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Stability 509

To further illustrate the Routh–Hurwitz criteria, consider a system governed by the
equation:

3
2
dx dx dx
a + a + a + ax = 0 (14.7.24)
0 3 1 2 2 3
dt dt dt
In this case the criteria of Eqs. (14.7.10) become:
a > 0, a > 0, a > 0, and a > 0 (14.7.25)
0 1 2 3

and

aa > a a (14.7.26)
12 0 3
Finally, consider a system governed by the equation:

3
2
4
dx dx dx dx
a + a + a + a + ak = 0 (14.7.27)
0 4 1 3 2 2 3 4
dt dt dt dt
Here, the Routh–Hurwitz criteria of Eqs. (14.7.10) become:
a > 0, a > 0, a > 0, a > 0, and a > 0 (14.7.28)
0 1 2 3 4

and

aa > a a and aa a > a a + a a (14.7.29)
2
2
12 0 3 12 3 0 3 1 4

14.8 Closure
This concludes our relatively brief introduction to stability and its associated computa-
tional procedures. Our discussions have been limited to infinitesimal stability. There are
other less stringent stability criteria, but these are beyond the limited scope of our discus-
sion. Those interested in these more advanced concepts and their associated procedures
may want to refer to the references or to more theoretical works devoted exclusively to
stability.

References
14.1. Meirovitch, L., Methods of Analytical Dynamics, McGraw-Hill, New York, 1970, pp. 222–224.
14.2. Di Stefano, J. J., Stubberud, A. R., and Williams, I. J., Theory and Problems of Feedback and Control
Systems, Schaum’s Outline Series, McGraw-Hill, New York, 1967, 114 ff.
14.3. Davis, S. A., Feedback and Control Systems, Simon & Shuster, New York, 1974, 262 ff.
14.4. Skelton, R. E., Dynamic Systems Control: Linear Systems Analysis and Synthesis, Wiley, New York,
1988, chap. 7.