Routh stability criterion

The Routh method can be used to find out if there are any roots of a polynomial in the RHP. It can be applied to either closedloop or openloop systems by using the appropnate characteristic equation. 

Assume the characteristic equation of interest is an Mh-order polynomial  

Then the first column of the array of Eq. (10.14) is examined. The number of sign changes of this first column is equal to the number of roots of the polynomial that are in the RHP. 

Thus for the system to be stable there can be no sign changes in the first column of the Routh array. 

Let us illustrate the application of the Routh stability criterion in some specific examples. 

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Hence, to give a GM of 2 and a PM of 50°, the controller gain must be set at 1.0. If it is doubled, i.e. multiplied by the GM, then the system just becomes unstable. Check using the Routh stability criterion ... 

The Routh stability criterion is quite useful, but it has definite limitations. It cannot handle systems with deadtime. It tells if the system is stable or unstable but it gives no information about how stable or unstable the system is. That is, if the test tells us that the system is stable, we do not know how close to instability it is. Another limitation of the Routh method is the need to express the character istic equation explicitly as a polynomial in s. This can become complex in high-order systems. 

KU. Use the Routh stability criterion to find the ultimate gain of the closedloop three-CSTR system with a PI controller ... 

If the process is openloop unstMe, Gm<.) will have one or more poles in the RHP, so F(,y = 1 + Gm( , B( will also have one or more poles in the RHP. We can find out how many poles there are by solving for the roots of the openloop characteristic equation or by using the Routh stability criterion on the openloop characteristic equation (the denominator of Gj, ). Once the number of poles P is known, the number of zeros can be found from Eq. (13.6). 

We could make a root locus plot in the U) plane. Or we could use the direct-substitution method (let U) = iv) to find the maximum stable value of. Let us use the Routh stability criterion. This criterion cannot be applied in the z plane because it gives the number of positive roots, not the number of roots outside the unit circle. The Routh array is... 

In Examples 11.6 to 11.8, the characteristic equations were either first- or second-order, and thus we could find the roots analytically. For higher-order polynomials, this is not possible, and numerical root-finding techniques (Chapra and Canale, 2010), also available in MATLAB and Mathematica, must be employed. An attractive alternative, the Routh stability criterion, is available to evaluate stability without requiring calculation of the roots of the characteristic equation. 

The Routh stability criterion is based on a characteristic equation that has the form... 

Note that the expressions in the numerators of Eqs. 11-94 to 11-97 are similar to the calculation of a determinant for a 2 X 2 matrix except that the order of subtraction is reversed. Having constructed the Routh array, we can now state the Routh stability criterion ... 

Routh Stability Criterion. A necessary and sufficient condition for all roots of the characteristic equation in Eq. 11-93 to have negative real parts is that all of the elements in the left column of the Routh array are positive. 

Next we present three examples that show how the Routh stability criterion can be applied. 

The direct-substitution method is related to the Routh stability criterion in Section 11.4.2. If the characteristic equation has a pair of roots on the imaginary axis, equidistant from the origin, and all other roots are in the left-hand plane, the single element in the next-to-last row of the Routh array will be zero. Then the location of the two imaginary roots can be obtained from the solution of the equation. 

If the process model is nonlinear, then advanced stability theory can be used (Khalil, 2001), or an approximate stability analysis can be performed based on a linearized transfer function model. If the transfer function model includes time delays, then an exact stability analysis can be performed using root-finding or, preferably, the frequency response methods of Chapter 14. A less desirable alternative is to approximate the terms and apply the Routh stability criterion. 

The Bode stability criterion has two important advantages in comparison with the Routh stability criterion of Chapter 11 ... 

It provides exact results for processes with time delays, while the Routh stability criterion provides only approximate results because of the polynomial approximation that must be substituted for the time delay. 

This chapter begins by presenting useful background information in Section J.l. The Bode and Nyquist stability criteria in Sections J.2 and J.3 are generally applicable and, unlike the Routh stability criterion of Chapter 11, provide exact results for systems with time delays. These stability criteria also provide measures of... 

The Routh stability criterion of Chapter 11 can be used because the characteristic equation is a polynomial in s. By trial and error, the maximum value is found to be... 

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