## Effect of adding a zero – example – Control System Design

Effects of the addition of zeros: the addition of a zero to the open loop transfer function has the effect of pulling the rootlocus to the left, with which the system tends to be more stable, and also accelerates the settlement of the response. The effect of such control is to introduce a degree of foresight to the system and speed up the transient response.

To illustrate this effect, let’s look at the following example:

`Example 1`

Suppose we are in the presence of a system with an unstable plant. An example of such a situation is the following:

Where G (s) is the transfer function of the plant and H (s) is the transfer function of the sensor used to assemble the closed-loop system, as shown in Figure 1:

Figure 1

We know from Block Algebra and Transfer Function theory that the open loop transfer function of this system Gd (s) is:

We also know that the rootlocus is drawn with the open loop transfer function of this system Gd (s), for which we can use the following command in Matlab:

Graph 1

Analysis: In graph 1 we can see that the system is unstable for all positive values ​​of gain K. That is, if we move through the blue and green curves, varying the value of K, as in graph 2, where K1 = 0.143; K2 = 3.66 and K1 = 30.5, respectively, we see that the poles of the system are located on the right side of the s-plane, and therefore it is an unstable system:

Let’s apply the principle of addition of a zero to the open loop transfer function for this case. We are going to add a zero at s = -0.5 (Figure 2), therefore the Gd (s) of the system is:

Figure 2

Let’s see the effect of adding a zero to the system by:

Graph 3

Analysis: In graph 3 we see that the LGR of the system has shifted to the left and that the system is stable for any positive value of the gain k, that is, that all the poles of the closed-loop system are located on the left side from plane s (Graph 4), an essential condition for the system to be stable:

Graph 4

Source:

Katsuhiko Ogata, Modern Control Engineering, pages 442-443.

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## 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…

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## 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:

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.

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

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## 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%
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.

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## 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:

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.

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);

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 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…

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## PID – Study of the proportional action with Matlab

To make the study of proportional action, consider the following system:

1. Implement the locus of the roots indicating the points of interest. We execute the following commands in Matlab:

>> s=tf(‘s’)

>> s1=(1)/(0.2*s+1)

>> s2=(2)/(0.1*s+1)

>> s3=(1)/(s+1)

>> G=s1*s2*s3

The direct transfer function G(s) is:Reshaping:If we add a proportional controller with a gain Kp, the direct transfer function G(s) is:Th closed-loop transfer function Gce(s) is:

The characteristic equation is:The characteristic equation in its form 1+G(s)H(s) is:

To obtain in Matlab the locus of the roots, we execute the following command:

>> rlocus(G)

We obtain:

Through this first exercise we get the locus of the roots of the system, we can observe the most immediate effect of applying a proportional controller: the displacement of the roots.

The change in the gain Kp allows us to change the value of the roots of the characteristic equation (when traveling through the blue, green and red lines of the locus in the previous graph, we see how the Kp gain changes), which is the same as changing the poles of the closed-loop transfer function. By changing these poles, we change the value of the damping coefficient ζ and the natural frequency ωn for a unit step input, thus adapting the transient response of the system to the design requirements that can be requested.

Note that the roots of the characteristic equation are in s=-10, s=-5 and s=-1, when Kp=0.

But in real terms Kp can not be zero, because in practice it means that we cancel the input to the system and thus, the output is zero as well. To represent the system operating without the proportional controller, we do Kp = 1. Under this condition, let’s see what is the value of ζ, as well as the value of three quantities of extreme importance for the designers: the overshoot Mp, the rise time Tr and the steady-state error ess. Then, suppose we want to modify this performance in terms of ζ and we vary Kp (we move the roots) until we achieve ζ = 0.5.

`Kp=1`

If Kp=1, then:

The closed-loop transfer function Gce(s) is:

We use the following commands in Matlab to know the values of ζ, the overshoot and the time of establishment:

> >Gce=feedback(G,1)

> >damp(Gce)

We obtain:

 Pole Damping Frequency -2.26e+00 + 2.82e+00i 6.26e-01 3.62e+00 -2.26e+00 – 2.82e+00i 6.26e-01 3.62e+00 -1.15e+01 1.00e+00 1.15e+01

Note that in the previous graph, Kp = Gain = 1. The dominant complex roots are s1 = -2.26 + j2.82 and s2 = -2.26-j2.82, so that, according to the graph, we can consider ζ = 0.626 when Kp = 1. Regarding the overshoot, the rise time and the steady-state error we use the following command:

>> stepinfo(Gce)

RiseTime: 0.5626

Overshoot: 7.5449

Peak: 0.7170

>> step(Gce)

If the input is the unit step and the output reaches the final value of c = 0.663, the steady-state error is ess = 1-0.663 = 0.337. We will see that despite varying the value of Kp and moving the roots, the proportional controller does not completely cancel out the steady-state error, it will always have a value different from zero, so an integral action is required to annul this error. On the other hand, the value of the overshoot is Mp = 7.17%, taking into account that the final value of the system is c = 0.663 and the maximum value reached is of c = 0.7170.

