Bode Diagram, Control System Analysis, Transfer function

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:

null

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:

null

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

null

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.

null

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:

null

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

null

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

Resolving problems!!

WhatsApp:  +34633129287  Immediate attention!!

Twitter: @dademuch

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contacto: España. +34633129287

Caracas, Quito, Guayaquil, Cuenca. 

WhatsApp:  +34633129287   +593998524011  

Twitter: @dademuch

FACEBOOK: DademuchConnection

email: dademuchconnection@gmail.com

 

Bode Diagram

The Bode Diagrams – Plotting of the frequency response of a control system.

The Bode Diagrams is the plotting of the frequency response of a system with separate magnitude and phase plots. The log-magnitude and phase frequency response curves as functions of log ω  are called Bode plots or Bode diagrams. Sketching Bode plots can be simplified because they can be approximated as a sequence of straight lines. Straight-line approximations simplify the evaluation of the magnitude and phase frequency response.

When plotting separate magnitude and phase plots, the magnitude curve can be plotted in decibels (dB) vs. log ω, where dB = 20 log M. Meanwhile, the phase curve is plotted as phase angle vs. log ω.

Example

Plot The Bode Diagram for the frequency response of the characterized by the system Transfer Function G(s):

null

null

In construction…

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

Resolving problems!!

WhatsApp:  +34633129287  Immediate attention!!

Twitter: @dademuch

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contacto: España. +34633129287

Caracas, Quito, Guayaquil, Cuenca. 

WhatsApp:  +34633129287   +593998524011  

Twitter: @dademuch

FACEBOOK: DademuchConnection

email: dademuchconnection@gmail.com

 

 

Control System Analysis, The Nyquist Criteria

Stability via the Nyquist Diagram – The Nyquist Criteria

The Nyquist criterion can tell us if the system is stable or unstable by determining how many closed-loop poles are in the right half-plane of the closed-loop system of Figure 1:

null

Figure 1

Consider the contour A defined in s-plane of Figure 2:

null

Figure 2

If a contour, A, that encircles the entire right half-plane of the root-locus of the system determined by the characteristic equation 1+ G(s)H(s), is mapped through G(s)H(s), then the number of closed-loop poles, Z, in the right half-plane equals the number of open-loop poles, P, that are in the right half-plane minus the number of counterclockwise revolutions, N, around -1 of the mapping; that is, Z:

null

Thus, to reach stability, Z must be equal to zero.

This mapping is called the Nyquist diagram, or Nyquist plot, of G(s)H(s).

To understand the Nyquist criteria for stability, we must first establish four important concepts that will be used during its application:

(1) the relationship between the poles of 1+ G(s)H(s) and the poles of G(s)H(s); (2) the relationship between the zeros of 1+ G(s)H(s) and the poles of the closed-loop transfer function (3) the concept of mapping points; and (4) the concept of mapping contours.

We could demonstrate that the poles of 1+ G(s)H(s) are the same as the
poles of G(s)H(s), the open-loop system, and (2) the zeros of  1+ G(s)H(s) are the
same as the poles of closed-loop transfer function of the system.

In construction…

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

Resolving problems!!

WhatsApp:  +34633129287  Immediate Attention!!

Twitter: @dademuch

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contacto: España. +34633129287

Caracas, Quito, Guayaquil, Cuenca. 

WhatsApp:  +34633129287   +593998524011  

Twitter: @dademuch

FACEBOOK: DademuchConnection

email: dademuchconnection@gmail.com

Control System Analysis, The Nyquist Criteria

Sketching the Nyquist Diagram

The Nyquist diagram is also known as “The Polar Trace” of a transfer function G(jω), is a graph of the magnitude of G(jω) against the phase angle of G(jω) in polar coordinates, according to ω variables of zero to infinity. Therefore, The Nyquist diagram is the geometric place of vectors:

null
conforming ω modified from zero to infinity. Note that, in polar graphs, the phase angles are positive (negative) if they are measured counterclockwise (clockwise) from the positive real axis.

The following Figure shows an example of a Nyquist Diagram:

null

All points of the polar trace of G(jω) represent the end point of a vector at a given value of ω. The projections of G(jω) on the real and imaginary axes are their real and imaginary components. The magnitude and phase angle of G(jω) must be calculated directly for each frequency ω in order to construct polar traces.

