# An Overview on Modelling of Complex Interconnected Nonlinear Systems

Authors: Mourad Elloumi; Hamdi Gassara; Omar Naifar
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper proposes new mathematical models of representation, which can describe the dynamic behavior of large-scale nonlinear systems, such as an extended mathematical model of Volterra series, interconnected Hammerstein structures, and interconnected Wiener structures. In this research, we focus on the class of large-scale nonlinear systems, which are composed of several interconnected nonlinear subsystems. In this context, a discrete nonlinear mathematical model with unknown time-varying parameters, mono-variable, characterizes each interconnected subsystem operating in a deterministic or stochastic environment. An illustrative numerical simulation example of two interconnected nonlinear processes is provided to prove the validity and the performance of the developed theoretical results.


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## 1. Introduction

In the literature, the description of a dynamical system by a mathematical model (MM) can be carried out from two different approaches: the first approach is based on a theoretical analysis which allows the system to be described by a MM based on the universal laws that govern it. As for the second, it is realized by experimental analysis which makes it possible to describe the system by a MM based on the results of the experimental tests carried out of this system. The first step in any system study undoubtedly requires representing it by a model that can characterize its dynamic behavior. Thus, this step is essential in the synthesis of a control scheme, with a view to solving either a tracking problem, or a regulation problem, or a joint tracking and regulation problem, of a dynamic system (mechanical, electrical, biological), since it determines the targeted control performance (rapidity, stability, accuracy). It can present difficulties of practical implementation, more particularly in the case of complex systems.In both the scientific and social sciences, the study of complex systems is becoming increasingly relevant. It is widely assumed that there is such a thing as a complex system, with many instances examined across a variety of fields. However, there is no succinct description of a complex system, much less one that is agreed upon by all scientists. Various attempts have been made to describe a complex system and examine a core set of characteristics that are generally identified with complex systems in the literature and by specialists. Some of these characteristics are neither required nor sufficient for complexity, while others are too imprecise or ambiguous to be analytically useful. To add mathematical development to the topic, various common measures of complexity are undertaken from the scientific literature, and taxonomy for them is offered, before claiming that the statistical complexity best reflects the qualitative idea of the order generated by complex systems. In this context, requirements as a characterization of complexity might be provided. These are qualitative requirements that may or may not be adequate for complexity when taken together. It is a ripe research field with a plethora of viewpoints to consider. With the increasing use of complex systems in real-world applications, especially biomedical, finance, and engineering, the study and analysis of complex behavior and dynamic response of complex systems have become increasingly essential. The development of the estimation and the control strategies for a complex system is based on the description of its dynamic behavior by a MM. This description may essentially be carried out using two ways of analysis, the theoretical technique and the experimental one. These MMs can be expressed by difference equations (IOMMs), transfer functions, or state equations (state MMs). The complex systems modelling (nonlinear, nonstationary, high dimensional) from a theoretical analysis can lead to a failure. Indeed, these systems are complex enough for us to be able to apply universal laws to them, in order to formulate theoretically MMs allowing them to describe correctly their dynamic behavior. However, the obtained MM, which is based on theoretical analysis, will not be useable in general for the synthesis of a control law, in particular a digital control law. This is due, on the one hand, to the equations resolution complexity of this MM and, on the other hand, to the disregard of disturbances acting on different points in the system. To overcome the problems relating to the modelling of a dynamic system based on a method of theoretical analysis, we seek to apprehend this system in a phenomenological way, by establishing a mathematical model from experimental analysis. Therefore, we seek to link the measured quantities of the system (input, output, state) by a certain combination.The study of dynamical high dimensional systems has attracted the attention of many researchers and automation engineers worldwide. Every large-scale system can be envisaged as a system consisting of a large number of interacting interconnected systems. Since such a system normally comprises several interconnected systems (power network system and set of coupled tanks), the formulation problem of their parametric estimation or their control is too intricate. Several studies dealing with different themes (modelling, identification, control, stability, and optimization) have been developed and published in the literature [1–3]. In fact, the study [1] is motivated by the desire to build decentralized control for a class of large-scale systems that do not meet the matching condition criterion. The author of [2] discusses the topic of implicit self-tuning control for a class of large-scale systems that have been deconstructed into linked subsystems. The authors look at plants with unknown characteristics that are characterized by a linear invariant or slowly variable model. In addition, the development of recursive estimation techniques for large-scale stochastic systems utilizing the maximum likelihood method was given in [3]. The findings of this research concentrated on large-scale linear systems that can be defined as either continuous or discrete MM. However, certain results concerning large-scale nonlinear systems have been developed and published [4–13]. Indeed, the author of [4] proposes a fault-tolerant control of a class of linked feedback linearizable nonlinear systems via a decentralized adaptive approximation architecture. An adaptive approximation strategy for decentralized fault-tolerant control for a class of nonlinear large-scale systems with unknown multiple time-delayed interaction faults is proposed in [5]. Using the input-output linearization idea, the author of [6] suggested a resilient adaptive fuzzy semidecentralized control for a class of large-scale nonlinear systems. The author of [7] investigates the topic of decentralized adaptive control in large-scale nonstrict-feedback nonlinear systems with a dynamic interaction and unmeasurable states, where the dynamic interaction is connected to both input and output items. A unique extended modal series approach for tackling the infinite horizon optimal control issue of nonlinear linked large-scale dynamic systems is presented in [8]. The infinite horizon nonlinear large-scale two-point boundary value problem (TPBVP), derived from Pontryagin’s maximum principle, is converted into a series of linear time-invariant TPBVPs using this approach. An adaptive fuzzy decentralized output-feedback control issue for a class of nonlinear large-scale systems is discussed in [9]. The parametric absolute stability of linked Lurie systems with several subsystems is studied in [10], where the parametric stability refers to the difficulty of determining the feasibility and stability of equilibrium states when the unknown parameters change. A decentralized fuzzy control problem for asymptotic stabilization of a class of nonlinear large-scale systems using an observer-based output-feedback method has been presented in [11]. A PD-type iterative learning control has been developed and applied for uncertain spatially interconnected systems [12]. Tao et al. were proposed a robust PD-type iterative learning control for discrete systems with multiple time delays subjected to polytopic uncertainty and restricted frequency domain [13]. The most of these works concerned the large-scale systems which can be described by a linear MM (input-output MM and state MM) with constant or slow time-varying parameters. However, a few results were published concerning the large-scale nonlinear systems which are described by nonlinear state MMs. Furthermore, we may use other MMs to characterize these nonlinear dynamic systems. The traditional structure relies on the nonlinear system’s approximation by the Volterra series. Other forms of representations, like input-output models and linked block models, allow us to characterize the dynamic behavior of considered systems.Consequently, we shall build a variety of nonlinear discrete MMs capable of describing the dynamic behavior of large-scale nonlinear systems in this study. The emphasis will be on the class of large-scale nonlinear systems that composed of several linked mono-variable nonlinear systems with unknown time-varying parameters. We suppose that these complex systems can operate in a deterministic or stochastic environment.The remainder of this research is organized as follows: Section 2 is devoted to the description of large-scale nonlinear systems by MMs in a series of functions, where two forms of MMs are developed. In Section 3, input-output MMs for modelling the dynamic behavior of linked nonlinear systems, operating in a deterministic or stochastic environment, are proposed. The modelling of linked nonlinear mono-variable systems, based on interconnected Hammerstein and Wiener structures, is derived in Section4. Finally, some simulation results and concluding remarks are provided in Sections 5 and 6.


## 2. MMs in Series of Functions

Serial models of functions are one of the MMs that may represent the dynamic behavior of a nonlinear system. This family of models allows its output at a given moment to be described by an infinite sum of polynomial functions dependent on the input at the same and previous instants. As a result, the Volterra series representation and the Volterra parametric representation may be extended to describe nonlinear systems with huge dimensions. These two representations are commonly used in the study and description of nonlinear systems [14–17], particularly those defined by discrete MMs. In fact, this is the second installment of a two-volume guidebook [14] that provides a detailed review of nonlinear dynamic system identification. Many elements of nonlinear processes are covered in the books, including modelling, parameter estimates, structure search, nonlinearity, and model validity testing. Not only nonparametric models but also parametric models with a restricted number of parameters are included in the book. The estimate of time-domain parameters is covered in depth, as well as frequency domain and power spectrum processes. This work is aimed towards postgraduate students, researchers, and engineers working in the field of nonlinear systems. There are numerous instances, case studies, and experimental identifications of genuine processes. The study [15] gives an overview of works in the field of mathematical modelling of nonlinear input-output dynamic systems with Volterra polynomials that were undertaken at systems. The author of [16] presents a method for identifying nonlinear aeroelastic systems based on the Volterra theory of nonlinear systems. The theory’s recent applicability to difficulties in computational and experimental aeroelasticity is discussed. The book [17] covers simple, brief, and easy-to-understand methods for identifying nonlinear systems, as well as new research discoveries in the field of adaptive nonlinear system identification. These approaches make use of adaptive filter algorithms, which are well-known for identifying linear systems. They can be used to simulate nonlinear systems that polynomials can efficiently model.


### 2.1. MM in a Series of Volterra

The Volterra series represent nonlinear MMs without output feedback. Thus, a nonlinear system can be described by the following MM in a series of Volterra:(1)yk=fuk−1,uk−2,…,uk−τ.The previous model can be extended to describe the dynamics of an INS.(2)yαk=fαuαk−1,uαk−2,…,uαk−τ,uβk−1,uβk−2,…,uβk−τ,where yαk and uαk represent, respectively, the output and input of the INS Sα, α=1,…,Ν; uβk indicates the inputs from the other connected subsystems Sβ, β=1,…,N;β≠α; Ν represents the number of INSs; and fα. is a nonlinear function, which is approximated by a polynomial for the case of the Volterra model.An approximation of the MM (2) allowing to describe an INS of orderM, which is composed of Ν INSs, can be obtained using the second-order nuclei, such as(3)yαk=μα+∑r=1Mζα,ruαk−r+∑r=1M∑s=1Mζα,rsuαk−ruαk−s,+∑β=1,β≠αN∑r=1Mζαβ,ruβk−r+∑β=1,β≠αN∑r=1M∑s=1Mζαβ,rsuαk−ruβk−s,where α,β=1,…,N;β≠α, μα is a constant, and ζα,r, ζα,rs, ζαβ,r, and ζαβ,rs are positive parameters. These parameters are called Volterra kernels.


### 2.2. Parametric Model of Volterra

This MM family is distinguished by linear feedback of outputs and a polynomial function of inputs. In the literature, the Volterra parametric model is used to describe a nonlinear system.(4)yk=fuk−1,uk−2,…,uk−m−∑h=1mahyk−h.Thereby, we propose the following parametric model of Volterra in order to describe the dynamic of an INS:(5)yαk=fαuαk−1,uαk−2,…,uαk−m,uβk−1,uβk−2,…,uβk−m,−∑h=1maα,hyαk−h+∑β=1,β≠αN∑h=1maαβ,hyβk−h,with uαk is the input and yαk is the output of the INS Sα; yβk and uβk represent, respectively, the outputs and inputs from the other INSs Sβ, β=1,…,N;β≠α; aα,h and aαβ,h are constant parameters; and m is a positive parameter which corresponds to the order of the considered system.In the case of an order systemM and a polynomial of degree 2, the expression of the output yαk is written in the following form:(6)yαk=μα−∑h=1Maα,hyαk−h+∑r=1Mζα,ruαk−r+∑r=1M∑s=1Mζα,rsuαk−ruαk−s,+∑β=1,β≠αN∑r=1Mζαβ,ruβk−r+∑β=1,β≠αN∑h=1Maαβ,hyβk−h+∑β=1,β≠αN∑r=1M∑s=1Mζαβ,rsuαk−ruβk−s.We can notice that these different representations of MMs become more and more complex by increasing the order of the system or the nonlinearity degree.


## 2.1. MM in a Series of Volterra

The Volterra series represent nonlinear MMs without output feedback. Thus, a nonlinear system can be described by the following MM in a series of Volterra:(1)yk=fuk−1,uk−2,…,uk−τ.The previous model can be extended to describe the dynamics of an INS.(2)yαk=fαuαk−1,uαk−2,…,uαk−τ,uβk−1,uβk−2,…,uβk−τ,where yαk and uαk represent, respectively, the output and input of the INS Sα, α=1,…,Ν; uβk indicates the inputs from the other connected subsystems Sβ, β=1,…,N;β≠α; Ν represents the number of INSs; and fα. is a nonlinear function, which is approximated by a polynomial for the case of the Volterra model.An approximation of the MM (2) allowing to describe an INS of orderM, which is composed of Ν INSs, can be obtained using the second-order nuclei, such as(3)yαk=μα+∑r=1Mζα,ruαk−r+∑r=1M∑s=1Mζα,rsuαk−ruαk−s,+∑β=1,β≠αN∑r=1Mζαβ,ruβk−r+∑β=1,β≠αN∑r=1M∑s=1Mζαβ,rsuαk−ruβk−s,where α,β=1,…,N;β≠α, μα is a constant, and ζα,r, ζα,rs, ζαβ,r, and ζαβ,rs are positive parameters. These parameters are called Volterra kernels.


