Open loop and closed loop transfer function – examples

To understand the concept of Transfer Function in open loop, or on the contrary, closed loop, we use a block diagram of a closed loop system, Figure 1:

Figure 1

Where G (s) is the transfer function of the plant and H (s) is the transfer function of the sensor. The sensor generates a signal B (s) that is fed back to the summing point, where it is compared with the reference signal R (s), generating a signal called the error signal E(s). Applying block algebra to Figure 1 we can clearly see that the output signal C (s) can be obtained by multiplying E (s) by G (s):

That is to say:

The function G (s) of equation (1) is known as the direct path transfer function (quotient between the output and the error signal):

Again applying block algebra to Figure 1 we can see that the feedback signal B (s) can be obtained by multiplying C (s) by H (s), that is:

That is:

The product G(s)H (s) from equation (2) is known as the open-loop transfer function (quotient between the feedback signal and the error signal):

Important notes:

• If the transfer function H (s) of the feedback path (FT of the sensor) is equal to one, H(s)=1, only in this case, the closed-loop transfer function is equal to the transfer function direct path;
• The direct path transfer function G (s) is also known simply as the Direct Transfer Function.

That is, if the system is represented by the DB in Figure 2:

Figure 2

Then the direct transfer function G(s) is also the open-loop function.

Once again, applying block algebra to Figure 1 we can see that the output signal C (s) can be obtained by multiplying G (s) by E (s), that is:

Solving for C (s), we obtain that:

From where:

The function C (s) / R (s) of equation (3) is known as the closed-loop transfer function (quotient between the output signal and the input signal):

Important note: Equation (3) allows us to obtain the Laplace transform of the output for any input, once we know what the closed-loop transfer function is, by:

Example:

 

I suggest to visit: Effect of adding a zero to a control system

Source:

1. Katsuhiko Ogata, Ingeniería de Control Moderno, páginas 65-66.

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Mass-spring-damper rotational system. Problems solved. Catalog 3

The transfer function of a rotational Mass-Spring-Damper System. 

In this PDF guide, the Transfer Function of the exercises that are most commonly used in the mass-spring-damper system classes that are in turn part of control systems, signals and systems, analysis of electrical networks with DC motor, is determined. electronic systems in mechatronics, etc. It is a good resource to also learn how to obtain the block diagram of the system, or the representation in state variables. Request via email – WhatsApp. Payment is provided by PayPal, Credit or debit card. Cost: € 15

Below, the statements of problems solved in this guide.

1. Given the System of Figure 22, find the transfer function Θ_(s)/T_(s).

2. Given the System of Figure 23, find the transfer function Θ_(s)/T_(s).

3. Given the System of Figure 24, find the transfer functions Θ_1(s)/T_(s)  and Θ_2(s)/T_(s).

4. Find the state-space representation of the system of the previous exercise, Figure 24, taking Θ_1(t)  as the output and T_(t) as the input. Build the block diagram of the system and determine the transfer function Θ_1(s)/T_(s).

5. Given the System of Figure 26, find the transfer function Θ_L(s)/T_m(s).

6. Given the System of Figure 27, find the transfer functions Θ_1(s)/T_m(s) and Θ_2(s)/T_m(s).

7. Given the System of Figure 28, find the transfer functions Θ_1(s)/T_(s) and Θ_2(s)/T_(s).

8. Given the System of Figure 29, find the transfer function Θ_2(s)/T_(s).

9. Given the System of Figure 30, find the transfer functions  Θ_1(s)/T_(s) and Θ_2(s)/T_(s) . Consider: k₁=9, k₂=3 N-m/rad, b₁=8, b₂=1 N-m-s/rad, J₁=5, J₂=3 Kg-m².

10. Find the state-space representation of the system of the previous exercise, Figure 30, taking Θ_2(t) as the output and T_(t) as the input. Direct Transform the state-space representation obtained into the transfer function Θ_2(s)/T_(s). Consider k₁=9, k₂=3 N-m/rad, b₁=8, b₂=1 N-m-s/rad, J₁=5, J₂=3 Kg-m².

11. Find the state-space representation of the system in Figure 32, taking Θ_2(t)  as the output and T_(t) as the input. Directly, using Matlab, Transform the state-space representation obtained into the transfer function  Θ_2(s)/T_(s). Consider k₁= k₂=1 N-m/rad, b₁= b₂=1 N-m/rad, J=1 Kg-m².

12. Given the System of Figure 33, find the transfer functions  Θ_1(s)/T_m(s) and Θ_2(s)/T_m(s).

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You may be also interested in:

• Mass-Spring-Damper System. Problem solved. Catalog 1
• Mass-Spring-Damper System. Problem solved. Catalog 2

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Mass-spring-damper Problems solved. Catalog 2

The transfer function of a Mass-Spring-Damper System. 

In this PDF guide, the Transfer Function of the exercises that are most commonly used in the mass-spring-damper system classes that are in turn part of control systems, signals and systems, analysis of electrical networks with DC motor, is determined. electronic systems in mechatronics, etc. It is a good resource to also learn how to obtain the block diagram of the system, or the representation in state variables. Request via email – WhatsApp. Payment is provided by PayPal, Credit or debit card. Cost: € 15

Below, the statements of problems solved in this guide.

