What is Routh Stability criteria?
The Routh stability criterion is a mathematical technique employed to ascertain whether all the roots of a polynomial lie on the left-hand side of the s-plane. It is a valuable tool in the assessment of stability for linear time-invariant systems. Typically, the characteristic equation for such systems is expressed as a polynomial equation.
Necessary and sufficient condition for Routh Stability
Using Routh stability criteria, we can derive a necessary and sufficient condition that we may not get by observing the characteristic equation alone.
Necessary but not sufficient condition by observing characteristic equation
$$1+K(s)G(s)H(s)=a_0s^n+a_1s^{n-1}+a_2s^{n-2}+…+a_{n-1}s^1+a_ns^0=0
$$
The presence of negative or zero coefficients within a characteristic polynomial serves as an indicator that the system is unstable or, at best, marginally stable. Therefore, for a system to be considered stable, it is imperative that all coefficients in its characteristic polynomial be positive. The occurrence of any zero or negative coefficient is an immediate indication of instability.
However, in cases where all coefficients are indeed positive, it does not guarantee the system’s stability. There may still exist roots in the right half of the complex plane or on the imaginary axis. To ensure that all roots possess negative real parts, it is necessary but not sufficient for all coefficients in the characteristic equation to be positive.
Necessary and sufficient condition using Routh table
The necessary and sufficient condition for establishing stability using Routh-Hurwitz criteria is that every element within the first column of the Routh array must possess a positive value or have the same sign. When this condition is not satisfied, the system is deemed unstable.
Additionally, the count of sign changes within the first column of the Routh array corresponds directly to the number of roots of the characteristic equation located in the right half of the s-plane.
Routh Array Method : Creating the Routh Array/Matrix
In the construction of the Routh array, the coefficients of the characteristic equation are organized into two rows. This arrangement starts with the coefficients of sn and sn-1 in that order, and then continues with the even-numbered and odd-numbered coefficients, as illustrated below:
| $$s^n$$ | $$a_0$$ | $$a_2$$ | $$a_4$$ | $$a_6$$ | $$\dots{}$$ |
|---|---|---|---|---|---|
| $$s^{n-1}$$ | $$a_1$$ | $$a_3$$ | $$a_5$$ | $$a_7$$ | $$\dots{}$$ |
The following rows are added subsequently to complete the Routh array:
| $$s^{n-2}$$ | $$b_1=\cfrac{a_1a_2-a_0a_3}{a_1}$$ | $$b_2=\cfrac{a_1a_4-a_0a_5}{a_1}$$ | $$b_3=\cfrac{a_1a_6-a_0a_7}{a_1}$$ | $$\dots{}$$ |
|---|---|---|---|---|
| $$s^{n-3}$$ | $$c_1=\cfrac{b_1a_3-b_2a_1}{b_1}$$ | $$c_2=\cfrac{b_1a_5-b_3a_1}{b_1}$$ | $$\dots{}$$ | $$\dots{}$$ |
| $$\dots{}$$ | $$\dots{}$$ | $$\dots{}$$ | $$\dots{}$$ | $$\dots{}$$ |
| $$s^0$$ | $$a_n$$ |
This process continues until there are no more ci elements remaining. Subsequently, the remaining rows are constructed in a similar manner, extending down to the s0-row. The entire array takes on a triangular shape, with the noteworthy observation that both the s1-row and the s0-row consist of just a single term. It’s important to emphasize that when constructing the Routh array, any absent terms are treated as zeroes. Additionally, throughout the process, you have the flexibility to multiply or divide all the elements in any row by a positive constant, a step taken to streamline the computational effort. This adjustment, however, does not alter the signs of the elements in the first column.
Routh-Hurwitz example
Let’s consider a fourth-order system with the characteristic equation
$$s^4+10s^3+18s^2+16s+5=0$$
| $$s^4$$ | $$1$$ | $$18$$ | $$5$$ |
|---|---|---|---|
| $$s^3$$ | $$10$$ | $$16$$ | $$0$$ |
| $$s^2$$ | $$ \cfrac{82}{5}=\cfrac{10\cdot{}18-1\cdot{}16}{10}$$ | $$5=\cfrac{10\cdot{}5-1\cdot{}0}{10}$$ | $$\dots{}$$ |
| $$s^1$$ | $$\cfrac{531}{41}=\cfrac{(82/5)\cdot{}16-10\cdot{}5}{(82/5)}$$ | $$0$$ | |
| $$s^0$$ | $$5=\cfrac{(531/41)\cdot{}5-(82/5)\cdot{}0}{(531/41)}$$ |
The elements of the first column of the above-mentioned table are all positive, and hence the system is stable.
Let’s consider another example :
$$3s^4+20s^3+5s^2+5s+2=0$$
| $$s^4$$ | $$3$$ | $$5$$ | $$2$$ |
|---|---|---|---|
| $$s^3$$ | $$20$$ | $$5$$ | $$0$$ |
| $$s^2$$ | $$\cfrac{85}{20}=\cfrac{20\cdot{}5-3\cdot{}5}{20}$$ | $$2=\cfrac{20\cdot{}2-3\cdot{}0}{20}$$ | $$0$$ |
| $$s^1$$ | $$-\cfrac{75}{17}=\cfrac{(85/20)\cdot{}5-20\cdot{}2}{(85/20)}$$ | $$0$$ | |
| $$s^0$$ | $$2$$ |
Analyzing the first column of the Routh array reveals two sign changes, one from 85/20 to -75/17 and another from -75/17 to 2. Consequently, the system being examined is unstable, featuring two poles situated in the right half of the s-plane. It’s important to note that the Routh stability criterion exclusively provides the count of roots within the right half of the s-plane. This method does not furnish details about the actual root values, nor does it differentiate between real and complex roots.