Identification of nonlinear continuous-time systems from noisy input/output observations
The identification and control of LTI systems (linear time-invariant systems) is well established among engineers. So, there is no surprise if one tries to model a nonlinear system with a bunch of LTI systems. There are various ways to do so but maybe LPV (linear parameter varying) modeling approaches receives more attention from the community. For LPV systems, there is (are) a parameter (p) which assigns an LTI system to any constant trajectory. There are two major LPV identification approaches: local and global. In the local LPV identification approach, an LTI model is obtained for some constant trajectories and then they are bound together by an interpolating function (scheduling function) (local LPV) to approximate the whole nonlinear system. However, it is possible to identify the system along variable trajectories and then estimate the whole system throughout the data coming from these experiments (global LPV). This is exactly what is intended in this project. The latter method is more laborious than the former because an LPV model should be approximated from the beginning. On the other hand, the global method needs fewer experiments to gather data.
In this project, we are going to model a nonlinear system as a global LPV model. The system is excited by the sum of two different signals simultaneously: a large-amplitude slow signal and a small-amplitude fast signal (in contrast to the large-amplitude signal). We have a conjecture that large input and output signals could be used as scheduling parameters. Also, as we know, an LPV system must be estimated for the global case. However, it is known that any LPV model is inherently an LTV model too. Therefore, the coefficients of that linear time-varying system must have something to do with the variable trajectory parameters (i.e. the large signal input & outputs). This proposed method has the extra merit that it doesn’t require an extra independent scheduling variable. Also, there is no difficulty in the measurement of the scheduling parameters as these are given by the input and output of the system.
The underlying math comes from the well-known Taylor expansion of a nonlinear function around a trajectory. In this case, the coefficients of the LPV/LTV system are obtained via linearization of the plant around the large signal trajectory. A slowly varying trajectory and The small signal assumption guaranties that a first order Taylor series expansion with coefficients solely dependent on large input/output signals is sufficient.
To validate this proposed method, we think about estimating the nonlinear system along a single variable trajectory. Then we expect that this estimated nonlinear system is also a good approximation along other trajectories, within the bounds of the estimation data set. Promising results have been obtained on simulations.