Unit 8
Lecture 2
8.3 Stability by the Method of Liapunov
Russian mathematician A.M. Liapunov has proposed a few theorems for the study of stability of the
system. The most popular among this is called the “Second Method of Liapunov” or “Direct Method
of Liapunov”. This method is very general in its formulation and can be used to study of stability of
linear or nonlinear systems. The method is called ‘direct’ method as it does not involve the solution
of the system differential equations and stability information is available without solving the
equations which is definitely an advantage for nonlinear systems. The stability information obtained
by this method is precise and involved no approximation.
First Method of Liapunov: The first method of Liapunov, though rarely talked about, is essentially
a theorem stating the conditions under which system stability information can be inferred by
examining the simplified equations obtained through local linearization. This theorem is applicable
only to autonomous systems.
8.4 Sign Definiteness
Let V(x1, x2, x3, ……. Xn) be a scalar function of the state variables x1, x2, x3, …….., xn. Then the
following definitions are useful for the discussion of Liapunov’s second method.
8.4.1 Scalar Functions:
A scalar function V(x) is said to be positive definite in a region Ω if V(x) > 0 for all nonzero states x
in the region Ω and V(0) = 0. A scalar function V(x) is said to be negative definite in a region Ω if
V(x) < 0 for all nonzero states x in the region Ω and V(0) = 0. A scalar function V(x) is said to be
positive semi-definite in a region Ω if it is positive for all states in the region Ω and except at the
origin and at certain other states, where it is zero. A scalar function V(x) is said to be negative semidefinite in a region Ω if it is negative for all states in the region Ω and except at the origin and at
certain other states, where it is zero. A scalar function V(x) is said to be indefinite if in the region Ω
it assumes both positive and negative values, no matter how small the region Ω is.
8.4.2 Sylvester’s Criteria for Definiteness:
A necessary and sufficient condition in order that the quadratic form xTAx, where A is an nxn real
symmetric matrix, be positive definite is that the determinant of A be positive and the successive
principal minors of the determinant of A be positive.
Dept. of EEE, NIT-Raichur
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Unit 8
Lecture 2
A necessary and sufficient condition in order that the quadratic form xTAx, where A is an nxn real
symmetric matrix, be negative definite is that the determinant of A be positive if n is even and
negative if n is odd, and the successive principal minors of even order be positive and the successive
principal minors of odd order be negative. i.e.,
A necessary and sufficient condition in order that the quadratic form xTAx, where A is an nxn real
symmetric matrix, be positive semi-definite is that the determinant of A be singular and the
successive principal minors of the determinant of A be nonnegative. i.e.,
A necessary and sufficient condition in order that the quadratic form xTAx, where A is an nxn real
symmetric matrix, be negative semi-definite is that the determinant of A be singular and all the
principal minors of even order be nonnegative and those of odd orders be non positive. i.e.,
Dept. of EEE, NIT-Raichur
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