Řešení stability nelineárních řídících systémů

Stability Analysis of Nonlinear Control Systems

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. But actual control systems usually contain some nonlinear elements. Three methods for stability analysis of nonlinear control systems will be introduced in this lecture: method of linearization, Lyapunov direct method and Popov criterion. Since stability analysis of nonlinear control systems is difficult task in engineering practice, these methods are made easier and tabulated. In the lecture we will 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. We can construct 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. Lyapunov direct method: Lyapunovs method is a very powerful tool for studying the stability of equilibrium points. However, there is drawback of the method that we should be aware of. There is no systematic method for finding a Lyapunov function for a given system. We would like to eliminate the first drawback of Lyapunovs method. This is the table of Lyapunovs functions . Popov criterion: The Popov criterion is considered as one of the most appropriate criteria for nonlinear systems and it can be compared with the Nyquist criterion for linear systems. The sufficient condition for stability of nonlinear circuit is that the plot of G(s) should lie entirely to the right of the Popov line which crosses the real axis at -1 divided k at a slope 1 divided q (q is an arbitrary real number) . We can construct the table that will allow us to directly determine the stability of the nonlinear circuit with the transfer function G(s) and the nonlinearity that satisfies the slope k.

Řídicí systémy v energetických zařízeních obvykle obsahují nelineární prvky. Proto není možno pro jejich vyšetřování použít lineárních metod, jmenovitě metod pro vyšetřování stability lineárních spojitých systémů. Metody pro vyšetřování nelineárních systémů patří k obtížným úkolům inženýrské praxe. Cílem tohoto příspěvku je jejich zjednodušení do té míry, aby se staly snadno dostupné a použitelné pro technickou praxi. Jsou zde uvedeny tři nejpoužívanější metody a všechny tři jsou dovedeny do přehledných tabulek, z kterých se dá zjistit, zda je či není vyšetřovaný systém stabilní a to bez speciálních znalostí ze řízení systémů.

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. But actual control systems usually contain some nonlinear elements. Three methods for stability analysis of nonlinear control systems will be introduced in this lecture: method of linearization, Lyapunov direct method and Popov criterion. Since stability analysis of nonlinear control systems is difficult task in engineering practice, these methods are made easier and tabulated. In the lecture we will 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. We can construct 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. Lyapunov direct method: Lyapunovs method is a very powerful tool for studying the stability of equilibrium points. However, there is drawback of the method that we should be aware of. There is no systematic method for finding a Lyapunov function for a given system. We would like to eliminate the first drawback of Lyapunovs method. This is the table of Lyapunovs functions . Popov criterion: The Popov criterion is considered as one of the most appropriate criteria for nonlinear systems and it can be compared with the Nyquist criterion for linear systems. The sufficient condition for stability of nonlinear circuit is that the plot of G(s) should lie entirely to the right of the Popov line which crosses the real axis at -1 divided k at a slope 1 divided q (q is an arbitrary real number) . We can construct the table that will allow us to directly determine the stability of the nonlinear circuit with the transfer function G(s) and the nonlinearity that satisfies the slope k.

Popovovo kritérium; Ljapunovo kritérium; linearizace+ přenos.

Popov criterion; Lyapunov criterion; linearization; transfer function.

2004

01.01.2004

Graz University of Technology

Graz

Summer School on Control Theory and Applications

29–29

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