`Find Kp to achieve ζ = 0.5.`

The locus of the roots allows us to vary the value of the gain Kp until reaching the requested damping, ζ = 0.5. We move on the geometric place of the roots in Matlab by clicking on the line of the dominant poles and dragging the point until it reaches the requested damping:

The previous graph shows us that we can obtain a ζ = 0.5 when the gain Kp has an approximate value of 1.46. If Kp = 1.46, the direct transfer function and the closed-loop transfer function are:

We confirm the value of the damping by:

>> Gce2=(146)/(s^3+16*s^2+65*s+196)

> >damp(Gce2)

 Pole Damping Frequency -2.04e+00 + 3.51e+00i 5.02e-01 4.05e+00 -2.26e+00  – 3.51e+00i 5.02e-01 4.05e+00 -1.19e+01 1.00e+00 1.19e+01

> >stepinfo(Gce2)

RiseTime: 0.4356

Overshoot: 15.0397

Peak: 0.8569

It is observed that the overshoot will be higher (from 7.5449 to 15.0397) after the compensation (change the Kp value from 1 to 1.46) because the damping ζ is lower (from 0.626 to 5.02), while the rise time Tr improves slightly (from 0.5626 to 0.4356)

The answers to the unit step input of both systems (before and after the compensation), can be observed by the following Matlab command:

>> step(Gce,Gce2)

The final value of the system after the compensation (in red color) is approximately c = 0.748, so the steady-state error in this case is slightly lower, ess = 1-0.748 = 0.252. It is clearly seen in the graph that the rise time is shorter after the compensation, but at the cost of a larger overshoot due to a smaller damping.

Another more sophisticated Matlab tool to design compensators is SISO Design Tool. It can be called with rltool.

`>> rltool`

The graphical user interface is opened (GUI).

Once there, we can import systems fron the Matlab console through file>import>G>browse>available models>G>import>close>ok.

Supongamos el requerimiento ζ=0.5. Placing the cursor on the locus, we right click and select design requirement>new>design requirement type>damping ratio>0.5>ok.

The lower legend is obtained by placing the course on the pink dot, a hand is formed and left click. We can vary the graph until we achieve approximately the desired damping. If we place the course on the left side of the graph (white color), the legend appears Loop gain changed to 1.47. That is to say Kp=1.47.

Although, to be more exact, the value of the gain is Kp = 1.4663. This value can be seen in the other window that opens simultaneously with the Editor: Control and estimation tools manager. There, when selecting the Compensator Editor tab, we can see that C = 1.4663. Therefore, the tool allows us to be much more specific in terms of the value of the gain.

Going back to the graphic editor (SISO design task), we select analysis>response to step command, we obtain the unit step response in a new window. Once there, right click, we select plot type>step. We can see the value of main characteristics of transient response with Kp=1.46:

characteristics>rise time

characteristics>peak response

`Respuesta transitoria para diferentes valores de Kp.`

To complete the study it is only necessary to assign several values to Kp and analyze the transient response as well as the steady-state error of the different systems, through the programming tools presented so far. It should be noted how sensitive the system is for very close Kp values. This is shown in the following graph where the answers to the unit step input appear simultaneously for different values of Kp:

Kp=0.2, Kp=0.5, Kp=1, Kp=1.5,  Kp=1.7, Kp=2.

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

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## Example 1 – PI Controller Design (Proportional-Integral) – Matlab

To better appreciate the effect of the PI controller, let’s look at the following example. Suppose we have the system of Figure 7-23.

The direct transfer function G(s) for this system is as follows:

Where K is the pre-amplifier constant.

Teh design specifications for this system are:

Donde:

• ess: steady-state error due to a parabolic input
• Mp: Maximum overshoot
• Tr: Rise time
• Ts: Settling time

The first thing we are going to do is analyze the steady-state error ess of the system before compensating, and see how much it meets or not with the first design requirement.

For a parabolic input we must work with the acceleration constant Ka:This means an infinite error for a parabolic input:To improve the steady-state error we incorporate a PI controller in the direct path of the system, which will now have the following transfer function:
By increasing the typology of the system from type 1 to type 2 we immediately improve the steady-state error. Now, the ess due to the parabolic input will be a constant:
That is to say:

This exercise we already worked with the PD controller. For that case we select a value of K=181.17 (Example 1 – PD Controller Design (Proportional-Diferential) – Matlab)

We allow ourselves the freedom to consider the same value for K in this case in order to maintain the transient response under acceptable and known conditions when applying the PD of Example 1. If necessary we will adjust the value of K later. We can appreciate that to achieve an ess as specified in the design, the larger K is, the smaller will have to be Ki, which may be convenient. With the values of K and ess we can calculate a first approximate value of Ki to meet the requirements:

The next thing we will do is analyze the stability of the system because the selection of the Kp and Ki parameters depends on it. We will apply the Routh-Hurwitz criterion to calculate the limit values of the mentioned parameters in such a way that the system remains stable. For this we need the characteristic equation that arises from the transfer function of the closed loop system Gce(s):

The characteristic equation of the system is:

With this equation we apply the Routh-Hurwitz criterion. In this way we discovered that the system is stable for the following range of values:

This result indicates that the Ki / Kp of the controller can not be very close to zero, so a value as low as Ki = 0.002215 is not convenient. Another criterion for selecting Ki / Kp is that it is convenient to select the zero added by the controller in s = -Ki / Kp so that it is located near the origin and far from the most significant poles of the system. Through the locus of the roots in Matlab we can see which are the most significant poles of the characteristic equation, assuming an acceptable relationship between the Ki and Kp parameters from the point of view of the previous stability study but relatively close to the origin, say Ki / Kp = 10, keeping Ki constant and varying Kp.

