Root Locus in control system
Table of Contents
Root Locus in control system- We can see the closed loop poles’ path in the root locus diagram. As a result, we can determine the type of control system. In order to determine the stability of the closed loop control system, we will use an open loop transfer function in this method.
Basics of Root Locus
By varying system gain K from zero to infinity, the Root locus is the location of the roots of the characteristic equation.
We know that, the characteristic equation of the closed loop control system is
1+G(s)H(s)=0
We can represent G(s)H(s) as
G(s)H(s)=K(N(s)/D(s))
Where,
- K represents the multiplying factor
- N(s) represents the numerator term having (factored) nth order polynomial of ‘s’.
- D(s) represents the denominator term having (factored) mth order polynomial of ‘s’.
Substitute, G(s)H(s) value in the characteristic equation.
1+k(N(s)/D(s))=0
⇒D(s)+KN(s)=0
Case 1 − K = 0
If K=0, then D(s)=0
That means, the closed loop poles are equal to open loop poles when K is zero.
Case 2 − K = ∞
Re-write the above characteristic equation as
K{(1/K)+(N(s)/D(s))}=0⇒{(1/K)+(N(s)/D(s))}=0
Substitute, K=∞ in the above equation.
N(s)/D(s) = 0 => N(s)=0
If K=∞, then N(s)=0. It means the closed loop poles are equal to the open loop zeros when K is infinity.
From above two cases, we can conclude that the root locus branches start at open loop poles and end at open loop zeros.
Angle Condition and Magnitude Condition
The angles are met by the points on the root locus branches. In order to determine whether a point exists on the root locus branch or not, the angle condition is used. Utilizing the magnitude condition, we can determine the value of K for the points on the root locus branches. Since the angle condition is satisfied, we can use the magnitude condition for the points.
Characteristic equation of closed loop control system is
1+G(s)H(s)=0
⇒G(s)H(s)=−1+j0
The phase angle of G(s)H(s) is
∠G(s)H(s)=tan−1(0/−1)=(2n+1)π
The angle condition is the point at which the angle of the open loop transfer function is an odd multiple of 1800.
Magnitude of G(s)H(s) is –
|G(s)H(s)|=√{(−1)2+02}=1
The magnitude condition is that the point (which satisfied the angle condition) at which the magnitude of the open loop transfer function is one.
The root locus is depicted graphically in the s-domain and has symmetrical properties about the real axis. Due to the fact that the open loop poles and zeroes are present in the s-domain and have values that can be either real or complex conjugate pairs. Let us talk about how to create (draw) the root locus in this chapter.
Rules for Construction of Root Locus in control system
Follow these rules for constructing a root locus.
Rule 1 − Locate the open loop poles and zeros in the ‘s’ plane.
Rule 2 − Find the number of root locus branches.
We are aware that the open loop zeros and poles mark the beginning and end of the root locus branches. In other words, the number of finite open loop poles P or zeros Z, whichever is greater, equals the number of root locus branches N.
Mathematically, we can write the number of root locus branches N as
N=P if P≥Z
N=Z if P<Z
Rule 3 − Identify and draw the real axis root locus branches.
A point is on the root locus if the angle of the open loop transfer function at that point is an odd multiple of 180. A point on the real axis is on the root locus branch if an odd number of the open loop poles and zeros are present to its left. The branch of points that meets this requirement is thus the true axis of the root locus branch.
Rule 4 − Find the centroid and the angle of asymptotes.
- If P=Z, then all the root locus branches start at finite open loop poles and end at finite open loop zeros.
- If P>Z , then Z number of root locus branches start at finite open loop poles and end at finite open loop zeros and P−Z number of root locus branches start at finite open loop poles and end at infinite open loop zeros.
- If P<Z , then P number of root locus branches start at finite open loop poles and end at finite open loop zeros and Z−P number of root locus branches start at infinite open loop poles and end at finite open loop zeros.
So, some of the root locus branches approach infinity, when P≠Z. Asymptotes give the direction of these root locus branches. The intersection point of asymptotes on the real axis is known as centroid.