Conceptually, the Nyquist diagram is plotted by substituting the points
of the contour that encloses the right half-plane into the function G(s)H(s).  This process is called mapping. Next Figure shows the process of mapping:

null

Consider the closed-loop control system of Figure 1:

null

Figure 1

Thus, in the Nyquist diagram, the contour that encloses the right half-plane, shown in Figure 2, can be mapped through the function G(s)H(s), derived from Figure 1,  by substituting points along the contour into G(s)H(s):

null

Figure 2

If a contour, A, that encircles the entire right half-plane of the root-locus of the system determined by the characteristic equation 1+ G(s)H(s), is mapped through G(s)H(s), then the number of closed-loop poles, Z, in the right half-plane equals the number of open-loop poles, P, that are in the right half-plane minus the number of counterclockwise revolutions, N, around -1+j0 of the mapping; that is, Z:

null

Thus, Z must be equal to zero to reach stability.

This mapping is called the Nyquist diagram, or Nyquist plot, of G(s)H(s). For more information an examples, see: The Nyquist Criteria

 

Example

Consider the control system whose block diagram and diagram are shown in the following Figure 3:

null

Figure 3

Conceptually, the Nyquist diagram is plotted by substituting the points of the contour shown in Figure 4(a) into G(s)H(s):

null

Each Pole and Zero term of G(s) shown in Figure 3(b) is a vector in Figure 4(a) and 4(b). The resultant vector, , found at any point along the contour is in general the product of the Zero vectors divided by the product of the Pole vectors (see Figure 4(c)). Thus, the magnitude of the resultant is the product of the Zero lengths divided by the product of the Pole lengths, and the angle of the resultant is the sum of the Zero angles minus the sum of the Pole angles.

null

Figure 4

The mapping from point A to point C can also be explained analytically. From
A to C the collection of points along the contour is imaginary. Hence, from A to C,
G(s)H(s)=G(s)*1=G(s)=G(jω), or from Figure 3(b):

nullAt zero frequency:

null

Thus, the Nyquist diagram starts at 50/3 at an angle of 0°. As ω increases the real part remains positive, and the imaginary part remains negative.

At null the real part becomes negative. At null, the Nyquist diagram crosses the negative real axis since the imaginary term goes to zero. The real value at the axis crossing, point Q in Figure 4(c), is -0.874. Continuing toward , the real part is negative, and the imaginary part is positive. At infinite frequency:

nullor approximately zero at 90°.

Around the infinite semicircle from point C to point D shown in Figure 4(b), the vectors rotate clockwise, each by 180°. Hence, the resultant undergoes a counterclockwise rotation of 3×180, starting at point C’ and ending at point D’ of Figure 4(c).

Nyquist diagram with Matlab

Consider the following open loop transfer function:

null

To create the Nyquist Diagram of the system, use the following commands in the command window of Matlab:

>> s=tf(‘s’)

>> G=1/(s^2+0.8*s+1)

>> nyquist(G)

This line of commands yields:

null

We can obtain information of points of interest in the Nyquist Diagram by cliking once over the contour. This yields:

null

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

I solve problems!!

WhatsApp:  +34633129287  Immediate Attention!!

Twitter: @dademuch

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contacto: España. +34633129287

Caracas, Quito, Guayaquil, Cuenca. 

WhatsApp:  +34633129287   +593998524011  

Twitter: @dademuch

FACEBOOK: DademuchConnection

email: dademuchconnection@gmail.com

 

 

 

Block Diagram, Control System Analysis

Definition of Electromechanical System

“Electromechanical Systems are those hybrid systems of mechanical and electrical variables.” Applications for electromechanical components cover a broad spectrum, from control systems for robots and star-trackers, to household appliances and hard disk position controls on a computer, or the control of DC motors in air conditioning systems for residential installations.

gettyimages-155388818-1024x1024
Detail of copper winding, stack and shaft of a electric permeant magnet motor for home appliances.

Figure 2.1 shows an electromechanical drive system. It consists of a power and energy source, a gate circuit for the converter, electronic converters (rectifier, inverter, electronic power controller), current sensors (shunts, current transformer, Hall sensor), voltage sensor (divider voltage, potential transformer), speed sensors (tachometers) and displacement sensors (encoders), three-phase rotary machines, gearboxes and specific loads (pump, fan, car, etc.). In Figure 2-1 all components, with the exception of gears, are represented by a Transfer Function (output variables as a function of time), while the gearbox is represented by a Characteristic Function (Xout output variable depending on the input variable Xin)

The electric machine is perhaps the best example of an electromechanical device because of the frequency with which it is used in numerous applications of daily life. An electric machine is a device that can convert mechanical energy into electrical energy (a hydroelectric plant, for example), or convert electrical energy into mechanical energy (a motor).

