The nonlocal reduction method in the theory of absolute stability of nonlinear systems. I
The following dynamic equations of control systems are considered: (1) \(\dot z=Az+b\phi(\sigma)\), \({\dot \sigma}=c^*z+\rho\phi(\sigma)\), where A is a constant \(n\times n\) matrix, b and c are constant n-vectors, \(\rho\) is a number and \(\phi(\sigma)\) is a continuously differentiable function. Two types of assumptions are made. On the one hand, in order to make the application of the Kalman-Yakubovich lemma possible, there is a number of assumptions concerning the transfer function of (1). On the other hand, it is assumed that the second-order equation (2) \({\ddot \theta}+a{\dot \theta}+\phi (\theta)=0\) is stable in the sense of Lagrange. Boundedness and stability results are proved by using several Lyapunov functions. Each Lyapunov function has the form \(z^*Hz-F_ k(\sigma)^ 2/2\), where H is a positive definite \(n\times n\) matrix and \(F_ k(\theta)\) is a suitably chosen solution of the equation (2)' \(dF/d\theta =-a-\phi(\theta)/F\), a being a nonnegative number. Thus, in many respects, the study of system (1) can be reduced to the study of the second-order equation (2) and to the checking of certain conditions imposed on the linear part of system (1). The paper has an appendix in which theorems of Barbashin-Krasovskii type for several Lyapunov functions are presented.
scientific article; zbMATH DE number 3882304
scientific article