We can calculate the centroid α by using this formula,
α=∑Real part of finite open loop poles−∑Real part of finite open loop zeros / (P−Z)
The formula for the angle of asymptotes θ is
θ={(2q+1)180}/(P−Z)
Where,
q=0,1,2,….,(P−Z)−1
Rule 5 − Find the intersection points of root locus branches with an imaginary axis.
So Using the Routh array method and special case (ii), we can determine the location of the root locus branch’s intersection with the imaginary axis and the value of K at that location.
- The root locus branch intersects the imaginary axis if all elements in any row of the Routh array are zero, and vice versa.
- Determine the row so that if we set the first element to zero, the elements in the entire row will also be set to zero. Determine K’s value for this combination.
- So Put this K value in the auxiliary equation as a replacement. You will discover where the root locus branch crosses a hypothetical axis.
Rule 6 − Find Break-away and Break-in points.
- There will be a break-away point between these two open loop poles if there is a real axis root locus branch between them.
- A break-in point will be present between two open loop zeros if there is a real axis root locus branch present between them.
Note − Break-away and break-in points exist only on the real axis root locus branches.
Follow these steps to find break-away and break-in points.
- Write K in terms of s from the characteristic equation 1+G(s)H(s)=0.
- Differentiate K with respect to s and make it equal to zero. Substitute these values of s in the above equation.
- The values of s for which the K value is positive are the break points.
Rule 7 − Find the angle of departure and the angle of arrival.
At the complex conjugate open loop poles and zeros, respectively, the angle of departure and the angle of arrival can be calculated.
The formula for the angle of departure ϕd is
ϕd=180−ϕ
The formula for the angle of arrival ϕa is
ϕa=180+ϕ
Where,
ϕ=∑ϕP−∑ϕZ
Example of Root Locus in control system
Let us now draw the root locus of the control system having open loop transfer function, G(s)H(s)=K/{s(s+1)(s+5)}
Step 1 − The given open loop transfer function has three poles at s=0,s=−1 and s=−5. It doesn’t have any zero. Therefore, the number of root locus branches is equal to the number of poles of the open loop transfer function.
N=P=3
So The three poles are located are shown in the above figure. The line segment between s=−1 and s=0 is one branch of root locus on real axis. So And the other branch of the root locus on the real axis is the line segment to the left of s=−5
Step 2 − We will get the values of the centroid and the angle of asymptotes by using the given formula.
Centroid α=−2
So The angle of asymptotes are θ=60,180 and 300
The centroid and three asymptotes are shown in the following figure.
Step 3 − Since two asymptotes have the angles of 60 and 300, two root locus branches intersect the imaginary axis. So By using the Routh array method and special case(ii), the root locus branches intersects the imaginary axis at j√5 and −j√5.
So There will be one break-away point on the real axis root locus branch between the poles s=−1 and s=0. By following the procedure given for the calculation of break-away point, we will get it as s=−0.473.
The root locus diagram for the given control system is shown in the following figure.
So This will allow you to observe the motion of the closed loop transfer function’s poles and draw the root locus diagram for any control system. So We can learn the range of K values for various types of damping from the root locus diagrams.
Effects of Adding Open Loop Poles and Zeros on Root Locus
The root locus can be shifted in ‘s’ plane by adding the open loop poles and the open loop zeros.
- So Some root locus branches will move toward the right half of the “s” plane if a pole is included in the open loop transfer function. The damping ratio δ decreases as a result. Which implies that the time domain specifications such as delay time td, rise time tr, and peak time tp decrease while the damped frequency ωd increases. But it affects the stability of the system.
- So The left half of the “s” plane will be where some of the root locus branches move if we include a zero in the open loop transfer function. Thus, it will improve the stability of the control system. The damping ratio δ rises in this situation. Which implies that the time domain specifications such as delay time td, rise time tr, and peak time tp increase while the damped frequency ωd decreases.
So, based on the requirement, we can include (add) the open loop poles or zeros to the transfer function.