First, we reshape the characteristic equation in its form 1+G(s)H(s):

Note that in this way the controller PI is adding a zero in s=-10.

We apply the following command in Matlab to obtain the locus of the roots of this compensated system:

>> s=tf(‘s’)

>> sys=(815265*(s+10))/(s^3+361.2*s^2)

>> rlocus(sys)

So we get:

As it can be seen, the most significant pole of the characteristic equation is located in s=-361. Therefore, the criterion that we must use to select s=-Ki/Kp is:With this result, the direct transfer function G(s):It would be simplified as:

The term Ki / Kp would be negligible compared to the magnitude of s when s assumes values along the root locus that corresponds to a convenient relative damping factor of 0.7 <ζ <0.1. Then, a zero cancels a pole at zero. The maximum overshoot must be equal to or less than 5%. This means that you want a relative damping factor ζ approximate to the following:

With the help of the locus we can locate the poles that correspond to this value of ζ:

According to the graph, the value of Kp required to obtain the desired damping factor is:

And so:

We also observe in the previous graph that if Kp = 0.0838, then the roots of the characteristic equation (the poles of the system) are in s1 = -175 + j184 and in s2 = -175-j184 If we look around these roots we can notice that the zero added by the controller at s = -10 is very close to the origin compared to the poles of s1 and s2, practically canceling a pole at the origin, thus ratifying the approach we made earlier for the direct transfer function of this system then of the compensation:

So:

We can observe the response to the step according to these partial results and the comparison of the compensated and uncompensated systems, by means of the following simulation:

>> Ga=(815265)/(s*(s+361.2))

>> Gd=(68319)/(s*(s+361.2))

>> Gce1=feedback(Ga,1)

>> Gce2=feedback(Gd,1)

>> step(Gce1,Gce2)

The previous graph, with the system after the compensation in red line, shows that the PI improves the steady-state error and reduces the overshoot, but at the expense of significantly increasing the rise time. The graph also shows that the maximum overshoot is 5%, therefore the requirement is met. It is necessary to note that another relationship can be selected for Ki and Kp that meets the requirement and still improve the overshoot, for example Ki / Kp = 5, or Ki / Kp = 2. You just have to pay attention to the issue of the stability of the system. With the calculated values, we re-calculate the overshoot, and we evaluate the rise time tr and the settlement time ts . The following command gives us the value of ζ and ωn, the relative damping factor and the natural frequency respectively.

>> damp(Gce2)

Pole: -1.81e+02 + 1.89e+02i /   -1.81e+02 + 1.89e+02i

Damping: 6.91e-01

Frequency: 2.61e+02

•  Maximum Overshoot (MP):

• Rise Time (Tr):

>> Gd=(68319)/(s*(s+361.2))

>> sys=feedback(Gd,1)

>> step(sys)

• Settling time (Ts):

Values that we can also see in the response to the step input graph generated previously:

We can conclude that the compensated system fulfills the requirements of the design, although it exceeds a little in the settling time. The latter can be reduced slightly with a Ki / Kp = 2 ratio, but at the expense of getting too close to the unstable area of the system

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

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Respuesta Transitoria de un Sistema de Control

Estabilidad de un sistema de control

Simulación de Respuesta Transitoria con Matlab – Introducción

## PID – Diseño y configuración del controlador.

`Configuración del controlador.`

El PID es uno de los controladores más ampliamente utilizados en los esquemas de compensación siguientes, algunas de las cuáles se ilustran en la Figura 10-2:

1. Compensación en serie (cascada),
2. Compensación mediante realimentación,
3. Compensación mediante realimentación de estado,
4. Compensación en serie-realimentada,

En general, la dinámica de un proceso lineal controlado puede representarse mediante el diagrama de la Figura 10-1.

El objetivo de diseño es que las variables controladas, representadas por el vector de salida y(t), se comporten de cierta forma deseada. El problema esencialmente involucra el determinar la señal de control u(t) dentro de un intervalo prescrito para que todas las especificaciones de diseño sean satisfechas (ver Estabilidad de un sistema de control, Respuesta transitoria, Error en estado estable).

El controlador PID aplica una señal al proceso de control mostrado en la Figura 10-1, que es una combinación proporcional, integral y derivada de las señal de actuación e(t). Debido a que estos componentes de la señal se pueden realizar y visualizar fácilmente en el dominio del tiempo, los controladores PID se diseñan comúnmente empleando métodos en el dominio del tiempo.

Después de que el diseñador ha seleccionado una configuración para el controlador, debe escoger además el tipo de controlador. En general, mientras más complejo el controlador, más costoso, menos confiable y más difícil de diseñar. Por ende, en la práctica se selecciona el tipo de controlador más simple que permita cumplir con las especificaciones de diseño, lo que involucra experiencia, intuición, arte y ciencia.

Las componentes integral y derivativo de un controlador PID tienen una implicación individual en el desempeño, y sus aplicaciones requieren un entendimiento de las bases de estos elementos. Por ello, se consideran por separado, iniciando con la porción PD.

