Kharitonov's theorem

Kharitonov's theorem This article includes a list of references, letture correlate o collegamenti esterni, ma le sue fonti rimangono poco chiare perché mancano di citazioni inline. Aiutaci a migliorare questo articolo introducendo citazioni più precise. (Marzo 2012) (Scopri come e quando rimuovere questo messaggio modello) 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 (cioè. if all roots 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.

Definition An interval polynomial is the family of all polynomials {stile di visualizzazione p(S)=a_{0}+un_{1}s^{1}+un_{2}s^{2}+...+un_{n}s^{n}} where each coefficient {stile di visualizzazione a_{io}in r} can take any value in the specified intervals {stile di visualizzazione l_{io}leq a_{io}leq u_{io}.} It is also assumed that the leading coefficient cannot be zero: {displaystyle 0notin [l_{n},tu_{n}]} .

Theorem An interval polynomial is stable (cioè. all members of the family are stable) if and only if the four so-called Kharitonov polynomials {stile di visualizzazione k_{1}(S)=l_{0}+l_{1}s^{1}+tu_{2}s^{2}+tu_{3}s^{3}+l_{4}s^{4}+l_{5}s^{5}+cdot } {stile di visualizzazione k_{2}(S)=u_{0}+tu_{1}s^{1}+l_{2}s^{2}+l_{3}s^{3}+tu_{4}s^{4}+tu_{5}s^{5}+cdot } {stile di visualizzazione k_{3}(S)=l_{0}+tu_{1}s^{1}+tu_{2}s^{2}+l_{3}s^{3}+l_{4}s^{4}+tu_{5}s^{5}+cdot } {stile di visualizzazione k_{4}(S)=u_{0}+l_{1}s^{1}+l_{2}s^{2}+tu_{3}s^{3}+tu_{4}s^{4}+l_{5}s^{5}+cdot } 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.

Riferimenti V. l. Kharitonov, "Asymptotic stability of an equilibrium position of a family of systems of differential equations", Differentsialnye uravneniya, 14 (1978), 2086-2088. (in russo) Academic home page of Prof. V. l. Kharitonov Categories: Control theoryTheorems about polynomialsTheorems in dynamical systemsCircuit theorems