Stability criterion

A necessary and sufficient condition for the real parts of all roots of an equation

$$ \tag{* } \lambda ^ {n} + a _ {1} \lambda ^ {n - 1 } + \dots + a _ {n} = 0 $$

to be negative.

A stability criterion is used in applying Lyapunov's theorem on the stability of the first approximation to a fixed point of an autonomous system of differential equations (cf. Lyapunov stability). The most commonly used stability criterion is the Routh–Hurwitz criterion or Hurwitz criterion: For the real parts of all roots of the equation (*) to be negative it is necessary and sufficient that the inequalities $ \Delta _ {i} > 0 $, $ i \in \{ 1 \dots n \} $, be satisfied, where

$$ \Delta _ {1} = a _ {1} ,\ \ \Delta _ {2} = \left | \begin{array}{ll} a _ {1} & 1 \\ a _ {3} &a _ {2} \\ \end{array} \ \right | ,\ \ \Delta _ {3} = \left | \begin{array}{lll} a _ {1} & 1 & 0 \\ a _ {3} &a _ {2} &a _ {1} \\ a _ {5} &a _ {4} &a _ {3} \\ \end{array} \ \right | \dots $$

are the principal diagonal minors of the matrix

$$ \left \| \begin{array}{lllllllll} a _ {1} & 1 & 0 & 0 & 0 & 0 &\cdot &\cdot & 0 \\ a _ {3} &a _ {2} &a _ {1} & 1 & 0 & 0 &\cdot &\cdot & 0 \\ a _ {5} &a _ {4} &a _ {3} &a _ {2} &a _ {1} & 1 &\cdot &\cdot & 0 \\ \cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot \\ 0 & 0 & 0 & 0 & 0 & 0 &\cdot &\cdot &a _ {n} \\ \end{array} \ \right \| $$

(on the main diagonal of this matrix there stand $ a _ {1} \dots a _ {n} $; for $ i > n $, $ a _ {i} = 0 $).

For $ n = 2 $ the Routh–Hurwitz stability criterion takes a particularly simple form: For the real parts of the roots of $ \lambda ^ {2} + a _ {1} \lambda + a _ {2} = 0 $ to be negative it is necessary and sufficient that the coefficients of the equation be positive: $ a _ {1} > 0 $, $ a _ {2} > 0 $.

For each $ n \in \mathbf N $, for the real parts of all roots of the equation (*) to be negative it is necessary (but for $ n > 2 $ not sufficient) that all coefficients of the equation be positive: $ a _ {i} > 0 $, $ i \in \{ 1 \dots n \} $. If at least one of the determinants $ \Delta _ {i} $, $ i \in \{ 1 \dots n \} $, is negative, then there is a root of (*) with positive real part (this assertion is used in applying Lyapunov's theorem on the instability of the first approximation to a fixed point of an autonomous system of differential equations, cf. Lyapunov stability). If $ \Delta _ {i} \geq 0 $ for all $ i \in \{ 1 \dots n \} $, but $ \Delta _ {i} = 0 $ for a certain $ i \in \{ 1 \dots n \} $, then the location of the roots of the equation (*) relative to the imaginary axis can also be described without finding the roots (cf. [5], [8], Chapt. XVI, Sect. 8).

Much simpler in applications is the Liénard–Chipart criterion: For the real parts of all roots of the equation (*) to be negative it is necessary and sufficient that the following inequalities hold: $ a _ {i} > 0 $, $ i \in \{ 1 \dots n \} $, $ \Delta _ {n - 2i + 1 } > 0 $, $ i \in \{ 1 \dots [ n/2] \} $( the determinant $ \Delta _ {i} $ is the same as in the Routh–Hurwitz criterion).

Hermite's criterion (historically the first, cf. [1], [10], Sect. 3.1) allows one to determine with the help of a finite number of arithmetic operations on the coefficients of (*) whether all roots of this equation have negative real parts. The Routh–Hurwitz criterion formulated above is a modification of Hermite's criterion found by A. Hurwitz. A Lyapunov stability criterion is also known (cf. [3], [8], Chapt. XVI, Sect. 5, [10], Sect. 3.5).

For a study of the stability of fixed points of differentiable mappings (autonomous systems with discrete time) as well as for a study of orbit stability of closed trajectories of autonomous systems of differential equations one has to apply necessary and sufficient conditions for the absolute values of all roots of the equation (*) to be less than one. This criterion is obtained from the above-mentioned stability criterion by the mapping $ \lambda \mapsto ( \lambda + 1)/( \lambda - 1) $ from the open unit disc onto the open left half-plane (cf. [10], Sect. 3.2).