## 2.2. Parametric Model of Volterra

This MM family is distinguished by linear feedback of outputs and a polynomial function of inputs. In the literature, the Volterra parametric model is used to describe a nonlinear system.(4)yk=fuk−1,uk−2,…,uk−m−∑h=1mahyk−h.Thereby, we propose the following parametric model of Volterra in order to describe the dynamic of an INS:(5)yαk=fαuαk−1,uαk−2,…,uαk−m,uβk−1,uβk−2,…,uβk−m,−∑h=1maα,hyαk−h+∑β=1,β≠αN∑h=1maαβ,hyβk−h,with uαk is the input and yαk is the output of the INS Sα; yβk and uβk represent, respectively, the outputs and inputs from the other INSs Sβ, β=1,…,N;β≠α; aα,h and aαβ,h are constant parameters; and m is a positive parameter which corresponds to the order of the considered system.In the case of an order systemM and a polynomial of degree 2, the expression of the output yαk is written in the following form:(6)yαk=μα−∑h=1Maα,hyαk−h+∑r=1Mζα,ruαk−r+∑r=1M∑s=1Mζα,rsuαk−ruαk−s,+∑β=1,β≠αN∑r=1Mζαβ,ruβk−r+∑β=1,β≠αN∑h=1Maαβ,hyβk−h+∑β=1,β≠αN∑r=1M∑s=1Mζαβ,rsuαk−ruβk−s.We can notice that these different representations of MMs become more and more complex by increasing the order of the system or the nonlinearity degree.


## 3. IOMMs

The use of input-output MMs to describe nonlinear systems is a popular strategy in industrial settings [15, 17]. In fact, the paper [15] provides an overview of work done at systems in the subject of mathematical modelling of nonlinear input-output dynamic systems with Volterra polynomials. Based on the Volterra theory of nonlinear systems, the author of [16] proposes a technique for finding nonlinear aeroelastic systems. The theory’s current relevance to computational and experimental aeroelasticity problems is reviewed. The book [17] discusses simple, concise, and simple-to-understand approaches for identifying nonlinear systems, as well as recent research findings in the field of adaptive nonlinear system identification. These methods employ adaptive filter techniques, which are well-known for finding linear systems. These methods can be used to simulate nonlinear systems whose models are approximated by polynomial functions.This paragraph is concerned with the description of large-scale nonlinear systems, which are made up of multiple linked mono-variable nonlinear systems functioning in a deterministic or stochastic environment [18]. We can differentiate three types of nonlinearities, which are as follows:(1)
Nonlinearity with respect to the parameters(2)
Nonlinearity with respect to the observations(3)
Nonlinearity with respect to the parameters and the observationsWe are particularly interested in input-output MMs with linear parameters and nonlinear data.


### 3.1. Nonlinearity with respect to the Inputs

In this part, we will present input-output MMs that can be used to characterize the dynamics of INSs with nonlinearity with respect to the inputs. We will concentrate on linked nonlinear dynamical systems that may be characterized by the class of deterministic or stochastic input-output MMs, which are nonlinear with respect to inputs, mono-variables, with time-varying parameters.


#### 3.1.1. Deterministic Input-Output MMs

We are interested here in the description of INSs by deterministic input-output MMs. In this context, we consider an INSSα, 1≤α≤N, coupled with other interconnected subsystems Sβ, β=1,…,N;β≠α, having a nonlinearity with respect to the inputs, which can be modeled by the following IOMM INDARMA (interconnected nonlinear deterministic autoregressive moving average) [18]:(7)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβk,+fguuαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,where yαk and uαk represent, respectively, the output and input of the INS Sα; uαk and uβk denote, respectively, the outputs and inputs from the other interconnected nonlinear subsystems Sβ, β=1,…,N;β≠α; dα presents the intrinsic delay of the considered system; tαβ and dαβ represent the delays of the interactions, which are relative to the outputs and the inputs of the other INSs Sβ; and Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials, defined as follows:(8)Aαq−1,k=1+aα,1kq−1+...+aα,nAαkq−nAa,Bαq−1,k=bα,1kq−1+...+bα,nB,akq−nBa,Bαβq−1,k=bαβ,1kq−1+...+bαβ,nBαβkq−nBαβ,and(9)Aαβq−1,k=1+aαβ,1kq−1+...+aαβ,nAαβkq−nAαβ,with α,β=1,…,N;β≠α, and nAα, nBα, nBαβ, and nAαβ are the orders of the polynomials Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k.The termfgu. represents a nonlinear function with nonlinearity degree p, which depends on the input sequences of the interconnected system (IS) Sα, 1≤α≤N, and the other INSs Sβ, β=1,…,N;β≠α, defined as follows [18]:(10)fgu.=∑r1=1ngαα,r1∑r2=1ngαα,r1r2gαα,r1r2kuαk−r1uαk−r2,+∑r1=1ngαα,r1∑r2=1ngαα,r1r2∑r3=1ngαα,r1r2r3gαα,r1r2r3kuαk−r1uαk−r2uαk−r3,+⋯+∑r1=1ngαα,r1∑r2=1ngαα,r1r2⋯∑rp=1ngαα,r1r2...rpgαα,r1...rpkuαk−r1…uαk−rp,+∑β=1,β≠αN∑r1=1ngαβ,r1∑r2=1ngαβ,r1r2gαβ,r1r2kuαk−r1uβk−r2,+∑β=1,β≠αN∑r1=1ngαβ,r1∑r2=1ngαβ,r1r2∑r3=1ngαβ,r1r2r3gαβ,r1r2r3kuαk−r1uαk−r2uβk−r3,+⋯+∑β=1,β≠αN∑r1=1ngαβ,r1∑r2=1ngαβ,r1r2⋯∑rp=1ngαβ,r1r2...rpgαβ,r1...rpkuαk−r1…uβk−rp.Note that each ISSα, α=1,…,N, is coupled with the outputs and the inputs of the other INSs Sβ, by the polynomials Aαβq−1,k and Bαβq−1,k.


#### 3.1.2. Stochastic Input-Output MMs

This subsection deals with the description of the INSs, which are described by stochastic IOMMs. Consider a stochastic INSSα, 1≤α≤N, which is coupled to other ISs Sβ, β=1,…,N;β≠α, nonlinear with respect to the inputs. This system can be qualified by the following mathematical input-output model INARMAX (interconnected nonlinear autoregressive moving average with exogenous) [18]:(11)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβk,+fguuαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,+fγueuαk−1,uαk−2,…,uαk−nα,eαk−1,eαk−2,…,eαk−nα+fceeαk−1,eαk−2,…,eαk−nα+Cαq−1eαk,where eαk represents the set of random variables acting on the IS Sα, which can be assimilated to a Gaussian distribution with zero mean and constant variance σα2, fgu. is a nonlinear function of degree p given by (10), Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials defined by (8), (9), (10), and (9), and Cαq−1 is a polynomial with constant parameters, defined as follows:(12)Cαq−1=1+cα,1q−1+...+cα,nCaq−nCα,where nCα denotes the order of Cαq−1.The termfγue. denotes a nonlinear function of degree p, which is determined by the IS’s input sequences Sα and the disturbance eαk. This function can be expressed as follows:(13)fγue.=∑r1=1nγαα,r1∑r2=1nγαα,r1r2γαα,r1r2uαk−r1eαk−r2+……+,∑r1=1nγαα,r1∑r2=1nγαα,r1r2⋯∑rp=1nγαα,r1r2...rpγαα,r1...rpuαk−r1...eαk−rp.The termfce. represents a nonlinear function of degree p, which depends only on the noise sequence eαk. This term is defined as follows:(14)fce.=∑r1=1ncαα,r1∑r2=1ncαα,r1r2cαα,r1r2eαk−r1eαk−r2+……+,∑r1=1ncαα,r1∑r2=1ncαα,r1r2⋯∑rp=1ncαα,r1r2...rpcαα,r1...rpeαk−r1...eαk−rp.As an example, we consider a large-scale nonlinear dynamic system composed of two INSsS1 and S2. Each subsystem can be modeled by the INARMAX mathematical model of the second order, nonlinear with respect to the inputs and having a degree of nonlinearity equal to 2. Thus, the output yαk of each INS Sα is demonstrated by the following expression:(15)yαk=−∑r=12aα,rkyαk−r+∑r=12bα,rkuαk−dα−r+∑r=12cα,reαk−r,+∑β=1,β≠α2∑r=12bαβ,rkuβk−dαβ−r+∑β=1,β≠α2∑r=12aαβ,rkyβk−tαβ−r,+∑r1=12∑r2=12gαα,r1r2kuαk−r1uαk−r2+∑β=1,β≠α2∑r1=12∑r2=12gαα,r1r2kuαk−r1uβk−r2,with α,β=1,2;β≠α.


### 3.2. Nonlinearity with respect to the Outputs

This section is dedicated to the description of INSs with nonlinearity in their outputs. This type of dynamical system may be characterized by input-output MMs that are nonlinear in terms of the outputs, mono-variable, deterministic, or stochastic and include time-varying parameters.


#### 3.2.1. Deterministic Input-Output MMs

Let us consider an INS operating in a deterministic environment, mono-variable and having a nonlinearity with respect to the outputs. The general structure of the considered system can be described by the following INDARMA mathematical model [18]:(16)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβk,+ffαβyyαk−1,yαk−2,…,yαk−nα,yβk−1,yβk−2,…,yβk−nα,−ffααyyαk−1,yαk−2,…,yαk−nα,where the terms ffααy. and ffαβy. are nonlinear functions of nonlinearity degree p, which depend on the output sequences of the IS Sα and the other ISs Sβ, respectively. These functions can be defined by the following expressions:(17)ffααy.=∑r1=1nfαα,r1∑r2=1nfαα,r1r2fαα,r1r2kyαk−r1yαk−r2,+∑r1=1nfαα,r1∑r2=1nfαα,r1r2∑r3=1nfαα,r1r2r3fαα,r1r2r3kyαk−r1yαk−r2yαk−r3,+…+∑r1=1nfαα,r1∑r2=1nfαα,r1r2⋯∑rp=1nfαα,r1r2...rpfαα,r1...rpkyαk−r1…yαk−rp,and(18)ffαβy.=∑β=1,β≠αN∑r1=1nfαβ,r1∑r2=1nfαβ,r1r2fαβ,r1r2kyαk−r1yβk−r2,+∑β=1,β≠αN∑r1=1nfαβ,r1∑r2=1nfαβ,r1r2∑r3=1nfαβ,r1r2r3fαβ,r1r2r3kyαk−r1yαk−r2yβk−r3,+…+∑β=1,β≠αN∑r1=1nfαβ,r1∑r2=1nfαβ,r1r2⋯∑rp=1nfαβ,r1r2...rpfαβ,r1...rpkyαk−r1…yβk−rp.


#### 3.2.2. Stochastic Input-Output MMs

We consider an INSSα, 1≤α≤N, which is coupled to other INSs Sβ, β=1,…,N;β≠α, exhibiting nonlinearity in outputs and working in a stochastic environment. We suppose that the noise operating on the investigated system is made up of a sequence of independent random variables with a zero mean and a finite variance, σα2. The general structure of this MM is given by the following expression [18]:(19)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβq−1,k,+ffαβyyαk−1,yαk−2,…,yαk−nα,yβk−1,yβk−2,…,yβk−nα,−ffααyyαk−1,yαk−2,…,yαk−nα+fceeαk−1,eαk−2,…,eαk−nα,+fλyeyαk−1,yαk−2,…,yαk−nα,eαk−1,eαk−2,…,eαk−nα+Cαq−1eαk,where Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials, defined by (8), (9), (10), and (9); Cαq−1 is a polynomial with constant parameters, given by (14); and the nonlinear functions fce., ffααy., and ffαβy. are defined by (16), (19), and (20), respectively.The termfλye. denotes a nonlinear function of degree p, which depends on the output sequences of the IS Sα and the noise eαk. This function can be defined as follows:(20)fλye.=∑r1=1nλαα,r1∑r2=1nλαα,r1r2λαα,r1r2yαk−r1eαk−r2+……+,∑r1=1nλαα,r1∑r2=1nλαα,r1r2…∑rp=1nλαα,r1r2...rpλαα,r1...rpyαk−r1…eαk−rp.For example, the following model corresponds to an IOMMINARMAX of the second order with a nonlinearity degree equal to 2, making it possible to describe the dynamic behavior of a large-scale nonlinear process composed of two interconnected nonlinear subsystemsS1 and S2. The output yαk of each interconnected nonlinear subsystem Sα is described as(21)yαk=−∑r=12aα,rkyαk−r−∑r1=12∑r2=12fαα,r1r2kyαk−r1yαk−r2,+∑r=12bα,rkuαk−dα−r+∑β=1,β≠α2∑r=12bαβ,rkuβk−dαβ−r,+∑β=1,β≠α2∑r=12aαβ,rkyβk−tαβ−r+∑β=1,β≠α2∑r1=12∑r2=12fαβ,r1r2kyαk−r1yβk−r2,+∑r=12cα,reαk−r,with α,β=1,2;β≠α.


### 3.3. Nonlinearity with respect to the Observations

In this part, we will create input-output MMs of representation that allow us to describe ISs that are nonlinear with respect to the observations, are mono-variable, and have unknown time-varying parameters [18].


