Routh-Hurwitz test

Examination of the characteristic equation indicates that it is not necessary to compute the actual values of the roots. All that is required is a knowledge of the location of the roots, i.e., if the roots lie on the right- or left-hand side of the imaginary axis. A simple test known as the Routh-Hurwitz test allows one to determine if any root is located on the right side of the imaginary axis, therefore rendering the system unstable. 

Chapter 15. The mathematical proof of the Routh-Hurwitz tests can be found in the classic book ... 

In the same way that the Routh-Hurwitz criterion offers a simple method of determining the stability of continuous systems, the Jury (1958) stability test is employed in a similar manner to assess the stability of discrete systems. 

Stability analysis methods Routh-Hurwitz criterion Apply the Routh test on the closed-loop characteristic polynomial to find if there are closed-loop poles on the right-hand-plane. 

We first introduce the time honored (/.< ., ancient ) Routh-Hurwitz criterion for stability testing. 

According to the first test of the Routh-Hurwitz criterion for stability (see Section 15.3), eq. (24.14) has at least one root with positive real part if any of its coefficients are negative. Thus the closed-loop behavior of the process is unstable if Kc i and Kc2 take on such values that make the last term of eq. (24.14) negative [all other terms in eq. (24.14) are always positive] ... 

Note. The allowable range of values for Kc and Kc2, which render stable responses when both loops are closed, can be found by applying the second test of the Routh-Hurwitz criterion. The reader is encouraged to complete this example and find the range of values of Kc and Kc2, which render stable responses. 

The criterion of stability for closed-loop systems does not require calculation of the actual values of the roots of the characteristic polynomial. It only requires that we know if any root is to the right of the imaginary axis. The Routh-Hurwitz procedure allows us to test if any root is to the right of the imaginary axis and thus reach quickly a conclusion as to the stability of the closed-loop system without computing the actual values of the roots. 

The condition under which the roots of the characteristic equation have negative real parts can be established using the Routh-Hurwitz criteria (11, 23]. A careful examination of these criteria indicate that for active control system considered herein, all but one of the conditions are satisfied automatically. The only condition to be satisfied is that the test function, T, should be positive [11, 23], where... 

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