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1. The document discusses the Routh-Hurwitz stability criterion, which is a test used to determine the stability of linear time-invariant systems by constructing a Routh array from the coefficients of the characteristic equation and analyzing it. 2. Special cases that can occur with the Routh array, such as a zero in the first column or an entire zero row, are explained along with their implications. 3. An important application of the Routh-Hurwitz criterion is determining the range of values for a system gain K that ensures stability.
Cse presentation
- 1. Gujarat Technical University BIRLA VISHWAKARMA MAHAVIDYALAYA ET Department ROUTH-STABILITY CRITERSION SUBMITED BY:- APAR TRIVEDI :130080112057 VATSAL BODIWALA :140083111002 Under the Guidance : Prof. Amit Chokasi ET department, BVM
- 2. CONTENT The concept of stability The Routh-Hurwitz stability criterion The relative stability of feedback systems Design examples and MATLAB simulation Summary
- 3. The concept of stability A stable system is a dynamic system with a bounded output to a bounded input (BIBO). absolute stability relative stability
- 4. Stability for LTI system A necessary and sufficient condition for a feedback system to be stable is that all the poles of the system transfer function have negative real parts. Impulse response approach to zero when time lead to infinite.
- 5. Routh-Hurwitz criterion Routh criterion: This criterion states that the number of roots of characteristic equation with positive real parts is equal to the number of changes in sign of the first column of the Routh array.
- 8. Routh-Hurwitz Stability Criterion – Generate Routh Table Given Routh Table
- 9. Routh-Hurwitz Stability Criterion – Generate Routh Table Routh Table The value in a row can be divided for easy calculation
- 10. Routh-Hurwitz Stability Criterion – Generate Routh Table Example Given
- 12. R-H Criterion – Special Cases 1. Zero in the first column 2. Zero in the entire row
- 13. Special Case 1: There are cases when the first element of the Routh array is zero and the rest of the row is non- zero. In such a case we cannot proceed to the next stage without making some modifications. Example: 𝑆5 +𝑆4 +2𝑆3 +2𝑆2 +3S+5=0 2 3 4 5 S S S S 0 1 1 2 2 2 0 5 3
- 14. Special Case 2: There are cases when all the elements of a row in the Routh array are zero. Obtaining a row of zeros implies one of the four conditions. i. Real roots but symmetrically located about the j-axis. ii. Conjugate roots on imaginary axis. iii. Symmetrically located complex roots about the j-axis. iv. Repeated conjugate roots on the j-axis.
- 16. 𝑆6 +2𝑆5 +8𝑆4 +12𝑆3 +20𝑆2 +16𝑆+16=0 𝑎0=1, 𝑎1 = 2, 𝑎2=8, 𝑎3=12, 𝑎4=20, 𝑎5=16, 𝑎6=16 The Routh array is, 𝑆3 row is zero, Take auxiliary equation using 𝑆4 row, A(s)=2𝑆4+12𝑆2+16 𝑑𝐴(𝑠) 𝑑𝑠 =8𝑆3+24S 3 4 5 6 S S S S 0 2 2 1 0 12 12 8 0 16 16 20 0 0 16
- 17. Application of R-H criterion One of the important application of the Routh array to determine the value of the gain K for stability. In many practical examples, an amplifier of gain K is introduced to control the overall system. 1 + k G(s) H(s) = 0 H(s) G(s)+ - K R(s) C(s)
- 18. ADVANTAGES Stability of the system can be determined without actually solving the characteristic equation. No evaluation of determinants is required which saves calculation time. It is not tedious or time consuming. It progresses systematically. For unstable system, it gives the number of roots of characteristic equation having positive real part.
- 19. ADVANTAGES Relative stability of the system can be ascertained. It helps in finding out the range of values of K for system stability. It helps in finding out intersection point of root locus with imaginary axis. By using this criterion, critical value of gain can be determined and hence the frequency of sustained oscillations.
- 20. DISADVANTAGES It is valid only for real coefficient of the characteristic equation. It does not provide exact locations of the closed- loop poles in left or right half of s-plane. It does not suggest methods of stabilizing an unstable system. Applicable only to linear systems.