Stability Design via Routh-Hurwitz

PROBLEM: Find the range of gain, K, for the system of Figure 6.10 that will cause the system to be stable, unstable, and marginally stable. Assume K > 0.

Question

Stability Design via Routh-Hurwitz

PROBLEM: Find the range of gain, K, for the system of Figure 6.10 that will cause the system to be stable, unstable, and marginally stable. Assume K > 0.

Accepted Answer

First find the closed-loop transfer function as

[latex]T(s)=\frac{K}{s^{3}+18s^{2}+77s+K }[/latex]

Next form the Routh table shown as Table 6.15.

[latex]s^{3}[/latex]	1	77
[latex]s^{2}[/latex]	18	K
[latex]s^{1}[/latex]	[latex]\frac{1386-K}{18}[/latex]
[latex]s^{0}[/latex]	K

Since K is assumed positive, we see that all elements in the first column are always positive except the [latex]s^{1}[/latex] row. This entry can be positive, zero, or negative, depending upon the value of K. If K < 1386, all terms in the first column will be positive, and since there
are no sign changes, the system will have three poles in the left half-plane and be stable.

If K > 1386, the [latex]s^{1}[/latex]term in the first column is negative. There are two sign changes, indicating that the system has two right-half-plane poles and one left half-plane pole,
which makes the system unstable.

If K = 1386, we have an entire row of zeros, which could signify jω poles. Returning to the [latex]s^{2}[/latex] row and replacing K with 1386, we form the even polynomial

[latex]P(s)=18s^{2} +1386[/latex]

Differentiating with respect to s, we have

[latex]\frac{dP(s)}{ds}=36s+0[/latex]

Replacing the row of zeros with the coefficients of Eq. (6.22), we obtain the Routh-Hurwitz table shown as Table 6.16 for the case of K = 1386.

[latex]s^{3}[/latex]	1	77
[latex]s^{2}[/latex]	18	1386
[latex]s^{1}[/latex]	[latex]\cancel{0} 36[/latex]
[latex]s^{0}[/latex]	1386

Since there are no sign changes from the even polynomial ( [latex]s^{2}[/latex]row) down to the bottom of the table, the even polynomial has
its two roots on the jω-axis of unit multiplicity. Since there are no sign changes above the even polynomial, the remaining root is in the left
half-plane. Therefore the system is marginally stable.

Students who are using MATLAB should now run ch6p2 in Appendix B.
You will learn how to set up a loop to search for the range of gain to yield stability. This exercise uses MATLAB to do Example 6.9.

Students who are performing the MATLAB exercises and want to explore the added capability of MATLAB’s Symbolic Math Toolbox should now run ch6sp2 in Appendix F at www.wiley.com/college/ nise. You will learn how to use the Symbolic Math Toolbox to calculate the values of cells in a Routh table even if the table contains
symbolic objects, such as a variable gain, K. You will see that the Symbolic Math Toolbox and MATLAB yield an alternative way to solve Example 6.9.

Question 6.9: Stability Design via Routh-Hurwitz PROBLEM: Find the range o...

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