Routh array

The complete Routh array analysis allows us to find, for example, the number of poles on the imaginary axis. Since BIBO stability requires that all poles lie in the left-hand plane, we will not bother with these details (which are still in many control texts). Consider the fact that we can calculate easily the exact roots of a polynomial with MATLAB, we use the Routh criterion to the extent that it serves its purpose.1 That would be to derive inequality criteria for proper selection of controller gains of relatively simple systems. The technique loses its attractiveness when the algebra becomes too messy. Now the simplified Routh-Hurwitz recipe without proof follows. 

If the characteristic polynomial passes the coefficient test, we then construct the Routh array to find the necessary and sufficient conditions for stability. This is one of the few classical techniques that we do not emphasize and the general formula is omitted. The array construction up to a fourth order polynomial is used to illustrate the concept. 

In the case of a second order system, the first column of the Routh array reduces to simply the coefficients of the polynomial. The coefficient test is sufficient in this case. Or we can say that both the coefficient test and the Routh array provide the same result. 

The two additional constraints from the Routh array are hence... 

With the Routh-Hurwitz criterion, we need immediately xr > 0 and Kc > 0. (The, v term requires Kc > -1, which is overridden by the last constant coefficient.) The Routh array for this third order polynomial is... 

The results are exact—we do not need to make approximations as we had to with root locus or the Routh array. The magnitude plot is the same as the first order function, but the phase lag increases without bound due to the dead time contribution in the second term. We will see that this is a major contribution to instability. On the Nyquist plot, the G(jco) locus starts at Kp on the real axis and then "spirals" into the origin of the s-plane. 

Ultimate gain and ultimate period (Pu = 2tt/(0u) that can be used in the Ziegler-Nichols continuous cycling relations. Result on ultimate gain is consistent with the Routh array analysis. Limited to relatively simple systems. 

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

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... 

The characteristic equation is already in the form of equation 7.122 and all the coefficients are positive. Thus, to examine the stability of the system further, the Routh array must be constructed, viz. ... 

A further example of the use of the Routh array in connection with discrete systems appears in Section 7.17.5.)... 

Place coeffieients of polynomial into a Routh array as follows ... 

Second test. If all coefficients ao, a 1, <2 2,..., a . 1, a are positive, then from the first test we cannot conclude anything about the location of the roots. Form the following array (known as the Routh array) ... 

If Kc > 0.5, the third element of the first column of the Routh array becomes negative. We have two sign changes in the elements of the first column therefore, we have two roots of the characteristic equation located to the right of the imaginary axis. 

The coefficients 2 and Kc/ti are the elements of the Routh array in the row which is located just before the row whose first column element is zero (i.e., the elements of the second row). 

Return to Example 15.4 and let t, = 0.1. Then the third element of the first column in the Routh array becomes... 

The Routh array has n + 1 rows, where n is the order of the characteristic equation, Eq. 11-93. The Routh array has a roughly triangular structure with only a single element in the last row. The first two rows are merely the coefficients in the characteristic equation, arranged according to odd and even powers of 5-. The elements in the remaining rows are calculated from the formulas... 

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. 

This example illustrates that stability limits for controller parameters can be derived analytically using the Routh array in other words, it is not necessary to specify a numerical value for Kc before performing the stability analysis. 

The necessary condition for stability is that each coefficient in this characteristic equation must be positive. This situation occurs if —0.5 < Kc <5.5. The Routh array is... 

In this example, the Routh array provides no additional information but merely confirms that the system with the Fade approximation is stable if -0.5 < Kc< 5.5. 

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. 

Derive the characteristic equation and construct the Routh array for a control system with the following... 

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