# Combining Augmented Error Modeling Technique and Block-Pulse Functions Method for Tracking Control Design of Nonlinear Polynomial Systems with Multiple Time-Delayed States Subject to Nonsymmetric Input Saturation

Authors: Bassem Iben Warrad
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

The present research work is intended to synthesize a novel tracking control strategy for a class of nonlinear polynomial systems characterized by multiple well-defined delays in state variables under the presence of nonsymmetric input saturation. The design strategy makes full use of an associate’s memory nonlinear state feedback control with integral-based actions. An original control scheme joining block-pulse functions method combined with the augmented error modeling technique is used to infer the controller’s tracking gains. The objective is to convert the investigated nonlinear algebraic problem governed by specifying constraints into a constrained linear one that can be solved in the constrained least square methodology. Detailed novel sufficient conditions proving the closed-loop augmented system’s practical stability are elaborated. The instance of a twofold inverted pendulums benchmark is considered so as to exhibit the benefits of the proposed control approach.


---

## Body

## 1. Introduction

The design of an output tracking control scheme for continuous-time nonlinear systems subject to time delays is one of the most investigated problems in nonlinear control research theory [1]. This topic is not a challenging one only because this class of time-delayed systems is highly confronted in the industrial process, but mainly due to the restrictions in proving the desired closed-loop performance objectives, in addition to the stability requirement. As needs be, the delicacy identified with the tracking control problem is intensively considered by researchers nowadays [2–5].In the framework of the Lyapunov theory, there are mainly two fundamental methodologies to synthesize a tracking control law for a particular class of nonlinear time-delayed systems. The first one is based on the Lyapunov–Krasovskii functionals, while the subsequent one exploits the Lyapunov–Razumikhin functionals. Both methodologies can provide an asymptotic tracking performance. Based on these Lyapunov functionals, most kinds of tracking control methods for various classes of nonlinear time-delayed systems have been widely developed, such as sliding mode control [6], model predictive control [7], fault-tolerant control [8], backstepping control [9, 10], dynamic surface control [11], H∞ control [12], and adaptive fuzzy or network control [13, 14].The significant drawbacks of those strategies originate from the fact that no generic algorithms exist to define a Lyapunov–Krasovskii/Razumikhin function candidate, as well as from the accompanied computational complexity of the existing tracking control methods for providing the controller’s gains.It is worth mentioning that all of the aforementioned control approaches are applicable for particular classes of nonlinear systems under the severe Lipschitz assumption and other additional constraints [6, 7], just to mention a few, cases of nonlinear systems with time delays designated by a particular differentiable form [10], cases of nonlinear systems with time delays represented in pure feedback form [9] or a lower-triangular form [11] or strict-feedback form [8], and cases of nonlinear systems with time delays modeled by advanced technique [12–14].On the other hand, it is recognized that the actuator saturation is a potential problem of all practical dynamic systems [15]. So, every control system has to face an input saturation, which can result in performance abasement such as oscillations, high overshoot, undershoot, lag, large settling time, and sometimes even instability. Thus, both actuator saturation and time delays are frequently encountered when modeling an engineering system. To design appropriate control laws for dealing with the presence of input saturation, three main strategies have been presented in the literature for continuous-time nonlinear systems affected by time delays, which can be cited as follows:(a)
The anti-windup compensation-based methods, in which an auxiliary design system is introduced to compensate the effect of input saturation. There are two schemes in this case, called one-step design scheme or two-step design scheme. For the last one, first a controller is designed without regarding control input nonlinearity, and then, an anti-windup compensator will be augmented in order to minimize the undesirable degradation of closed-loop performance caused by input saturation. The one-step design scheme deals with saturation effect by simultaneously designing the controller and its associated anti-windup compensation. Recently, a few exceptional works have been developed in which the tracking control design problems have been investigated. In [16], a full-order dynamic internal model control-based anti-windup compensation architecture was proposed for stable nonlinear time-delayed systems, satisfying the Lipschitz condition, under actuator saturation. A local decoupling anti-windup compensation design was suggested using the delay-range-dependent approach, for compensation of the undesirable saturation effects. In [17], by employing the concept of decoupled architecture-based design, a dynamic anti-windup compensation applicable to both stable or unstable nonlinear time-delayed systems was proposed, by reformulating the Lipschitz nonlinearities of the considered class of system via linear parameter-varying models. In [18], a novel approach was proposed for designing a static anti-windup compensation for two classes of delayed nonlinear systems under input actuator saturation. Based on the Wirtinger-based inequality, a delay-range-dependent technique was exploited to derive a condition for calculating the anti-windup compensation gains. By using the quadratic Lyapunov–Krasovskii functional, Wirtinger-based inequality, sector conditions, bound on delays, time-varying range of delay, and H∞ gain reduction, several conditions in terms of LMIs are derived to guarantee local and global stabilization and performance of the overall closed-loop system. It should be mentioned that all of these methods can only handle the nonlinear time-delayed systems under the severe Lipschitz condition.(b)
The convexity-based methods, in which the input constraints are considered at the controller design stage in order to adjust the controller gains according to the saturation levels. In this case, there are two approaches. The first one is to use a polytopic representation of the saturation function. Based on this representation and using the Takagi–Sugeno modeling, some interesting works were devoted only to the stabilization control problem for saturated T-S fuzzy time-delayed systems by means of delay-dependent Lyapunov–Krasovskii functional [19] or delay-independent Lyapunov–Krasovskii functional [20]. Another way to take the saturation effects into account relying on the sector nonlinearity approach is adopted to derive Popov and Circle criteria [21]. Lately, this approach was effectively applied to tackle the stabilization control problem for Lipschitz nonlinear time-delayed systems subject to symmetric input saturation [22], as well as the tracking control problem for lower-triangular nonlinear time-delayed systems subject to nonsymmetric input saturation [23, 24]. It is important to highlight that these methods, despite that they are restricted to specific classes of nonlinear time-delayed systems, provide an attractive region of initial states that ensure the asymptotic stability of the considered closed-loop system.(c)
The approximation-based intelligent control methods, in which the nonsmooth nonlinear input saturation is approximated by a smooth function. Generally, two commonly used saturated functions are the logistic sigmoid and the hyperbolic tangent. By using neural network or fuzzy approximation technique to relax the growth assumptions and employing Lyapunov–Krasovskii functionals to deal with time delay, an increasing attention has been paid to the adaptive tracking control design for saturated nonlinear time-delayed systems in the triangular form [25, 26] or strict-feedback form [27, 28]. A main limitation of the mentioned works is that the required number of neural-network rules or fuzzy rules to achieve given accuracy leads to increasing complexities of the procedure of computing tracking control law.Up to our knowledge, the time-varying set point tracking control problem for a particular class of multi-input multi-output nonlinear time-delayed systems, namely nonlinear polynomial system with multiple time-delayed states, subject to nonsymmetric input saturation is not yet investigated until now. It must be noted that such polynomial model is considered as a single unified form to represent the nonlinear dynamics of analytical systems. In fact, any analytical nonlinear model can be represented by a polynomial one utilizing the Kronecker tonsorial power [29] and the Taylor series expansion as main powerful analytical tools. Therefore, this kind of model can accurately describe the dynamical behavior of a large set of physical processes and real industrial plants, including but not limited to the following: electrical machines [30], robot manipulators [31], power systems [32], and interconnected inverted pendulums [33]. This advantage is justified by its capacity (i) to preserve the high-order model as well as its original nonlinearity, which leads to an accurate controller’s gains when computing the control law, and (ii) to enlarge the operating region approximation around the equilibrium point.Due to the necessity of bringing remedy to the drawbacks of the exiting control methods based on the Lyapunov theory, increasing interests are showed up as of late toward the improvement of numerical approximation methods based on the modeling concept of nonlinear polynomial systems. Some works are devoted to the tracking control problem for such undelayed system in the case of a step input [31] as well as in the case of a time-varying set point input [34]. For the subclass of the nonlinear polynomial system under the presence of a single time-delayed state, whose value is supposed to be constant and known, the tracking control problem has been addressed in [35] using block-pulse functions as a tool of approximation. Nevertheless, it is notable that up to now, none published research in which any one of the control design problems for the nonlinear polynomial system with or without time delays under input saturation is studied.Motivated by the two concerns: firstly, to tackle the output tracking control problem for the nonlinear polynomial system with multiple time-delayed states subject to nonsymmetric input saturation; secondly, to surmount the drawbacks of previous studies building through a Lyapunov–Krasovskii or Razumikhin function approach, whose cannot tackle the mentioned class of nonlinear time-delayed systems, then it might be beneficial to exploit simultaneously the augmented error modeling technique and numerical method tools. This is justified by the fact that the static errors are reduced effectively by increasing the state vector through a new output tracking error vector, as well as by the conversion of the posed saturated tracking control problem into solving a constrained linear system of algebraic equations, depending on the parameters of the feedback regulator.Proceeding from the above idea, a new algebraic scheme for the development of a memory polynomial state feedback controller with an integral-action-based compensator for continuous-time nonlinear polynomial system affected by multiple time delays in the states and under nonsymmetric input saturation is presented in the current study. Right off the bat, the modeling technique of augmented error is performed to determine an equivalent augmented form for the closed-loop controlled system, which permits to achieve high trajectory tracking performance [34, 36]. Then, the block-pulse functions basis as an approximation tool as well as their operational matrices are exploited to transform the augmented state-space model of controlled saturated process into an equivalent constrained linear model of algebraic equations, which depends only on the controller gains. Thus, the proposed computation method in this work is simple as well as swift.Walsh functions [37], block-pulse functions [38], and Haar wavelet functions [39] are considered as the most impressive types of piecewise constant basis functions. Contrary to them, block-pulse functions are featured by their simple structure and direct implementation [40]. This is generally because of their significant properties, that is, labeled disjointedness, fulfillment, and orthogonality. Thus, they have been exploited widely as a valuable tool in the synthesis, analysis, estimation, identification, and different problems related to the area of control and systems [41–43].The main purpose of this work is to substitute both coefficients of delayed and undelayed augmented state vectors by those of reference models issued from projecting the state-space model over a basis of block-pulse functions. By applying the operational matrices of the considered basis accompanied by the use of explicit arithmetical control operations, it comes out a constrained linear system of algebraic equations, which can be solved by the means of constrained least squares optimization. It is worth noting that the insertion of the imposed limits by the actuator saturation on the control input is translated here through the expansion of the saturated input control over the considered block-pulse functions basis. As an original result, novel sufficient conditions are determined for practical stability guarantees of the augmented closed-loop system under the presence of input saturation constraint. Consequently, the significant contribution of this study can be summed up as follows:(1)
This work is addressing for the first time, the output trajectory tracking control problem for the class of nonlinear polynomial systems with multiple time-delayed states, subject to nonsymmetric input saturation. For this reason, a new combined block-pulse functions method joined to the augmented error modeling technique is applied. This scheme helps in designing memory polynomial state feedback controllers with integral actions.(2)
To our knowledge, the proposed method for the introduction of the imposed limits on the control input by the actuator saturation in the procedure of computing tracking control law for the addressed class of saturated nonlinear time-delayed systems is developed for the first time in this study. This method has the ability to treat both nonsymmetric or symmetric input saturation, which can be evaluated as an important advantage.(3)
Dissimilar to the based Lyapunov–Krasovskii/Razumikhin function tracking method and its related computational intricacy, the proposed controller’s gains in this work are derived by means of constrained least squares optimization. This is established throughout algebraic operations where sufficient stability conditions of the controlled augmented system are defined explicitly. These conditions are defined without precedent for this study.Following this introduction, a short description of block-pulse functions and their fundamentals are presented. Next, the closed-loop system model is described leading to the problem statement. The essential outcomes are given in the subsequent section. The efficiency and performance of the developed strategy are evaluated on a twofold inverted pendulums benchmark.Notation. During the whole of this study, the n dimensional Euclidean space is denoted by ℝn, while ℝn×m is referred to the set of all real matrices with n rows and m columns, whereas ℝn×mni+j stands for the set of all real matrices with n rows and m×ni+j columns, where i and j are two natural numbers. In denotes the identity matrix of size n×n, whereas 0n×m denotes the zero matrix of size n×m. AT corresponds to the transpose of the matrix A. eip denotes the p dimensional unit vector which has 1 in the ith element and zero elsewhere. The adopted vector norm is the Euclidean norm, and the matrix norm is the corresponding induced norm.


