Vyšetřování stability nelineárních řídících systémů metodou linearizace

Stability analysis of nonlinear control systems using linearization

conference paper

en

The most powerful methods of systems analysis have been developed for linear control systems. For a linear control system, all the relationships between the variables are linear differential equations, usually with constant coefficients. Actual control systems usually contain some nonlinear elements. In the following we show how the equations for nonlinear elements may be linearized. But the result is applicable only in a small enough region. When all the roots of the characteristic equation are located in the left half-plane, the system is stable. However that linearization fails when Re si ˇÜ 0 for all i, with Re si = 0 for some i. The table includes the nonlinear equations and their the linear approximation. Then it is easy to find out if the nonlinear system is or is not stable; the task that usually ranks among the difficult task in engineering practice.

Lineární diskrétní systém je asymptoticky stabilní, když póly přenosové funkce nebo kořeny charakteristické rovnice leží uvnitř jednotkové kružnice.Když jednoduché póly jsou umístěny na jednotkové kružnici, je obvod kriticky stabilní.Pro násobné kořeny na jednotkové kružnici je ovšem obvod nestabilní. Čtyři metody pro vyšetřování stabilty diskrétních řídících systémů jsou uvedeny v tomto příspěvku: – algebraická kritéria stability; – frekvenční metody; – metody kořenového hodografu; – vyšetřování stability bilineární transformací. Protože vyšetřování stability diskrétních systémů je obtížná úloha v inženýrské praxi, tento článek dává pro praxi použitelné řešení.

The most powerful methods of systems analysis have been developed for linear control systems. For a linear control system, all the relationships between the variables are linear differential equations, usually with constant coefficients. Actual control systems usually contain some nonlinear elements. In the following we show how the equations for nonlinear elements may be linearized. But the result is applicable only in a small enough region. When all the roots of the characteristic equation are located in the left half-plane, the system is stable. However that linearization fails when Re si ˇÜ 0 for all i, with Re si = 0 for some i. The table includes the nonlinear equations and their the linear approximation. Then it is easy to find out if the nonlinear system is or is not stable; the task that usually ranks among the difficult task in engineering practice.

linearization, nonlinear system, equilibrium points, phase-plane trajectory

2004

01.01.2004

DELTA

Zakopane

83-89772-00-0

Proceedings of 5th International Carpathian Control Conference

5