`Diseño con el controlador PD`

La Figura 10-3 muestra el diagrama de bloques de un sistema de control realimentado que deliberadamente tiene una planta prototipo de segundo orden con la siguiente función de transferencia Gp(s):

El controlador en serie es del tipo proporcional-derivativo (PD) con la función de transferencia:

Por lo tanto, la señal de control U(s) aplicada a la planta es:

en donde Kp y Kd son las constantes proporcional y derivativa respectivamente, mientras E(s) es la señal de error. La realización del controlador PD mediante circuitos electrónicos se muestra en la Figura 10-4:

La función de transferencia del circuito de la Figura 10-4 es:

Al comparar con la Figura 10-3:

La función de transferencia directa del compensador mostrado en la Figura 10-3 es:

lo cual muestra que el control PD equivale a añadir un cero simple en s=-Kp/Kd a la función de transferencia directa. El efecto de control PD sobre la respuesta transitoria de un sistema de control se puede investigar al referirse a la respuesta en tiempo del sistema como se muestra en la Figura 10-5:

Se supone que la respuesta al escalón unitario de un sistema estable con el control proporcional es solamente como la que se presenta en la Figura 10-5(a). Se observa un sobrepaso máximo relativamente grande y un poco oscilatorio. La señal de error e(t) correspondiente, que es la diferencia entre la entrada r(t) escalón unitario y la salida y(t), y la derivada de dicho error en el tiempo, se muestran en las Figuras (b) y (c) respectivamente.

Durante el intervalo 0<t<t1, la señal de error es positiva, el sobrepaso es grande y se observa gran oscilación en la salida debido a la falta de amortiguamiento en este período. Durante el intervalo t1<t<t2, la señal de error es negativa, la salida se invierte y tiene un sobrepaso negativo. Este comportamiento se alterna sucesivamente hasta que la amplitud del error se reduce con cada oscilación, y la salida se establece eventualmente en su valor final. Se observa que el controlador PD puede añadir amortiguamiento a un sistema y reduce el sobrepaso máximo, pero no afecta el estado estable directamente.

`Ejemplo. `

Para apreciar mejor el efecto del controlador PD, veamos el siguiente ejemplo. Supongamos que tenemos el sistema de la Figura 7-23.

La función de transferencia directa G(s) de este sistema viene dada por la siguiente expresión:

Donde K es la constante del preamplificador.

Las especificaciones de diseño para este sistema son las siguientes:

Donde:

• ess: Error en estado estable debido a una entrada de rampa unitaria
• Mp: Sobrepaso máximo
• Tr: Tiempo de levantamiento
• Ts: Tiempo de asentamiento
1. Selección del valor de K

Lo primero que vamos a hacer es hallar K para cumplir con el primer requerimiento de diseño, error en estado estable ess debido a una entrada rampa:

(Para repasar el concepto de error en estado estable ver Error en estado estable de un sistema de control)

1.a Hallar la constante de velocidad Kv porque es la relacionada a una entrada rampa:

1.b Hallar ess en función de K:

1.c Hallar para ess=0.000433:

Con este valor de K, la función de transferencia directa G(s) es:

2. Cálculo de sobrepaso

Veamos ahora como queda el sobrepaso para el valor de K obtenido.

(Para un repaso del concepto de sobrepaso y la respuesta transitoria ver Respuesta Transitoria de un Sistema de Control)

2.a La función de transferencia de lazo cerrado Gce(s) es:

2.b Hallamos a partir de aquí el factor de amortiguamiento relativo ζ y la frecuencia natural del sistema ωn.

2.c Con estos valores, hallamos el sobrepaso máximo Mp:

En porcentaje:

Este valor supera la exigencia de la especificación, por lo que se considera insertar un controlador PD en la trayectoria directa del sistema con el fin de mejorar el amortiguamiento y ajustar el sobrepaso máximo a la especificación de diseño exigida, manteniendo sin embargo el error en estado estable en 0.000433.

`3. Diseño en el dominio del tiempo del controlador PD`

Añadiendo el controlador Gc(s) de la Figura 10-3 a la trayectoria directa del sistema aeronáutico, y asignando K=185.4503, la función de transferencia directa G(s) del sistema de control de posición de la aeronave es:

Mientras, la función de transferencia a lazo cerrado Gce(s) es:

Esta última ecuación muestra los efectos del controlador PD sobre la función de transferencia de lazo cerrado del sistema al cual se aplica:

1. Añadir un cero en s=-Kp/Kd
2. Incrementar el “término asociado al amortiguamiento”, el cual es el coeficiente de s en el denominador de Gce(s). Es decir, de 361.2 hasta 361.2 + 834526.56Kd

3.a Selección de Kp

Para asegurarnos de que se mantenga el error en estado estable para una entrada rampa de acuerdo con las especificaciones, evaluamos dicho error y seleccionamos un valor para Kp:

Al elegir Kp igual a uno, mantenemos el mismo valor para Kv que se tenía antes de añadir el controlador. Es decir, mantenemos el valor del error en estado estable para entrada rampa tal como lo exige la especificación de diseño. Entonces:3.b Selección de Kd

De acuerdo con la ecuación de sobrepaso máximo:

El sobrepaso máximo depende del factor de amortiguamiento relativo ζ. La ecuación característica del sistema es:

Donde:

Deducimos la expresión para el factor de amortiguamiento relativo ζ:

Este resultado muestra claramente el efecto positivo de Kd sobre el amortiguamiento. Sin embargo, se debe resaltar el hecho de que la función de transferencia directa G(s) ya no representa un sistema prototipo de segundo orden, por lo que la respuesta transitoria también se verá afectada por el cero en s=-Kp/Kd.