For the study of electromechanical systems from the point of view of control engineering, we have decided to focus our attention on DC motors, especially armature-controlled DC servo motors, as they are components intensively used in emerging industries that combine electromechanical engineering with Telematics, as is the case with Robotics and Drones technology. And because, precisely, these areas, together with that of electric vehicles and industry 4.0, are initiating a paradigm shift in all areas of life.

null

We are dedicated to developing the mathematical model of an electromechanical system with DC motor, as well as the characteristics of this system when it is part of an open loop or closed loop control system (Servomotors). We also provide numerous examples of how to determine and use the Transfer Function of an electromechanical system to analyze its stability and its response over time (transient and steady state).

And gradually we will cover these industries more specifically, with great potential for innovation and future labor demand.

NEXT:

Sources:

  1. Control Systems Engineering, Nise
  2. Sistemas de Control Automatico Benjamin C Kuo
  3. Modern_Control_Engineering, Ogata 4t
  4. Libro Rashid – Power Electronic Handbook p 663-666
  5. Getty Images

Literature review by:

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

WhatsApp:  +34633129287  

Twitter: @dademuch

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contacto: España. +34633129287

Caracas, Quito, Guayaquil, Cuenca. 

WhatsApp:  +34633129287   +593998524011  

Twitter: @dademuch

FACEBOOK: DademuchConnection

email: dademuchconnection@gmail.com

Block Diagram, Control System Analysis

Open-loop control system – Electromechanical.

A control system can be open loop or closed loop. To understand this difference we must pay attention to the effect that the output has on the system control action (Ogata, 1998). If the output influences the control action, the system is closed loop. On the other hand, if the output does not affect the control action, we are in the presence of an open-loop control system.

The controlled variable is the quantity or condition that is measured or controlled. The manipulated variable, or control variable, is the quantity or condition that the controller modifies to affect the value of the controlled variable.

To better understand the concept of an open-loop system, consider the following scheme, which represents a very frequent and basic component in every electromechanical system, a Potentiometer:

null
Figure 1

In practice, the operation of this system is simple. Needle position B (control variable) depends on angular displacement desplazamiento Θi(t) (system input). The position of the needle determines a voltage Vo (t) (system output, controlled variable) that can have a value between +50 and -50 volts. In this system, the output does not affect the control action, which is the mechanical movement of the hand (controller). Therefore, it is an open loop system, which we can represent by the following block diagram:

null
Figure 2

If we wanted to configure the system of Figure 2 as a closed loop system, we would have to measure the output, first, and compare it with the reference signal, secondly, so that a Controller executes the controlling action based on the result of this comparison. This process could be represented by the following diagram:

null
Figure 3

Quite often, the Potentiometer in Figure 1 is the component that activates a DC Motor as shown in the following example:

null
                                 Figure 4

The system of Figure 4 is another example of an open-loop electromechanical system, which involves a greater number of components, including the use of a DC Motor and a Gearbox that allows to transform a rotational movement into a translational displacement, but in which the output does not influence the controlling action.

The following system, on the other hand, also has a Potentiometer that measures the displacement at the output and this measure influences the control action:

null
                                                        Figure 5

The system of Figure 5 is a closed loop electromechanical system that compares the output voltage c with the input voltage r. This comparison is manifested as a voltage difference ev=r-c that then feeds a Differential Amplifier, which in turn activates a DC Motor that, through a Gear system, moves the Potentiometer c. This process is repeated until ev=0, that is, until r = c. In other words, the system looks for the output to match the input, so this system is called the automatic input follower system, Position Control System or Servosystem.

When a DC Motor is part of a Servo System, it is called Servo Motor. The Position Control System is one of the most used essential mechanisms in engineering, hence its great importance.

Sources:

  1. Control Systems Engineering, Nise
  2. Sistemas de Control Automatico Benjamin C Kuo
  3. Modern_Control_Engineering, Ogata 4t
  4. Libro Rashid – Power Electronic Handbook p 663-666
  5. Getty Images

Literature review made by:

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

WhatsApp:  +34633129287  

Twitter: @dademuch

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contacto: España. +34633129287

Caracas, Quito, Guayaquil, Cuenca. 

WhatsApp:  +34633129287   +593998524011  

Twitter: @dademuch

FACEBOOK: DademuchConnection

email: dademuchconnection@gmail.com

Control System Analysis, PID, PID Control

PI Controller – Proportional Integral – Control System

Steady-state error can be improved by placing an open-loop pole at the origin,
because this increases the system type by one
. For example, a Type 0 system
responding to a step input with a finite error, will responds with zero error if the system
type is increased by one. But, we want to do this without affecting the transient response.

