Routh's criterion Routh's criterion

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Routh Hurwitz Stability Criterion

The physical meaning is that there are symmetrically located roots of the characteristic equation in the s plane. Thus the criterion provides a way to determine if the equations of motion of a linear system have only stable solutions, without solving the system directly.

In order to construct the routh array follow these steps: Routh Stability Criterion This criterion is also known as modified Hurwitz Criterion of stability of the system. The general form of determinant is given below: The Routh test is an efficient recursive algorithm that English mathematician Edward John Routh proposed in to determine whether all the roots of the characteristic polynomial of a linear system have negative real parts.

The transfer function of the feedforward pass of a feedback system is and the feedback gain is negative feedback. The system will be stable if and only if the value of each determinant is greater than zero, i.

Mathematically we write as second element Similarly, we can calculate all the elements of the fourth row. The overall transfer function is therefore: Note that row 3 is divided by 2 to become row 3' without affecting the result. Multiply a0 with the diagonally opposite element of next to next column i.

The elements of third row can be calculated as: On solving the limit at every element if we will get positive limiting value then we will say the given system is stable otherwise in all the other condition we will say the given system is not stable.

The elements from the third row on are computed based on the determinant of a 2 by 2 array composed of the two elements of the first column of the previous two rows and the two elements of the subsequent columns.