Best Linear Approximation of Nonlinear Continuous-Time Systems Subject to Process Noise and Operating in Feedback

Abstract

In many engineering applications the level of nonlinear distortions in frequency response function (FRF) measurements is quantified using specially designed periodic excitation signals called random phase multisines and periodic noise. The technique is based on the concept of the best linear approximation (BLA) and it allows one to check the validity of the linear framework with a simple experiment. Although the classical BLA theory can handle measurement noise only, in most applications the noise generated by the system -called process noise -is the dominant noise source. Therefore, there is a need to extend the existing BLA theory to the process noise case. In this paper we study in detail the impact of the process noise on the BLA of nonlinear continuous-time systems operating in a closed loop. It is shown that the existing nonparametric estimation methods for detecting and quantifying the level of nonlinear distortions in FRF measurements are still applicable in the presence of process noise. All results are also valid for discrete-time systems and systems operating in open loop.

Best Linear Time-Varying Approximation of a General Class of Nonlinear Time-Varying Systems

“…The process noise n p (t) acts on each nonlinear block NL p , and the equivalence of Fig. 4 can then be stated [7], [8]. Hence, it holds that…”

“…where E{y p (t)|u(t)} is the expected value of y p (t) for a fixed input signal u(t). The process noise w p (t) and the stochastic nonlinear distortions y s,p (t) are mutually uncorrelated and are uncorrelated with -but not independent of -the input u(t) [7], [8]. Contrary to the linear case, w p (t) depends -in general -on the input u(t).…”

“…However, in many applications where the nonlinearities are not too strong, linear models are preferred as these are more simple to use. In response, the concept 'best linear approximation' was introduced in [6] and, next, generalized to the process noise case in [7] and [8]. Also, the system's dynamics are not always time-invariant.…”

Control-Oriented Neural State-Space Models for State-Feedback Linearization and Pole Placement

“…where matrices B0 and Ĉ0 are defined using classic linear model estimation techniques such as best linear approximation (BLA, see e.g. [21]). The matrix  is computed before training and fixed during the learning stage in such a way that the resulting closed-loop tends to be as close as possible to the dynamics imposed in the linear case through the pole-placement:…”

Frequency Response Function Measurements via Local Rational Modeling, Revisited

“…If the system behaves nonlinearly, then the proposed procedure estimates the best linear approximation [see [25] for the definition of the best linear approximation and the class of nonlinear systems considered]. Its variance depends on the input-output measurement noise, the process noise, and the nonlinear distortions due to the nonlinear input-output and nonlinear input-process noise interactions [see [25] for the details].…”