Root locus analysis for linear control systems

You will receive an overview of the rules governing the root locus. More details will be provided in the next lectures.

In this lecture the first two rules of root locus are introduced, namely symmetry w.r.t. the real axis and the number of branches. After completing this lecture you can identify the root locus sketches drawn incorrectly and you will be ready to learn the next rules.

You will be able to determine from where the root locus branches originate and where they end up.

You will learn how the root locus branches behave as they approach the infinity. The point of abscissa and the angle of the asymptotes with the real axis will be explained in details.

You will be able to determine which parts of the real axis belong to root locus for a given system.

You will learn about the break-away and break-in points, when they exist and how to find them.

You will learn to calculate the angles of arrival at zeros and angles of departures from poles. You will also know when they are necessary to calculate, and when you can find their values directly by looking at the root locus sketch.

In this video you will see how the departure and arrival angles are found by applying the phase condition.

Sometimes, some of the root locus branches have an intersection with the imaginary axis. In this lecture, you will learn when this happens and how to find the intersection point on the imaginary axis, and also how to determine the value of the gain (Rho) at that point. This is also quite important regarding the changes in the stability of the system.

The stability of the closed-loop system could be analysed based on the root locus sketch. In this lecture you will learn this and find the range of values for Rho such that the system remains stable.

In this lecture you will learn how to draw the root locus for a system with a controller in which the varying parameter is not directly the gain of the controller, but possibly the location of its pole or zero.

In this example, you will learn drawing root locus for a system with a single pole at s = -4, Gp(s) = 1/(s+4). Then you will learn how to analyse the stability of the closed-loop system and how to find the value of Rho for which the closed-loop system pole is located at some specific position (s=-6).

In this example, you will learn drawing root locus for a system with a pole at s = -2 and a zero at s = -4, Gp(s) = 2(s+4)/(s+2). Then you will learn how to analyse the stability of the closed-loop system and how to find the value of Rho for which the closed-loop system pole is located at some specific position (s=-3).

In this example, you will learn drawing root locus for a system with a pole at s = +1 and a pole at s = -3, Gp(s) = 2/(s-1)(s+3). Then you will learn how to analyse the stability of the closed-loop system and how to find the value of Rho for which the closed-loop system pole is located at some specific position (s=-1+j).

In this example, you will learn drawing root locus for a system with three poles at s = 0, s = -1 and s = -3, Gp(s) = 5/s(s+1)(s+3). Then you will learn how to analyse the stability of the closed-loop system and how to find the value of Rho for which the closed-loop system pole is located at some specific position (s=-4).

In this example, you will learn drawing root locus for a system with a zero at s = -2 and two poles at s = -1, Gp(s) = 4/(s+2)(s+1)^2. Then you will learn how to analyse the stability of the closed-loop system and how to find the value of Rho for which the closed-loop system pole is located at some specific position (s=-5). You will also learn how to find the location of the second closed-loop pole.

In this example, you will learn drawing root locus for a system with a zero at s = +1 and two poles at s = -1-2j and s = -1+2j, Gp(s) = 2(s-1)/(s^2+2s+5). Then you will learn how to analyse the stability of the closed-loop system and how to find the value of Rho for which the closed-loop system pole is located at some specific position (s=-2).

• Draw the root locus sketch for a system with two poles -2 at and two zeros at -1 (Gc(s) = (s+1)^2/(s+2)^2).

• Discuss the stability of the closed-loop system referring to the root locus sketch.

Here you will find the solution of Exercise 1.

• Draw the root locus sketch for a system with three poles 0 at and one zero at -2 (Gc(s) = (s+2)/s^3).

• Discuss the stability of the closed-loop system referring to the root locus sketch.

Here you will find the solution of Exercise 1.

Congratulations! Now you are ready to learn designing controllers using root locus. Stay tuned for my next course on this topic.