Control Systems: Characteristic Equation

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Questions and Answers

What is obtained by setting the denominator of the right-hand side of Equation (6–1) equal to zero?

The poles of the open-loop system

The characteristic equation of the closed-loop system (correct)

The root locus of the system

The transfer function of the system

What can be extended to include the transport lag e–Ts?

The root locus method

The analysis of the open-loop system

The characteristic equation of the system

The analysis of the closed-loop system (correct)

What is represented by the root locus?

The loci of the open-loop poles as the gain K is varied

The loci of the closed-loop poles as the gain K is varied (correct)

The characteristic equation of the system

The transfer function of the system

What is required to begin sketching the root loci of a system?

The location of the poles and zeros of G(s)H(s) Signup and view all the answers

What is obtained by equating the angles and magnitudes of both sides of Equation (6–2)?

The angle and magnitude conditions Signup and view all the answers

What is the direction in which the angles of the complex quantities originating from the open-loop poles and open-loop zeros are measured?

Counterclockwise direction Signup and view all the answers

What are the values of s that fulfill both the angle and magnitude conditions?

The roots of the characteristic equation Signup and view all the answers

What is the characteristic equation of the system if G(s)H(s) involves a gain parameter K?

1 + KG(s)H(s) = 0 Signup and view all the answers

What is the condition for the sum of the angles of G(s)H(s) at a test point?

∑G(s)H(s)∑ = 1 Signup and view all the answers

What is the angle of G(s)H(s) when G(s)H(s) = -1?

180° Signup and view all the answers

What is the magnitude of G(s)H(s) for a given system?

KB1 / (A1A2A3A4) Signup and view all the answers

Why do we only need to construct the upper half of the root loci?

Because the open-loop poles and zeros are always located symmetrically about the real axis. Signup and view all the answers

What is the purpose of using graphical computation combined with inspection in constructing root-locus plots?

To enhance understanding of how the closed-loop poles move in the complex plane. Signup and view all the answers

What is the angle of G(s)H(s) at a test point s in terms of the angles measured from the open-loop poles and zero?

f1 - u1 - u2 - u3 - u4 Signup and view all the answers

What is the condition for G(s)H(s) to be equal to -1?

1 + G(s)H(s) = 0 Signup and view all the answers

Why is it necessary to use the same divisions on the abscissa as on the ordinate axis when sketching the root locus on graph paper?

To ensure accurate measurements of angles and magnitudes. Signup and view all the answers

Study Notes

Characteristic Equation and Root Locus

The characteristic equation is obtained by setting the denominator of the right-hand side of Equation (6–1) equal to zero, resulting in Equation (6–2).
The characteristic equation can be split into two equations: the angle condition (Equation 6–3) and the magnitude condition (Equation 6–4).
The values of s that fulfill both the angle and magnitude conditions are the roots of the characteristic equation, or the closed-loop poles.

Root Locus

The root locus is a locus of the points in the complex plane satisfying the angle condition alone.
The roots of the characteristic equation (the closed-loop poles) corresponding to a given value of the gain can be determined from the magnitude condition.
The root loci for a system are the loci of the closed-loop poles as the gain K is varied from zero to infinity.

Application of Angle and Magnitude Conditions

To apply the angle and magnitude conditions, the location of the poles and zeros of G(s)H(s) must be known.
The angles of the complex quantities originating from the open-loop poles and open-loop zeros to the test point s are measured in the counterclockwise direction.

Example of G(s)H(s) with a Gain Parameter

If G(s)H(s) involves a gain parameter K, the characteristic equation may be written as Equation (6–5).
The root loci for the system are the loci of the closed-loop poles as the gain K is varied from zero to infinity.

Graphical Construction of Root Loci

The root loci can be constructed graphically using the angle and magnitude conditions.
Because the open-loop complex-conjugate poles and complex-conjugate zeros, if any, are always located symmetrically about the real axis, the root loci are always symmetrical with respect to this axis.
Therefore, only the upper half of the root loci needs to be constructed, and the mirror image of the upper half can be drawn in the lower-half s plane.

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Description

Learn about the characteristic equation of a closed-loop system, obtained by setting the denominator of the right-hand side of Equation (6-1) equal to zero.