#### 3.3.1. Deterministic Input-Output MMs

We consider a dynamical system, which is composed ofN ISs, working in a predictable environment and being nonlinear with regard to the observations. The input-output MMINDARMA, making it possible to describe the considered system, is given as follows [18]:(22)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβk,+fguuαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,+ffαβyyαk−1,yαk−2,…,yαk−nα,yβk−1,yβk−2,…,yβk−nα,−ffααyyαk−1,yαk−2,…,yαk−nα+fhℓuyyαk−1,yαk−2,…,yαk−nα,uαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,yβk−1,yβk−2,…,yβk−nα,where Aαq−1,k, Bαq−1,k,Aαβq−1,k, and Bαβq−1,k are polynomials defined by (8), (9), (10), and (9), respectively; fgu., ffααy., and ffαβy. are nonlinear functions given by (12), (19), and (20), respectively, and fλhluy. is described by the following nonlinear function:(23)fhℓuy.=∑β=1N∑r1=1nhαβ,r1∑r2=1nhαβ,r1r2hαβ,r1r2kuαk−r1yβk−r2,+∑β=1N∑r1=1nhαβ,r1∑r2=1nhαβ,r1r2∑r3=1nhαβ,r1r2r3hαβ,r1r2r3kuαk−r1uαk−r2yβk−r3+…+,+∑β=1N∑r1=1nhαβ,r1∑r2=1nhαβ,r1r2⋯∑rp=1nhαβ,r1r2...rphαβ,r1...rpkuαk−r1…yβk−rp,+∑β=1,β≠αN∑r1=1nℓαβ,r1∑r2=1nℓαβ,r1r2ℓαβ,r1r2kyαk−r1uβk−r2,+∑β=1,β≠αN∑r1=1nℓαβ,r1∑r2=1nℓαβ,r1r2∑r3=1nℓαβ,r1r2r3ℓαβ,r1r2r3kyαk−r1yαk−r2uβk−r3+…+,+∑β=1,β≠αN∑r1=1nℓαβ,r1∑r2=1nℓαβ,r1r2⋯∑rp=1nℓαβ,r1r2...rpℓαβ,r1...rpkyαk−r1…uβk−rp.


#### 3.3.2. Stochastic Input-Output MMs

Let us consider an INSSα, 1≤α≤N, which is coupled to the other IS Sβ, β=1,…,N;β≠α. This system is nonlinear with respect to the observations and can be described by the class of IOMMs. The considered structure of the INARMAX MM is given by the following expression [18]:(24)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk,+∑β=1,β≠αNq−tαβAαβq−1,kyβk+Cαq−1eαk,+fguuαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,+ffαβyyαk−1,yαk−2,…,yαk−nα,yβk−1,yβk−2,…,yβk−nα,−ffααyyαk−1,yαk−2,…,yαk−nα+fhℓuyyβk−1,yβk−2,…,yβk−nα,uαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,yβk−1,yβk−2,…,yβk−nα,+fγueuαk−1,uαk−2,…,uαk−nα,eαk−1,eαk−2,…,eαk−nα,+fλyeyαk−1,yαk−2,…,yαk−nα,eαk−1,eik−2,…,eαk−nα+fceeαk−1,eαk−2,…,eαk−nα,where fgu., fβue., fce., ffααy., ffαβy., , and fhℓuy. are nonlinear functions defined by (12), (15), (16), (19), (20), (22), and (23), respectively.For the reason of simplicity, we consider the following dynamical system, which consists ofN INSs, running in a deterministic environment and defined by an IOMM INDARMA with a nonlinearity degree equal to 2, such as(25)Aα1q−1yαk+Aα2q1−1,q2−1yα2k=Bα1q−1uαk+Bα2q1−1,q2−1uα2k+Aαβ1q−1yβk,+Bαβ1q−1uβk+Aαβ2q1−1,q2−1yαkyβk,+Bαβ2q1−1,q2−1uαkuβk+Fαβq1−1,q2−1uαkyβk,+Hαβq1−1,q2−1yαkuβk,where(26)Bα1q−1=∑r=1nBα1bα1,rq−r,Aα1q−1=1+∑r=1nAα1aα1,rq−r,Bα2q1−1,q2−1uα2k=∑r=1nBα21∑s=1nBα22bα2,rsuαk−ruαk−s,Aα2q1−1,q2−1yα2k=∑r=1nAα21∑s=1nAα22aα2,rsyαk−ryαk−s,Bαβ1q−1uβk=∑β=1,β≠αN∑r=1nBαβ1bαβ1,ruβk−r,Aαβ1q−1yβk=∑β=1,β≠αN∑r=1nAαβ1aαβ1,ryβk−r,Bαβ2q1−1,q2−1uαkuβk=∑β=1,β≠αN∑r=1nBαβ21∑s=1nBαβ22bαβ2,rsuαk−ruβk−s,Aαβ2q1−1,q2−1yαkyβk=∑β=1,β≠αN∑r=1nAαβ21∑s=1nAαβ22aαβ2,rsyαk−ryβk−s,Fαβq1−1,q2−1uαkyβk=∑β=1N∑r=1nFαβ1∑s=1nFαβ2fαβ,rsuαk−ryβk−s,and(27)Hαβq1−1,q2−1yαkuβk=∑β=1,β≠αN∑r=1nHαβ1∑s=1nHαβ2hαβ,rsyαk−ruβk−s,with α,β=1,…,N;β≠α.From the developed MM, we can distinguish special cases of input-output MM, based on various representations, such as the following:(1)
Serial model of Volterra:(28)Aα1q−1=0,Aα2q1−1,q2−1=0,Aαβ1q−1=0,Aαβ2q1−1,q2−1=0,Fαβq1−1,q2−1=0,Hαβq1−1,q2−1=0.(2)
Parametric model of Volterra:(29)Aα2q1−1,q2−1=0,Aαβ2q1−1,q2−1=0,Fαβq1−1,q2−1=0,Hαβq1−1,q2−1=0.(3)
Bilinear model:(30)Aα2q1−1,q2−1=0,Bα2q1−1,q2−1=0.(4)
Linear model with respect to the input signal:(31)Bα2q1−1,q2−1=0,Bαβ2q1−1,q2−1=0,Fαβq1−1,q2−1=0,Hαβq1−1,q2−1=0.(5)
Linear model with respect to the output signal:(32)Aα2q1−1,q2−1=0,Aαβ2q1−1,q2−1=0,Fαβq1−1,q2−1=0,Hαβq1−1,q2−1=0.For example, we consider a large-scale nonlinear process, which is constituted of two INSS1 and S2. Each interconnected subsystem can be defined by the INDARMA mathematical model of the second order with time-varying parameters and having a nonlinearity degree equal to 2.Figure1 shows the interaction structure diagram of the considered nonlinear process:Figure 1
Interaction structure diagram of the considered process.Thus, the outputyαk of the INS Sα can be expressed as(33)yαk=−∑r=12aα,rkyαk−r−∑r1=12∑r2=12fαα,r1r2kyαk−r1yαk−r2,+∑r=12bα,rkuαk−dα−r+∑β=1,β≠α2∑r=12bαβ,rkuβk−dαβ−r,+∑β=1,β≠α2∑r=12aαβ,rkyβk−tαβ−r+∑r1=12∑r2=12gαα,r1r2kuαk−r1uαk−r2,+∑β=1,β≠α2∑r1=12∑r2=12gαβ,r1r2kuαk−r1uβk−r2+∑β=1,β≠α2∑r1=12∑r2=12fαβ,r1r2kyαk−r1yβk−r2,+∑β=12∑r1=12∑r2=12hαβ,r1r2kuαk−r1yβk−r2+∑β=1,β≠α2∑r1=12∑r2=12ℓαβ,r1r2kyαk−r1uβk−r2,with α,β=1,2;β≠α.We notice that these different representations of developed MMs become more and more complex by increasing the nonlinearity degreep and/or the order of the IS.


## 3.1. Nonlinearity with respect to the Inputs

In this part, we will present input-output MMs that can be used to characterize the dynamics of INSs with nonlinearity with respect to the inputs. We will concentrate on linked nonlinear dynamical systems that may be characterized by the class of deterministic or stochastic input-output MMs, which are nonlinear with respect to inputs, mono-variables, with time-varying parameters.


### 3.1.1. Deterministic Input-Output MMs

We are interested here in the description of INSs by deterministic input-output MMs. In this context, we consider an INSSα, 1≤α≤N, coupled with other interconnected subsystems Sβ, β=1,…,N;β≠α, having a nonlinearity with respect to the inputs, which can be modeled by the following IOMM INDARMA (interconnected nonlinear deterministic autoregressive moving average) [18]:(7)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβk,+fguuαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,where yαk and uαk represent, respectively, the output and input of the INS Sα; uαk and uβk denote, respectively, the outputs and inputs from the other interconnected nonlinear subsystems Sβ, β=1,…,N;β≠α; dα presents the intrinsic delay of the considered system; tαβ and dαβ represent the delays of the interactions, which are relative to the outputs and the inputs of the other INSs Sβ; and Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials, defined as follows:(8)Aαq−1,k=1+aα,1kq−1+...+aα,nAαkq−nAa,Bαq−1,k=bα,1kq−1+...+bα,nB,akq−nBa,Bαβq−1,k=bαβ,1kq−1+...+bαβ,nBαβkq−nBαβ,and(9)Aαβq−1,k=1+aαβ,1kq−1+...+aαβ,nAαβkq−nAαβ,with α,β=1,…,N;β≠α, and nAα, nBα, nBαβ, and nAαβ are the orders of the polynomials Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k.The termfgu. represents a nonlinear function with nonlinearity degree p, which depends on the input sequences of the interconnected system (IS) Sα, 1≤α≤N, and the other INSs Sβ, β=1,…,N;β≠α, defined as follows [18]:(10)fgu.=∑r1=1ngαα,r1∑r2=1ngαα,r1r2gαα,r1r2kuαk−r1uαk−r2,+∑r1=1ngαα,r1∑r2=1ngαα,r1r2∑r3=1ngαα,r1r2r3gαα,r1r2r3kuαk−r1uαk−r2uαk−r3,+⋯+∑r1=1ngαα,r1∑r2=1ngαα,r1r2⋯∑rp=1ngαα,r1r2...rpgαα,r1...rpkuαk−r1…uαk−rp,+∑β=1,β≠αN∑r1=1ngαβ,r1∑r2=1ngαβ,r1r2gαβ,r1r2kuαk−r1uβk−r2,+∑β=1,β≠αN∑r1=1ngαβ,r1∑r2=1ngαβ,r1r2∑r3=1ngαβ,r1r2r3gαβ,r1r2r3kuαk−r1uαk−r2uβk−r3,+⋯+∑β=1,β≠αN∑r1=1ngαβ,r1∑r2=1ngαβ,r1r2⋯∑rp=1ngαβ,r1r2...rpgαβ,r1...rpkuαk−r1…uβk−rp.Note that each ISSα, α=1,…,N, is coupled with the outputs and the inputs of the other INSs Sβ, by the polynomials Aαβq−1,k and Bαβq−1,k.


### 3.1.2. Stochastic Input-Output MMs

This subsection deals with the description of the INSs, which are described by stochastic IOMMs. Consider a stochastic INSSα, 1≤α≤N, which is coupled to other ISs Sβ, β=1,…,N;β≠α, nonlinear with respect to the inputs. This system can be qualified by the following mathematical input-output model INARMAX (interconnected nonlinear autoregressive moving average with exogenous) [18]:(11)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβk,+fguuαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,+fγueuαk−1,uαk−2,…,uαk−nα,eαk−1,eαk−2,…,eαk−nα+fceeαk−1,eαk−2,…,eαk−nα+Cαq−1eαk,where eαk represents the set of random variables acting on the IS Sα, which can be assimilated to a Gaussian distribution with zero mean and constant variance σα2, fgu. is a nonlinear function of degree p given by (10), Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials defined by (8), (9), (10), and (9), and Cαq−1 is a polynomial with constant parameters, defined as follows:(12)Cαq−1=1+cα,1q−1+...+cα,nCaq−nCα,where nCα denotes the order of Cαq−1.The termfγue. denotes a nonlinear function of degree p, which is determined by the IS’s input sequences Sα and the disturbance eαk. This function can be expressed as follows:(13)fγue.=∑r1=1nγαα,r1∑r2=1nγαα,r1r2γαα,r1r2uαk−r1eαk−r2+……+,∑r1=1nγαα,r1∑r2=1nγαα,r1r2⋯∑rp=1nγαα,r1r2...rpγαα,r1...rpuαk−r1...eαk−rp.The termfce. represents a nonlinear function of degree p, which depends only on the noise sequence eαk. This term is defined as follows:(14)fce.=∑r1=1ncαα,r1∑r2=1ncαα,r1r2cαα,r1r2eαk−r1eαk−r2+……+,∑r1=1ncαα,r1∑r2=1ncαα,r1r2⋯∑rp=1ncαα,r1r2...rpcαα,r1...rpeαk−r1...eαk−rp.As an example, we consider a large-scale nonlinear dynamic system composed of two INSsS1 and S2. Each subsystem can be modeled by the INARMAX mathematical model of the second order, nonlinear with respect to the inputs and having a degree of nonlinearity equal to 2. Thus, the output yαk of each INS Sα is demonstrated by the following expression:(15)yαk=−∑r=12aα,rkyαk−r+∑r=12bα,rkuαk−dα−r+∑r=12cα,reαk−r,+∑β=1,β≠α2∑r=12bαβ,rkuβk−dαβ−r+∑β=1,β≠α2∑r=12aαβ,rkyβk−tαβ−r,+∑r1=12∑r2=12gαα,r1r2kuαk−r1uαk−r2+∑β=1,β≠α2∑r1=12∑r2=12gαα,r1r2kuαk−r1uβk−r2,with α,β=1,2;β≠α.