However, if we add a pole at the origin to increase the system type, the angular contribution of the open-loop poles at hypothetical point A is no longer 180, and the root locus no longer goes through point A, as shown in Figure 1.a and 1.b:

Figure 1.

To solve the problem, we also add a zero close to the pole at the origin, as shown
in Figure 2:

Figure 2.

Now the angular contribution of the compensator zero and compensator pole cancel out, point A is still on the root locus, and the system type has been increased. That is how we can improve the steady-state error without affecting the transient response.

A compensator with a pole at the origin and a zero close to the pole is called an ideal integral compensator, or Proportional-plus-Integral PI compensator, which transfer function Gc(s)  is:

Next example allows to find how PI compensation works.

For control system of Figure 3, it is required to reduce steady-state error to zero, through a PI controller, keeping damping at ξ=0.173. The plant transfer function is G(s) and its original controller is represented by the gain k:

Figure 3.

The first step is to evaluate the system before the compensation, then to find the location of the two closed-loop second-order dominant poles  in order to get the damping requiered by the design specifications.

Figure 4 shows the Root-Locus of the system before compensation:

>> sgrid(z,0)
>> s=tf(‘s’);
>> G=1/((s+1)*(s+2)*(s+10));
>> rlocus(G);

Figure 4.

Using the damping line in Matlab, we can find the intersection point between the root-locus and the value ξ=0.173as we can see in Figure 5:

>> z=0.173;
>> sgrid(z,0)

Figure 5.

The intersection of Figure 5 shows us that adjusting the gain to k=165 of the original controller, we obtain the damping requiered: ξ=0.173. We also see in Figure 5 that the closed-loop second-order dominant poles s1 and s2, before compensation are:

Now we look for the third pole in the root locus. In Figure 6 we must set the same gain k=165 at the third pole line, in consequence s3 is located at:

Figure 6.

With k=165 we calculate the steady-state error e1(∞) for a step input, before compensation:

Where kp1 the position constant before compensation:

Where kG(s) is the system forward transfer function multiplied by the adjusted gain, before compensation, as in Figure 3. Therefore:

We add a PI controller in cascade into the system, as in Figure 7:

Figure 7.

Here, we have matched the gain constant of the compensator with the original gain constant, that is to say k=ki. The constant a is determined by the location of compensator zero, wich must be near the compensator pole. That is why we set the compensator zero at s=-0.1 , that is to say  a=0.1. The root locus of this compensated system is in Figure 8:

>> G=(s+0.1)/(s*(s+1)*(s+2)*(s+10));
>> rlocus(G);

Figure 8.

In view of the fact that we want to maintain the transient response as unchanged as possible, in Figure 9 we draw the damping line in the root locus and search for the point of intersection between the lines of the root locus and ξ=0.173:

>> z=0.173;
>> sgrid(z,0);

Figure 9.

Adjusting the gain to k=159 in Figure 9, we obtain the damping ξ=0.173. We see that closed-loop second-order dominant poles s1 and s2, after compensation, are:

Looking for the third pole in the root locus,  we must set the gain k=159 at the third pole line. After that, s3 is located at:

These results show that approximately the values ​​of the 3 poles before and after the PI compensation have been conserved, indicating a similar transient response after correcting the error in steady state from 0.108 to 0, as shwon later.

The forward transfer function G2(s)  of the system after compensation is:

One more time, we calculate steady-state error e2(∞) for a step input, after compensation:

In consequence:

Figure 10 compares the step response of the closed-loop system  before and after compensatio PI:

>> G1=165/((s+1)*(s+2)*(s+10));
>> sys_antes=feedback(G1,1);
>> G2=(159*(s+0.1))/(s*(s+1)*(s+2)*(s+10));
>> sys_despues=feedback(G2,1);
>> step(G1,G2)

Figure 10.

Figure 10 shows that through PI compensation we have managed to improve the steady-state error without considerably modifying the transient response of the original system.

Compensación en Cascada - Lag Compensation

In construction…

Source :

  1. Control Systems Engineering, Nise

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

Twitter: @dademuch

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contact: Caracas, Quito, Guayaquil, Cuenca. telf – 0998524011

WhatsApp: +593998524011   +593981478463 

Twitter: @dademuch

FACEBOOK: DademuchConnection

email: dademuchconnection@gmail.com

 

Control System Analysis, PID Control

Design via Root Locus – Improving Steady-state error via Cascade Compensation

We discuss two ways to improve the steady-state error of a feedback control system using cascade compensation. One objective of this design is to improve the steady-state error without appreciably affecting the transient response.