## 2. Block-Pulse Functions and their Properties

### 2.1. Block-Pulse Functions

Over the interval0,T, an N set of block-pulse functions (BPFs) can be defined as follows [38]:(1)φit=1,iTN≤t≤i+1TN,i=0,…,N−1,0,else where,where for each i∈0,…,N−1, φit represents the i-th block-pulse functions.The principal attributes of block-pulse functions are as follows.


#### 2.1.1. Disjointness

The block-pulse functions are disjoint with each other in the interval0,T:(2)∀i,j∈0,…,N−1,φitφjt=φitfori=j0fori≠j.


#### 2.1.2. Orthogonality

The block-pulse functions are orthogonal with each other in the interval0,T:(3)∀i,j∈0,…,N−1,∫0Tφitφjtdt=TNfori=j0fori≠j.The orthogonal property of block-pulse functions is the basis of expanding functions into their block-pulse series. Thus, any absolutely integrable functionft of the Lebesgue measure in the interval 0,T may be expanded over a BPFs basis as follows:(4)ft≅∑i=0N−1fiφit=FNSNt,where the block-pulse coefficient vector FN is given by(5)FN=f0⋯fN−1,and the block-pulse vector basis of order N is given by(6)SNt=φ0t⋯φN−1tT,with fi is the block-pulse coefficient computing as follows:(7)∀i∈0,…,N−1,fi=NT∫0Tftφitdt=NT∫iT/Ni+1T/Nftdt.


#### 2.1.3. Completeness

The block-pulse function set is complete when the orderN approaches infinity. This means that for any square-integrable function ft of the Lebesgue measure in the interval 0,T, we have(8)∫0Tf2tdt=limN⟶+∞∑i=0N−1fi2∫0Tφi2tdt.


### 2.2. Error in BPFs Approximation

Assuming thatft is a differentiable function, where his first derivative on the time interval 0,T is bounded, that is to say(9)∃M≻0,suchf′t≤M.Over every subintervaliT/N,i+1T/N, the residual error is given by(10)eit=fiφit−ft=fi−ft=NT∫iT/Ni+1T/Nfτdτ−ft.Furthermore, the total error can be defined as follows:(11)et=∑i=0N−1eit,t∈0,T,which can be bounded as follows [43]:(12)et≤MNT312,allowing to conclude that(13)limN⟶+∞et=0.Consequently, over a BPFs basis with a high orderN, we can represent exactly the function ft.


### 2.3. BPFs Expansion of a Vector Function

A vector functionxt of n dimensional components which are square-integrable with respect to Lebesgue measure in the interval 0,T can be expressed practically by a finite series of block-pulse functions as follows:(14)xt≅∑i=0N−1xiφit=X1,NSNt,with X1,N=x0,…,xN−1, where xi are the block-pulse coefficients of vector function xt, as computed through the orthogonality property of the block-pulse functions as follows:(15)xi=NT∫iTNi+1TNxtdt.


### 2.4. BPFs Expansion of a Time-Delayed Vector Function

Assume that the vectorxt has its initial as(16)xt=ζt−τ≤t≤0,with 0<τ<T. Then, for t∈0,T, the time-delayed vector xt−τ is presented by(17)xt−τ=ζt−τfor0≤t≤τxt−τforτ<t≤T.As a result, the expansion of time-delayed vectorxt−τ over the BPFs basis is given by [44](18)xt−τ≅∑i=0N−1xi∗τφit=X1,N∗τSNt,with(19)X1,N∗τ=x0∗τ,…xN−1∗τ,(20)xi∗τ=NT∫iT/Ni+1T/Nxt−τdt=ζi∗τfori≤μxi−μfori>μ,where μ is the number of BPFs considered over 0≤t≤τ, and(21)ζi∗τ=NT∫iT/Ni+1T/Nζt−τdt.


### 2.5. Operational Matrix of Integration

For the BPFs basis, an operational matrix of integration is defined in [44] by(22)∫0tSNtdt=PNSNt,where PN=T/2N12…201…2⋮⋱⋱20…01.


### 2.6. Operational Matrix of Product

The elementary matrix can be defined as follows:(23)Ei,jp×q=eip⊗ejqT,where ⊗ is the symbol of the Kronecker product [29].Based on the disjointness property of BPFs, the operational matrix of product is given by [35](24)SNt⊗SNt=E1,1N×N⋮EN,NN×NSNt=MNSNt.


