Nonparametric Estimation of Time-Varying Systems Using 2-D Regularization

In this paper a nonparametric time-domain estimation method of linear time-varying systems from measured noisy data is presented. The challenge with time-varying systems is that the time-varying two-dimensional (2-D) impulse response functions (IRFs) are not uniquely determined from a single set of input and output signals as in the case of linear time-invariant systems. Due to this nonuniqueness, the number of possible solutions is growing quadratically with the number of samples. To decrease the degrees of freedom, user-defined (adjustable) constraints will be imposed. In this particular case, it is imposed that the entire 2-D IRF is smooth. This is implemented by a 2-D kernel-based regularization. This regularization is applied over the system time (the direction of the impulse responses) and simultaneously over the global time (representing the system memory). To illustrate the efficiency of the proposed method, it is demonstrated on a measurement example as a conceptual instrument for measuring and analyzing time-varying systems.