Sketching Root Locus with Matlab – Control Systems

The Root Locus graphically displayed both transient response and stability information. 

The locus can be sketched quickly to get a general idea of the changes in transient response generated by changes in gain. Specific points on the locus also can be found accurately to give quantitative design information.

The root locus typically allows us to choose the proper loop gain to meet a transient response specification. As the gain is varied, we move through different regions of response. Setting the gain at a particular value 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.

We can use Matlab and its Control System Toolbox to plot The Root Locus of a control system, given its characteristic equation or its forward transfer function.

1) As a first example, consider the system which characteristic equation is as follows:

The factored form of this characteristic equation is:

That is to say:

Where G(s)H(s) is the open-loop forward transfer function for a system represented by its block diagram as follows:

The root locus sketching looks as follows:

As it was said before, we can get the root locus for G(s)H(s) with Matlab using the following simple commands:

>> s=tf(‘s’)

>> sys=(1)/(s*(s+3)*(s^2+2*s+2))

>> rlocus(sys)

We obtain the next chart:

When gain K=8.1,  s=0+j1.09. We can see it by clicking on the lower intersection with the imaginary axis. Doing similarly, we can find the value of K at s=0-j1.09.

We can also see in the locus of the roots provided by Matlab for this system, the following:

1. The geometric places are symmetrical with respect to the real axis.
2. The four points at the root locus where K=0 (the poles, where the geometric places begin) are s=0, -3, -1+j, -1-j. Those where K=∞ (the zeros, where the geometric places finish) are s=∞, ∞ , ∞ , and ∞.
3. The maximum between n and m is 4, thus we say that the root locus has 4 branches, labeled by green colour ( -3), blue (0), celeste ( -1-j) and red ( -1+j).
4. The number of asymptotes is 4, (n-m=4). Since the number of finite poles exceeds the number of finite zeros, the locus of the roots approaches to s=∞ along the asymptotes.
5. The angles and centroid of the asymptotes are given below:

Once we sketch the root locus, we may want to accurately locate points on the root locus as well as find their associated gain. For example, we might want to know the exact coordinates of the root locus as it crosses the radial line representing 20% overshoot. Further, we also may want the value of gain at that point. Let us to illustrate that with the next example:

2) Given a unity feedback system that has the forward transfer function:

a. Sketch the root locus.
b. Find the imaginary-axis crossing.
c. Find the gain, K, at the jv-axis crossing.
d. Find the break-in point.
e. Find the point where the locus crosses the 0.5 damping ratio line.
f. Find the gain at the point where the locus crosses the 0.5 damping ratio line.
g. Find the range of gain, K, for which the system is stable.

a. Sketch the root locus.

>> numg=poly([2 4]);
>> deng=[1 6 25];
>> G=tf(numg,deng)

G

s^2 – 6 s + 8
————–
s^2 + 6 s + 25

>> rlocus(G);

b. Find the imaginary-axis crossing.

We can see in the chart that when the geometric place cross by the imaginary axe, the poles are at:

c. Find the gain, K, at the jv-axis crossing.

According to the previous graph:

The previous graph also tells us that the system is unstable (negative damping) when:

d. Find the break-in point.

e. Find the point where the locus crosses the 0.5 damping ratio line.
>> z=0.5;
>> sgrid(z,0)

f. Find the gain at the point where the locus crosses the 0.5 damping ratio line.

According to the previous graph:

g. Find the range of gain, K, for which the system is stable.

Previously, we have seen that the system is stable when:

Proportional Control

3. Find the gain to meet the damping ξ= 0.5, given the forward transfer function for a control system with unitary feedback:

We already know that this control system has the following configuration:

By adding to this original system a controller to manipulate the K gain of its Root Locus until obtaining the desired damping factor, we will be executing a Proportional Control Action. It is customary to add said controller in the region of low power, in series and just before the plant, or just before the direct transfer function, as illustrated in the following figure:

Adding a proportional controller of gain Kp, lThe forward transfer function G(s) is:

When K_P=1, we have the original system.

We will use the following commands of Matlab, rltool:

>> numg=100;
>> deng=[1 16 65 50];
>> G=tf(numg,deng)

G =

100
————————
s^3 + 16 s^2 + 65 s + 50

>> rltool(G)

Doing right click:

Here we get ζ=0.5 when K_P=1.46

Comparing the system before and after the compensation:

>> numc=100*1.46;
>> Gc=tf(numc,deng)

Gc =

146
————————
s^3 + 16 s^2 + 65 s + 50

>> sys_before=feedback(G,1);
>> sys_after=feedback(Gc,1);
>> step(sys_before,sys_after)

In blue, the response of the system to a step input before compensation is shown, when K_P=1. In red, after compensation, when K_P=1.46. The rise time is better (from 0.5626 s to 0.4356 s), but overshot is bigger due to less damping ζ (from 0.626 to 0.502). Steady-state error is also better.

We can see this data using:

>> stepinfo(sys_before)

RiseTime: 0.5626
SettlingTime: 1.7487
Overshoot: 7.5449

>> stepinfo(sys_after)

RiseTime: 0.4356
SettlingTime: 2.0551
Overshoot: 15.0397

>> damp (sys_before)

Damping= 6.26e-01

>> damp (sys_after)

Damping=5.02e-01

Or doing right click on the Root Locus, and then design requirement>new>design requirement type>damping ratio>0.5>ok. 

Conclusion

It is confirmed by these examples that the locus of the roots is a powerful method of analysis and design for the stability and transient response of a control system (Evans, 1948, 1950).

The feedback control systems are difficult to understand from the qualitative point of view, so that this understanding depends largely on mathematics. The root locus is the graphic technique that gives us that qualitative description of the performance of the control system we are designing.

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. That is what the following topic is about:

NEXT: Design via Root Locus – Improving Transient Response via Cascade Compensation

Sources:

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

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

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