1. Given the System of Figure 12, find the transfer functions Y_1(s)/U_(s) and Y_2(s)/U_(s).

2. Given the System of Figure 13, find the transfer functions Y_1(s)/U_(s) and Y_2(s)/U_(s).

3. Given the System of Figure 14, find the transfer functions X_1(s)/U_(s)  and  X_2(s)/U_(s).

4. Given the System of Figure 15, find the transfer function X_2(s)/U_(s). Consider k₁=1, k₂= 15 N/m, b₁=4, b₂= 16 N-s/m, m₁= 8, m₂=3  Kg.

5. Given the System of Figure 16, find the transfer function X_3(s)/U_(s). Consider k₁=5, k₂= 4, k₃= 4  N/m, b₁=2, b₂= 2, b₃= 3  N-s/m, m₁= 4, m₂=5, m₃=5  Kg.

6.Given the System of Figure 17, find the transfer function X_1(s)/U_(s). Consider k₁=k₂= 1 N/m, b₁= b₂= b₃= 1  N-s/m, m₁= 2, m₂=1, m₃=1  Kg. The same exercise is solved with state variables in the next number._.

7. Find the state-space representation of the system of the previous exercise, Figure 17, taking x_1(t) as the output and u_(t) as the input. Transform the state-space representation obtained in the transfer function X_1(s)/U_(s). Consider k₁=k₂= 1 N/m, b₁= b₂= b₃= 1  N-s/m, m₁= 2, m₂=1, m₃=1  Kg.

8. Given the System of Figure 19, find the transfer function Y_h(s)/f_up(s). Consider k_h=7, k_s=8, k_ave=5  N/m, b_f=3, b_h= 10  N-s/m, m_h=1, m_f=2 Kg.

9. Given the System of Figure 20, find the transfer functions X_2(s)/U_(s) and X_3(s)/U_(s). Consider k₁=1, k₂=2, k₃=3, k₄=4 N/m, b₁=2,b₂= 1,b₃= 3 N-s/m, m₁=2,m₂=1,m₃=3  Kg.

10. Find the state-space representation of the system in Figure 21, taking x_3(t) as the output and u_(t) as the input. Transform the state-space representation obtained in the transfer function X_3(s)/U_(s). Consider k=2 N/m, b₁=b₂=b₃=b₄=b₅=1 N-s/m, m₁=2,m₂=1,m₃=1  Kg.

11. Determine the transfer function X_1(s)/ F_(s) and the block diagram of the system in Figure 22:

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You can be interested in:

• • Mass-Spring-Damper System. Problem solved. Catalog 1
  • Rotational Mass-Spring-Damper System. Problem solved. Catalog 3

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Obtaining Transfer Function from Bode Diagram

Bode plots are a convenient presentation of the frequency response data for
the purpose of estimating the transfer function. These plots allow parts of the
transfer function to be determined and extracted, leading the way to further
refinements to find the remaining parts of the transfer function.

Although experience and intuition are invaluable in the process, the following steps are still offered as a guideline:

1. Look at the Bode magnitude and phase plots and estimate the pole-zero configuration of the system. Look at the initial slope on the magnitude plot to determine system type. Look at phase excursions to get an idea of the difference between the number of poles and the number of zeros.
2. See if portions of the magnitude and phase curves represent obvious first- or second-order pole or zero frequency response plots.
3. See if there is any telltale peaking or depressions in the magnitude response plot that indicate an underdamped second-order pole or zero, respectively.
4. If any pole or zero responses can be identified, overlay appropriate ±20 or ±40 dB/decade lines on the magnitude curve or ±45°/decade lines on the phase curve and estimate the break frequencies.For second-order poles or zeros, estimate the damping ratio and natural frequency from the standard curves.
5. Form a transfer function of unity gain using the poles and zeros found. Obtain the frequency response of this transfer function and subtract this response from the previous frequency response (Franklin, 1991). You now have a frequency response of reduced complexity from which to begin the process again to extract more of the system’s poles and zeros. A computer program such as MATLAB is of invaluable help for this step.

Example

Find the transfer function of the subsystem whose Bode plots are shown in Figure 1:

Figure 1

Let us first extract the underdamped poles that we suspect, based on the peaking in the magnitude curve.We estimate the natural frequency to be near the peak frequency, or approximately 5 rad/s. From Figure 1, we see a peak of about 6.5 dB, which translates into a damping ratio of about ζ=0,24. The unity gain second-order function is thus:

The frequency response plot of this function is made and subtracted from the previous
Bode plots to yield the response in Figure 2:

Figure 2

Overlaying a -20 dB/decade line on the magnitude response and a -45°/decade line on the phase response, we detect a final pole. From the phase response, we estimate the break frequency at 90 rad/s. Subtracting the response of G2(s)=90/(s+90) from the previous response yields the response in Figure 3.

Figure 3

Figure 3 has a magnitude and phase curve similar to that generated by a lag function. We draw a -20 dB/decade line and fit it to the curves. The break frequencies are read from the figure as 9 and 30 rad/s. A unity gain transfer function containing a pole at -9 and a zero at -30 is G3(s)=0.3(s+30)/(s+9). Upon subtraction of G1(s)G2(s)G3(s), we find the magnitude frequency response flat ±1 dB and the phase response flat at -3± 5°. We thus conclude that we are finished extracting dynamic transfer functions as:

It is interesting to note that the original curve was obtained from the function:

Sources:

1. Modern_Control_Engineering, Ogata 4t
2. Control Systems Engineering, Nise
3. Sistemas de Control Automatico, Kuo

Literature review by:

Prof. Larry Francis Obando – Technical Specialist – Educational Content Writer

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