Fundamental Lemma for Data-Driven Analysis of Linear Parameter-Varying Systems

On a Continuous-Time Version of Willems' Lemma

“…has a solution α(t), then this solution must also satisfy (12). Now, the solution of a differential equation of the form ( 14) is known to exist and be unique for arbitrary initial conditions if the elements of the matrix Θ(t)…”

“…Remark 3: Although we expressed (12) as an implicit differential equation, in practice a solution α(t) can be easily obtained by solving the ODE (14).…”

“…In [10], [11], the databased state estimation problem was addressed. Besides these applications, the data-based representation of dynamical systems has been extended to linear time-varying systems [12], [13], to stochastic systems [14], and for classes of nonlinear systems [4], [15], [16]. A maximum likelihood approach was used to determine optimal system representations from noisy data in [17].…”

Data-Driven Control of Linear Time-Varying Systems

“…Challenges associated with direct datadriven control in the presence of noise are addressed for certain classes of control problems involving LTI systems in [13]- [16]. Inspired by the promising results related to LTI systems, extensions have been proposed for data from multiple data sets [17], certain classes of nonlinear systems [18]- [21], linear parameter-varying systems [22] and switched systems [23].…”

“…Both the cases of noise-free and noisy data are considered. In contrast to the related results for linear parameter-varying [22] and switched systems [23], the presented data-driven methods are applicable to (linear) arbitrarily time-varying systems and do not rely on any assumptions or prior knowledge of the system structure or parameter variation. However, it is shown how such knowledge can be exploited to simplify the analysis and derive asymptotic guarantees for the special case of periodically time-varying systems.…”