Aplicaremos ahora el método del lugar geométrico de la raíces a la ecuación característica para examinar el efecto de variar Kd, mientras se mantiene constante el valor de Kp=1.

Si deseamos obtener un Mp=5% tal y como se pide en las especificaciones de diseño, eso significa obtener un factor de amortiguamiento relativo igual a lo siguiente:

La ecuación característica del sistema y su forma 1+G(s)H(s) son:

Utilizando el siguiente comando en Matlab obtenemos el lugar geométrico de las raíces para G(s)H(s):

>> s=tf(‘s’)

>> sys=(834526.56*s)/(s^2+361.2*s+834526.56)

>> rlocus(sys)

La gráfica siguiente muestra como mejora el factor de amortiguamiento relativo ζ a medida que aumenta la ganancia Kd:

Mientras, en la gráfica siguiente se muestra que para lograr un factor de amortiguamiento relativo ζ=0.69 o mejor que ese, lo cual significa un sobrepaso menor de 5% como se especifica, es necesario tener una ganancia mínima Kd= 0.00108:

Sin embargo, antes de seleccionar un valor definitivo para Kd debemos observar el cumplimiento de los otros requerimientos de diseño.

3.c Evaluación de Tr y Ts según Kd y Kp calculados.

Analizamos a continuación el valor del tiempo de levantamiento Tr para el valor de ζ=0.69 , Kd= 0.00108 y Kp= 1,  utilizando la función de transferencia a lazo cerrado del sistema Gce(s)  y el gráfico de respuesta a la entrada escalón generado por el siguiente comando en Matlab:

>> s=tf(‘s’)

>>sys=(834526.56*(1+0.00108*s))/(s^2+(361.2+834526.56*0.00108)*s+834526.56)

sys =     (901.3 s + 8.345e05) / (s^2 + 1262 s + 8.345e05)

> step(sys)

Utilizando la gráfica para la salida C(t) del sistema a una entrada escalón para un valor determinado del factor de amortiguamiento relativo (ζ=0.69). Para hallar Tr, restamos los tiempos para los cuáles C(t)=0.9 C(t)=0.1:

La gráfica anterior nos permite determinar el valor de Tr para un valor de ζ=0.69 de la siguiente manera:

Podemos ver que este valor cumple con el requerimiento de que Tr≤0.005 s. Veamos ahora que pasa con Ts. Utilizando el criterio del 2% podemos calcular Ts mediante la siguiente fórmula:

Así vemos que el factor de amortiguamiento ζ=0.69 genera un Ts que no cumple con la condición de un Ts menor o igual a 0.005 s. Sin embargo, aumentando Kd mejoramos ζ logrando satisfacer dicha condición. Para ser más específicos, despejamos ζ a partir del valor máximo aceptado para Ts:

Utilizamos nuevamente el lugar geométrico de las raíces para determinar el valor de Kd que se corresponde con el de ζ=0.8757:

Si el valor de Kd=0.00148 y mantenemos el valor de Kp=1, la función de transferencia directa es:

Mientras, la función de transferencia a lazo cerrado del sistema en estudio es la siguiente:

Para esta función de transferencia revisamos los valores de sobrepaso Mp y tiempo de levantamiento Tr para asegurarnos que cumplen con las especificaciones de diseño:

Por tanto, el valor de Kd debe tener un valor mínimo de:

Y nuestro controlador PD puede tener entonces la siguiente función de transferencia:

Escrito por: Larry Francis Obando – Technical Specialist – Educational Content Writer.

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

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

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 Atención: Si lo que Usted necesita es resolver con urgencia un problema de “Sistema Masa-Resorte-Amortiguador” (encontrar la salida X(t), gráficas en Matlab del sistema de 2do Orden y parámetros relevantes, etc.), o un problema de “Sistema de Control Electromecánico” que involucra motores, engranajes, amplificadores diferenciales, etc…para entregar a su profesor en dos o tres días, o con mayor urgencia…o simplemente necesita un asesor para resolver el problema y estudiar para el próximo examen…envíeme el problema…Yo le resolveré problemas de Sistemas de Control, le entrego la respuesta en digital y le brindo una video-conferencia para explicarle la solución…incluye además simulación en Matlab.

Respuesta Transitoria de un Sistema de Control

Estabilidad de un sistema de control

Simulación de Respuesta Transitoria con Matlab – Introducción

## PID – Basic Control System Actions

NEXT: PID – Effect of integrative and derivative control actions.

### Introduction

An automatic controller compares the real value of the output of a plant with the input reference (the desired value), determines the deviation and produces a control signal that will reduce the deviation to zero or a small value. The way in which the automatic controller produces the control signal is called control action.

Classification of industrial controls

According to their control actions, industrial controllers are classified as:

1. Two-position (On / Off)
2. Proportional
3. Integrals
4. Proportional-Integrals
5. Proportional-Derivatives
6. Proportional-Integrals-Derivatives

Almost all industrial controllers use electricity as an energy source or a pressurized fluid, such as oil or air. The controllers can also be classified, according to the type of energy they use in their operation, like pneumatic, hydraulic or electronic. The type of controller that is used must be decided based on the nature of the plant and operational conditions, including considerations such as safety, cost, availability, reliability, precision, weight and size.