Improving Transient Response - Compensation

We have seen before that setting the gain at a particular value on the root locus yields the transient response dictated by the poles at that point on the root locus. Thus, we are limited to those responses that exist along the root locus. (See Sketching Root Locus with Matlab – Control Systems)

Unfortunately, most of the time the overshoot specification for designing control systems exceed the posibilities of the current root locus. What can we do then?

Rather than change the existing system, we augment, or compensate, the system with additional poles and zeros, so that the compensated system has a root locus that goes through the desired pole location for some value of gain. One of the advantages of compensating a system in this way is that additional poles and zeros can be added at the low-power end of the system before the plant. We should evaluate the transient response through simulation after the design is complete to be sure the requirements have been met.

There are two configurations of compensation mostly used in control systems design: cascade compensation and feedback compensation. These methods are modeled in Figure 1 and Figure 2:

Figure 1. Cascade Compensation of a control system.

With cascade compensation, the compensating network, G1(s), is placed at the low-power end of the forward path in cascade with the plant, Figure 1.

Figure 2. Feedback Compensation of a control system.

With feedback compensation, the compensator, H1(s), is placed in the feedback path, Figure 2.

Both methods change the open-loop poles and zeros, thereby creating a new root locus that goes through the desired closed-loop pole location.

Cascade Compensation - PI Controller

Steady-state error can be improved by placing an open-loop pole at the origin,
because this increases the system type by one. For example, a Type 0 system
responding to a step input with a finite error, will responds with zero error if the system
type is increased by one. But, we want to do this without affecting the transient response.

However, if we add a pole at the origin to increase the system type, the angular contribution of the open-loop poles at hypothetical point A is no longer 180, and the root locus no longer goes through point A, as shown in Figure 3.a and 3.b:

Figure 3.

To solve the problem, we also add a zero close to the pole at the origin, as shown
in Figure 4:

Figure 4.

Now the angular contribution of the compensator zero and compensator pole cancel out, point A is still on the root locus, and the system type has been increased. That is how we can improve the steady-state error without affecting the transient response.

A compensator with a pole at the origin and a zero close to the pole is called an ideal integral compensator, or Proportional-plus-Integral PI compensator, which transfer function Gc(s)  is:

Next example allows to find how PI compensation works.

For control system of Figure 5, it is required to reduce steady-state error to zero, through a PI controller, keeping damping at ξ=0.173. The plant transfer function is G(s) and its original controller is represented by the gain k:

Figure 5.

The first step is to evaluate the system before the compensation, then to find the location of the two closed-loop second-order dominant poles  in order to get the damping requiered by the design specifications.

Figure 6 shows the Root-Locus of the system before compensation:

>> sgrid(z,0)
>> s=tf(‘s’);
>> G=1/((s+1)*(s+2)*(s+10));
>> rlocus(G);

Figure 6.

Using the damping line in Matlab, we can find the intersection point between the root-locus and the value ξ=0.173as we can see in Figure 7:

>> z=0.173;
>> sgrid(z,0)

Figure 7.

The intersection of Figure 7 shows us that adjusting the gain to k=165 of the original controller, we obtain the damping requiered: ξ=0.173. We also see in Figure 7 that the closed-loop second-order dominant poles s1 and s2, before compensation are:

Now we look for the third pole in the root locus. In Figure 8 we must set the same gain k=165 at the third pole line, in consequence s3 is located at:

Figure 8.

With k=165 we calculate the steady-state error e1(∞) for a step input, before compensation:

Where kp1 the position constant before compensation:

Where kG(s) is the system forward transfer function multiplied by the adjusted gain, before compensation, as in Figure 5. Therefore:

We add a PI controller in cascade into the system, as in Figure 9:

Figure 9.

Here, we have matched the gain constant of the compensator with the original gain constant, that is to say k=ki. The constant a is determined by the location of compensator zero, wich must be near the compensator pole. That is why we set the compensator zero at s=-0.1 , that is to say  a=0.1. The root locus of this compensated system is in Figure 10:

>> G=(s+0.1)/(s*(s+1)*(s+2)*(s+10));
>> rlocus(G);

Figure 10.

In view of the fact that we want to maintain the transient response as unchanged as possible, in Figure 11 we draw the damping line in the root locus and search for the point of intersection between the lines of the root locus and ξ=0.173:

>> z=0.173;
>> sgrid(z,0);

Figure 11.