## 2.1. Block-Pulse Functions

Over the interval0,T, an N set of block-pulse functions (BPFs) can be defined as follows [38]:(1)φit=1,iTN≤t≤i+1TN,i=0,…,N−1,0,else where,where for each i∈0,…,N−1, φit represents the i-th block-pulse functions.The principal attributes of block-pulse functions are as follows.


### 2.1.1. Disjointness

The block-pulse functions are disjoint with each other in the interval0,T:(2)∀i,j∈0,…,N−1,φitφjt=φitfori=j0fori≠j.


### 2.1.2. Orthogonality

The block-pulse functions are orthogonal with each other in the interval0,T:(3)∀i,j∈0,…,N−1,∫0Tφitφjtdt=TNfori=j0fori≠j.The orthogonal property of block-pulse functions is the basis of expanding functions into their block-pulse series. Thus, any absolutely integrable functionft of the Lebesgue measure in the interval 0,T may be expanded over a BPFs basis as follows:(4)ft≅∑i=0N−1fiφit=FNSNt,where the block-pulse coefficient vector FN is given by(5)FN=f0⋯fN−1,and the block-pulse vector basis of order N is given by(6)SNt=φ0t⋯φN−1tT,with fi is the block-pulse coefficient computing as follows:(7)∀i∈0,…,N−1,fi=NT∫0Tftφitdt=NT∫iT/Ni+1T/Nftdt.


### 2.1.3. Completeness

The block-pulse function set is complete when the orderN approaches infinity. This means that for any square-integrable function ft of the Lebesgue measure in the interval 0,T, we have(8)∫0Tf2tdt=limN⟶+∞∑i=0N−1fi2∫0Tφi2tdt.


## 2.1.1. Disjointness

The block-pulse functions are disjoint with each other in the interval0,T:(2)∀i,j∈0,…,N−1,φitφjt=φitfori=j0fori≠j.


## 2.1.2. Orthogonality

The block-pulse functions are orthogonal with each other in the interval0,T:(3)∀i,j∈0,…,N−1,∫0Tφitφjtdt=TNfori=j0fori≠j.The orthogonal property of block-pulse functions is the basis of expanding functions into their block-pulse series. Thus, any absolutely integrable functionft of the Lebesgue measure in the interval 0,T may be expanded over a BPFs basis as follows:(4)ft≅∑i=0N−1fiφit=FNSNt,where the block-pulse coefficient vector FN is given by(5)FN=f0⋯fN−1,and the block-pulse vector basis of order N is given by(6)SNt=φ0t⋯φN−1tT,with fi is the block-pulse coefficient computing as follows:(7)∀i∈0,…,N−1,fi=NT∫0Tftφitdt=NT∫iT/Ni+1T/Nftdt.


## 2.1.3. Completeness

The block-pulse function set is complete when the orderN approaches infinity. This means that for any square-integrable function ft of the Lebesgue measure in the interval 0,T, we have(8)∫0Tf2tdt=limN⟶+∞∑i=0N−1fi2∫0Tφi2tdt.


## 2.2. Error in BPFs Approximation

Assuming thatft is a differentiable function, where his first derivative on the time interval 0,T is bounded, that is to say(9)∃M≻0,suchf′t≤M.Over every subintervaliT/N,i+1T/N, the residual error is given by(10)eit=fiφit−ft=fi−ft=NT∫iT/Ni+1T/Nfτdτ−ft.Furthermore, the total error can be defined as follows:(11)et=∑i=0N−1eit,t∈0,T,which can be bounded as follows [43]:(12)et≤MNT312,allowing to conclude that(13)limN⟶+∞et=0.Consequently, over a BPFs basis with a high orderN, we can represent exactly the function ft.


## 2.3. BPFs Expansion of a Vector Function

A vector functionxt of n dimensional components which are square-integrable with respect to Lebesgue measure in the interval 0,T can be expressed practically by a finite series of block-pulse functions as follows:(14)xt≅∑i=0N−1xiφit=X1,NSNt,with X1,N=x0,…,xN−1, where xi are the block-pulse coefficients of vector function xt, as computed through the orthogonality property of the block-pulse functions as follows:(15)xi=NT∫iTNi+1TNxtdt.


## 2.4. BPFs Expansion of a Time-Delayed Vector Function

Assume that the vectorxt has its initial as(16)xt=ζt−τ≤t≤0,with 0<τ<T. Then, for t∈0,T, the time-delayed vector xt−τ is presented by(17)xt−τ=ζt−τfor0≤t≤τxt−τforτ<t≤T.As a result, the expansion of time-delayed vectorxt−τ over the BPFs basis is given by [44](18)xt−τ≅∑i=0N−1xi∗τφit=X1,N∗τSNt,with(19)X1,N∗τ=x0∗τ,…xN−1∗τ,(20)xi∗τ=NT∫iT/Ni+1T/Nxt−τdt=ζi∗τfori≤μxi−μfori>μ,where μ is the number of BPFs considered over 0≤t≤τ, and(21)ζi∗τ=NT∫iT/Ni+1T/Nζt−τdt.


## 2.5. Operational Matrix of Integration

For the BPFs basis, an operational matrix of integration is defined in [44] by(22)∫0tSNtdt=PNSNt,where PN=T/2N12…201…2⋮⋱⋱20…01.


## 2.6. Operational Matrix of Product

The elementary matrix can be defined as follows:(23)Ei,jp×q=eip⊗ejqT,where ⊗ is the symbol of the Kronecker product [29].Based on the disjointness property of BPFs, the operational matrix of product is given by [35](24)SNt⊗SNt=E1,1N×N⋮EN,NN×NSNt=MNSNt.


## 3. Problem Formulation and System Description

Consider a continuous-time nonlinear polynomial systems with known and fixed multiple time delays in statesS, which are described by the following state equation:(25)Sx˙t=∑i=1q1Aixit+∑j=1s1∑l=1v1Djlxjt−τl+∑i=1q1∑j=1s1∑l=1v1Gijlxit⊗xjt−τl+∑i=1q2Biut⊗xit+∑j=1s2∑l=1v2Cjlut⊗xjt−τl+∑i=1q2∑j=1s2∑l=1v2Qijlut⊗xit⊗xjt−τl+Butyt=Cxtxt=ζtfort∈−τ¯,0,where ut=u1t,…,umtT∈ℝm is the input vector, xt∈ℝn is the nondelayed state vector, xt−τl∈ℝn is the delayed state vector with τl denoted time delay, and yt∈ℝp is the output vector. The continuous vector-valued function ζt denotes the initial data, where τ¯=maxτl for each l∈1…v, with v=maxvi for each i∈1,2.In (25), Ai∈ℝn×ni, Djl∈ℝn×nj, Gijl∈ℝn×ni+j, Bi∈ℝn×mni, Cjl∈ℝn×mnj, Qijl∈ℝn×mni+j, B∈ℝn×m, C∈ℝp×n, and xi is the ith Kronecker power of the state vector x, defined by x0=1,xi+1=xi⊗x∀i>0.


### 3.1. Assumption

The consider system (25) is locally controllable around x0=ζ0 where its n state components are all physically measurable.


### 3.2. Recommended Strategy

Control objective involves the tracking of outputs of saturated nonlinear time delays system for given reference input vectoryct∈ℝp. For this purpose, we intend to design a memory polynomial state feedback controller accompanied by compensator gains:(26)ut=N¯Rt−∑i=1q1Kixit−∑j=1s1∑l=1v1Ljlxjt−τl,with(27)Rt=etT,e1tT,…ew−1tTT,where(28)et=∫0t…∫0t︸wtimesycσ−yσdσw,with w is a positive integer which depends on the type of reference input vector. In (26), N¯∈ℝm×wp, Ki∈ℝm×ni, Ljl∈ℝm×nj, and eht are the h−th derivative of et.Each element of the actuator vectoruit should satisfy the following constraint:(29)−v¯i≤uit≤u¯iwith0≺u¯iand0≺v¯i,which is equivalent to(30)−vmax≤ut≤umax,where umax=u¯1,…,u¯mT and vmax=v¯1,…,v¯mT are given positive reals vectors.The constrained inputs, denotedsatuit, are saturating functions and defined for each i∈1…m as follows:(31)satuit=u¯iifuit≻u¯iuitif−v¯i≤uit≤u¯i−v¯iifuit≺−v¯i,with(32)satut=satu1t…satumtT.