## 3.1.1. Deterministic Input-Output MMs

We are interested here in the description of INSs by deterministic input-output MMs. In this context, we consider an INSSα, 1≤α≤N, coupled with other interconnected subsystems Sβ, β=1,…,N;β≠α, having a nonlinearity with respect to the inputs, which can be modeled by the following IOMM INDARMA (interconnected nonlinear deterministic autoregressive moving average) [18]:(7)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβk,+fguuαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,where yαk and uαk represent, respectively, the output and input of the INS Sα; uαk and uβk denote, respectively, the outputs and inputs from the other interconnected nonlinear subsystems Sβ, β=1,…,N;β≠α; dα presents the intrinsic delay of the considered system; tαβ and dαβ represent the delays of the interactions, which are relative to the outputs and the inputs of the other INSs Sβ; and Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials, defined as follows:(8)Aαq−1,k=1+aα,1kq−1+...+aα,nAαkq−nAa,Bαq−1,k=bα,1kq−1+...+bα,nB,akq−nBa,Bαβq−1,k=bαβ,1kq−1+...+bαβ,nBαβkq−nBαβ,and(9)Aαβq−1,k=1+aαβ,1kq−1+...+aαβ,nAαβkq−nAαβ,with α,β=1,…,N;β≠α, and nAα, nBα, nBαβ, and nAαβ are the orders of the polynomials Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k.The termfgu. represents a nonlinear function with nonlinearity degree p, which depends on the input sequences of the interconnected system (IS) Sα, 1≤α≤N, and the other INSs Sβ, β=1,…,N;β≠α, defined as follows [18]:(10)fgu.=∑r1=1ngαα,r1∑r2=1ngαα,r1r2gαα,r1r2kuαk−r1uαk−r2,+∑r1=1ngαα,r1∑r2=1ngαα,r1r2∑r3=1ngαα,r1r2r3gαα,r1r2r3kuαk−r1uαk−r2uαk−r3,+⋯+∑r1=1ngαα,r1∑r2=1ngαα,r1r2⋯∑rp=1ngαα,r1r2...rpgαα,r1...rpkuαk−r1…uαk−rp,+∑β=1,β≠αN∑r1=1ngαβ,r1∑r2=1ngαβ,r1r2gαβ,r1r2kuαk−r1uβk−r2,+∑β=1,β≠αN∑r1=1ngαβ,r1∑r2=1ngαβ,r1r2∑r3=1ngαβ,r1r2r3gαβ,r1r2r3kuαk−r1uαk−r2uβk−r3,+⋯+∑β=1,β≠αN∑r1=1ngαβ,r1∑r2=1ngαβ,r1r2⋯∑rp=1ngαβ,r1r2...rpgαβ,r1...rpkuαk−r1…uβk−rp.Note that each ISSα, α=1,…,N, is coupled with the outputs and the inputs of the other INSs Sβ, by the polynomials Aαβq−1,k and Bαβq−1,k.


## 3.1.2. Stochastic Input-Output MMs

This subsection deals with the description of the INSs, which are described by stochastic IOMMs. Consider a stochastic INSSα, 1≤α≤N, which is coupled to other ISs Sβ, β=1,…,N;β≠α, nonlinear with respect to the inputs. This system can be qualified by the following mathematical input-output model INARMAX (interconnected nonlinear autoregressive moving average with exogenous) [18]:(11)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβk,+fguuαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,+fγueuαk−1,uαk−2,…,uαk−nα,eαk−1,eαk−2,…,eαk−nα+fceeαk−1,eαk−2,…,eαk−nα+Cαq−1eαk,where eαk represents the set of random variables acting on the IS Sα, which can be assimilated to a Gaussian distribution with zero mean and constant variance σα2, fgu. is a nonlinear function of degree p given by (10), Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials defined by (8), (9), (10), and (9), and Cαq−1 is a polynomial with constant parameters, defined as follows:(12)Cαq−1=1+cα,1q−1+...+cα,nCaq−nCα,where nCα denotes the order of Cαq−1.The termfγue. denotes a nonlinear function of degree p, which is determined by the IS’s input sequences Sα and the disturbance eαk. This function can be expressed as follows:(13)fγue.=∑r1=1nγαα,r1∑r2=1nγαα,r1r2γαα,r1r2uαk−r1eαk−r2+……+,∑r1=1nγαα,r1∑r2=1nγαα,r1r2⋯∑rp=1nγαα,r1r2...rpγαα,r1...rpuαk−r1...eαk−rp.The termfce. represents a nonlinear function of degree p, which depends only on the noise sequence eαk. This term is defined as follows:(14)fce.=∑r1=1ncαα,r1∑r2=1ncαα,r1r2cαα,r1r2eαk−r1eαk−r2+……+,∑r1=1ncαα,r1∑r2=1ncαα,r1r2⋯∑rp=1ncαα,r1r2...rpcαα,r1...rpeαk−r1...eαk−rp.As an example, we consider a large-scale nonlinear dynamic system composed of two INSsS1 and S2. Each subsystem can be modeled by the INARMAX mathematical model of the second order, nonlinear with respect to the inputs and having a degree of nonlinearity equal to 2. Thus, the output yαk of each INS Sα is demonstrated by the following expression:(15)yαk=−∑r=12aα,rkyαk−r+∑r=12bα,rkuαk−dα−r+∑r=12cα,reαk−r,+∑β=1,β≠α2∑r=12bαβ,rkuβk−dαβ−r+∑β=1,β≠α2∑r=12aαβ,rkyβk−tαβ−r,+∑r1=12∑r2=12gαα,r1r2kuαk−r1uαk−r2+∑β=1,β≠α2∑r1=12∑r2=12gαα,r1r2kuαk−r1uβk−r2,with α,β=1,2;β≠α.


## 3.2. Nonlinearity with respect to the Outputs

This section is dedicated to the description of INSs with nonlinearity in their outputs. This type of dynamical system may be characterized by input-output MMs that are nonlinear in terms of the outputs, mono-variable, deterministic, or stochastic and include time-varying parameters.


### 3.2.1. Deterministic Input-Output MMs

Let us consider an INS operating in a deterministic environment, mono-variable and having a nonlinearity with respect to the outputs. The general structure of the considered system can be described by the following INDARMA mathematical model [18]:(16)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβk,+ffαβyyαk−1,yαk−2,…,yαk−nα,yβk−1,yβk−2,…,yβk−nα,−ffααyyαk−1,yαk−2,…,yαk−nα,where the terms ffααy. and ffαβy. are nonlinear functions of nonlinearity degree p, which depend on the output sequences of the IS Sα and the other ISs Sβ, respectively. These functions can be defined by the following expressions:(17)ffααy.=∑r1=1nfαα,r1∑r2=1nfαα,r1r2fαα,r1r2kyαk−r1yαk−r2,+∑r1=1nfαα,r1∑r2=1nfαα,r1r2∑r3=1nfαα,r1r2r3fαα,r1r2r3kyαk−r1yαk−r2yαk−r3,+…+∑r1=1nfαα,r1∑r2=1nfαα,r1r2⋯∑rp=1nfαα,r1r2...rpfαα,r1...rpkyαk−r1…yαk−rp,and(18)ffαβy.=∑β=1,β≠αN∑r1=1nfαβ,r1∑r2=1nfαβ,r1r2fαβ,r1r2kyαk−r1yβk−r2,+∑β=1,β≠αN∑r1=1nfαβ,r1∑r2=1nfαβ,r1r2∑r3=1nfαβ,r1r2r3fαβ,r1r2r3kyαk−r1yαk−r2yβk−r3,+…+∑β=1,β≠αN∑r1=1nfαβ,r1∑r2=1nfαβ,r1r2⋯∑rp=1nfαβ,r1r2...rpfαβ,r1...rpkyαk−r1…yβk−rp.


### 3.2.2. Stochastic Input-Output MMs

We consider an INSSα, 1≤α≤N, which is coupled to other INSs Sβ, β=1,…,N;β≠α, exhibiting nonlinearity in outputs and working in a stochastic environment. We suppose that the noise operating on the investigated system is made up of a sequence of independent random variables with a zero mean and a finite variance, σα2. The general structure of this MM is given by the following expression [18]:(19)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβq−1,k,+ffαβyyαk−1,yαk−2,…,yαk−nα,yβk−1,yβk−2,…,yβk−nα,−ffααyyαk−1,yαk−2,…,yαk−nα+fceeαk−1,eαk−2,…,eαk−nα,+fλyeyαk−1,yαk−2,…,yαk−nα,eαk−1,eαk−2,…,eαk−nα+Cαq−1eαk,where Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials, defined by (8), (9), (10), and (9); Cαq−1 is a polynomial with constant parameters, given by (14); and the nonlinear functions fce., ffααy., and ffαβy. are defined by (16), (19), and (20), respectively.The termfλye. denotes a nonlinear function of degree p, which depends on the output sequences of the IS Sα and the noise eαk. This function can be defined as follows:(20)fλye.=∑r1=1nλαα,r1∑r2=1nλαα,r1r2λαα,r1r2yαk−r1eαk−r2+……+,∑r1=1nλαα,r1∑r2=1nλαα,r1r2…∑rp=1nλαα,r1r2...rpλαα,r1...rpyαk−r1…eαk−rp.For example, the following model corresponds to an IOMMINARMAX of the second order with a nonlinearity degree equal to 2, making it possible to describe the dynamic behavior of a large-scale nonlinear process composed of two interconnected nonlinear subsystemsS1 and S2. The output yαk of each interconnected nonlinear subsystem Sα is described as(21)yαk=−∑r=12aα,rkyαk−r−∑r1=12∑r2=12fαα,r1r2kyαk−r1yαk−r2,+∑r=12bα,rkuαk−dα−r+∑β=1,β≠α2∑r=12bαβ,rkuβk−dαβ−r,+∑β=1,β≠α2∑r=12aαβ,rkyβk−tαβ−r+∑β=1,β≠α2∑r1=12∑r2=12fαβ,r1r2kyαk−r1yβk−r2,+∑r=12cα,reαk−r,with α,β=1,2;β≠α.


## 3.2.1. Deterministic Input-Output MMs

Let us consider an INS operating in a deterministic environment, mono-variable and having a nonlinearity with respect to the outputs. The general structure of the considered system can be described by the following INDARMA mathematical model [18]:(16)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβk,+ffαβyyαk−1,yαk−2,…,yαk−nα,yβk−1,yβk−2,…,yβk−nα,−ffααyyαk−1,yαk−2,…,yαk−nα,where the terms ffααy. and ffαβy. are nonlinear functions of nonlinearity degree p, which depend on the output sequences of the IS Sα and the other ISs Sβ, respectively. These functions can be defined by the following expressions:(17)ffααy.=∑r1=1nfαα,r1∑r2=1nfαα,r1r2fαα,r1r2kyαk−r1yαk−r2,+∑r1=1nfαα,r1∑r2=1nfαα,r1r2∑r3=1nfαα,r1r2r3fαα,r1r2r3kyαk−r1yαk−r2yαk−r3,+…+∑r1=1nfαα,r1∑r2=1nfαα,r1r2⋯∑rp=1nfαα,r1r2...rpfαα,r1...rpkyαk−r1…yαk−rp,and(18)ffαβy.=∑β=1,β≠αN∑r1=1nfαβ,r1∑r2=1nfαβ,r1r2fαβ,r1r2kyαk−r1yβk−r2,+∑β=1,β≠αN∑r1=1nfαβ,r1∑r2=1nfαβ,r1r2∑r3=1nfαβ,r1r2r3fαβ,r1r2r3kyαk−r1yαk−r2yβk−r3,+…+∑β=1,β≠αN∑r1=1nfαβ,r1∑r2=1nfαβ,r1r2⋯∑rp=1nfαβ,r1r2...rpfαβ,r1...rpkyαk−r1…yβk−rp.


