Root neighborhoods, generalized lemniscates, and robust stability of dynamic systems



A root neighborhood (or pseudozero set) of a degree-n polynomial p(z) is the set of all complex numbers that are the roots of polynomials whose coefficients differ from those of p(z), under a specified norm in \({\mathbb{C}^{n+1}}\) , by no more than a fixed amount \({\epsilon}\) . Root neighborhoods corresponding to commonly used norms are bounded by higher-order algebraic curves called generalized lemniscates. Although it may be neither convenient nor useful to derive their implicit equations, such curves are amenable to graphical analysis by means of simple contouring algorithms. Root neighborhood methods offer advantages over alternative approaches (the Kharitonov theorems and their generalizations) for investigating the robust stability of dynamic systems with uncertain parameters, since they offer valuable insight concerning which roots of the characteristic polynomial will become unstable first, and the relative importance of parameter variations on the root locations—and hence speed and damping of the system response. We derive a generalization of root neighborhoods to the case of polynomial coefficients having an affine linear dependence on a set of complex uncertainty parameters, bounded under a general weighted norm, and we discuss their applications to robust stability problems. The methods are illustrated by several computed examples.


Polynomial roots Root neighborhoods Pseudozero sets Spectral sets Generalized lemniscate Kharitonov theorem Robust stability 


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  1. 1.
    Ackermann J. (1993). Robust Control: Systems with Uncertain Physical Parameters. Springer, London MATHGoogle Scholar
  2. 2.
    Barmish B.R. (1994). New Tools for Robustness of Linear Systems. Macmillan, New York MATHGoogle Scholar
  3. 3.
    Barmish B.R. and Tempo R. (1991). On the spectral set for a family of polynomials. IEEE Trans. Automat. Contr.\CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Bose N.K. and Shi Y.Q. (1987). A simple general proof of Kharitonov’s generalized stability criterion. IEEE Trans. Circ. Syst. 34: 1233–1237 CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Cerone V. (1997). A fast technique for the generation of the spectral set of a polytope of polynomials. Automatica 33: 277–280 CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Chapellat H., Bhattacharyya S.P. and Dahleh M. (1990). Robust stability of a family of disc polynomials. Int. J. Contr. 51: 1353–1362 MathSciNetMATHGoogle Scholar
  7. 7.
    Chesi and G. (2002). Complete characterization of the spherical spectral set. IEEE Trans. Automat. Contr. 47: 1875–1879 CrossRefGoogle Scholar
  8. 8.
    Collins G.E. and Hong H. (1991). Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comp. 12: 299–328 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Davis P.J. (1975). Interpolation and Approximation. Dover (reprint), New York MATHGoogle Scholar
  10. 10.
    Farouki R.T. and Han C.Y. (2004). Robust plotting of generalized lemniscates. Appl. Numer. Math. 51: 257–272 CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Gantmacher, F.R.: The Theory of Matrices, vol. 2. Chelsea, New York (1960)Google Scholar
  12. 12.
    Hoffman J.W., Madden J.J. and Zhang H. (2003). Psuedozeros of multivariate polynomials. Math. Comp. 72: 975–1002 CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Kharitonov V.L. (1978). Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differensial’nye Uravneniya 14: 1483–1485 MathSciNetGoogle Scholar
  14. 14.
    Kharitonov V.L. (1978). On a generalization of a stability criterion. Izv. Akad. Nauk. Kazakh. SSR Ser. Fiz. Mat. 1: 53–57 MathSciNetGoogle Scholar
  15. 15.
    Marden M. (1966). Geometry of Polynomials. American Mathematical Society, Providence MATHGoogle Scholar
  16. 16.
    Mosier R.G. (1986). Root neighborhoods of a polynomial. Math. Comp. 47: 265–273 CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Polyak B.T. and Tsypkin Ya. Z. (1991). Robust stability under complex perturbations of parameters. Automat. Rem. Contr. 52: 1069–1077 MathSciNetMATHGoogle Scholar
  18. 18.
    Stetter, H.J.: Polynomials with coefficients of limited accuracy. In: Computer Algebra in Scientific Computing—CASC ’99 (Munich), pp. 409–430. Springer, Berlin Heidelberg New York (1999)Google Scholar
  19. 19.
    Stetter H.J. (1999). The nearest polynomial with a given zero and similar problems. SIGSAM Bull. 33(4): 2–4 CrossRefMATHGoogle Scholar
  20. 20.
    Toh K-C. and Trefethen L.N. (1994). Pseudozeros of polynomials and pseudospectra of companion matrices. Numer. Math. 68: 403–425 CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Zhang H. (2001). Numerical condition of polynomials in different forms. Elect. Trans. Numer. Anal. 12: 66–87 MATHGoogle Scholar

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© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Mechanical and Aeronautical EngineeringUniversity of CaliforniaDavisUSA

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