### 3.3. Control Objective

The undelayed augmented state vectorXt as well as the delayed augmented state vector Xt−τl can be expressed using the augmented error modeling technique, as follows:(33)Xt=xtT,RtTT,(34)Xt−τl=xt−τlT,Rt−τlTT.When taking into consideration that(35)xt=In0n,wpXt=ΨXt,(36)xit=ΨiXit,(37)xt−τl=ΨXt−τl,(38)xit−τl=ΨiXit−τl.Then, the augmented system subject to the constrained input controlSa can be described by the following state equation:(39)SaX˙t=∑i=1q1A¯iXit+∑j=1s1∑l=1v1D¯jlXjt−τl+∑i=1q1∑j=1s1∑l=1v1G¯ijlXit⊗Xjt−τl+∑i=1q2B¯isatut⊗Xit+∑j=1s2∑l=1v2C¯jlsatut⊗Xjt−τl+∑i=1q2∑j=1s2∑l=1v2Q¯ijlsatut⊗Xit⊗Xjt−τl+B¯satut+E¯yctYt=C¯XtXt=ζ¯t=ζtT01,wpTfort∈−τ¯,0,with A¯1=A10n,p0n,p……0n,p0p,n0p,pIp0p,p…0p,p0p,n0p,p0p,pIp⋱⋮⋮⋮⋮⋱⋱0p,p0p,n0p,p0p,p…0p,pIp−C0p,p0p,p……0p,p,E¯=0n+w−1p,pIp,C¯=C0p,wp,B¯=B0wp,m, and for each i∈2…q1, j∈1…s1 and l∈1…v1(39a)A¯i=Ai0wp,niΨi,D¯jl=Djl0wp,njΨj,G¯ijl=Gijl0wp,ni+jΨi+j,and for each i∈1…q2, j∈1…s2 and l∈1…v2B¯i=Bi0wp,mniIm⊗Ψi,C¯jl=Cjl0wp,mnjIm⊗Ψj,Q¯ijl=Qijl0wp,mni+jIm⊗Ψi+j.Assuming that the obtained augmented systemSa is locally controllable around X0=ζ¯0, then the dynamical behavior of the closed-loop augmented system Sa must exactly match the behavior of a linear reference model Sr, which is represented also by its state equations:(40)SrX˙rt=EXrt+FyctYrt=ZXrtXrt=0n+wp,1,where Xrt∈ℝn+wp and Yrt∈ℝp.The output vectorYrt of the above reference model, which should be adequately designed, must track perfectly the reference input vector yct.Remark 1.
In order to make the adopted control strategy (26) in the adequate form, corresponding to the augmented system (39), it should be written as follows:(41)ut=−∑i=1q1K¯iXit−∑j=1s1∑l=1v1L¯jlXjt−τl,with(42)K¯1=K1−N¯,K¯i=KiΨi,∀i∈2…q1,(43)L¯jl=LjlΨj∀j∈1…s1,∀l∈1…v1.Remark 2.
The structure of the adopted strategy of control is perfectly adequate to the structure of the considered state model. This is justified by the following facts:(i)
The compensator gainN¯ is designed to eliminate the steady-state error for a time-varying set point reference input vector:(44)e∞=yc∞−y∞=0.(ii)
The control gainsKi and Ljl for each i∈1,…,q1,j∈1…s1,l∈1…v1 are designed to ensure the stability performance of the following closed-loop nonlinear polynomial time delays system under input control saturation u¯t:(45)x˙t=∑i=1q1Aixit+∑j=1s1∑l=1v1Djlxjt−τl+∑i=1q1∑j=1s1∑l=1v1Gijlxit⊗xjt−τl+∑i=1q2Bisatu¯t⊗xit+∑j=1s2∑l=1v2Cjlsatu¯t⊗xjt−τl+∑i=1q2∑j=1s2∑l=1v2Qijlsatu¯t⊗xit⊗xjt−τl+Bsatu¯t,with(46)u¯t=−∑i=1q1Kixit−∑j=1s1∑l=1v1Ljlxjt−τl.


## 3.1. Assumption

The consider system (25) is locally controllable around x0=ζ0 where its n state components are all physically measurable.


## 3.2. Recommended Strategy

Control objective involves the tracking of outputs of saturated nonlinear time delays system for given reference input vectoryct∈ℝp. For this purpose, we intend to design a memory polynomial state feedback controller accompanied by compensator gains:(26)ut=N¯Rt−∑i=1q1Kixit−∑j=1s1∑l=1v1Ljlxjt−τl,with(27)Rt=etT,e1tT,…ew−1tTT,where(28)et=∫0t…∫0t︸wtimesycσ−yσdσw,with w is a positive integer which depends on the type of reference input vector. In (26), N¯∈ℝm×wp, Ki∈ℝm×ni, Ljl∈ℝm×nj, and eht are the h−th derivative of et.Each element of the actuator vectoruit should satisfy the following constraint:(29)−v¯i≤uit≤u¯iwith0≺u¯iand0≺v¯i,which is equivalent to(30)−vmax≤ut≤umax,where umax=u¯1,…,u¯mT and vmax=v¯1,…,v¯mT are given positive reals vectors.The constrained inputs, denotedsatuit, are saturating functions and defined for each i∈1…m as follows:(31)satuit=u¯iifuit≻u¯iuitif−v¯i≤uit≤u¯i−v¯iifuit≺−v¯i,with(32)satut=satu1t…satumtT.


## 3.3. Control Objective

The undelayed augmented state vectorXt as well as the delayed augmented state vector Xt−τl can be expressed using the augmented error modeling technique, as follows:(33)Xt=xtT,RtTT,(34)Xt−τl=xt−τlT,Rt−τlTT.When taking into consideration that(35)xt=In0n,wpXt=ΨXt,(36)xit=ΨiXit,(37)xt−τl=ΨXt−τl,(38)xit−τl=ΨiXit−τl.Then, the augmented system subject to the constrained input controlSa can be described by the following state equation:(39)SaX˙t=∑i=1q1A¯iXit+∑j=1s1∑l=1v1D¯jlXjt−τl+∑i=1q1∑j=1s1∑l=1v1G¯ijlXit⊗Xjt−τl+∑i=1q2B¯isatut⊗Xit+∑j=1s2∑l=1v2C¯jlsatut⊗Xjt−τl+∑i=1q2∑j=1s2∑l=1v2Q¯ijlsatut⊗Xit⊗Xjt−τl+B¯satut+E¯yctYt=C¯XtXt=ζ¯t=ζtT01,wpTfort∈−τ¯,0,with A¯1=A10n,p0n,p……0n,p0p,n0p,pIp0p,p…0p,p0p,n0p,p0p,pIp⋱⋮⋮⋮⋮⋱⋱0p,p0p,n0p,p0p,p…0p,pIp−C0p,p0p,p……0p,p,E¯=0n+w−1p,pIp,C¯=C0p,wp,B¯=B0wp,m, and for each i∈2…q1, j∈1…s1 and l∈1…v1(39a)A¯i=Ai0wp,niΨi,D¯jl=Djl0wp,njΨj,G¯ijl=Gijl0wp,ni+jΨi+j,and for each i∈1…q2, j∈1…s2 and l∈1…v2B¯i=Bi0wp,mniIm⊗Ψi,C¯jl=Cjl0wp,mnjIm⊗Ψj,Q¯ijl=Qijl0wp,mni+jIm⊗Ψi+j.Assuming that the obtained augmented systemSa is locally controllable around X0=ζ¯0, then the dynamical behavior of the closed-loop augmented system Sa must exactly match the behavior of a linear reference model Sr, which is represented also by its state equations:(40)SrX˙rt=EXrt+FyctYrt=ZXrtXrt=0n+wp,1,where Xrt∈ℝn+wp and Yrt∈ℝp.The output vectorYrt of the above reference model, which should be adequately designed, must track perfectly the reference input vector yct.Remark 1.
In order to make the adopted control strategy (26) in the adequate form, corresponding to the augmented system (39), it should be written as follows:(41)ut=−∑i=1q1K¯iXit−∑j=1s1∑l=1v1L¯jlXjt−τl,with(42)K¯1=K1−N¯,K¯i=KiΨi,∀i∈2…q1,(43)L¯jl=LjlΨj∀j∈1…s1,∀l∈1…v1.Remark 2.
The structure of the adopted strategy of control is perfectly adequate to the structure of the considered state model. This is justified by the following facts:(i)
The compensator gainN¯ is designed to eliminate the steady-state error for a time-varying set point reference input vector:(44)e∞=yc∞−y∞=0.(ii)
The control gainsKi and Ljl for each i∈1,…,q1,j∈1…s1,l∈1…v1 are designed to ensure the stability performance of the following closed-loop nonlinear polynomial time delays system under input control saturation u¯t:(45)x˙t=∑i=1q1Aixit+∑j=1s1∑l=1v1Djlxjt−τl+∑i=1q1∑j=1s1∑l=1v1Gijlxit⊗xjt−τl+∑i=1q2Bisatu¯t⊗xit+∑j=1s2∑l=1v2Cjlsatu¯t⊗xjt−τl+∑i=1q2∑j=1s2∑l=1v2Qijlsatu¯t⊗xit⊗xjt−τl+Bsatu¯t,with(46)u¯t=−∑i=1q1Kixit−∑j=1s1∑l=1v1Ljlxjt−τl.