## 3.2.2. Stochastic Input-Output MMs

We consider an INSSα, 1≤α≤N, which is coupled to other INSs Sβ, β=1,…,N;β≠α, exhibiting nonlinearity in outputs and working in a stochastic environment. We suppose that the noise operating on the investigated system is made up of a sequence of independent random variables with a zero mean and a finite variance, σα2. The general structure of this MM is given by the following expression [18]:(19)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβq−1,k,+ffαβyyαk−1,yαk−2,…,yαk−nα,yβk−1,yβk−2,…,yβk−nα,−ffααyyαk−1,yαk−2,…,yαk−nα+fceeαk−1,eαk−2,…,eαk−nα,+fλyeyαk−1,yαk−2,…,yαk−nα,eαk−1,eαk−2,…,eαk−nα+Cαq−1eαk,where Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials, defined by (8), (9), (10), and (9); Cαq−1 is a polynomial with constant parameters, given by (14); and the nonlinear functions fce., ffααy., and ffαβy. are defined by (16), (19), and (20), respectively.The termfλye. denotes a nonlinear function of degree p, which depends on the output sequences of the IS Sα and the noise eαk. This function can be defined as follows:(20)fλye.=∑r1=1nλαα,r1∑r2=1nλαα,r1r2λαα,r1r2yαk−r1eαk−r2+……+,∑r1=1nλαα,r1∑r2=1nλαα,r1r2…∑rp=1nλαα,r1r2...rpλαα,r1...rpyαk−r1…eαk−rp.For example, the following model corresponds to an IOMMINARMAX of the second order with a nonlinearity degree equal to 2, making it possible to describe the dynamic behavior of a large-scale nonlinear process composed of two interconnected nonlinear subsystemsS1 and S2. The output yαk of each interconnected nonlinear subsystem Sα is described as(21)yαk=−∑r=12aα,rkyαk−r−∑r1=12∑r2=12fαα,r1r2kyαk−r1yαk−r2,+∑r=12bα,rkuαk−dα−r+∑β=1,β≠α2∑r=12bαβ,rkuβk−dαβ−r,+∑β=1,β≠α2∑r=12aαβ,rkyβk−tαβ−r+∑β=1,β≠α2∑r1=12∑r2=12fαβ,r1r2kyαk−r1yβk−r2,+∑r=12cα,reαk−r,with α,β=1,2;β≠α.


## 3.3. Nonlinearity with respect to the Observations

In this part, we will create input-output MMs of representation that allow us to describe ISs that are nonlinear with respect to the observations, are mono-variable, and have unknown time-varying parameters [18].


### 3.3.1. Deterministic Input-Output MMs

We consider a dynamical system, which is composed ofN ISs, working in a predictable environment and being nonlinear with regard to the observations. The input-output MMINDARMA, making it possible to describe the considered system, is given as follows [18]:(22)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβk,+fguuαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,+ffαβyyαk−1,yαk−2,…,yαk−nα,yβk−1,yβk−2,…,yβk−nα,−ffααyyαk−1,yαk−2,…,yαk−nα+fhℓuyyαk−1,yαk−2,…,yαk−nα,uαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,yβk−1,yβk−2,…,yβk−nα,where Aαq−1,k, Bαq−1,k,Aαβq−1,k, and Bαβq−1,k are polynomials defined by (8), (9), (10), and (9), respectively; fgu., ffααy., and ffαβy. are nonlinear functions given by (12), (19), and (20), respectively, and fλhluy. is described by the following nonlinear function:(23)fhℓuy.=∑β=1N∑r1=1nhαβ,r1∑r2=1nhαβ,r1r2hαβ,r1r2kuαk−r1yβk−r2,+∑β=1N∑r1=1nhαβ,r1∑r2=1nhαβ,r1r2∑r3=1nhαβ,r1r2r3hαβ,r1r2r3kuαk−r1uαk−r2yβk−r3+…+,+∑β=1N∑r1=1nhαβ,r1∑r2=1nhαβ,r1r2⋯∑rp=1nhαβ,r1r2...rphαβ,r1...rpkuαk−r1…yβk−rp,+∑β=1,β≠αN∑r1=1nℓαβ,r1∑r2=1nℓαβ,r1r2ℓαβ,r1r2kyαk−r1uβk−r2,+∑β=1,β≠αN∑r1=1nℓαβ,r1∑r2=1nℓαβ,r1r2∑r3=1nℓαβ,r1r2r3ℓαβ,r1r2r3kyαk−r1yαk−r2uβk−r3+…+,+∑β=1,β≠αN∑r1=1nℓαβ,r1∑r2=1nℓαβ,r1r2⋯∑rp=1nℓαβ,r1r2...rpℓαβ,r1...rpkyαk−r1…uβk−rp.


### 3.3.2. Stochastic Input-Output MMs

Let us consider an INSSα, 1≤α≤N, which is coupled to the other IS Sβ, β=1,…,N;β≠α. This system is nonlinear with respect to the observations and can be described by the class of IOMMs. The considered structure of the INARMAX MM is given by the following expression [18]:(24)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk,+∑β=1,β≠αNq−tαβAαβq−1,kyβk+Cαq−1eαk,+fguuαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,+ffαβyyαk−1,yαk−2,…,yαk−nα,yβk−1,yβk−2,…,yβk−nα,−ffααyyαk−1,yαk−2,…,yαk−nα+fhℓuyyβk−1,yβk−2,…,yβk−nα,uαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,yβk−1,yβk−2,…,yβk−nα,+fγueuαk−1,uαk−2,…,uαk−nα,eαk−1,eαk−2,…,eαk−nα,+fλyeyαk−1,yαk−2,…,yαk−nα,eαk−1,eik−2,…,eαk−nα+fceeαk−1,eαk−2,…,eαk−nα,where fgu., fβue., fce., ffααy., ffαβy., , and fhℓuy. are nonlinear functions defined by (12), (15), (16), (19), (20), (22), and (23), respectively.For the reason of simplicity, we consider the following dynamical system, which consists ofN INSs, running in a deterministic environment and defined by an IOMM INDARMA with a nonlinearity degree equal to 2, such as(25)Aα1q−1yαk+Aα2q1−1,q2−1yα2k=Bα1q−1uαk+Bα2q1−1,q2−1uα2k+Aαβ1q−1yβk,+Bαβ1q−1uβk+Aαβ2q1−1,q2−1yαkyβk,+Bαβ2q1−1,q2−1uαkuβk+Fαβq1−1,q2−1uαkyβk,+Hαβq1−1,q2−1yαkuβk,where(26)Bα1q−1=∑r=1nBα1bα1,rq−r,Aα1q−1=1+∑r=1nAα1aα1,rq−r,Bα2q1−1,q2−1uα2k=∑r=1nBα21∑s=1nBα22bα2,rsuαk−ruαk−s,Aα2q1−1,q2−1yα2k=∑r=1nAα21∑s=1nAα22aα2,rsyαk−ryαk−s,Bαβ1q−1uβk=∑β=1,β≠αN∑r=1nBαβ1bαβ1,ruβk−r,Aαβ1q−1yβk=∑β=1,β≠αN∑r=1nAαβ1aαβ1,ryβk−r,Bαβ2q1−1,q2−1uαkuβk=∑β=1,β≠αN∑r=1nBαβ21∑s=1nBαβ22bαβ2,rsuαk−ruβk−s,Aαβ2q1−1,q2−1yαkyβk=∑β=1,β≠αN∑r=1nAαβ21∑s=1nAαβ22aαβ2,rsyαk−ryβk−s,Fαβq1−1,q2−1uαkyβk=∑β=1N∑r=1nFαβ1∑s=1nFαβ2fαβ,rsuαk−ryβk−s,and(27)Hαβq1−1,q2−1yαkuβk=∑β=1,β≠αN∑r=1nHαβ1∑s=1nHαβ2hαβ,rsyαk−ruβk−s,with α,β=1,…,N;β≠α.From the developed MM, we can distinguish special cases of input-output MM, based on various representations, such as the following:(1)
Serial model of Volterra:(28)Aα1q−1=0,Aα2q1−1,q2−1=0,Aαβ1q−1=0,Aαβ2q1−1,q2−1=0,Fαβq1−1,q2−1=0,Hαβq1−1,q2−1=0.(2)
Parametric model of Volterra:(29)Aα2q1−1,q2−1=0,Aαβ2q1−1,q2−1=0,Fαβq1−1,q2−1=0,Hαβq1−1,q2−1=0.(3)
Bilinear model:(30)Aα2q1−1,q2−1=0,Bα2q1−1,q2−1=0.(4)
Linear model with respect to the input signal:(31)Bα2q1−1,q2−1=0,Bαβ2q1−1,q2−1=0,Fαβq1−1,q2−1=0,Hαβq1−1,q2−1=0.(5)
Linear model with respect to the output signal:(32)Aα2q1−1,q2−1=0,Aαβ2q1−1,q2−1=0,Fαβq1−1,q2−1=0,Hαβq1−1,q2−1=0.For example, we consider a large-scale nonlinear process, which is constituted of two INSS1 and S2. Each interconnected subsystem can be defined by the INDARMA mathematical model of the second order with time-varying parameters and having a nonlinearity degree equal to 2.Figure1 shows the interaction structure diagram of the considered nonlinear process:Figure 1
Interaction structure diagram of the considered process.Thus, the outputyαk of the INS Sα can be expressed as(33)yαk=−∑r=12aα,rkyαk−r−∑r1=12∑r2=12fαα,r1r2kyαk−r1yαk−r2,+∑r=12bα,rkuαk−dα−r+∑β=1,β≠α2∑r=12bαβ,rkuβk−dαβ−r,+∑β=1,β≠α2∑r=12aαβ,rkyβk−tαβ−r+∑r1=12∑r2=12gαα,r1r2kuαk−r1uαk−r2,+∑β=1,β≠α2∑r1=12∑r2=12gαβ,r1r2kuαk−r1uβk−r2+∑β=1,β≠α2∑r1=12∑r2=12fαβ,r1r2kyαk−r1yβk−r2,+∑β=12∑r1=12∑r2=12hαβ,r1r2kuαk−r1yβk−r2+∑β=1,β≠α2∑r1=12∑r2=12ℓαβ,r1r2kyαk−r1uβk−r2,with α,β=1,2;β≠α.We notice that these different representations of developed MMs become more and more complex by increasing the nonlinearity degreep and/or the order of the IS.


## 3.3.1. Deterministic Input-Output MMs

We consider a dynamical system, which is composed ofN ISs, working in a predictable environment and being nonlinear with regard to the observations. The input-output MMINDARMA, making it possible to describe the considered system, is given as follows [18]:(22)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk+∑β=1,β≠αNq−tαβAαβq−1,kyβk,+fguuαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,+ffαβyyαk−1,yαk−2,…,yαk−nα,yβk−1,yβk−2,…,yβk−nα,−ffααyyαk−1,yαk−2,…,yαk−nα+fhℓuyyαk−1,yαk−2,…,yαk−nα,uαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,yβk−1,yβk−2,…,yβk−nα,where Aαq−1,k, Bαq−1,k,Aαβq−1,k, and Bαβq−1,k are polynomials defined by (8), (9), (10), and (9), respectively; fgu., ffααy., and ffαβy. are nonlinear functions given by (12), (19), and (20), respectively, and fλhluy. is described by the following nonlinear function:(23)fhℓuy.=∑β=1N∑r1=1nhαβ,r1∑r2=1nhαβ,r1r2hαβ,r1r2kuαk−r1yβk−r2,+∑β=1N∑r1=1nhαβ,r1∑r2=1nhαβ,r1r2∑r3=1nhαβ,r1r2r3hαβ,r1r2r3kuαk−r1uαk−r2yβk−r3+…+,+∑β=1N∑r1=1nhαβ,r1∑r2=1nhαβ,r1r2⋯∑rp=1nhαβ,r1r2...rphαβ,r1...rpkuαk−r1…yβk−rp,+∑β=1,β≠αN∑r1=1nℓαβ,r1∑r2=1nℓαβ,r1r2ℓαβ,r1r2kyαk−r1uβk−r2,+∑β=1,β≠αN∑r1=1nℓαβ,r1∑r2=1nℓαβ,r1r2∑r3=1nℓαβ,r1r2r3ℓαβ,r1r2r3kyαk−r1yαk−r2uβk−r3+…+,+∑β=1,β≠αN∑r1=1nℓαβ,r1∑r2=1nℓαβ,r1r2⋯∑rp=1nℓαβ,r1r2...rpℓαβ,r1...rpkyαk−r1…uβk−rp.


