Control Engineering Lecture# 10 & th April’2008

Stability Analysis BIBO: Bounded Input Bounded Output systems. BIBO: Bounded Input Bounded Output systems. For LTI systems this requires that all poles of the closed-loop For LTI systems this requires that all poles of the closed-loop transfer function lie in the left half of the complex plane. transfer function lie in the left half of the complex plane. Determine if the transfer function has any poles either on the imaginary axis or in the right half of the s-plane. Determine if the transfer function has any poles either on the imaginary axis or in the right half of the s-plane. Routh-Hurwitz criterion: determine if any roots of a polynomial lie outside the lelf half of the complex plane. It does not find the exact locations of the roots. Routh-Hurwitz criterion: determine if any roots of a polynomial lie outside the lelf half of the complex plane. It does not find the exact locations of the roots. Other methods find the exact locations of the roots. For first and second order systems, analytical method can be used. For higher order systems, computer programs or simulation are required. Other methods find the exact locations of the roots. For first and second order systems, analytical method can be used. For higher order systems, computer programs or simulation are required.

Some basic results : Some basic results : Second order system: Second order system:

We see that the coefficients of the polynomial are given by: We see that the coefficients of the polynomial are given by: Suppose that all the roots are real and on the left half plane, then all coefficients of the polynomial are positive. Suppose that all the roots are real and on the left half plane, then all coefficients of the polynomial are positive. If all the roots are real and in the left half plane then no coefficient can be zero. If all the roots are real and in the left half plane then no coefficient can be zero. The only case for which a coefficient can be negative is when there is at least one root in the right half plane. The only case for which a coefficient can be negative is when there is at least one root in the right half plane.

The above is also true for complex roots. The above is also true for complex roots. 1) If any coefficient is equal to zero, then not all roots are in 1) If any coefficient is equal to zero, then not all roots are in the left half plane. the left half plane. 2) If any coefficient is negative, then at least one root is in 2) If any coefficient is negative, then at least one root is in the right half plane. the right half plane. 3) The converse of rule 2) is not always true. 3) The converse of rule 2) is not always true. Example: Example:

Routh-Hurwitz Stability Criterion All the coefficients must be positive if all the roots are in the left half plane. Also it is necessary that all the coefficients for a stable system be nonzero. All the coefficients must be positive if all the roots are in the left half plane. Also it is necessary that all the coefficients for a stable system be nonzero. These requirements are necessary but not sufficient. That is we know the system is unstable if they are not satisfied; yet if they are satisfied, we must proceed further to ascertain the stability of the system. These requirements are necessary but not sufficient. That is we know the system is unstable if they are not satisfied; yet if they are satisfied, we must proceed further to ascertain the stability of the system. For example, For example, The Routh-Hurwitz is a necessary and sufficient criterion for the stability of linear systems. The Routh-Hurwitz is a necessary and sufficient criterion for the stability of linear systems.

The Routh-Hurwitz criterion applies to a polynomial of the form: The Routh-Hurwitz criterion applies to a polynomial of the form: The Routh-Hurwitz array: The Routh-Hurwitz array:

Columns of s are only for accounting. Columns of s are only for accounting. The b row is calculated from the two rows above it. The b row is calculated from the two rows above it. The c row is calculated from the two rows directly above it. The c row is calculated from the two rows directly above it. Etc… Etc… The equations for the coefficients of the array are: The equations for the coefficients of the array are: Note: the determinant in the expression for the ith coefficient in a row is formed from the first column and the (i+1)th column of the two preceding rows. Note: the determinant in the expression for the ith coefficient in a row is formed from the first column and the (i+1)th column of the two preceding rows.

The number of polynomial roots in the right half plane is equal to the number of sign changes in the first column of the array. The number of polynomial roots in the right half plane is equal to the number of sign changes in the first column of the array. Example: Example: Since there are two sign changes on the first column, there are two roots of the polynomial in the right half plane: system is unstable. Since there are two sign changes on the first column, there are two roots of the polynomial in the right half plane: system is unstable. Note: The Routh-Hurwitz criterion shows only the stability of the system, it does not give the locations of the roots, therefore no information about the transient response of a stable system is derived from the R-H criterion. Also it gives no information about the steady state response. Obviously other analysis techniques in addition to the R- H criterion are needed. Note: The Routh-Hurwitz criterion shows only the stability of the system, it does not give the locations of the roots, therefore no information about the transient response of a stable system is derived from the R-H criterion. Also it gives no information about the steady state response. Obviously other analysis techniques in addition to the R- H criterion are needed.

From the equations, the array cannot be completed if the first element in a row is zero. Because the calculations require divisions by zero. We have 3 cases: From the equations, the array cannot be completed if the first element in a row is zero. Because the calculations require divisions by zero. We have 3 cases: Case 1: none of the elements in the first column of the array is zero. This is the simplest case. Follow the algorithm as shown in the previous slides. Case 1: none of the elements in the first column of the array is zero. This is the simplest case. Follow the algorithm as shown in the previous slides. Case 2: The first element in a row is zero, with at least one nonzero element in the same row. In this case, replace the first element which is zero by a small number. All the elements that follow will be functions of. After all the elements are calculated, the signs of the elements in the first column are determined by letting approach zero. Case 2: The first element in a row is zero, with at least one nonzero element in the same row. In this case, replace the first element which is zero by a small number. All the elements that follow will be functions of. After all the elements are calculated, the signs of the elements in the first column are determined by letting approach zero. Example: Example:

There are 2 sign changes regardless of is positive or negative. Therefore the system is unstable. There are 2 sign changes regardless of is positive or negative. Therefore the system is unstable.

Case 3: All elements in a row are zero. Case 3: All elements in a row are zero. Example: Example: Here the array cannot be completed because of the zero element in the first column. Here the array cannot be completed because of the zero element in the first column. Another example: Another example:

Case 3 polynomial contains an even polynomial as a factor. It is called the auxiliary polynomial. In the first example, the auxiliary polynomial is Case 3 polynomial contains an even polynomial as a factor. It is called the auxiliary polynomial. In the first example, the auxiliary polynomial is and in the second example, auxiliary polynomial is and in the second example, auxiliary polynomial is Case 3 polynomial may be analyzed as follows: Case 3 polynomial may be analyzed as follows: Suppose that the row of zeros is the row, then the Suppose that the row of zeros is the row, then the auxiliary polynomial is differentiated with respect to s, auxiliary polynomial is differentiated with respect to s, and the coefficients of the resulting polynomial used to replace the zeros in the row. The calculation of the array then continues as in the case 1. and the coefficients of the resulting polynomial used to replace the zeros in the row. The calculation of the array then continues as in the case 1. Example: Example: