Algebraic stability criteria and symbolic derivation of stability conditions for feedback control systems

Abstract

This article provides algebraic settings of the stability criteria of Nyquist and Popov and the circle criterion for closed-loop linear control systems with linear or nonlinear feedback whose transfer functions are rational ones with integer coefficients. The proposed settings make use of algebraic methods of parametric curve implicitisation, real root isolation, symbolic integration and quantifier elimination and allow one to derive exact stability conditions for feedback control systems with symbolic computation. An example is presented to illustrate the algebraic approach and its effectiveness. Some numerical stability results obtained previously are confirmed.

Keywords:

• feedback control
• stability condition
• Nyquist criterion
• Popov criterion
• circle criterion
• algebraic method
• symbolic computation

Acknowledgements

The author wishes to thank the referees for their helpful comments. This work has been supported by the ANR-NSFC project EXACTA (ANR-09-BLAN-0371-01).

Notes

Notes

1. The implicit curve F(x, y) = 0 may contain (real) points that are not on the Nyquist curve of G(jw) because the Nyquist curve is only part of the corresponding parametric curve for which w ∈ ℝ. For this example, the point (α₃, 0) is on the implicit curve, but not on the Nyquist curve because it corresponds to . The occurrence of extraneous points like (α₃, 0) here may lead to extra computation, but does not affect the final result.

2. See http://www.usna.edu/Users/cs/qepcad/B/QEPCAD.html