## 4. Major Outcomes

### 4.1. Proposed Tracking Control Approach for Nonlinear Polynomial Time Delays System under Input Saturation

Let us define the following matrices which depend only on control parameters:(47)∀i∈1…q1,∀j∈1…s1,∀l∈1…v1,Mi=A¯i−B¯K¯i,Hjl=D¯jl−B¯L¯jl.(48)∀i∈1…q1,∀j∈1…q2,L¯ij=B¯jK¯i⊗In+wpj,(49)∀j∈1…s1,∀l∈1…v1,∀i∈1…s2,∀h∈1…v2,V¯jlih=C¯ihL¯jl⊗In+wpi,(50)∀i∈1…q1,∀h∈1…q2,∀j∈1…s2,∀l∈1…v2,U¯ijl=C¯jlK¯i⊗In+wpj,Y¯ihjl=Q¯hjlK¯i⊗In+wph+j,(51)∀j∈1…s1,∀l∈1…v1,∀i∈1…q2,∀h∈1…s2,∀c∈1…v2N¯jli=B¯iL¯jl⊗In+wpi,Z¯jlihc=Q¯ihcL¯jl⊗In+wpi+h.By substituting the constrained inputs controlsatut by the unconstrained inputs control ut in state equation (39) and by taking into account the relations (41) and (47)–(51), it comes out:(52)X˙t=∑i=1q1MiXit+∑j=1s1∑l=1v1HjlXjt−τl+∑i=1q1∑j=1s1∑l=1v1G¯ijlXit⊗Xjt−τl+E¯yct−∑i=1q1∑j=1q2L¯ijXit⊗Xjt−∑j=1s1∑l=1v1∑i=1q2N¯jliXjt−τl⊗Xit−∑i=1q1∑j=1s2∑l=1v2U¯ijlXit⊗Xjt−τl−∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihXjt−τl⊗Xit−τh−∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlXit⊗Xht⊗Xjt−τl−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcXjt−τl⊗Xit⊗Xht−τc.With respect tot over the interval 0,T, the integration of equation (52) leads to(53)Xt−X0=∑i=1q1Mi∫0tXiσdσ+∑j=1s1∑l=1v1Hjl∫0tXjσ−τldσ+∑i=1q1∑j=1s1∑l=1v1G¯ijl∫0tXiσ⊗Xjσ−τldσ+E¯∫0tycσdσ−∑i=1q1∑j=1q2L¯ij∫0tXiσ⊗Xjσdσ−∑j=1s1∑l=1v1∑i=1q2N¯jli∫0tXjσ−τl⊗Xiσdσ−∑i=1q1∑j=1s2∑l=1v2U¯ijl∫0tXiσ⊗Xjσ−τldσ−∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlih∫0tXjσ−τl⊗Xiσ−τhdσ−∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjl∫0tXiσ⊗Xhσ⊗Xjσ−τldσ−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihc∫0tXjσ−τl⊗Xiσ⊗Xhσ−τcdσ.Over the basis of the block-pulse functions truncated to an orderN, the expansion of the fixed reference input vector yct can be done as follows:(54)yct=Y1,cNSNt,where Y1,cN denote reference input coefficients, computing from the scalar product (15).We indicate that the essential key is based on the equalization of the undelayed state vector of the controlled augmented system and the state vector of the reference model. By way of explanation∀t∈0,T:(55)Xt=Xrt=X1,rNSNt,where X1,rN denote state coefficients, computing from the scalar product (15).As a result, the delayed state vector of the controlled augmented systemXt−τl, for each l∈1…v, can be expressed as follows:(56)Xt−τl=ζ¯tfor0≤t≤τlXt−τl=Xrt−τlforτl<t≤T,and then, its expansion over the considered basis leads to(57)Xt−τl=X1,rN∗τlSNt,where X1,rN∗τl denote the delayed state coefficients, computing from the scalar product (20).Using the operational matrix of product, theith Kronecker power of Xt could be expressed as follows:(58)X2t=Xt⊗Xt=X1,rNSNt⊗X1,rNSNt=X1,rN⊗X1,rNSNt⊗SNt=X1,rN⊗X1,rNMNSNt=X2,rNSNt⋮Xq1t=Xq1−1t⊗Xt=Xq1−1,rNSNt⊗X1,rNSNt=Xq1−1,rN⊗X1,rNSNt⊗SNt=Xq1−1,rN⊗X1,rNMNSNt=Xq1,rNSNt.In the same way, thejth Kronecker power of Xt−τl, for each l∈1…v1, could be expressed as follows:(59)X2t−τl=Xt−τl⊗Xt−τl=X1,rN∗τlSNt⊗X1,rN∗τlSNt=X1,rN∗τl⊗X1,rN∗τlSNt⊗SNt=X1,rN∗τl⊗X1,rN∗τlMNSNt=X2,rN∗τlSNt⋮Xs1t−τl=Xs1−1t−τl⊗Xt−τl=Xs1−1,rN∗τlSNt⊗X1,rN∗τlSNt=Xs1−1,rN∗τl⊗X1,rN∗τlSNt⊗SNt=Xs1−1,rN∗τl⊗X1,rN∗τlMNSNt=Xs1,rN∗τlSNt.In addition to this, for eachi∈1…q1, j∈1…s1, and l∈1…v1, the Kronecker product terms Xit⊗Xjt−τl can be expressed as follows:(60)Xit⊗Xjt−τl=Xi,rNSNt⊗Xj,rN∗τlSNt=Xi,rN⊗Xj,rN∗τlSNt⊗SNt=Xi,rN⊗Xj,rN∗τlMNSNt.As well, for eachi∈1…q1 and j∈1…q2, the Kronecker product terms Xit⊗Xjt can be also expressed as follows:(61)Xit⊗Xjt=Xi,rNSNt⊗Xj,rNSNt=Xi,rN⊗Xj,rNSNt⊗SNt=Xi,rN⊗Xj,rNMNSNt.As well, for eachi∈1…q2, j∈1…s1, and l∈1…v1, the Kronecker product terms Xjt−τl⊗Xit can be also expressed as follows:(62)Xjt−τl⊗Xit=Xj,rN∗τlSNt⊗Xi,rNSNt=Xj,rN∗τl⊗Xi,rNSNt⊗SNt=Xj,rN∗τl⊗Xi,rNMNSNt.As well, for eachi∈1…q1, j∈1…s2, and l∈1…v2, the Kronecker product terms Xit⊗Xjt−τl can be also expressed as follows:(63)Xit⊗Xjt−τl=Xi,rNSNt⊗Xj,rN∗τlSNt=Xi,rN⊗Xj,rN∗τlSNt⊗SNt=Xi,rN⊗Xj,rN∗τlMNSNt.As well, for eachi∈1…s2, j∈1…s1, l∈1…v1, and h∈1…v2, the Kronecker product terms Xjt−τl⊗Xit−τh can be also expressed as follows:(64)Xjt−τl⊗Xit−τh=Xj,rN∗τlSNt⊗Xi,rN∗τhSNt=Xj,rN∗τl⊗Xi,rN∗τhSNt⊗SNt=Xj,rN∗τl⊗Xi,rN∗τhMNSNt.As well, for eachi∈1…q1, h∈1…q2, j∈1…s2, and l∈1…v2, the Kronecker product terms Xit⊗Xht⊗Xjt−τl can be also expressed as follows:(65)Xit⊗Xht⊗Xjt−τl=Xi,rNSNt⊗Xh,rNSNt⊗Xj,rN∗τlSNt=Xi,rN⊗Xh,rN⊗Xj,rN∗τlSNt⊗SNt⊗SNt=Xi,rN⊗Xh,rN⊗Xj,rN∗τlIN⊗MNMNSNt.As well, for eachi∈1…q2, h∈1…s2, j∈1…s1, l∈1…v1, and c∈1…v2, the Kronecker product terms Xjt−τl⊗Xit⊗Xht−τc can be also expressed as follows:(66)Xjt−τl⊗Xit⊗Xht−τc=Xj,rN∗τlSNt⊗Xi,rNSNt⊗Xh,rN∗τcSNt=Xj,rN∗τl⊗Xi,rN⊗Xh,rN∗τcSNt⊗SNt⊗SNt=Xj,rN∗τl⊗Xi,rN⊗Xh,rN∗τcIN⊗MNMNSNt.For simplicity, let us denote from relations (60)–(66) the following constants matrices:(67)∀i∈1…q1,∀j∈1…s1,∀l∈1…v1,Gijl=Xi,rN⊗Xj,rN∗τlMN,(68)∀i∈1…q1,∀j∈1…q2,ℒij=Xi,rN⊗Xj,rNMN,(69)∀j∈1…s1,∀l∈1…v1,∀i∈1…s2,∀h∈1…v2,Vjlih=Xj,rN∗τl⊗Xi,rN∗τhMN,(70)∀i∈1…q1,∀h∈1…q2,∀j∈1…s2,∀l∈1…v2,Uijl=Xi,rN⊗Xj,rN∗τlMN,Yihjl=Xi,rN⊗Xh,rN⊗Xj,rN∗τlIN⊗MNMN,(71)∀j∈1…s1,∀l∈1…v1,∀i∈1…q2,∀h∈1…s2,∀c∈1…v2Njli=Xj,rN∗τl⊗Xi,rNMN,Zjlihc=Xj,rN∗τl⊗Xi,rN⊗Xh,rN∗τcIN⊗MNMN.Then, it results from the expansion of the whole state model (53) over the considered orthogonal functions basis:(72)X1,rNSNt−X0NSNt=∑i=1q1MiXi,rN∫0tSNσdσ+∑j=1s1∑l=1v1HjlXj,rN∗τl∫0tSNσdσ+∑i=1q1∑j=1s1∑l=1v1G¯ijlGijl∫0tSNσdσ+E¯Y1,cN∫0tSNσdσ−∑i=1q1∑j=1q2L¯ijℒij∫0tSNσdσ−∑j=1s1∑l=1v1∑i=1q2N¯jliNjli∫0tSNσdσ−∑i=1q1∑j=1s2∑l=1v2u¯ijlUijl∫0tSNσdσ−∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihVjlih∫0tSNσdσ−∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlYihjl∫0tSNσdσ−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcZjlihc∫0tSNσdσ.where X0N=x00n,1…0n,10wp,10wp,1…0wp,1∈Rn+wp×N.The application of the integration operational matrixPN, defined by relation (22), leads