Kharitonov's theorem
Encyclopedia
Kharitonov's theorem is a result used in control theory
to assess the stability of a dynamical system
when the physical parameters of the system are not known precisely. When the coefficients of the characteristic polynomial
are known, the Routh-Hurwitz stability criterion
can be used to check if the system is stable (i.e. if all root
s have negative real parts). Kharitonov's theorem can be used in the case where the coefficients are only known to be within specified ranges. It provides a test of stability for a so-called interval polynomial, while Routh-Hurwitz is concerned with an ordinary polynomial
.
where each coefficient
can take any value in the specified intervals
It is also assumed that the leading coefficient cannot be zero:
.
are stable.
What is somewhat surprising about Kharitonov's result is that although in principle we are testing an infinite number of polynomials for stability, in fact we need to test only four. This we can do using Routh-Hurwitz or any other method. So it only takes four times more work to be informed about the stability of an interval polynomial than it takes to test one ordinary polynomial for stability.
Kharitonov's theorem is useful in the field of robust control
, which seeks to design systems that will work well despite uncertainties in component behavior due to measurement errors, changes in operating conditions, equipment wear and so on.
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
to assess the stability of a dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
when the physical parameters of the system are not known precisely. When the coefficients of the characteristic polynomial
Characteristic polynomial
In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
are known, the Routh-Hurwitz stability criterion
Routh-Hurwitz stability criterion
The Routh–Hurwitz stability criterion is a necessary and sufficient method to establish the stability of a single-input, single-output , linear time invariant control system. More generally, given a polynomial, some calculations using only the coefficients of that polynomial can lead to the...
can be used to check if the system is stable (i.e. if all root
Root
In vascular plants, the root is the organ of a plant that typically lies below the surface of the soil. This is not always the case, however, since a root can also be aerial or aerating . Furthermore, a stem normally occurring below ground is not exceptional either...
s have negative real parts). Kharitonov's theorem can be used in the case where the coefficients are only known to be within specified ranges. It provides a test of stability for a so-called interval polynomial, while Routh-Hurwitz is concerned with an ordinary polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
.
Definition
An interval polynomial is the family of all polynomialswhere each coefficient
can take any value in the specified intervalsIt is also assumed that the leading coefficient cannot be zero:
.Theorem
An interval polynomial is stable (i.e. all members of the family are stable) if and only if the four so-called Kharitonov polynomialsare stable.
What is somewhat surprising about Kharitonov's result is that although in principle we are testing an infinite number of polynomials for stability, in fact we need to test only four. This we can do using Routh-Hurwitz or any other method. So it only takes four times more work to be informed about the stability of an interval polynomial than it takes to test one ordinary polynomial for stability.
Kharitonov's theorem is useful in the field of robust control
Robust control
Robust control is a branch of control theory that explicitly deals with uncertainty in its approach to controller design. Robust control methods are designed to function properly so long as uncertain parameters or disturbances are within some set...
, which seeks to design systems that will work well despite uncertainties in component behavior due to measurement errors, changes in operating conditions, equipment wear and so on.