a high pass filter (HPF)

Effect on the system:

• Rise time and settling time decreases and Bandwidth increases
• The transient response becomes faster
• The steady-state response is not affected
• Improves the stability

Given circuit is a lag compensator and transfer function is given as

\(\frac{{{{\rm{V}}_0}\left( {\rm{s}} \right)}}{{{{\rm{V}}_{{\rm{in}}}}\left( {\rm{s}} \right)}} = \frac{{\left( {1 + \frac{1}{{\rm{s}}}} \right)}}{{10 + \frac{1}{{\rm{s}}}}} = \frac{{{\rm{s}} + 1}}{{10{\rm{s}} + 1}} = \frac{{\left( {1 + {\rm{sT}}} \right)}}{{\left( {1 + {\rm{\beta sT}}} \right)}}\)

On comparison we get, \(T = 1\)

\(\beta T = 10\; \Rightarrow \;\beta = 10\)

The frequency at which maximum lead occurs is \({{\rm{\omega }}_{\rm{m}}} = \frac{1}{{{\rm{T}}\sqrt {\rm{\beta }} }}\)

Proportional + Derivate:

The additive combination of proportional & Derivative control is known as P-D control.

The overall transfer function for a PD controller is given by:

\({G_C}\left( s \right) = \frac{{U\left( s \right)}}{{E\left( s \right)}} = {K_P} + s{K_D}\)

PD controller is nothing but a differentiator (or) a High Pass Filter.

The frequency of noise is very high. So this high pass filter will allow noise into the system which results in noise amplification.

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. the transient response is improved
• Improves gain margin, phase margin, and resonant peak
• Increases the input noise
• Improves the stability

Effects of Proportional Integral (PI) controllers:

• Increases the type of the system by one
• Rise time and settling time increase 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