Figure 5-1 shows a typical configuration for an Industrial Control System:

The previous figure consists of a Block Diagram for an industrial control system composed of an automatic controller, an actuator, a plant and a sensor (measuring element). The controller detects the error signal, which is usually at a very low power level, and amplifies it to a sufficiently high level. The output of an automatic controller feeds an actuator that can be a pneumatic valve or an electric motor. The actuator is a power device that produces the input for The plant according to the control signal, so that the output signal approaches the reference input signal. The sensor, or measurement element, is a device that converts an output variable, such as a displacement, into another manageable variable, such as a voltage, that can be used to compare the output with the reference input signal. This element is in the feedback path of the closed-loop system. The setpoint of the controller must be converted into a reference input with the same units as the feedback signal from the sensor or from the measuring element.

Two positions control (On / Off).

In a two position control system, the acting element only has two fixed positions that, in many cases, are simply turned on and off. The On/Off control is relatively simple and cheap, which is why it is extensively used in industrial and domestic control systems.

Suppose that the output signal of the controller is u(t) and that the error signal is e(t). In the control of two positions, the signal u(t) remains at a value either maximum or minimum, depending on whether the error signal is positive or negative. In this way,

where U1 y U2 are constants. Very often, the minimum value of U2 is zero or –U1.

It is common for two-position controllers to be electrical devices, in which case an electrical valve operated by solenoids is widely used. Pneumatic proportional controllers with very high gains function as two-position controllers and are sometimes referred to as two-position pneumatic controllers.

Figures 5-3 (a) and (b) show the block diagrams for two controllers of two positions The range in which the error signal must move before the commutation is called differential gap. In Figure 5-3 (b) a differential gap is indicated. Such a gap causes the output of the controller u (t) to retain its present value until the error signal has moved slightly beyond zero. In some cases, the differential gap is the result of unintentional friction and a lost movement; however, it is often intentionally caused to avoid too frequent operation of the on and off mechanism.

Proportional control action.

For a controller with proportional control action, the relationship between the controller output u (t) and the error signal e (t) is:

or, in quantities transformed by the Laplace method:

where Kp is considered proportional gain.

Whatever the actual mechanism and the form of the operating power, the controller

proportional is, in essence, an amplifier with an adjustable gain. A block diagram of such a controller is presented in Figure 5-6.

Integral control action.

In a controller with integral control action, the value of the controller output u (t) is changed to a ratio proportional to the error signal e (t). That is to say,

O well:

where Ki is an adjustable constant. The transfer function of the integral controller is:

If the value of e (t) is doubled, the value of u (t) varies twice as fast. For an error of zero, the value of u (t) remains stationary. Sometimes, the integral control action is called adjustment control (reset). Figure 5-7 shows a block diagram of such controller.

Integral-proportional control action.
The control action of a proportional-integral controller (PI) is defined by:

or the transfer function of the controller, which is:

where Kp is the proportional gain and Ti is called integral time. Both Kp and Ti are adjustable. Integral time adjusts the integral control action, while a change in the value of Kp affects the integral and proportional parts of the control action.

The inverse of the integral time Ti is called the readjustment speed. The rate of readjustment is the number of times per minute that the proportional part of the control action is doubled. The rate of readjustment is measured in terms of the repetitions per minute. The Figure 5-8 (a) shows a block diagram of a PI controller. If the error signal e (t) is a unit step function, as shown in Figure 5-8 (b), the controller output u (t) becomes what is shown in Figure 5-8 (c).

Proportional-derivative control action.
The control action of a proportional-derivative (PD) controller is defined by:

The transfer function is:

where Kp is the proportional gain and Td is a constant called derivative time. Both Kp and Td are adjustable. The derivative control action, sometimes called speed control, occurs where the magnitude of the controller output is proportional to the rate of change of the error signal. The derivative time Td is the time interval during which the action of the velocity advances the effect of the proportional control action.

Figure 5-9 (a) shows a block diagram of a PD controller. If the error signal e (t) is a unit ramp function as shown in Figure 5-9 (b), the controller output u (t) becomes that shown in Figure 5-9 (c). ). The derivative control action has a forecast nature. However, it is obvious that a derivative control action never foresees an action that has never occurred.

Although the derivative control action has the advantage of being forecast, it has the disadvantages that it amplifies the noise signals and can cause a saturation effect in the actuator. Note that the derivative control action is never used alone, because it is only effective during transient periods.

Proportional-Integral-derivative (PID) control action.

The combination of a proportional control action, an integral control action and a derivative control action is called proportional-integral-derivative (PID) control action.

This combined action has the advantages of each of the three individual control actions. The equation of a controller with this combined action is obtained by:

The transfer function is:

where Kp is the proportional gain, Ti is the integral time and Td is the derivative time. The block diagram of a PID controller appears in Figure 5-10 (a). If e (t) is a unit ramp function, like the one shown in Fig. 5-10 (b), the controller output u (t) becomes that of Fig. 5-10 (c).

Effects of the sensor on the performance of the system.

