Routh Hurwitz Stability Test & Analysis of Closed-loop System Root Locus Analysis

Given Transfer Function of a system Characteristic polynomial of the system: ROUTH HURWITZ TABLE

Characteristic Polynomial: EXAMPLE: Characteristic Polynomial: ROUTH HURWITZ TABLE: 2 SIGN CHANGES IN FIRST COLUMN. THEREFORE TWO UNSTABLE POLES

ONE SIGN CHANGE IN FIRST COLUMN. THEREFORE ONE UNSTABLE POLE

Y(s) = KP G(s) [R(s) – H(s)Y(s)] CLOSED-LOOP SYSTEM R(s) Y(s) E(s) U(s) E(s) = R(s) – H(s)Y(s) U(s) = KP E(s) = KP [R(s) – H(s)Y(s)] Y(s) = G(s) U(s) Y(s) = KP G(s) [R(s) – H(s)Y(s)]

ROOTS (poles) OF THE CLOSED-LOOP SYSTEM The poles of the closed-loop system now depend on the gain Kp and the compensator H(s) as well as the poles of the original plant i.e. G(s) We want a quick way to determine what will happen to the closed-loop poles when the gain Kp is increased from zero to infinity. A ROOT LOCUS sketch enables us to see the migration of the closed-loop poles.

THE ROOT-LOCUS Let Since Equation 1 Kp = 0, Equation 1 becomes D(s) = 0, The poles (roots) of the closed-loop system are the poles of the transmittance GH Kp = infinity, Equation 1 becomes N(s) = 0, The poles (roots) of the closed-loop system are the ZEROS of the transmittance GH

Example to illustrate that the roots of the closed-loop go from the POLES of GH to the ZEROS of GH Poles of GH : s =0, s = -1, s = -4 ZEROS of GH: s = -0.5 + j, s = -0.5 - j

The root locus on the previous slide is reproduced here K = 0 Real axis segment Real axis segment A real axis segment of the root locus exists if the total number of poles + zeros to the right of a point are an ODD number

Centroid = n = number of poles of GH = 3 m = number of zeros of GH = 2 Asymptotic angles If n – m < 2 then asymptotic angles are meaningless

Poles of GH at s = 0, s = -1, s = -4 Zeros of GH at s = -6, s = -0.5 +j, s = -0.5 - j

Poles of GH at s = -5, s = -1 + j, s = -1 - j Zeros at infinity System remains stable for a range of gain Kp from zero to 72

Centroid = n = number of poles of GH = 3 m = number of zeros of GH = 0 Asymptotic angles Draw straight lines starting from the centroid at -2.33 making angles of 60, -60 and 180. The locii are asymptotic to these straight lines

What is G(s)H(s) from the root-locus?

ROOT LOCUS SKETCHING 1. Form the product G(s)H(s) 2. Draw the POLES and ZEROS of G(s)H(s) 3. Draw the REAL-AXIS SEGMENTS of the root locus 4. Find the CENTROID 5. Find the ASYMPTOTIC ANGLES (when locii go to zeros at infinity) i is an integer 0, 1, 2, 3, -----

6. Find the range of gain Kp for which the system remains stable, since stability is determined by the closed-loop poles Use the Routh Hurwitz table to find the gain Kp that gives a table row of all zeros, (boundary of stability)

Closed-loop Poles Kcritical = 74 The third row can be made all zero

Kcritical = 74 s = jω Therefore, at Kcriticical the poles are To find the value of the poles at cross over, let Therefore, at Kcriticical the poles are

System remains stable for a range of gain Kp from zero to 72

G(s) H(s) Kp CLOSED-LOOP SYSTEM Exercise to see how the poles of this closed-loop system move with increasing K G(s) H(s) Kp CLOSED-LOOP SYSTEM R(s) Y(s) E(s) U(s) Perform Root Locus Analysis on the product GH

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