Adjusting the gain to k=159 in Figure 11, we obtain the damping ξ=0.173. We see that closed-loop second-order dominant poles s1 and s2, after compensation, are:

Looking for the third pole in the root locus,  we must set the gain k=159 at the third pole line. After that, s3 is located at:

These results show that approximately the values ​​of the 3 poles before and after the PI compensation have been conserved, indicating a similar transient response after correcting the error in steady state from 0.108 to 0, as shwon later.

The forward transfer function G2(s)  of the system after compensation is:

One more time, we calculate steady-state error e2(∞) for a step input, after compensation:

In consequence:

Figure 12 compares the step response of the closed-loop system  before and after compensatio PI:

>> G1=165/((s+1)*(s+2)*(s+10));
>> sys_antes=feedback(G1,1);
>> G2=(159*(s+0.1))/(s*(s+1)*(s+2)*(s+10));
>> sys_despues=feedback(G2,1);
>> step(G1,G2)

Figure 12.

Figure 12 shows that through PI compensation we have managed to improve the steady-state error without considerably modifying the transient response of the original system.

Compensación en Cascada - Lag Compensation

In construction…

Source :

  1. Control Systems Engineering, Nise

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

Twitter: @dademuch

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contact: Caracas, Quito, Guayaquil, Cuenca. telf – 0998524011

WhatsApp: +593998524011   +593981478463 

Twitter: @dademuch

FACEBOOK: DademuchConnection

email: dademuchconnection@gmail.com

 

Análisis de sistemas de control, Lugar geométrico de las raíces, PID Control

Design a PD compensator to yield a 16% overshoot – Control system

Given the system of Figure 1, design a PD compensator to yield a 16% overshoot, with a threefold reduction in settling time (one-third of the uncompensated system’s settling time).

Figure 1

Let us first evaluate the performance of the uncompensated system. The root locus for the uncompensated system is shown in Figure 2:

>> s=tf(‘s’);
>> G=1/(s*(s+4)*(s+6));

Figure 2

Since 16% overshoot is equivalent to ξ=0.504, we search along that damping ratio line in Figure 3:

>> z=0.504;
>> sgrid(z,0);

Figure 3

According to Figure 3, adjusting the gain to k=43.4 we get ξ=0.504 and a natural frequency ω=2.39 rad/s. 

Based upon a second-order approximation, we can use the 2% criteria and calculate the settling-time Ts1 before the compensation, as a function of the naural frequency ω  and the damping ξ, by means of the following equation:

Simulation of Figure 3 generates the necessary values for equation (1), so that:

In the other hand, the value of the factor ω*ξ =1.2045 matches the real part σ  of closed-loop second-order dominant poles, as we can see in Figure 3 or by the following command in Matlab, taking into consideration that the straight-forward transfer function is now G1:

>> G1=43.4/(s*(s+4)*(s+6));
>> sys_antes=feedback(G1,1)

>> damp(sys_antes)

The desig requirements ask for an 16% overshoot and a reduction of the settling-time of 1/3 after compensation. So, the settling-time Ts2 after compensation is:

Using equation (1) we can know the value of the factor ω*ξ  after compensation:

That is to say, the real part of second-order dominant poles after compensation is σ=3.6137. To find the imaginary part wd we use the root-locus of  Figure 4:

Figure 4.

Consequently, after compensation the second-order dominant poles must be located at   p=-3.6137+j6.1940.

Now, to evaluate the whole system we will use point p as a test point.

PD compensation consists of a cascaded controller with a Gc(s) transfer funcion that is:

The configuration of such a controller is:

Figure 5.

Next step is to design the location of Zero zc using the test point and finding the equivalent values for k1 and k2.

The result is the sum of the angles to the design point of all the poles and zeros of the compensated system except for those of the compensator zero itself. The difference between the result obtained and 180 is the angular contribution required of the compensator zc es:

The geometry is shown in Figura 6, where we can get the real part of zc by means of the following formula:

Figure 6.

From where:

Now, we study the root-locus of Figure 7, where the forward-path transfer function is G2:

>> G2=(s+3.006)/(s*(s+4)*(s+6));
>> rlocus(G2)

Figure 7.

According to Figure 8, adjusting the gain k=47.4 we keep ξ=0.504, an overshoot 16%,  the second-order dominant pole s=-3.6137+j6.1940, at a natural frequency ω=7.17 rad/s.

>> z=0.504;
>> sgrid(z,0);

Figure 8.

