The Loewner framework for nonlinear identification and reduction of Hammerstein cascaded dynamical systems

We present an algorithm for data‐driven identification and reduction of nonlinear cascaded systems with Hammerstein structure. The proposed algorithm relies on the Loewner framework (LF) which constitutes a non‐intrusive algorithm for identification and reduction of dynamical systems based on interpolation. We address the following problem: the actuator (control input) enters a static nonlinear block. Then, this processed signal is used as an input for a linear time‐invariant system (LTI). Additionally, it is considered that the orders of the linear transfer function and of the static nonlinearity are not a priori known.


Introduction
In some engineering applications that deal with the study of dynamical control systems, the control input enters the differential equations in a nonlinear fashion [5]. It is of interest to identify the hidden nonlinearity while at the same time reduction is needed for robust simulations and control design [1]. The LF [2][3][4] constitutes a non-intrusive method that uses only inputoutput data. The matrix pencil composed of two Loewner matrices reveals the minimality (in terms of McMillan degree) of the LTI system. By means of a singular value decomposition (SVD), one can find left and right projection matrices that are used to construct a low order model.
The Hammerstein system is characterized by two blocks connected in series, where the static nonlinear (memoryless) block is followed by a linear time-invariant system (LTI) as in Fig. 1. The scalar control input-u(t) is used as an argument to the static nonlinearity-F and then the signal F(u(t)) passes through a linear time-invariant (LTI) system. The static polynomial map approximates other non-polynomial maps (Taylor series expansion) s.a. tanh(·), exp(·), etc. The aim is to identify the cascaded system by estimating the coefficients of the polynomial map k i , i = 1, 2, . . . , n and the hidden LTI system by using only input-output data (u(t), y(t)) , t ≥ 0.
y(t) output The steady state output solution can be computed explicitly with the convolution integral 1 , the impulse response h(t), t ≥ 0 and the linear transfer function H(jω), jω ∈ C of the LTI as: Let the singleton real input be defined as u(t) = A cos(ωt) = αe jωt + αe −jωt with the amplitude α = A/2, the imaginary unit j, the driving frequency ω > 0 and time t ≥ 0. By substituting the above input in Eq. (1) and by making use of the binomial theorem, we conclude that: Section 17: Applied and numerical linear algebra At frequency ω the th harmonic is computed by applying the single-sided Fourier transform in Eq. (2) as: The Loewner-Hammerstein identification method As we have computed the total output of the Hammerstein cascaded system, we proceed with the method of determining the unknowns from input-output data. The symmetry in Eq. (3) allows the cancellation of the unknown contribution of the transfer function. Thus, we first determine the unknown coefficients k i , and afterwards, we fit the LTI system by means of the LF. For this purpose, it is important to define the following invariant frequency quantities λ p,q .

Definition 2.1 (Frequency invariant quantities)
The Y p,q denotes the q th harmonic at p frequency.
The entries λ p,q are independent of ω.
The above harmonic map allows the construction of the following linear system. Due to the mixing linearities (i.e. k 1 u(t) and (u h)(t)), we can fix k 1 to an arbitrary value. For p = 1 and q = 2, . . . , n results:  Finally, as we have identify the scaled (k 1 arbitrary) coefficient vector k = (k 1 , k 2 , . . . , k n ), we can transform the above harmonic map into a measurement map for the linear transfer function as H(j ω) = Y ω, / n ≤i =0 k i φ i, . The identification and reduction of the LTI system is done by applying the LF [2][3][4].

Algorithm 1: Hammerstein identification with the Loewner framework
Input: Apply signals u(t) = α cos(ω i t) with driving frequencies ω i , i = 1, . . . , n where n is the maximum nonzero harmonic index. Output: An identified Hammerstein system. 1 Apply FT and measure U (jω i ), Y 1 st (jω i ), Y 2 nd (2jω i ), . . . , Y n th (njω i ) from the power spectrum.; 2 Fix k 1 to an arbitrary value and determine the scaled coefficient vector k = (k 1 , k 2 , . . . , kn) by solving the system in Eq. (5).; 3 Estimate the measurements of the linear transfer function from H(j ω) = Y ω, / n ≤i =0 k i φ i, . ; 4 Apply the linear Loewner framework for identification and reduction of the LTI.