## 3.3.2. Stochastic Input-Output MMs

Let us consider an INSSα, 1≤α≤N, which is coupled to the other IS Sβ, β=1,…,N;β≠α. This system is nonlinear with respect to the observations and can be described by the class of IOMMs. The considered structure of the INARMAX MM is given by the following expression [18]:(24)Aαq−1,kyαk=q−dαBαq−1,kuαk+∑β=1,β≠αNq−dαβBαβq−1,kuβk,+∑β=1,β≠αNq−tαβAαβq−1,kyβk+Cαq−1eαk,+fguuαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,+ffαβyyαk−1,yαk−2,…,yαk−nα,yβk−1,yβk−2,…,yβk−nα,−ffααyyαk−1,yαk−2,…,yαk−nα+fhℓuyyβk−1,yβk−2,…,yβk−nα,uαk−1,uαk−2,…,uαk−nα,uβk−1,uβk−2,…,uβk−nα,yβk−1,yβk−2,…,yβk−nα,+fγueuαk−1,uαk−2,…,uαk−nα,eαk−1,eαk−2,…,eαk−nα,+fλyeyαk−1,yαk−2,…,yαk−nα,eαk−1,eik−2,…,eαk−nα+fceeαk−1,eαk−2,…,eαk−nα,where fgu., fβue., fce., ffααy., ffαβy., , and fhℓuy. are nonlinear functions defined by (12), (15), (16), (19), (20), (22), and (23), respectively.For the reason of simplicity, we consider the following dynamical system, which consists ofN INSs, running in a deterministic environment and defined by an IOMM INDARMA with a nonlinearity degree equal to 2, such as(25)Aα1q−1yαk+Aα2q1−1,q2−1yα2k=Bα1q−1uαk+Bα2q1−1,q2−1uα2k+Aαβ1q−1yβk,+Bαβ1q−1uβk+Aαβ2q1−1,q2−1yαkyβk,+Bαβ2q1−1,q2−1uαkuβk+Fαβq1−1,q2−1uαkyβk,+Hαβq1−1,q2−1yαkuβk,where(26)Bα1q−1=∑r=1nBα1bα1,rq−r,Aα1q−1=1+∑r=1nAα1aα1,rq−r,Bα2q1−1,q2−1uα2k=∑r=1nBα21∑s=1nBα22bα2,rsuαk−ruαk−s,Aα2q1−1,q2−1yα2k=∑r=1nAα21∑s=1nAα22aα2,rsyαk−ryαk−s,Bαβ1q−1uβk=∑β=1,β≠αN∑r=1nBαβ1bαβ1,ruβk−r,Aαβ1q−1yβk=∑β=1,β≠αN∑r=1nAαβ1aαβ1,ryβk−r,Bαβ2q1−1,q2−1uαkuβk=∑β=1,β≠αN∑r=1nBαβ21∑s=1nBαβ22bαβ2,rsuαk−ruβk−s,Aαβ2q1−1,q2−1yαkyβk=∑β=1,β≠αN∑r=1nAαβ21∑s=1nAαβ22aαβ2,rsyαk−ryβk−s,Fαβq1−1,q2−1uαkyβk=∑β=1N∑r=1nFαβ1∑s=1nFαβ2fαβ,rsuαk−ryβk−s,and(27)Hαβq1−1,q2−1yαkuβk=∑β=1,β≠αN∑r=1nHαβ1∑s=1nHαβ2hαβ,rsyαk−ruβk−s,with α,β=1,…,N;β≠α.From the developed MM, we can distinguish special cases of input-output MM, based on various representations, such as the following:(1)
Serial model of Volterra:(28)Aα1q−1=0,Aα2q1−1,q2−1=0,Aαβ1q−1=0,Aαβ2q1−1,q2−1=0,Fαβq1−1,q2−1=0,Hαβq1−1,q2−1=0.(2)
Parametric model of Volterra:(29)Aα2q1−1,q2−1=0,Aαβ2q1−1,q2−1=0,Fαβq1−1,q2−1=0,Hαβq1−1,q2−1=0.(3)
Bilinear model:(30)Aα2q1−1,q2−1=0,Bα2q1−1,q2−1=0.(4)
Linear model with respect to the input signal:(31)Bα2q1−1,q2−1=0,Bαβ2q1−1,q2−1=0,Fαβq1−1,q2−1=0,Hαβq1−1,q2−1=0.(5)
Linear model with respect to the output signal:(32)Aα2q1−1,q2−1=0,Aαβ2q1−1,q2−1=0,Fαβq1−1,q2−1=0,Hαβq1−1,q2−1=0.For example, we consider a large-scale nonlinear process, which is constituted of two INSS1 and S2. Each interconnected subsystem can be defined by the INDARMA mathematical model of the second order with time-varying parameters and having a nonlinearity degree equal to 2.Figure1 shows the interaction structure diagram of the considered nonlinear process:Figure 1
Interaction structure diagram of the considered process.Thus, the outputyαk of the INS Sα can be expressed as(33)yαk=−∑r=12aα,rkyαk−r−∑r1=12∑r2=12fαα,r1r2kyαk−r1yαk−r2,+∑r=12bα,rkuαk−dα−r+∑β=1,β≠α2∑r=12bαβ,rkuβk−dαβ−r,+∑β=1,β≠α2∑r=12aαβ,rkyβk−tαβ−r+∑r1=12∑r2=12gαα,r1r2kuαk−r1uαk−r2,+∑β=1,β≠α2∑r1=12∑r2=12gαβ,r1r2kuαk−r1uβk−r2+∑β=1,β≠α2∑r1=12∑r2=12fαβ,r1r2kyαk−r1yβk−r2,+∑β=12∑r1=12∑r2=12hαβ,r1r2kuαk−r1yβk−r2+∑β=1,β≠α2∑r1=12∑r2=12ℓαβ,r1r2kyαk−r1uβk−r2,with α,β=1,2;β≠α.We notice that these different representations of developed MMs become more and more complex by increasing the nonlinearity degreep and/or the order of the IS.


## 4. MMs in Connected Blocks

The linked block MMs explain the dynamic behavior of a nonlinear system composed of a linear dynamic element and a nonlinear static element. This form of MM is widely used in a variety of industrial applications for the description of nonlinear systems with tiny dimensions [17, 19–22]. In fact, the authors of [19] provided an approach for identifying nonlinear dynamic systems using extended Hammerstein and Wiener models. Following that, the author of [20] refined a strategy for identifying the Hammerstein model. Similarly, the authors of [21] produced promising findings for Hammerstein system identification. Around ten years later, the authors of [17] proposed an adaptive nonlinear system identification approach to the Volterra and Wiener Models. The author of [22] proposes a significant improvement in the identification of Hammerstein–Wiener models. The use of MMs to describe this type of dynamic system in linked blocks simplifies the construction of parametric estimates and control strategies. In linked blocks, there are two types of MMs, Hammerstein MM and Wiener MM.Two structures of linked block MMs are developed in this part for the description of mono-variable INSs. This type of system can be represented by discrete MMs of Hammerstein or Wiener, which can be deterministic or stochastic, and has unknown time-varying parameters.


### 4.1. Interconnected Hammerstein MMs

A Hammerstein MMs description of an interconnected nonlinear dynamic system relates to the interconnection of many MM structures, each of which consists of a static nonlinear portion followed by a dynamic linear part [23]. This family of models includes two types of MMs: a deterministic Hammerstein MM, in which an IDARMA input-output model defines the dynamic linear component of the investigated system, and a stochastic Hammerstein MM, in which an IARMAX input-output model describes the dynamic linear part.


#### 4.1.1. Deterministic Interconnected Hammerstein MMs

The structure of an INSSα, 1≤α≤N, operating in a deterministic environment and that can be defined by Hammerstein MM, is represented in Figure 2.Figure 2
Deterministic structure of the interconnected Hammerstein MM.Figure2 depicts the dynamic linear component of Hammerstein MM, which is characterized by the following formula [23]:(34)Aαq−1,kyαk=Bαq−1,khαuαk+∑β=1,β≠αNBαβq−1,khβuβk+∑β=1,β≠αNAαβq−1,khβyβk,where yαk and hαuαk denote, respectively, the output and the input of the dynamic linear block of the IS Sα; yβk, uβk, and uαk are the inputs of the static nonlinear blocks; hβuβk and hβyβk represent the inputs of the dynamic linear blocks of the other ISs Sβ, β=1,…,N;β≠α; and Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials, defined by (8), (9), (10), and (9). We must note that the INS Sα, 1≤α≤N, is linked to other INS Sβ, β=1,…∞,N;β≠α, by the polynomials Aαβq−1,k and Bαβq−1,k.The following equations represent the static nonlinear sections of the analyzed Hammerstein MM:(35)hαuαk=fhαuαuαk,(36)hβuβk=fhβuβuβk,(37)hβyβk=fhβyβyβk,where fhαuα., fhβuβ., and fhβyβ. represent nonlinear functions.Equations (40), (41), and (42) can be approximated by the following functions, such as(38)hαuαk=∑r1=1p1ηα,r1uαr1k+Δhαuαuαk,(39)hβuβk=∑r2=1p2λβ,r2uβr2k+Δhβuβuβk,(40)hβyβk=∑r3=1p3γβ,r3yβr3k+Δhβyβyβk,where Δhαuαuαk, Δhβuβuβk, and Δhβyβyβk represent the approximation errors of nonlinear functions fhαuα., fhβuβ., and fhβyβ., respectively, which can be assimilated to a disturbance acting on the output of the INS Sα, ηα,r1, λα,r2, and γβ,r3, rt=1,…,pt, t=1,2,3 are unknown parameters, and pt denotes the degree of nonlinearity. Note that the variances values of these approximation errors depend on the chosen of the nonlinearity degrees values pt for the nonlinear functions.From (40), which are related to the linear and the nonlinear parts of the Hammerstein model, we can describe the considered system by the following expression:(41)yαk=−∑r=1nAαaα,rkyαk−r+∑s=1nBα∑r1=1p1bα,skηα,r1uαr1k−s+∑s=1nBαΔhαuαuαk−s,+∑β=1,β≠αN∑s=1nBαβ∑r2=1p2bαβ,skλβ,r2uβr2k−s+∑s=1nBαβΔhβuβuβk−s,+∑β=1,β≠αN∑s=1nAαβ∑r3=1p3aαβ,skγβ,r3yβr3k−s+∑s=1nAijΔhβyβyβk−s.


#### 4.1.2. Stochastic Interconnected Hammerstein MMs

This second form of Hammerstein MM is distinguished by IARMAX input-output MM, which describes the dynamic linear component of the system under consideration [23]. We suppose that there is a disturbance operating on the output of the considered system and that it may be characterized by a moving average MM.As a result, Figure3 depicts the Hammerstein MM’s structure:Figure 3
Stochastic structure of the interconnected Hammerstein MM.The following formula describes the dynamic linear component of the examined Hammerstein MM [23]:(42)Aαq−1,kyαk=Bαq−1,khαuαk+∑β=1,β≠αNBαβq−1,khβuβk+∑β=1,β≠αNAαβq−1,khβyβk+Cαq−1eαk,where hαuαk, hβuβk, and hβyβk represent the outputs of the static nonlinear blocks of the considered MM, which are defined by equations (43), (44), and (45); eαk designates the set of disturbances acting on the output of the IS, which consists of an independent random variables sequence with zero mean and constant variance σα2; and Cαq−1 is a polynomial with constant parameters, given by (12).Taking into account the polynomials (43), (44), and (45), the output yαk, which is defined by (47), can be written as [26](43)yαk=−∑r=1nAαaα,rkyαk−r+∑s=1nBα∑r1=1p1bα,skηα,r1uαr1k−s+∑s=1nBαΔhαuαuαk−s,+∑β=1,β≠αN∑s=1nBαβ∑r2=1p2bαβ,skλβ,r2uβr2k−s+∑s=1nBαβΔhβuβuβk−s,+∑β=1,β≠αN∑s=1nAαβ∑r3=1p3aαβ,skγβ,r3yβr3k−s+∑s=1nAαβΔhβyβyβk−s+∑r=1nCαcα,reαk−r+eαk.Other forms of Hammerstein MMs may be distinguished in order to represent the dynamics of INSs, depending on different configurations of static nonlinear elements.For reason of simplicity, we assume that the polynomialsBαq−1,k, Aαq−1,k, Aαβq−1,k, Bαβq−1,k, and Cαq−1 of Hammerstein MMs, which are given by (41) and (43), have the same order nα (nα=nAα=nBα=nBαβ=nAαβ). We make also a choice of the nonlinearity degrees pt, t=1,2,3, in such a way that the approximation errors Δhαuαuαk, Δhβuβuβk, and Δhβyβyβk become negligible (Δhαuαuαk=Δhβuβuβk=Δhβyβyβk=0). These assumptions allow us to further simplify the formulation of the parametric estimation and the control problems for the INSs, which are described by the two types of the developed Hammerstein MMs, which are given by (46) and (48).


### 4.2. Interconnected Wiener MMs

The creation of Wiener MMs for characterizing the INSs is discussed in this section. This model’s class relates to the interconnection of numerous MM structures, each of which has a dynamic linear and a static nonlinear component. In this class of MMs, we may distinguish between two forms of Wiener MMs: deterministic Wiener MMs and stochastic Wiener MMs [24].


#### 4.2.1. Deterministic Interconnected Wiener MM

This section is intended for Wiener MMs [24] to describe INSs working in a deterministic environment. As a result, we investigate a nonlinear time-varying system made up of deterministic ISs. The deterministic Wiener MM may be used to explain each IS, and its structure is depicted in Figure 4.Figure 4
Deterministic structure of the interconnected Wiener MM.The previous structure, which is illustrated by Figure4, can be expressed by the following MM [24]:(44)Aαq−1,kwαk=Bαq−1,kuαk+∑β=1,β≠αNBαβq−1,kuβk+∑β=1,β≠αNAαβq−1,kyβk,where wαk and uαk are, respectively, the output and input of the dynamic linear block of the IS Sα, yβk, and uβk denote, respectively, the outputs and inputs, which arise from the other ISs Sβ, β=1,…,N;β≠α, and Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials, as given by (8), (9), (10), and (9), respectively.The following equation describes the static nonlinear component of the considered MM:(45)yαk=fwαwαk,where fwα. represents the nonlinear function.The following polynomial can be used to approximate (45):(46)yαk=∑r=1pηα,rwαrk+Δyαwαk,where p represents the nonlinearity degree of the nonlinear function, which can be chosen in an appropriate way; ηα,r, r=1,…,p, are unknown parameters; and Δyαwαk signifies the nonlinear function’s approximation error. This approximation error, which is dependent on the nonlinearity degree p chosen, might be compared to noise operating on the output of the IS in question. For an appropriate choice of the nonlinearity degree value p, this approximation error Δyαwαk can be neglected.The output of the ISyαk can be written as follows, taking into consideration the dynamic linear component of the investigated Wiener MM, as stated by (44) [24]:(47)yαk=∑r=1pηα,r−∑s=1nAαaα,skwαk−s+∑h=1nBαbα,hkuαk−h,+∑β=1,β≠αNBαβq−1,kuβk+∑β=1,β≠αNAαβq−1,kyβkr+Δyαwαk.