With this new data, we evaluate the settling-time Ts2 after compensation:

It shows that we have achieved the design goal. Figure 9 compares the response of the closed-loop system to an step input before and after  PD compensation:

>> G=43.4/(s*(s+4)*(s+6));
>> G3=(47.4*(s+3.006))/(s*(s+4)*(s+6));
>> sys_before=feedback(G,1);
>> sys_after=feedback(G3,1);
>> step(sys_before,sys_after)

Figure 9.

The response of Figure 9 shows a considerable improvement in the settling-time and, in general, the compensation allows a faster system with an overshoot that does not vary much. Before compensation,  Ts=3.4712 s. After compensation, Ts=1.1527 s.

An alternative design process in Matlab

Use MATLAB, the Control System Toobox, and the following steps to use SISOTOOL to perform the design of last Example.

  1. Type sisotool in the MATLAB Command Window.
  2. Select Import in the File menu of the SISO Design for SISO Design Task Window.
  3. In the Data field for G, type zpk([],[0,-4,-6],1) and hit ENTER on the keyboard. Click OK.
  4. On the Edit menu choose SISO Tool Preferences . . . and select Zero/pole/gain: under the Options tab. Click OK.
  5. Right-click on the root locus white space and choose Design Requirements/New . . .
  6. Choose Percent overshoot and type in 16. Click OK.
  7. Right-click on the root locus white space and choose Design Requirements/New . . .
  8. Choose Settling time and click OK.
  9. Drag the settling time vertical line to the intersection of the root locus and 16%
    overshoot radial line.
  10. Read the settling time at the bottom of the window.
  11. Drag the settling time vertical line to a settling time that is 1/3 of the value
    found in Step 9.
  12. Click on a red zero icon in the menu bar. Place the zero on the root locus real axis by clicking again on the real axis.
  13. Left-click on the real-axis zero and drag it along the real axis until the root locus intersects the settling time and percent overshoot lines.
  14. Drag a red square along the root locus until it is at the intersection of the root locus,
    settling time line, and the percent overshoot line.
  15. Click the Compensator Editor tab of the Control and Estimation Tools Manager window to see the resulting compensator, including the gain.

Source:

  1. Control Systems Engineering, Nise

Written by: Larry Francis Obando – Technical Specialist – Educational Content Writer.

Mentoring Académico / Empresarial / Emprendedores

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, Caracas.

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contact: Caracas, Quito, Guayaquil, Cuenca – Telf. 00593998524011

WhatsApp: +593998524011    /    +593981478463

Twitter: @dademuch

FACEBOOK: DademuchConnection

email: dademuchconnection@gmail.com

Control System Analysis, Electrical Engineer, PID Control

Design via Root Locus – Improving Transient Response via Cascade compensation

In this article, we discuss the PD controller and Lead Compensation, two ways to improve the transient response of a feedback control system by using cascade compensation. Typically, the objective is to design a response that has a desirable percent overshoot and a shorter settling time than the uncompensated system.

Improving Transient Response - Compensation

We have seen before that setting the gain at a particular value on the root locus yields the transient response dictated by the poles at that point on the root locus. Thus, we are limited to those responses that exist along the root locus. (See Sketching Root Locus with Matlab – Control Systems)

Unfortunately, most of the time the overshoot specification for designing control systems exceed the posibilities of the current root locus. What can we do then?

Rather than change the existing system, we augment, or compensate, the system with additional poles and zeros, so that the compensated system has a root locus that goes through the desired pole location for some value of gain. One of the advantages of compensating a system in this way is that additional poles and zeros can be added at the low-power end of the system before the plant. We should evaluate the transient response through simulation after the design is complete to be sure the requirements have been met.

There are two configurations of compensation mostly used in control systems design: cascade compensation and feedback compensation. These methods are modeled in Figure 1 and Figure 2:

Figure 1. Cascade Compensation of a control system.

With cascade compensation, the compensating network, G1(s), is placed at the low-power end of the forward path in cascade with the plant, Figure 1.

Figure 2. Feedback Compensation of a control system.

With feedback compensation, the compensator, H1(s), is placed in the feedback path, Figure 2.

Both methods change the open-loop poles and zeros, thereby creating a new root locus that goes through the desired closed-loop pole location.

Cascade Compensation - PD controller

As we said before, sometimes poles and zeros must be added in the forward path to produce a new open-loop function whose root locus goes through the design point on the s-plane, in order to meet design requirements. One way to speed up the original system that generally works is to add a single zero to the forward path.

This zero can be represented by a cascade compensator whose transfer function Gc(s) is:


This function, the sum of a differentiator s and a pure gain Zc, is called an ideal derivative compensation, or Proportional-Derivative PD controller. In summary, transient responses unattainable by a simple gain adjustment (proportional controller) can be obtained by augmenting the system’s zeros with an ideal derivative controller.