Since the dynamic and static characteristics of the sensor or measuring element affects the indication of the actual value of the output variable, the sensor fulfills a function important to determine the overall performance of the control system. As usual, the sensor determines the transfer function in the feedback path. If the time constants of a sensor are negligible compared to other constants of time of the control system, the sensor transfer function simply it becomes a constant. Figures 5-11 (a), (b) and (c) show diagrams of automatic controller blocks with a first-order sensor, an overdamped second-order sensor and a second-order underdamped sensor, respectively. Often the response of a thermal sensor is of the overdamped second order type.

NEXT: PID – Effect of integrative and derivative control actions.

Source:

Literature review by Larry Francis Obando – Technical Specialist – Educational Content Writer

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

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

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 – 0998524011

WhatsApp: +593984950376

## Acciones Básicas de Sistemas de Control – PID

Un controlador automático compara el valor real de la salida de una planta con la entrada de referencia (el valor deseado), determina la desviación y produce una señal de control que reducirá la desviación a cero o a un valor pequeño. La manera en la cual el controlador automático produce la señal de control se denomina acción de control.

`Clasificación de los controladores industriales.`

De acuerdo con sus acciones de control, los controladores industriales se clasifican en:

1. De dos posiciones o de encendido y apagado (On/Off)
2. Proporcionales
3. Integrales
4. Proporcionales-Integrales (PI)
5. Proporcionales-Derivativos (PD)
6. Proporcionales-Integrales-Derivativos (PID)

Casi todos los controladores industriales emplean como fuente de energía la electricidad o un fluido presurizado, tal como el aceite o el aire. Los controladores también pueden clasificarse, de acuerdo con el tipo de energía que utilizan en su operación, como neumáticos, hidráulicos o electrónicos. El tipo de controlador que se use debe decidirse con base en la naturaleza de la planta y las condiciones operacionales, incluyendo consideraciones tales como seguridad, costo, disponibilidad, confiabilidad, precisión, peso y tamaño.

La Figura 5-1 muestra la configuración típica de un Sistema de Control Industrial:

La figura anterior consiste en un Diagrama de Bloques para un sistema de control industrial compuesto por un controlador automático, un actuador, una planta y un sensor (elemento de medición). El controlador detecta la señal de error, que por lo general, está en un nivel de potencia muy bajo, y la amplifica a un nivel lo suficientemente alto. La salida de un controlador automático alimenta a un actuador que puede ser una válvula neumática o un motor eléctrico. El actuador es un dispositivo de potencia que produce la entrada para La planta de acuerdo con la señal de control, a fin de que la señal de salida se aproxime a la señal de entrada de referencia. El sensor, o elemento de medición, es un dispositivo que convierte la variable de salida, tal como un desplazamiento, en otra variable manejable, tal como un voltaje, que pueda usarse para comparar la salida con la señal de entrada de referencia. Este elemento está en la trayectoria de realimentación del sistema en lazo cerrado. El punto de ajuste del controlador debe convertirse en una entrada de referencia con las mismas unidades que la señal de realimentación del sensor o del elemento de medición.

Los sistemas de control realimentados son difíciles de comprender desde el punto de vista cualitativo, por lo que dicha comprensión depende en gran medida de las matemáticas. El Lugar Geométrico de las Raíces (LGR) es la técnica gráfica que nos da esa descripción cualitativa sobre el rendimiento del sistema de control que estamos diseñando. Para pasar de una vez a la práctica del análisis y diseño de sistemas de control, ver: El lugar geométrico de las raíces con Matlab

`Acción de control de dos posiciones o de encendido y apagado (on/off).`

En un sistema de control de dos posiciones, el elemento de actuación sólo tiene dos posiciones fijas que, en muchos casos, son simplemente encendido y apagado. El control de dos posiciones o de encendido y apagado es relativamente simple y barato, razón por la cual su uso es extendido en sistemas de control tanto industriales como domésticos.

Supongamos que la señal de salida del controlador es u(t) y que la señal de error es e(t). En el control de dos posiciones, la señal u(t) permanece en un valor ya sea máximo o mínimo, dependiendo de si la señal de error es positiva o negativa. De este modo,

donde U1 y U2 son constantes. Por lo general, el valor mínimo de U2 es cero ó U1.

Es común que los controladores de dos posiciones sean dispositivos eléctricos, en cuyo caso se usa extensamente una válvula eléctrica operada por solenoides. Los controladores neumáticos proporcionales con ganancias muy altas funcionan como controladores de dos posiciones y, en ocasiones, se denominan controladores neumáticos de dos posiciones.

Las Figuras 5-3(a) y (b) muestran los diagramas de bloques para dos controladores de dos posiciones. El rango en el que debe moverse la señal de error antes de que ocurra la conmutación se denomina brecha diferencial. En la Figura 5-3(b) se señala una brecha diferencial. Tal brecha provoca que la salida del controlador u(t) conserve su valor presente hasta que la señal de error se haya desplazado ligeramente más allá de cero. En algunos casos, la brecha diferencial es el resultado de una fricción no intencionada y de un movimiento perdido; sin embargo, con frecuencia se provoca de manera intencional para evitar una operación demasiado frecuente del mecanismo de encendido y apagado.

`Acción de control proporcional. `

Para un controlador con acción de control proporcional, la relación entre la salida del controlador u(t) y la señal de error e(t) es:

o bien, en cantidades transformadas por el método de Laplace:

donde Kp se considera la ganancia proporcional.

