In the field of dynamic system identification, where the information about the underlying dynamic system model is absent, modelling nonlinear dynamics in a nonparametric way with Volterra series is a common approach. Volterra series is similar to Taylor series in structure, but also adds the benefit of capturing memory effects.
In discrete time, single input: - single output:
relation of a stable nonlinear time invariant dynamic system is given by the discrete time Volterra series, where
represents the Volterra kernel, as:
In our system identification, where we use order 2 Volterra series with
and the constant term
is taken to 0, we can rewrite this as,
At each order of nonlinearity, Volterra kernels describe the dynamics of the system and constitute a canonical representation of the nonlinear dynamic system.
The main challenge of Volterra models is the exponentially growing number of kernel coefficients as the model order increases. An order Volterra model with
parameters to estimate. For multiple input multiple output dynamic system, the numbers of parameters to estimate increases to
, where
denotes the number of inputs.
One way to simplify kernel estimation is through expanding them on a orthogonal Laguerre basis. Once the kernels are estimated, Principal Dynamic Mode (PDM) can be used to characterize and explain the dynamics of the underlying nonlinear dynamic system.