Let´s use the Root Locus of Figure 3 to find out how a PD controller works. There, we have the Root Locus of a control system which forward transfer function G(s) with unitary feedback is:

If K=1, the commands in Matlab would be:

>> s=tf(‘s’);
>> G=1/((s+1)*(s+2)*(s+5));
>> rlocus(G);

Figure 3. Root Locus for G(s)

Suppose that we want to operate the system of Figure 3 with a damping ratio ξ=0.4. Figure 4 shows that we can get this damping ratio with a proportional compensator, setting the gain K=23.7:

>> z=0.4;
>> sgrid(z,0);

Use right click to select the damping:

Figure 4. Location in the RL of a gain K=23.7 and ξ=0.4

Figure 5 shows the Step Response of the closed-loop system for Kp=23.7 and ξ=0.4, and the values of the main parameters:

>> G1=23.7/((s+1)*(s+2)*(s+5));
>> sys1=feedback(G1,1);
>> step(sys1);
>> stepinfo(sys1)

Figure 5. Step response of the closed-loop uncompensated system 

Suppose now that we want to mantain the damping ratio ξ=0.4, improving rise time and settling time, making the system faster. That would be imposible using only a proportional controller because we are limited by the Root Locus according to Figures 3 and 4.

The uncompensated system of Figure 3 could becomes a compensated system by the addition of a compensating zero at -2, in Figure 6, using a cascade compensator whose transfer function Gc(s) is:

>> G2=((s+2))/((s+1)*(s+2)*(s+5));
>> rlocus(G2);

Figure 6. Root Locus for the compensated system.

Figure 7 shows that we can get a damping ratio ξ=0.4. setting the gain K=51.2:

>> z=0.4;
>> sgrid(z,0);

Use right click to select the damping:

Figure 7. Location in the RL of  ξ=0.4

Figure 8 shows the Step Response of the closed-loop system for Kp=51.2 and ξ=0.4, and the values of the main parameters:

>> G3=(51.2*(s+2))/((s+1)*(s+2)*(s+5));
>> sys2=feedback(G3,1);
>> step(sys2);
>> stepinfo(sys2)

Figure 8. Step response of the closed-loop compensated system

Mantaining the same damping ratio ξ=0.4, Rise Time has improved (from 0.6841 s to 0.1955 s) and Settling Time has improved (from 3.7471 s to 1.1218 s). However, Overshoot has increased (from 23.3070 to 25.3568) and also the Peak has increased (from 0.8672 to 1.1420). Figure 9 compares graphically both of the responses, before and after the PD compensation:

>>step(sys1, sys2)

Figure 9. Step response of Compensated Vs. Uncompensated System.

Figure 9 also shows that the final value is closer to the reference value (1), so the steady-state error has improved with PD compensation (from 0.297 to 0.088). However, readers must not assume that, in general, improvement in transient response always yields an improvement in steady-state error.

Now that we have seen what PD compensation can do, we are ready to design our own PD compensator to meet a transient response specification.

1) Given the system of Figure 10, design PD compensator to yield a 16% overshoot, with a threefold reduction in settling time.

Figure 10.

In construction…

How do we implement the PD controller?

The PD compensator used to improve the transient response is implemented with a proportional-plus-derivative (PD) controller. In Figure 11 the transfer function of the controller is:

Figure 11. Implementation of Proportional-plus-Derivative (PD) controller.

Lead Compensation

Just as the active ideal integral compensator can be approximated with a passive lag
network, an active ideal derivative compensator can be approximated with a passive
lead compensator. When passive networks are used, a single zero cannot be
produced; rather, a compensator zero and a pole result. However, if the pole is
farther from the imaginary axis than the zero, the angular contribution of the
compensator is still positive and thus approximates an equivalent single zero. In
other words, the angular contribution of the compensator pole subtracts from the
angular contribution of the zero but does not preclude the use of the compensator to improve transient response, since the net angular contribution is positive, just as for a single PD controller zero.

The advantages of a passive lead network over an active PD controller are that
(1) no additional power supplies are required and (2) noise due to differentiation is
reduced.

In construction…

Source:

  1. Control Systems Engineering, Nise

Written by:

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

Twitter: @dademuch

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contact: Caracas, Quito, Guayaquil, Cuenca. telf – 0998524011

WhatsApp: +593998524011   +593981478463 

Twitter: @dademuch

FACEBOOK: DademuchConnection

email: dademuchconnection@gmail.com