References

[1]	C. Hermite, "Sur le nombre des racines d'une équation algébrique comprise entre des limites donnés" J. Reine Angew. Math. , 52 (1856) pp. 39–51
[2]	E.J. Routh, "A treatise on the stability of a given state of motion" , Macmillan (1877)
[3]	A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)
[4]	A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt" Math. Ann. , 46 (1895) pp. 273–284
[5]	L. Orlando, "Sul problema di Hurwitz relative alle parti reali delle radici di un'equazione algebrica" Math. Ann. , 71 (1911) pp. 233–245
[6]	A. Liénard, M.H. Chipart, "Sur le signe de la partie réelle des racines d'une équation algébrique" J. Math. Pure Appl. (6) , 10 (1914) pp. 291–346
[7]	N.G. Chetaev, "Stability of motion" , Moscow (1965) (In Russian)
[8]	F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian)
[9]	B.P. Demidovich, "Lectures on the mathematical theory of stability" , Moscow (1967) (In Russian)
[10]	E. Jury, "Inners and stability of dynamic systems" , Wiley (1974)

Comments

See also Mikhailov criterion, which is equivalent to the Routh–Hurwitz criterion, but formulated in terms of the curve obtained from (*) by letting $ \lambda $ vary over the positive imaginary axis.

In control theory (robust control) one is often concerned with the stability of a whole family of polynomials rather than a single one. Stability results pertaining to this situation are generally known as Kharitonov-type theorems.

The original Kharitonov theorem, [a1], [a2], can be stated as follows. Let $ P( s; q) $ be the family of polynomials

$$ P( s; q) = q _ {0} + q _ {1} s + \dots + q _ {n} s ^ {n} , $$

where each $ q _ {i} $ ranges over a given closed interval $ [ q _ {i} ^ {-} , q _ {i} ^ {+} ] $. Form the four polynomials

$$ K _ {1} ( s) = \ q _ {0} ^ {-} + q _ {1} ^ {-} s + q _ {2} ^ {+} s ^ {2} + $$

$$ + q _ {3} ^ {+} s ^ {3} + q _ {4} ^ {-} s ^ {4} + q _ {5} ^ {-} s ^ {5} + q _ {6} ^ {+} s ^ {6} + \dots , $$

$$ K _ {2} ( s) = q _ {0} ^ {+} + q _ {1} ^ {+} s + q _ {2} ^ {-} s ^ {2} + $$

$$ + q _ {3} ^ {-} s ^ {3} + q _ {4} ^ {+} s ^ {4} + q _ {5} ^ {+} s ^ {5} + q _ {6} ^ {-} s ^ {6} + \dots , $$

$$ K _ {3} ( s) = q _ {0} ^ {+} + q _ {1} ^ {-} s + q _ {2} ^ {-} s ^ {2} + $$

$$ + q _ {3} ^ {+} s ^ {3} + q _ {4} ^ {+} s ^ {4} + q _ {5} ^ {-} s ^ {5} + q _ {6} ^ {-} s ^ {6} + \dots , $$

$$ K _ {4} ( s) = q _ {0} ^ {-} + q _ {1} ^ {+} s + q _ {2} ^ {+} s ^ {2} + $$

$$ + q _ {3} ^ {-} s ^ {3} + q _ {4} ^ {-} s ^ {4} + q _ {5} ^ {+} s ^ {5} + q _ {6} ^ {+} s ^ {6} + \dots . $$

Then every polynomial $ P( s; q) $, $ q _ {i} ^ {-} \leq q _ {i} \leq q _ {i} ^ {+} $, has its zeros strictly in the left half-plane if and only if the four polynomials $ K _ {i} ( s) $, $ i = 1 \dots 4 $, have this property.

There is a large variety of similar theorems applying to other regions of allowed zeros, otherwise shaped families of polynomials (than cubes such as above), and discrete-time stability. Cf. [a3] for a survey.

References

[a1]	V.L. Kharitonov, "Asymptotic stability of an equilibrium position of a family of systems of linear differential equations" Diff. Uravn. , 14 : 11 (1978) pp. 1483–1485 (In Russian)
[a2]	V.L. Kharitonov, "On a generalization of a stability criterion" Akad. Nauk KazakhsSSR, Fiz.-Mat. , 1 (1978) pp. 53–57 (In Russian)
[a3]	B.R. Barmish, B. Ross, "New tools for robustness analysis" , Proc. 27-th IEEE CDC , IEEE (1988) pp. 1–6