Feedback Control Sytem (FCS) Lecture19-20 Routh-Herwitz Stability Criterion Dr. Imtiaz Huain email: imtiaz.huain@faculty.muet.edu.pk URL :http://imtiazhuainkalwar.weebly.com/
Stability of Higher Order Sytem 5 2 3 1 10 3 5 1 1 R C ) ( ) ( ) ( 2) 3 ( 1) 10( ) ( ) ( 2 3 4 2 2 R C Un-Stable Stable
Routh-Hurwitz Stability Criterion It i a method for determining continuou ytem tability. The Routh-Hurwitz criterion tate that the number of root of the characteritic equation with poitive real part i equal to the number of change in ign of the firt column of the Routh array.
Routh-Hurwitz Stability Criterion Thi method yield tability information without the need to olve for the cloed-loop ytem pole. Uing thi method, we can tell how many cloed-loop ytem pole are in the left half-plane, in the right half-plane, and on the jw-axi. (Notice that we ay how many, not where.) The method require two tep: 1. Generate a data table called a Routh table. 2. interpret the Routh table to tell how many cloed-loop ytem pole are in the LHP, the RHP, and on the jw-axi.
Routh Stability Condition If the cloed-loop tranfer function ha all pole in the left half of the -plane, the ytem i table. Thu, a ytem i table if there are no ign change in the firt column of the Routh table. The Routh-Hurwitz criterion declare that the number of root of the polynomial that are lie in the right half-plane i equal to the number of ign change in the firt column. Hence the ytem i untable if the pole lie on the right hand ide of the -plane.
Generating a baic Routh Table Only the firt 2 row of the array are obtained from the characteritic eq. the remaining are calculated a follow;
Example#1 Conider the following characteritic equation: Develop Routh array and determine the tability of the ytem.
Conider the following ytem: Example#2 Develop Routh array and determine the tability of the ytem.
Example#3 Find the tability of the continue ytem having the characteritic polynomial of a third order ytem i given below The Routh array i Becaue TWO change in ign appear in the firt column, we find that two root of the characteritic equation lie in the right hand ide of the -plane. Hence the ytem i untable.
Example#4 Determine a rang of value of a ytem parameter K for which the ytem i table. The Routh table of the given ytem i computed and hown i the table below; For ytem tability, it i neceary that the condition 8 k >0, and 1 + k > 0, mut be atified. Hence the rang of value of a ytem parameter k mut lie between -1 and 8 (i.e., -1 < k < 8).
Special Cae Cae-1: Zero in the firt column If firt element of a row i zero, diviion by zero would be required to form the next row. To avoid thi phenomenon, zero i replaced by a very mall number (ay є).
Example#5
Example#5 Determine the tability of the ytem having a characteritic equation given below; The Routh array i hown in the table; Where There are TWO ign change due to the large negative number in the firt column, Therefore the ytem i untable, and two root of the equation lie in the right half of the -plane.
Example#6 Determine the range of parameter K for which the ytem i untable. The Routh array of the above characteritic equation i hown below; Where Therefore, for any value of K greater than zero, the ytem i untable. Alo, becaue the lat term in the firt column i equal to K, a negative value of K will reult in an untable ytem. Conequently, the ytem i untable for all value of gain K.
Cae-I: Stability via Revere Coefficient (Phillip, 1991). A polynomial that ha the reciprocal root of the original polynomial ha it root ditributed the ame right half-plane, left half plane, or imaginary axi becaue taking the reciprocal of the root value doe not move it to another region. If we can find the polynomial that ha the reciprocal root of the original, it i poible that the Routh table for the new polynomial will not have a zero in the firt column. The polynomial with reciprocal root i a polynomial with the coefficient written in revere order. Thi method i uually computationally eaier than the epilon method.
Determine the tability of the of the cloed-loop tranfer function; Table-1: The complete Routh table i formed by uing the denominator of the characteritic equation T(). Table-2: how the firt column of Table-1 along with the reulting ign for choice of ε poitive and ε negative. A zero appear only in the firt column (the 3 row). Next replace the zero by a mall number, ε, and complete the table. Aume a ign, poitive or negative, for the quantity ε. When quantity ε i either poitive or negative, in both cae the ign in the firt column of Routh table i change twice. Hence, the ytem i untable and ha two pole in the right half-plane.
Example-7: Determine the tability of the cloed-loop tranfer function; Firt write a polynomial that ha the reciprocal root of the denominator of T(). Thi polynomial i formed by writing the denominator of T() in revere order. Hence, The Routh table i Since there are TWO ign change, the ytem i untable and ha TWO right-halfplane pole. Thi i the ame a the reult obtained in the previou Example. Notice that Table doe not have a zero in the firt column.
Cae-II: Entire Row i Zero. Sometime while making a Routh table, we find that an entire row conit of zero. Thi happen becaue there i an even polynomial that i a factor of the original polynomial. Thi cae mut be handled differently from the cae of a zero in only the firt column of a row.
Cae-II: Entire Row i Zero. The characteritic equation q() of the ytem i q = 3 + 2 2 + 4 + 8
Example-8: Determine the tability of the ytem. The characteritic equation q() of the ytem i Where K i an adjutable loop gain. The Routh array i then; For a table ytem, the value of K mut be;
Example-8: Determine the tability of the ytem. Alo, when K = 8, we obtain a row of zero (Cae-II). The auxiliary polynomial, U(), i the equation of the row preceding the row of Zero. The U() in thi cae, obtained from the 2 row. The order of the auxiliary polynomial i alway even and indicate the number of ymmetrical root pair.
Example-8: Determine the tability of the ytem. The auxiliary polynomial, U(), can be obtain a;
Example#9 Conider the following characteritic equation. Determine the range of K for tability.
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