Koopman Form of Nonlinear Systems with Inputs

Abstract

The Koopman framework proposes a linear representation of finite dimensional nonlinear systems through a generally infinite dimensional globally linear representation. Originally, the Koopman formalism has been described for autonomous systems and the extension for actuated continuous-time systems with a linear input or a control affine form has only recently been addressed. However, such a derivation for discrete-time systems has not yet been developed. Thus, a particular Koopman form is generally assumed, predominantly a linear time invariant (LTI) model, as it facilitates the use of control techniques such as linear quadratic regulation and model predictive control. However, we show that this assumption is insufficient to capture the dynamics of the underlying nonlinear system. In the present paper, we systematically investigate and analytically derive lifted forms under inputs for a general class of nonlinear systems in both continuous and discrete time, using the fundamental theorem of calculus. We prove that the resulting lifted representations give linear state-space Koopman models where the input matrix becomes state (and input, for the discrete-time case)-dependent, hence it can be seen as a specially structured linear parameter-varying (LPV) description of the underlying system. We also provide error bounds on how much the parameter variation contributes and how well the system behaviour can be approximated by an LTI Koopman representation. The introduced theoretical insight greatly helps for performing proper model structure selection in system identification with Koopman models as well as making a proper choice for LTI or LPV techniques for the control of nonlinear systems through the Koopman approach.

Finite Dimensional Koopman Form of Polynomial Nonlinear Systems

“…with the autonomous part given by ( 10) and g : R nx → R nx×nu and u ∈ U ⊆ R nu . To obtain the lifted representation, one can use the sequential method described in (Iacob et al, 2022). First, an exact lifting of the autonomous part is assumed to exist, i.e.…”

“…with B(x) = ∂Φ ∂x (x)g(x). As described in (Iacob et al, 2022), one can further express (25) as a linear parameter varying (LPV) Koopman representation by introducing a scheduling map p = µ(z), where z = Φ(x) and defining B z • z = B. Then, the LPV Koopman model is described by: ż = Az + B z (p)u, (26) with z(0) = Φ(x(0)).…”

Data-Driven Control of Nonlinear Systems: Beyond Polynomial Dynamics

“…As described in Section 2, the choice of a suitable lifting function Φ is crucial for a small truncation error and therefore an accurate bilinear representation. How to select this function for a general nonlinear system is still an open research question (Iacob et al, 2022), where input-dependent liftings Ψ(x, u) are proposed instead of solely state-dependent liftings Φ(x).…”