#### 4.2.2. Stochastic Interconnected Wiener MM

In this part, we suppose that the output of the considered system is subjected to noise, which is composed of an independent random variable sequence. As a result, the Wiener MM’s dynamic linear portion is of type IARMAX.The structure of this model is depicted in Figure5.(48)Aαq−1,kwαk=Bαq−1,kuαk+∑β=1,β≠αNBαβq−1,kuβk+∑β=1,β≠αNAαβq−1,kyβk+Cαq−1eαk,where eαk designates all disturbances acting on the IS Sα, Cαq−1 is a polynomial defined by (12), and yαk corresponds to the output of the static nonlinear part of the stochastic Wiener MM, which is given by (46).Figure 5
Stochastic structure of the interconnected Wiener MM. The following formula describes the dynamic linear portion of Wiener MM [24].The output of the systemyαk may be represented in the following manner based on the dynamic linear component of the investigated Wiener MM, as given by (48).(49)yαk=∑r=1pηα,r−∑s=1nAαaα,skwαk−s+∑h=1nBαbα,hkuαk−h+∑β=1,β≠αNBαβq−1,kuβk,+∑β=1,β≠αNAαβq−1,kyβk+∑t=1nCαcα,teαk−t+eαkr+Δyαwαk.In the description of the INSs, we may use a variety of Wiener MMs, which are based on various forms of the static nonlinear portion.We assume that the polynomialsAαq−1,k, Bαq−1,k, Bαβq−1,k, Aαβq−1,k, and Cαq−1 intervening in the two types of Wiener MMs, given by (47) and (49), have the same order nα (nα=nAα=nBα=nAαβ=nBαβ). We also assume that the approximation error Δyαwαk is negligible, to reduce the formulation of the parametric estimation issue for large-scale dynamical systems defined by the Wiener MMs created.


## 4.1. Interconnected Hammerstein MMs

A Hammerstein MMs description of an interconnected nonlinear dynamic system relates to the interconnection of many MM structures, each of which consists of a static nonlinear portion followed by a dynamic linear part [23]. This family of models includes two types of MMs: a deterministic Hammerstein MM, in which an IDARMA input-output model defines the dynamic linear component of the investigated system, and a stochastic Hammerstein MM, in which an IARMAX input-output model describes the dynamic linear part.


### 4.1.1. Deterministic Interconnected Hammerstein MMs

The structure of an INSSα, 1≤α≤N, operating in a deterministic environment and that can be defined by Hammerstein MM, is represented in Figure 2.Figure 2
Deterministic structure of the interconnected Hammerstein MM.Figure2 depicts the dynamic linear component of Hammerstein MM, which is characterized by the following formula [23]:(34)Aαq−1,kyαk=Bαq−1,khαuαk+∑β=1,β≠αNBαβq−1,khβuβk+∑β=1,β≠αNAαβq−1,khβyβk,where yαk and hαuαk denote, respectively, the output and the input of the dynamic linear block of the IS Sα; yβk, uβk, and uαk are the inputs of the static nonlinear blocks; hβuβk and hβyβk represent the inputs of the dynamic linear blocks of the other ISs Sβ, β=1,…,N;β≠α; and Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials, defined by (8), (9), (10), and (9). We must note that the INS Sα, 1≤α≤N, is linked to other INS Sβ, β=1,…∞,N;β≠α, by the polynomials Aαβq−1,k and Bαβq−1,k.The following equations represent the static nonlinear sections of the analyzed Hammerstein MM:(35)hαuαk=fhαuαuαk,(36)hβuβk=fhβuβuβk,(37)hβyβk=fhβyβyβk,where fhαuα., fhβuβ., and fhβyβ. represent nonlinear functions.Equations (40), (41), and (42) can be approximated by the following functions, such as(38)hαuαk=∑r1=1p1ηα,r1uαr1k+Δhαuαuαk,(39)hβuβk=∑r2=1p2λβ,r2uβr2k+Δhβuβuβk,(40)hβyβk=∑r3=1p3γβ,r3yβr3k+Δhβyβyβk,where Δhαuαuαk, Δhβuβuβk, and Δhβyβyβk represent the approximation errors of nonlinear functions fhαuα., fhβuβ., and fhβyβ., respectively, which can be assimilated to a disturbance acting on the output of the INS Sα, ηα,r1, λα,r2, and γβ,r3, rt=1,…,pt, t=1,2,3 are unknown parameters, and pt denotes the degree of nonlinearity. Note that the variances values of these approximation errors depend on the chosen of the nonlinearity degrees values pt for the nonlinear functions.From (40), which are related to the linear and the nonlinear parts of the Hammerstein model, we can describe the considered system by the following expression:(41)yαk=−∑r=1nAαaα,rkyαk−r+∑s=1nBα∑r1=1p1bα,skηα,r1uαr1k−s+∑s=1nBαΔhαuαuαk−s,+∑β=1,β≠αN∑s=1nBαβ∑r2=1p2bαβ,skλβ,r2uβr2k−s+∑s=1nBαβΔhβuβuβk−s,+∑β=1,β≠αN∑s=1nAαβ∑r3=1p3aαβ,skγβ,r3yβr3k−s+∑s=1nAijΔhβyβyβk−s.


### 4.1.2. Stochastic Interconnected Hammerstein MMs

This second form of Hammerstein MM is distinguished by IARMAX input-output MM, which describes the dynamic linear component of the system under consideration [23]. We suppose that there is a disturbance operating on the output of the considered system and that it may be characterized by a moving average MM.As a result, Figure3 depicts the Hammerstein MM’s structure:Figure 3
Stochastic structure of the interconnected Hammerstein MM.The following formula describes the dynamic linear component of the examined Hammerstein MM [23]:(42)Aαq−1,kyαk=Bαq−1,khαuαk+∑β=1,β≠αNBαβq−1,khβuβk+∑β=1,β≠αNAαβq−1,khβyβk+Cαq−1eαk,where hαuαk, hβuβk, and hβyβk represent the outputs of the static nonlinear blocks of the considered MM, which are defined by equations (43), (44), and (45); eαk designates the set of disturbances acting on the output of the IS, which consists of an independent random variables sequence with zero mean and constant variance σα2; and Cαq−1 is a polynomial with constant parameters, given by (12).Taking into account the polynomials (43), (44), and (45), the output yαk, which is defined by (47), can be written as [26](43)yαk=−∑r=1nAαaα,rkyαk−r+∑s=1nBα∑r1=1p1bα,skηα,r1uαr1k−s+∑s=1nBαΔhαuαuαk−s,+∑β=1,β≠αN∑s=1nBαβ∑r2=1p2bαβ,skλβ,r2uβr2k−s+∑s=1nBαβΔhβuβuβk−s,+∑β=1,β≠αN∑s=1nAαβ∑r3=1p3aαβ,skγβ,r3yβr3k−s+∑s=1nAαβΔhβyβyβk−s+∑r=1nCαcα,reαk−r+eαk.Other forms of Hammerstein MMs may be distinguished in order to represent the dynamics of INSs, depending on different configurations of static nonlinear elements.For reason of simplicity, we assume that the polynomialsBαq−1,k, Aαq−1,k, Aαβq−1,k, Bαβq−1,k, and Cαq−1 of Hammerstein MMs, which are given by (41) and (43), have the same order nα (nα=nAα=nBα=nBαβ=nAαβ). We make also a choice of the nonlinearity degrees pt, t=1,2,3, in such a way that the approximation errors Δhαuαuαk, Δhβuβuβk, and Δhβyβyβk become negligible (Δhαuαuαk=Δhβuβuβk=Δhβyβyβk=0). These assumptions allow us to further simplify the formulation of the parametric estimation and the control problems for the INSs, which are described by the two types of the developed Hammerstein MMs, which are given by (46) and (48).


## 4.1.1. Deterministic Interconnected Hammerstein MMs

The structure of an INSSα, 1≤α≤N, operating in a deterministic environment and that can be defined by Hammerstein MM, is represented in Figure 2.Figure 2
Deterministic structure of the interconnected Hammerstein MM.Figure2 depicts the dynamic linear component of Hammerstein MM, which is characterized by the following formula [23]:(34)Aαq−1,kyαk=Bαq−1,khαuαk+∑β=1,β≠αNBαβq−1,khβuβk+∑β=1,β≠αNAαβq−1,khβyβk,where yαk and hαuαk denote, respectively, the output and the input of the dynamic linear block of the IS Sα; yβk, uβk, and uαk are the inputs of the static nonlinear blocks; hβuβk and hβyβk represent the inputs of the dynamic linear blocks of the other ISs Sβ, β=1,…,N;β≠α; and Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials, defined by (8), (9), (10), and (9). We must note that the INS Sα, 1≤α≤N, is linked to other INS Sβ, β=1,…∞,N;β≠α, by the polynomials Aαβq−1,k and Bαβq−1,k.The following equations represent the static nonlinear sections of the analyzed Hammerstein MM:(35)hαuαk=fhαuαuαk,(36)hβuβk=fhβuβuβk,(37)hβyβk=fhβyβyβk,where fhαuα., fhβuβ., and fhβyβ. represent nonlinear functions.Equations (40), (41), and (42) can be approximated by the following functions, such as(38)hαuαk=∑r1=1p1ηα,r1uαr1k+Δhαuαuαk,(39)hβuβk=∑r2=1p2λβ,r2uβr2k+Δhβuβuβk,(40)hβyβk=∑r3=1p3γβ,r3yβr3k+Δhβyβyβk,where Δhαuαuαk, Δhβuβuβk, and Δhβyβyβk represent the approximation errors of nonlinear functions fhαuα., fhβuβ., and fhβyβ., respectively, which can be assimilated to a disturbance acting on the output of the INS Sα, ηα,r1, λα,r2, and γβ,r3, rt=1,…,pt, t=1,2,3 are unknown parameters, and pt denotes the degree of nonlinearity. Note that the variances values of these approximation errors depend on the chosen of the nonlinearity degrees values pt for the nonlinear functions.From (40), which are related to the linear and the nonlinear parts of the Hammerstein model, we can describe the considered system by the following expression:(41)yαk=−∑r=1nAαaα,rkyαk−r+∑s=1nBα∑r1=1p1bα,skηα,r1uαr1k−s+∑s=1nBαΔhαuαuαk−s,+∑β=1,β≠αN∑s=1nBαβ∑r2=1p2bαβ,skλβ,r2uβr2k−s+∑s=1nBαβΔhβuβuβk−s,+∑β=1,β≠αN∑s=1nAαβ∑r3=1p3aαβ,skγβ,r3yβr3k−s+∑s=1nAijΔhβyβyβk−s.


## 4.1.2. Stochastic Interconnected Hammerstein MMs

This second form of Hammerstein MM is distinguished by IARMAX input-output MM, which describes the dynamic linear component of the system under consideration [23]. We suppose that there is a disturbance operating on the output of the considered system and that it may be characterized by a moving average MM.As a result, Figure3 depicts the Hammerstein MM’s structure:Figure 3
Stochastic structure of the interconnected Hammerstein MM.The following formula describes the dynamic linear component of the examined Hammerstein MM [23]:(42)Aαq−1,kyαk=Bαq−1,khαuαk+∑β=1,β≠αNBαβq−1,khβuβk+∑β=1,β≠αNAαβq−1,khβyβk+Cαq−1eαk,where hαuαk, hβuβk, and hβyβk represent the outputs of the static nonlinear blocks of the considered MM, which are defined by equations (43), (44), and (45); eαk designates the set of disturbances acting on the output of the IS, which consists of an independent random variables sequence with zero mean and constant variance σα2; and Cαq−1 is a polynomial with constant parameters, given by (12).Taking into account the polynomials (43), (44), and (45), the output yαk, which is defined by (47), can be written as [26](43)yαk=−∑r=1nAαaα,rkyαk−r+∑s=1nBα∑r1=1p1bα,skηα,r1uαr1k−s+∑s=1nBαΔhαuαuαk−s,+∑β=1,β≠αN∑s=1nBαβ∑r2=1p2bαβ,skλβ,r2uβr2k−s+∑s=1nBαβΔhβuβuβk−s,+∑β=1,β≠αN∑s=1nAαβ∑r3=1p3aαβ,skγβ,r3yβr3k−s+∑s=1nAαβΔhβyβyβk−s+∑r=1nCαcα,reαk−r+eαk.Other forms of Hammerstein MMs may be distinguished in order to represent the dynamics of INSs, depending on different configurations of static nonlinear elements.For reason of simplicity, we assume that the polynomialsBαq−1,k, Aαq−1,k, Aαβq−1,k, Bαβq−1,k, and Cαq−1 of Hammerstein MMs, which are given by (41) and (43), have the same order nα (nα=nAα=nBα=nBαβ=nAαβ). We make also a choice of the nonlinearity degrees pt, t=1,2,3, in such a way that the approximation errors Δhαuαuαk, Δhβuβuβk, and Δhβyβyβk become negligible (Δhαuαuαk=Δhβuβuβk=Δhβyβyβk=0). These assumptions allow us to further simplify the formulation of the parametric estimation and the control problems for the INSs, which are described by the two types of the developed Hammerstein MMs, which are given by (46) and (48).