to(73)X1,rNSNt−X0NSNt=∑i=1q1MiXi,rNPNSNt+∑j=1s1∑l=1v1HjlXj,rN∗τlPNSNt+∑i=1q1∑j=1s1∑l=1v1G¯ijlGijlPNSNt+E¯Y1,cNPNSNt−∑i=1q1∑j=1q2L¯ijLijPNSNt−∑j=1s1∑l=1v1∑i=1q2N¯jliNjliPNSNt−∑i=1q1∑j=1s2∑l=1v2U¯ijlUijlPNSNt−∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihVjlihPNSNt−∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlYihjlPNSNt−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcZjlihcPNSNt.By removing the block-pulse vector basisSNt in both sides of the last relation (73), we attain the following algebraic equation:(74)X1,rN−X0N=∑i=1q1MiXi,rNPN+∑j=1s1∑l=1v1HjlXj,rN∗τlPN+∑i=1q1∑j=1s1∑l=1v1G¯ijlGijlPN+E¯Y1,cNPN−∑i=1q1∑j=1q2L¯ijℒijPN−∑j=1s1∑l=1v1∑i=1q2N¯jliNjliPN−∑i=1q1∑j=1s2∑l=1v2U¯ijlUijlPN−∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihVjlihPN−∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlYihjlPN−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcZjlihcPN.By taking into account the relations (47)–(51), it results the following algebraic relation, where the only unknowns are the control law parameters:(75)X1,rN−X0N=∑i=1q1A¯i−B¯K¯iXi,rNPN+∑j=1s1∑l=1v1D¯jl−B¯L¯jlXj,rN∗τlPN+∑i=1q1∑j=1s1∑l=1v1G¯ijlGijlPN+E¯Y1,cNPN−∑i=1q1∑j=1q2B¯jK¯i⊗In+wpjℒijPN−∑j=1s1∑l=1v1∑i=1q2B¯iL¯jl⊗In+wpiNjliPN−∑i=1q1∑j=1s2∑l=1v2C¯jlK¯i⊗In+wpjUijlPN−∑j=1s1∑l=1v1∑i=1s2∑h=1v2C¯ihL¯jl⊗In+wpiVjlihPN−∑i=1q1∑h=1q2∑j=1s2∑l=1v2Q¯hjlK¯i⊗In+wph+jYihjlPN−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Q¯ihcL¯jl⊗In+wpi+hZjlihcPN.Using the vectorization operatorvec (see Appendix B in [34]) allows us to develop the last relation (75) in the following form:(76)β=−∑i=1q1χivecK¯i−∑j=1s1∑l=1v1ηjlvecL¯jl−∑i=1q1∑j=1q2ξijvecK¯i⊗In+wpj−∑j=1s1∑l=1v1∑i=1q2σjlivecL¯jl⊗In+wpi−∑i=1q1∑j=1s2∑l=1v2ζijlvecK¯i⊗In+wpj−∑j=1s1∑l=1v1∑i=1s2∑h=1v2μjlihvecL¯jl⊗In+wpi−∑i=1q1∑h=1q2∑j=1s2∑l=1v2ϖihjlvecK¯i⊗In+wph+j−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2κjlihcvecL¯jl⊗In+wpi+h,with β=vecX1,rN−vecX0N−∑i=1q1PNT⊗A¯ivecXi,rN−PNT⊗E¯vecY1,cN−∑j=1s1∑l=1v1PNT⊗D¯jlvecXj,rN∗τl−∑i=1q1∑j=1s1∑l=1v1PNT⊗G¯ijlvecGijl, where for each i∈1…q1, j∈1…s1, l∈1…v1: χi=Xi,rNPNT⊗B¯,ηjl=Xj,rN∗τlPNT⊗B¯ and for each i∈1…q1 and j∈1…q2ξij=ℒijPNT⊗B¯j and for each j∈1…s1, l∈1…v1, i∈1…s2, and h∈1…v2μjlih=VjlihPNT⊗C¯ih and for each i∈1…q1, h∈1…q2, j∈1…s2, l∈1…v2: ζijl=UijlPNT⊗C¯jl,ϖihjl=YihjlPNT⊗Q¯hjl and for each j∈1…s1, l∈1…v1, i∈1…q2, h∈1…s2, c∈1…v2: σjli=NjliPNT⊗B¯i,κjlihc=ZjlihcPNT⊗Q¯ihc.With the famous property of the vectorization operator, given for each matrixA∈ℝm×n by(77)vecA⊗Ip=vecE11m×n⊗Ip⋮…⋮vecEm1m×n⊗Ip⋮vecE22m×n⊗Ip⋮…⋮vecEm2m×n⊗Ip⋮…⋮vecE1nm×n⊗Ip⋮…⋮vecEmnm×n⊗Ip=Πm,nIpvecA,where Eijm×n is the elementary matrix defined in relation (23), it results the following outcome from the previous relation (76):(78)β=−∑i=1q1χivecK¯i−∑j=1s1∑l=1v1ηjlvecL¯jl−∑i=1q1∑j=1q2ξijΠm,n+wpiIn+wpjvecK¯i−∑j=1s1∑l=1v1∑i=1q2σjliΠm,n+wpjIn+wpivecL¯jl−∑i=1q1∑j=1s2∑l=1v2ζijlΠm,n+wpiIn+wpjvecK¯i−∑j=1s1∑l=1v1∑i=1s2∑h=1v2μjlihΠm,n+wpjIn+wpivecL¯jl−∑i=1q1∑h=1q2∑j=1s2∑l=1v2ϖihjlΠm,n+wpiIn+wph+jvecK¯i−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2κjlihcΠm,n+wpjIn+wpi+hvecL¯jl.It should be borne in mind that our goal is the gain determination of the adopted control law. Thus, from relations (42) and (43), we can infer that(79)vecK¯1=Imn0mwp,mnvecK1−0mn,mwpImwpvecN¯,and for each i∈2…q1, j∈1…s1, l∈1…v1:(80)vecK¯i=ΨiT⊗ImvecKi,vecL¯jl=ΨjT⊗ImvecLjl.Consequently, relation (78) can be transformed into the following form:(81)β=α1vecN¯−∑i=1q1χ¯ivecKi−∑j=1s1∑l=1v1η¯jlvecLjl,with α1=α1∗0mn,mwpImwp,χ¯1=χ¯1∗Imn0mwp,mn where(82)α1∗=χ1+∑j=1q2ξ1jΠm,n+wpIn+wpj+∑j=1s2∑l=1v2ζ1jlΠm,n+wpIn+wpj+∑h=1q2∑j=1s2∑l=1v2ϖ1hjlΠm,n+wpIn+wph+j,χ¯1∗=χ1+∑j=1q2ξ1jΠm,n+wpIn+wpj+∑j=1s2∑l=1v2ζ1jlΠm,n+wpIn+wpj+∑h=1q2∑j=1s2∑l=1v2ϖ1hjlΠm,n+wpIn+wph+j,and for each i∈2…q1, j∈1…s1, l∈1…v1: χ¯i=χ¯i∗ΨiT⊗Im,η¯jl=η¯jl∗ΨjT⊗Im where χ¯i∗=χi+∑j=1q2ξijΠm,n+wpiIn+wpj+∑j=1s2∑l=1v2ζijlΠm,n+wpiIn+wpj+∑h=1q2∑j=1s2∑l=1v2ϖihjlΠm,n+wpiIn+wph+jη¯jl∗=ηjl+∑i=1q2σjliΠm,n+wpjIn+wpi+∑i=1s2∑h=1v2μjlihΠm,n+wpjIn+wpi+∑i=1q2∑h=1s2∑c=1v2κjlihcΠm,n+wpjIn+wpi+h.Lettingα2=−χ¯1χ¯2…χ¯q1 and ∀j∈1…s1,γj=−η¯j1η¯j2…η¯jv1then, from equation (81), it comes out(83)β=α1vecN¯+α2θK+∑j=1s1γjθLj,with θK=vecK1⋮vecKq1 and ∀j∈1…s1,θLj=vecLj1⋮vecLjv1 which is equivalent to(84)β=α1vecN¯+α2θK+α3θL,where α3=γ1γ2…γs1andθL=θL1⋮θLs1.It is appreciable to systematize such problem as follows:(85)Aθ=B,where constant matrix A, constant vector B, and vector of searched parameters θ are given by(86)A=α1α2α3,B=β,θ=vecN¯θKθL.Now, by taking into account that the control lawut has to respect upper and lower bounds (30), and then from relation (41), we get the two following inequalities:(87)−∑i=1q1K¯iXit−∑j=1s1∑l=1v1L¯jlXjt−τl≤umax∑i=1q1K¯iXit+∑j=1s1∑l=1v1L¯jlXjt−τl≤vmax.Undertaken the equality (55), the expansion of the latter inequalities over the considered block-pulse functions basis and the use of the vec operator yield to the following inequality system:(88)Cθ≤d,where constant matrix C and constant vector d are given as follows:(89)C=−D1−D2−D3D1D2D3,d=vecUmax,NvecVmax,N.with(90)D1=−X1,rNT⊗Im0mn,mwpImwp,D2=X1,rNT⊗ImImn0mwp,mnD¯2,D3=d1…ds1,where(91)D¯2=Ψ2X2,rNT⊗Im…Ψq1Xq1,rNT⊗Im,dj=ΨjXj,rN∗τ1T⊗Im…ΨjXj,rN∗τv1T⊗Im,and Umax,N and Vmax,N denote the upper bound coefficients, which are computed from the scalar product (15).In the end, the posed control problem has been effectively reduced to an overdetermined linear system subject to an inequality constraint, which can be formulated from equality (85) and inequality (88) as follows:(92a)Aθ=B,subject to(92b)Cθ≤d.The above constrained linear system is ofn+wpN algebraic equations subject to 2mN algebraic constraints with respect to mwp+∑i=1q1ni+v1∑j=1s1nj unknowns.This kind of system can be easily solved in the constrained least square sense, by the formulation as a quadratic programming problem as follows:(93a)minθAθ−B,subject to(93b)Cθ≤d.