Cualquiera que sea el mecanismo real y la forma de la potencia de operación, el controlador proporcional es, en esencia, un amplificador con una ganancia ajustable. En la figura 5-6 se presenta un diagrama de bloques de tal controlador.

Para ver los efectos de aplicar un bloque proporcional a un sistema de control, ver: Control Proporcional de un Sistema de Control – PID.

`Acción de control integral. `

En un controlador con acción de control integral, el valor de la salida del controlador u(t) se cambia a una razón proporcional a la señal de error e(t). Es decir,

o bien:

en donde Ki es una constante ajustable. La función de transferencia del controlador integral es:

Si se duplica el valor de e(t), el valor de u(t) varía dos veces más rápido. Para un error de cero, el valor de u(t) permanece estacionario. En ocasiones, la acción de control integral se denomina control de reajuste (reset). La Figura 5-7 muestra un diagrama de bloques de tal controlador.

Para ver los efectos de aplicar un bloque integrador a un sistema de control, ver: PID – Estudio de la acción integral con Matlab

`Acción de control integral-proporcional. `

La acción de control de un controlador proporcional-integral (PI) se define mediante:

o la función de transferencia del controlador, la cual es:

en donde Kp es la ganancia proporcional y Ti se denomina tiempo integral. Tanto Kp como Ti son ajustables. El tiempo integral ajusta la acción de control integral, mientras que un cambio en el valor de Kp afecta las partes integral y proporcional de la acción de control.

El inverso del tiempo integral Ti se denomina velocidad de reajuste. La velocidad de reajuste es la cantidad de veces por minuto que se duplica la parte proporcional de la acción de control. La velocidad de reajuste se mide en términos de las repeticiones por minuto. La Figura 5-8(a) muestra un diagrama de bloques de un controlador proporcional más integral. Si la señal de error e(t) es una función escalón unitario, como se aprecia en la Figura 5-8(b), la salida del controlador u(t) se convierte en lo que se muestra en la Figura 5-8(c).

Para ver un ejemplo de diseño de un controlador PI, ver: Ejemplo 1 – Diseño de un controlador PI (Proporcional-Integral) – Matlab

`Acción de control proporcional-derivativa (PD)`

La acción de control de un controlador proporcional-derivativa (PD) se define mediante:

la función de transferencia es:

en donde Kp es la ganancia proporcional y Td es una constante denominada tiempo derivativo. Tanto Kp como Td son ajustables. La acción de control derivativa, en ocasiones denominada control de velocidad, ocurre donde la magnitud de la salida del controlador es proporcional a la velocidad de cambio de la señal de error. El tiempo derivativo Td es el intervalo de tiempo durante el cual la acción de la velocidad hace avanzar el efecto de la acción de control proporcional.

La Figura 5-9(a) muestra un diagrama de bloques de un controlador PD. Si la señal de error e(t) es una función rampa unitaria como se aprecia en la Figura 5-9(b), la salida del controlador u(t) se convierte en la que se muestra en la Figura 5-9(c). La acción de control derivativa tiene un carácter de previsión. Sin embargo, es obvio que una acción de control derivativa nunca prevé una acción que no ha ocurrido.

Aunque la acción de control derivativa tiene la ventaja de ser de previsión, tiene las desventajas de que amplifica las señales de ruido y puede provocar un efecto de saturación en el actuador. Observe que la acción de control derivativa no se usa nunca sola, debido a que sólo es eficaz durante periodos transitorios.

Para ver un ejemplo de diseño de un controlador PD, ver: Ejemplo 1 – Diseño de un controlador PD (Proporcional-Diferencial)

`Acción de control proporcional-Integral-derivativa (PID)`

La combinación de una acción de control proporcional, una acción de control integral y una acción de control derivativa se denomina acción de control proporcional-integral-derivativa (PID).

Esta acción combinada tiene las ventajas de cada una de las tres acciones de control individuales. La ecuación de un controlador con esta acción combinada se obtiene mediante:

la función de transferencia es:

en donde Kp es la ganancia proporcional, Ti es el tiempo integral y Td es el tiempo derivativo. El diagrama de bloques de un controlador PID aparece en la figura 5-10(a). Si e(t) es una función rampa unitaria, como la que se observa en la Figura 5-10(b), la salida del controlador u(t) se convierte en la de la Figura 5-10(c).

`Efectos del sensor sobre el desempeño del sistema.`

Dado que las características dinámica y estática del sensor o del elemento de medición afecta la indicación del valor real de la variable de salida, el sensor cumple una función importante para determinar el desempeño general del sistema de control. Por lo general, el sensor determina la función de transferencia en la trayectoria de realimentación. Si las constantes de tiempo de un sensor son insignificantes en comparación con otras constantes de tiempo del sistema de control, la función de transferencia del sensor simplemente se convierte en una constante. Las Figuras 5-11(a), (b) y (c) muestran diagramas de bloques de controladores automáticos con un sensor de primer orden, un sensor de segundo orden sobreamortiguado y un sensor de segundo orden subamortiguado, respectivamente. Con frecuencia la respuesta de un sensor térmico es del tipo de segundo orden sobreamortiguado.

Fuente:

1. Ingenieria de Control Moderna, 3° ED. – Katsuhiko Ogata,

Revisión hecha por:

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

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 +34 633129287

Caracas, Quito, Guayaquil, Cuenca

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