## 4.2. Interconnected Wiener MMs

The creation of Wiener MMs for characterizing the INSs is discussed in this section. This model’s class relates to the interconnection of numerous MM structures, each of which has a dynamic linear and a static nonlinear component. In this class of MMs, we may distinguish between two forms of Wiener MMs: deterministic Wiener MMs and stochastic Wiener MMs [24].


### 4.2.1. Deterministic Interconnected Wiener MM

This section is intended for Wiener MMs [24] to describe INSs working in a deterministic environment. As a result, we investigate a nonlinear time-varying system made up of deterministic ISs. The deterministic Wiener MM may be used to explain each IS, and its structure is depicted in Figure 4.Figure 4
Deterministic structure of the interconnected Wiener MM.The previous structure, which is illustrated by Figure4, can be expressed by the following MM [24]:(44)Aαq−1,kwαk=Bαq−1,kuαk+∑β=1,β≠αNBαβq−1,kuβk+∑β=1,β≠αNAαβq−1,kyβk,where wαk and uαk are, respectively, the output and input of the dynamic linear block of the IS Sα, yβk, and uβk denote, respectively, the outputs and inputs, which arise from the other ISs Sβ, β=1,…,N;β≠α, and Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials, as given by (8), (9), (10), and (9), respectively.The following equation describes the static nonlinear component of the considered MM:(45)yαk=fwαwαk,where fwα. represents the nonlinear function.The following polynomial can be used to approximate (45):(46)yαk=∑r=1pηα,rwαrk+Δyαwαk,where p represents the nonlinearity degree of the nonlinear function, which can be chosen in an appropriate way; ηα,r, r=1,…,p, are unknown parameters; and Δyαwαk signifies the nonlinear function’s approximation error. This approximation error, which is dependent on the nonlinearity degree p chosen, might be compared to noise operating on the output of the IS in question. For an appropriate choice of the nonlinearity degree value p, this approximation error Δyαwαk can be neglected.The output of the ISyαk can be written as follows, taking into consideration the dynamic linear component of the investigated Wiener MM, as stated by (44) [24]:(47)yαk=∑r=1pηα,r−∑s=1nAαaα,skwαk−s+∑h=1nBαbα,hkuαk−h,+∑β=1,β≠αNBαβq−1,kuβk+∑β=1,β≠αNAαβq−1,kyβkr+Δyαwαk.


### 4.2.2. Stochastic Interconnected Wiener MM

In this part, we suppose that the output of the considered system is subjected to noise, which is composed of an independent random variable sequence. As a result, the Wiener MM’s dynamic linear portion is of type IARMAX.The structure of this model is depicted in Figure5.(48)Aαq−1,kwαk=Bαq−1,kuαk+∑β=1,β≠αNBαβq−1,kuβk+∑β=1,β≠αNAαβq−1,kyβk+Cαq−1eαk,where eαk designates all disturbances acting on the IS Sα, Cαq−1 is a polynomial defined by (12), and yαk corresponds to the output of the static nonlinear part of the stochastic Wiener MM, which is given by (46).Figure 5
Stochastic structure of the interconnected Wiener MM. The following formula describes the dynamic linear portion of Wiener MM [24].The output of the systemyαk may be represented in the following manner based on the dynamic linear component of the investigated Wiener MM, as given by (48).(49)yαk=∑r=1pηα,r−∑s=1nAαaα,skwαk−s+∑h=1nBαbα,hkuαk−h+∑β=1,β≠αNBαβq−1,kuβk,+∑β=1,β≠αNAαβq−1,kyβk+∑t=1nCαcα,teαk−t+eαkr+Δyαwαk.In the description of the INSs, we may use a variety of Wiener MMs, which are based on various forms of the static nonlinear portion.We assume that the polynomialsAαq−1,k, Bαq−1,k, Bαβq−1,k, Aαβq−1,k, and Cαq−1 intervening in the two types of Wiener MMs, given by (47) and (49), have the same order nα (nα=nAα=nBα=nAαβ=nBαβ). We also assume that the approximation error Δyαwαk is negligible, to reduce the formulation of the parametric estimation issue for large-scale dynamical systems defined by the Wiener MMs created.


## 4.2.1. Deterministic Interconnected Wiener MM

This section is intended for Wiener MMs [24] to describe INSs working in a deterministic environment. As a result, we investigate a nonlinear time-varying system made up of deterministic ISs. The deterministic Wiener MM may be used to explain each IS, and its structure is depicted in Figure 4.Figure 4
Deterministic structure of the interconnected Wiener MM.The previous structure, which is illustrated by Figure4, can be expressed by the following MM [24]:(44)Aαq−1,kwαk=Bαq−1,kuαk+∑β=1,β≠αNBαβq−1,kuβk+∑β=1,β≠αNAαβq−1,kyβk,where wαk and uαk are, respectively, the output and input of the dynamic linear block of the IS Sα, yβk, and uβk denote, respectively, the outputs and inputs, which arise from the other ISs Sβ, β=1,…,N;β≠α, and Aαq−1,k, Bαq−1,k, Bαβq−1,k, and Aαβq−1,k are time-varying polynomials, as given by (8), (9), (10), and (9), respectively.The following equation describes the static nonlinear component of the considered MM:(45)yαk=fwαwαk,where fwα. represents the nonlinear function.The following polynomial can be used to approximate (45):(46)yαk=∑r=1pηα,rwαrk+Δyαwαk,where p represents the nonlinearity degree of the nonlinear function, which can be chosen in an appropriate way; ηα,r, r=1,…,p, are unknown parameters; and Δyαwαk signifies the nonlinear function’s approximation error. This approximation error, which is dependent on the nonlinearity degree p chosen, might be compared to noise operating on the output of the IS in question. For an appropriate choice of the nonlinearity degree value p, this approximation error Δyαwαk can be neglected.The output of the ISyαk can be written as follows, taking into consideration the dynamic linear component of the investigated Wiener MM, as stated by (44) [24]:(47)yαk=∑r=1pηα,r−∑s=1nAαaα,skwαk−s+∑h=1nBαbα,hkuαk−h,+∑β=1,β≠αNBαβq−1,kuβk+∑β=1,β≠αNAαβq−1,kyβkr+Δyαwαk.


## 4.2.2. Stochastic Interconnected Wiener MM

In this part, we suppose that the output of the considered system is subjected to noise, which is composed of an independent random variable sequence. As a result, the Wiener MM’s dynamic linear portion is of type IARMAX.The structure of this model is depicted in Figure5.(48)Aαq−1,kwαk=Bαq−1,kuαk+∑β=1,β≠αNBαβq−1,kuβk+∑β=1,β≠αNAαβq−1,kyβk+Cαq−1eαk,where eαk designates all disturbances acting on the IS Sα, Cαq−1 is a polynomial defined by (12), and yαk corresponds to the output of the static nonlinear part of the stochastic Wiener MM, which is given by (46).Figure 5
Stochastic structure of the interconnected Wiener MM. The following formula describes the dynamic linear portion of Wiener MM [24].The output of the systemyαk may be represented in the following manner based on the dynamic linear component of the investigated Wiener MM, as given by (48).(49)yαk=∑r=1pηα,r−∑s=1nAαaα,skwαk−s+∑h=1nBαbα,hkuαk−h+∑β=1,β≠αNBαβq−1,kuβk,+∑β=1,β≠αNAαβq−1,kyβk+∑t=1nCαcα,teαk−t+eαkr+Δyαwαk.In the description of the INSs, we may use a variety of Wiener MMs, which are based on various forms of the static nonlinear portion.We assume that the polynomialsAαq−1,k, Bαq−1,k, Bαβq−1,k, Aαβq−1,k, and Cαq−1 intervening in the two types of Wiener MMs, given by (47) and (49), have the same order nα (nα=nAα=nBα=nAαβ=nBαβ). We also assume that the approximation error Δyαwαk is negligible, to reduce the formulation of the parametric estimation issue for large-scale dynamical systems defined by the Wiener MMs created.


## 5. Case Study

In this section, we present a numerical example to illustrate the feasibility and effectiveness of the developed theoretical results. This example corresponds to a stochastic large-scale nonlinear system, composed of two interconnected subsystems S1 and S2, and can be described by the class of interconnected Hammerstein MM.Figure6 shows the general structure interaction of the considered process.Figure 6
General structure interaction of the considered process.The system outputyik, i=1,2, can be expressed as(50)yik=−ai,1kyik−1−ai,2kyik−2+αi,1uik−1+bi,2kαi,1uik−2,+αi,2ui2k−1+bi,2kαi,2ui2k−2+βj,1ujk−1+bij,2kβj,1ujk−2,+βj,2uj2k−1+bij,2kβj,2uj2k−2+eik+ci,1eik−1,where the relative data are selected as follows: a1,1k=−0.88+0.03sin0.2k, c1,1=0.25, α1,1=0.32, a1,2k=0.45+0.02cos0.2k, b1,2k=0.32+0.02sin0.2k, α1,2=0.23, β2,1=0.33, β2,2=0.22, b12,2k=0.33+0.03sin0.2k, a2,1k=−0.85+0.03sin0.2k, c2,1=0.27, α2,1=0.31, a2,2k=0.42+0.02cos0.2k, b2,2k=0.45+0.04sin0.2k, α2,2=0.21, β1,1=0.33, β1,2=0.24, b21,2k=0.43+0.03sin0.2k. Adding that the input uik that applied to the INS is a high level pseudo-random binary sequence [-1.5, +1.5], and the variances values of the noise sequence eik,i=1,2 are σ12=0.0937 and σ22=0.0853.Some results of this simulation example of the considered system are given. Thereby, Figures7 and 8 illustrate the evolution of the prediction error, the parametric distance, and their overall variances for each interconnected nonlinear subsystem.Figure 7
Evolution of the prediction errorε1k, the parametric distance d1k, and their overall variances σε12k and σd12k.Figure 8
Evolution of the prediction errorε2k, the parametric distance d2k, and their overall variances σε22k and σd22k.The obtained results indicate the good quality of the estimate, based on an iterative algorithm. This estimate quality ensures the exact choice of the parametric estimation algorithm and the mathematical model of representation that describe the best behavior of the considered interconnected nonlinear system, despite the parameters variations, the presence of interactions signals, and disturbances interim on each system output.Note that the mathematical model of representation, which is also called the mathematical model of control or mathematical model of behavior, corresponds to the mathematical model of the ″black box″. Let’s add that the mathematical model of representation is most often described by difference equations; this, therefore, corresponds to an input-output type mathematical model. However, the parameters involved in this type of mathematical model have no physical meaning. In fact, the theoretical mathematical model is much richer in physical meaning than the representation model, since it contains all the useful information about the real process.It should also be noted that the mathematical model of representation is the most currently used in the control of dynamic systems, in particular in the synthesis of numerical control laws. However, the choice of a mathematical model, from this set of models, may depend on several criteria, such as the type of application, the desired performance indices, and the strategy of the control law envisaged. Therefore, we must classify mathematical models according to their ability to approximate the system. Thus, they can be classified from the simplest mathematical model to the most complex mathematical model. The selection of a “good” mathematical model for the formulation of the control law is made according to the targeted control objectives (accuracy, robustness, and rapidity).


## 6. Conclusions

This paper has proposed new input-output MMs of representation, which can describe the dynamic behavior of large-scale nonlinear systems, such as the extended MM of Volterra series, the interconnected Hammerstein structures, and the interconnected Wiener structures. In this research, we considered the class of large-scale nonlinear systems, which are decomposed into several interconnected nonlinear subsystems. Each interconnected subsystem is described by discrete nonlinear MM, mono-variable, operating in a deterministic or stochastic environment, and with unknown time-varying parameters. An illustrative numerical simulation example of two interconnected nonlinear processes was treated to test the performance and the effectiveness of the developed theoretical results.Let us note that, in certain practical situations, the formulation of a representation mathematical model based on experimental method, describing an industrial system, becomes difficult or impossible in certain cases due to the difficulty in carrying out or analyzing experimental tests on the considered process (unmeasurable inputs and outputs variables and dangerous experimental measurements of certain real process). Besides, it can be remarked that the developed mathematical model becomes more complex with the increase of the dimension and the nonlinearity degree of the considered system. In this case, the synthesis of the adequate control design will be difficult.In future works, we will address to develop extended versions of different methods, which permit us to estimate the structure variables and the parameters of the interconnected nonlinear systems. The formulation of the control problem of this class of dynamical systems will be investigated in future research by developing various controllers based on different control approaches. [25].



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