### 4.2. Practical Stability Problem Analysis

Presently, the control parametersN¯, Ki, and Ljl can be found by solving the resulted quadratic programming problem (93a)-(93b) for fixed reference input yct. Thereafter, we intend to analyze the practical stability of the closed-loop augmented system (39).For this purpose, let us consider the following constants matrices:(94)∀i∈1…q1,∀j∈1…s1,∀l∈1…v1,ℳi=A¯i−12B¯K¯i,ℋjl=D¯jl−12B¯L¯jl.Definition 1.
The controlled system (39) is called practically stable if every bounded input vector produces a bounded states vector. In other words, there exist 0<R0<r, such that (see [45]):(95)X0<R0⇒Xt<r,∀t≥0.Theorem 1.
The controlled system (39) is practical stable if all eigenvalues of matrix ℳ1 have a strictly negative real part and if(96)λ1∑l=1v1ℋ1l+ω1<0,(97)X0<R0,where R0>0, λ1>0 and ω1<0 are given scalars satisfying:(98)eℳ1t≤λ1eω1t,∀t≥0,(99)R0=R1λ1,with R1 is the unique positive solution of the following equation:(100)fR1+λ1∑l=1v1ℋ1l+ω1=0,where(101)fR1=λ1∑i=2q1MiR1i−1+∑j=2s1∑l=1v1HjlR1j−1+∑i=1q1∑j=1s1∑l=1v1G¯ijlR1i+j−1+λ12∑i=1q1∑j=1q2L¯ijR1i+j−1+∑j=1s1∑l=1v1∑i=1q2N¯jliR1j+i−1+∑i=1q1∑j=1s2∑l=1v2U¯ijlR1i+j−1+λ12∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihR1j+i−1+∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlR1i+h+j−1+λ12∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcR1j+i+h−1+∑i=1q2∑j=1q1B¯iK¯jR1i+j−1+λ12∑i=1q2∑j=1s1∑l=1v1B¯iL¯jlR1i+j−1+∑j=1s2∑l=1v2∑i=1q1C¯jlK¯iR1i+j−1+λ12∑j=1s2∑l=1v2∑i=1s1∑h=1v1C¯jlL¯ihR1i+j−1+∑i=1q2∑j=1s2∑l=1v2∑h=1q1Q¯ijlK¯hR1i+j+h−1+λ12∑i=1q2∑j=1s2∑l=1v2∑h=1s1∑c=1v1Q¯ijlL¯hcR1i+j+h−1.Proof 1.
By taking into account thatsatut=u¯t+1/2ut with u¯t=satut−1/2ut, the augmented closed-loop system can be written from relations (39), (41), (94), and (48)–(51) as follows:(102)X˙t−M1Xt=∑i=2q1MiXit+∑l=1v1H1lXt−τl+∑j=2s1∑l=1v1HjlXjt−τl+B¯u¯t+∑i=1q1∑j=1s1∑l=1v1G¯ijlXitXjt−τl+E¯yct−12∑i=1q1∑j=1q2L¯ijXit⊗Xjt−12∑j=1s1∑l=1v1∑i=1q2N¯jliXjt−τl⊗Xit−12∑i=1q1∑j=1s2∑l=1v2U¯ijlXit⊗Xjt−τl−12∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihXjt−τl⊗Xit−τh−12∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlXit⊗Xht⊗Xjt−τl−12∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcXjt−τl⊗Xit⊗Xht−τc+∑i=1q2B¯iu¯t⊗Xit+∑j=1s2∑l=1v2C¯jlu¯t⊗Xjt−τl+∑i=1q2∑j=1s2∑l=1v2Q¯ijlu¯t⊗Xit⊗Xjt−τl.
By respecting thatℳ1e−ℳ1t=e−ℳ1tℳ1 which can be deduced from the series definition, we get the following:(103)ddte−ℳ1tXt=−ℳ1e−ℳ1tXt+e−ℳ1tX˙t=e−ℳ1tX˙t−ℳ1Xt.Thus, by integrating the last differential equation from 0 to t, it comes out:(104)e−ℳ1tXt−X0=∫0te−ℳ1σX˙σ−ℳ1Xσdσ,or equivalently(105)Xt=eℳ1tX0+∫0teℳ1t−σX˙σ−ℳ1Xσdσ.
Since all eigenvalues of matrixℳ1 must have the negative real part, then there exist two reals λ1>0 and ω1<0 such that relation (98) holds on. Consequently, the state vector Xt can be bounded as:(106)Xt≤λ1eω1tX0+λ1eω1t∫0te−ω1σX˙σ−ℳ1Xσdσ,which is similar to the following:(107)e−ω1tXt≤λ1X0+λ1∫0te−ω1σX˙σ−ℳ1Xσdσ.
Let us suppose that(108)Xt<R.
Through the following matrix norm properties:(109)Xit<Ri−1Xt,Xjt−τl<Rj−1Xt−τlXit⊗Xjt−τl<Ri+j−1Xt−τl,Xit⊗Xjt<Ri+j−1XtXjt−τl⊗Xit<Rj+i−1XtXiσ⊗Xhσ⊗Xjσ−τl<Ri+h+j−1Xt−τlXjσ−τl⊗Xiσ⊗Xhσ−τc<Rj+i+h−1Xt−τc.then we can deduce, from inequality (107) and relation (102), the following expression:(110)e−ω1tXt<λ1X0+λ1∑i=2q1MiRi−1∫0te−ω1σXσdσ+λ1∑l=1v1H1l∫0te−ω1σXσ−τldσ+λ1∑j=2s1∑l=1v1HjlRj−1∫0te−ω1σXσ−τldσ+λ1∑i=1q1∑j=1s1∑l=1v1G¯ijlRi+j−1∫0te−ω1σXσ−τldσ+λ1E¯δ∫0te−ω1σdσ+λ1B¯δ¯∫0te−ω1σdσ+λ12∑i=1q1∑j=1q2L¯ijRi+j−1∫0te−ω1σXσdσ+λ12∑j=1s1∑l=1v1∑i=1q2N¯jliRj+i−1∫0te−ω1σXσdσ+λ12∑i=1q1∑j=1s2∑l=1v2U¯ijlRi+j−1∫0te−ω1σXσ−τldσ+λ12∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihRj+i−1∫0te−ω1σXσ−τhdσ+λ12∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlRi+h+j−1∫0te−ω1σXσ−τldσ+λ12∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcRj+i+h−1∫0te−ω1σXσ−τcdσ+λ1∑i=1q2B¯iRi∫0te−ω1σu¯σdσ+λ1∑j=1s2∑l=1v2C¯jlRj∫0te−ω1σu¯σdσ+λ1∑i=1q2∑j=1s2∑l=1v2Q¯ijlRi+j∫0te−ω1σu¯σdσ.where δ>0 satisfying yct<δ, and δ¯=maxvmax,umax.
