** Steady-state error can be improved by placing an open-loop pole at the origin,**. For example, a Type 0 system

because this increases the system type by one

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

*, as shown in Figure 1.a and 1.b:*

**A**

**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

**is:**

*G*_{c(s)}

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

**and its original controller is represented by the gain**

*G*_{(s)}*:*

**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=0.173***, *as 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:

*. We also see in Figure 5 that the closed-loop second-order dominant poles*

**ξ=0.173****and**

*s*_{1}**, before compensation are:**

*s*_{2}

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

**is located at:**

*s*_{3}

**Figure 6.**

With *k=165* we calculate the steady-state error ** e_{1(∞)}** for a step input, before compensation:

Where ** k_{p1}** 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=k_{i}**. The constant

**is determined by the location of compensator zero, wich must be near the compensator pole. That is why we set the compensator zero at**

*a***, that is to say**

*s=-0.1***. The root locus of this compensated system is in Figure 8:**

*a=0.1*>> 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

*. We see that closed-loop second-order dominant poles*

**ξ=0.173****and**

*s*_{1}**, after compensation, are:**

*s*_{2}

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

**is located at:**

*s*_{3}

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

*as shwon later.*

**0,**The forward transfer function ** G_{2(s)}** of the system after compensation is:

One more time, we calculate steady-state error ** e_{2(∞)}** 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 :

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

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