## 4 Parallel Cascade Identification  
The PCI technique is based on the idea of modeling the non-linear system input/output relation of alternating dynamic linear (L) and static non-linear (N) elements by summing parallel cascades. The model built has a finite number of parallel LN cascade paths, where each path consists of a dynamic linear element followed by a static non-linearity. The static non-linearity can be a polynomial. The model output is the sum of the outputs of the parallel branches, as shown in Figure 3.  
[PERSON] in 1910 proved that in continuous time, a finite memory non-linear system whose output is a continuous mapping of its input can be uniformly approximated over a uniformly bounded equicontinuous set of inputs to an arbitrary degree of precision by a Volterra series of sufficient but finite order [28]. The Volterra series represents a functional expansion of a dynamic, non-linear, time-invariant functional. The Volterra series is commonly used in system identification. [PERSON] [29] showed that any discrete-time Volterra series with limited memory could be uniformly estimated by a limited sum of parallel LNL cascades, where the static non-linearities N are exponentials and logarithmic functions. [PERSON] [28] showed that any discrete-time finite memory non-linear system having a finite-order Volterra series representation can be exactly represented by a finite number of parallel LN cascade paths, where the N are polynomials. In practice, additional LN elements may be added in any cascade path to increase accuracy [28]. A major benefit of this technique is its independence of a Gaussian or white input, but it identifies separate L and N elements and may change depending on the statistical properties of the input chosen [28]. One cascade can be found at a time, and the non-linearities in the models are localized in static functions. This reduces the computation, as higher-order non-linearitiesare approximated using higher degree polynomials in the cascades rather than higher-order kernels in a Volterra series approximation. The technique begins by estimating the non-linear system by a first such cascade. The residual (i.e., the difference between the system output and the cascade outputs) is treated as the output of a new non-linear system; a second cascade is found to estimate the latter system, and thus the parallel array can augment one cascade at a time. Consider an unknown dynamic non-linear system with accessible input \(x(n)\) and output \(y(n)\) where \(n=0,\ldots,\ldots,T\), \(T\) is the length of the dataset or record used for the training. Under wide conditions, one can model the system using parallel cascade supposing that the output can depend on delayed input values \(x(n-j)\), for \(j=0,\ldots,R\). Here \(R\) is the maximum lag or delay and \((R+1)\) is the memory length (since the series output \(y(n)\) depends on input delays from 0 to \(R\) lags or delay). The maximum degree of non-linearity required for a good approximation of the system is \(D\). The polynomial degree \(D\) cannot exceed \((T-R)\) since there are \((D+1)\) coefficients to estimate in the polynomial, and there would be exactly \((T-R+1)\) points available for the estimation. However, a much smaller value is in practice used for the polynomial degree \(D\), and its value is application dependent. Figure 4 shows the main steps of the PCI algorithm. The PCI technique can be explained in the following five steps:  
Figure 4: Step-by-step implementation of PCI algorithm. Figure 3: Illustration of the parallel cascade identification. The first cascade output of the non-linear dynamic system is \(z1(n)\), as shown in Figure 4b, and it is estimated by a cascade of a dynamic linear \((L)\) followed by a static non-linear \((N)\) element. 2. Then, compute the first residual as shown in Figure 4c. \[y_{1}(n)=y(n)-z_{1}(n)\] (1)
3. Figure 4d shows the estimation of the new non-linear system having input \(x(n)\) and output \(y1(n)\) by a cascade of \(L2\) followed by \(N2\). 4. Compute the second residual.