On macromodeling of nonlinear systems with time periodic coefficients
The model order reduction (MOR) is a process which consists in replacing a large dynamical system by an equivalent small-scale system that can approximate the dynamics of the initial system: this macromodel is asked to preserve the desired stability and bifurcation characteristics of the original large-scale system. This paper suggests four practical order reduction techniques: a linear projection method, a nonlinear projection method based on singular perturbation, a post-processing approach and one technique using the time periodic invariant manifold. The linear and nonlinear systems discussed in this paper have time periodic coefficients. Given a periodic \(n\times n\)-matrix \(A\), with primitive period \(T,\) the linear dynamical evolution system \[ \dot{x}(t)=A(t)x(t),\quad A(t+T)=A(t), \] where \(x\) is an \(n\) vector is transformed using the Lyapunov-Floquet (L-F) transformation such that the linear parts of the new set of equations become time-invariant. Given a \(T\) time-periodic nonlinear function \(f(x,t),\) the time periodic manifold reduction allows one to study the critical dynamics of a nonlinear system \[ \dot{x}(t)=A(t)x(t)+f(x,t),\quad A(t+T)=A(t), \] with much smaller dimension than the original. It is based on the center manifold theory. Also, the numerical error due to linear approximation analysis is provided and the authors discuss advantages and limitations of the techniques by means of a practical example consisting of a system modeling the dynamics of two inverted pendulums moving in the horizontal plane with time-dependent load acting on each pendulum.
scientific article; zbMATH DE number 5069042
scientific article