Now by taking into account the following special property of function saturation (see [46]):(111)u¯t=satut−12ut≤12ut,which can be written from relation (41) and condition (108) as follows:(112)u¯t<12∑i=1q1K¯iXit+12∑j=1s1∑l=1v1L¯jlXjt−τl<12∑i=1q1K¯iRi−1Xt+12∑j=1s1∑l=1v1L¯jlRj−1Xt−τl.
It results the following inequality:(113)e−ω1tXt<λ1X0+λ1∑i=2q1MiRi−1∫0te−ω1σXσdσ+λ1∑l=1v1H1l∫0te−ω1σXσ−τhdσ+λ1∑j=2s1∑l=1v1HjlRj−1∫0te−ω1σXσ−τhdσ+λ1∑i=1q1∑j=1s1∑l=1v1G¯ijlRi+j−1∫0te−ω1σXσ−τhdσ+λ1E¯δ∫0te−ω1σdσ+λ1B¯δ¯∫0te−ω1σdσ+λ12∑i=1q1∑j=1q2L¯ijRi+j−1∫0te−ω1σXσdσ+λ12∑j=1s1∑l=1v1∑i=1q2N¯jliRj+i−1∫0te−ω1σXσdσ+λ12∑i=1q1∑j=1s2∑l=1v2U¯ijlRi+j−1∫0te−ω1σXσ−τldσ+λ12∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihRj+i−1∫0te−ω1σXσ−τhdσ+λ12∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlRi+h+j−1∫0te−ω1σXσ−τldσ+λ12∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcRj+i+h−1∫0te−ω1σXσ−τcdσ+λ12∑i=1q2∑j=1q1B¯iK¯jRi+j−1∫0te−ω1σXσdσ+λ12∑i=1q2∑j=1s1∑l=1v1B¯iL¯jlRi+j−1∫0te−ω1σXσ−τldσ+λ12∑j=1s2∑l=1v2∑i=1q1C¯jlK¯iRi+j−1∫0te−ω1σXσdσ+λ12∑j=1s2∑l=1v2∑i=1s1∑h=1v1C¯jlL¯ihRi+j−1∫0te−ω1σXσ−τhdσ+λ12∑i=1q2∑j=1s2∑l=1v2∑h=1q1Q¯ijlK¯hRi+j+h−1∫0te−ω1σXσdσ+λ12∑i=1q2∑j=1s2∑l=1v2∑h=1s1∑c=1v1Q¯ijlL¯hcRi+j+h−1∫0te−ω1σXσ−τcdσ.
Let us introduce the following function:(114)zt=supρ∈t−τ¯,tXρe−ω1t∀t≥0.then under considering that ∀t≥0 and l∈1…v: Xt≤supρ∈t−τ¯,tXρ and Xt−τl≤supρ∈t−τ¯,tXρ it turns out, from inequality (113), that(115)e−ω1tXt<λ1X0+fR+λ1∑l=1v1ℋ1l∫0tzσdσ+Ω∫0te−ω1σdσ,where function f. is given by relation (101) and Ω=λ1E¯δ+λ1B¯δ¯.
Now, by introducing a new functiongt as the right-hand member of inequality (115):(116)gt=λ1X0+fR+λ1∑l=1v1ℋ1l∫0tzσdσ+Ω∫0te−ω1σdσ.we can show that(117)∀t≥0,zt<gt.
By deriving the functiongt:(118)dgtdt=fR+λ1∑l=1v1ℋ1lzt+Ωe−ω1t,it comes out from inequality (117) that(119)dgtdt<fR+λ1∑l=1v1ℋ1lgt+Ωe−ω1t.
By integrating the last inequality from 0 to t, it follows, based on the use of the Bellman-Gronwall inequality (see appendix D in [34]) that(120)gt<efR+λ1∑l=1v1ℋ1ltg0+ΩfR+λ1∑l=1v1ℋ1l+ω1efR+λ1∑l=1v1ℋ1lt−e−ω1t,where g0=λ1X0.
Inequalities (117) and (120) imply(121)zt<λ1X0efR+λ1∑l=1v1ℋ1lt+ΩfR+λ1∑l=1v1ℋ1l+ω1efR+λ1∑l=1v1ℋ1lt−e−ω1t.
From the definition of functionzt, we can derived that(122)Xt<λ1X0efR+λ1∑l=1v1ℋ1l+ω1t+ΩfR+λ1∑l=1v1ℋ1l+ω1efR+λ1∑l=1v1ℋ1l+ω1t−1.
If condition (96) holds on, then for(123)fR+λ1∑l=1v1ℋ1l+ω1<0.which is verified for each radius R satisfying(124)R<R1,where R1 is the unique positive solution of the equality (100), we can make out the following result:(125)Xt<r=λ1X0−ΩfR+λ1∑l=1v1ℋ1l+ω1.
At this point, in order to make sure the hypothesis (108) for all t, it is enough to have(126)λ1X0<R1.
Consequent, from inequality (125) and condition (126), it allows that(127)X0<R0=R1λ1⇒Xt<r=λ1R0−ΩfR+λ1∑l=1v1ℋ1l+ω1.
Hence, the controlled system (39) is practically stable.


## 4.1. Proposed Tracking Control Approach for Nonlinear Polynomial Time Delays System under Input Saturation

Let us define the following matrices which depend only on control parameters:(47)∀i∈1…q1,∀j∈1…s1,∀l∈1…v1,Mi=A¯i−B¯K¯i,Hjl=D¯jl−B¯L¯jl.(48)∀i∈1…q1,∀j∈1…q2,L¯ij=B¯jK¯i⊗In+wpj,(49)∀j∈1…s1,∀l∈1…v1,∀i∈1…s2,∀h∈1…v2,V¯jlih=C¯ihL¯jl⊗In+wpi,(50)∀i∈1…q1,∀h∈1…q2,∀j∈1…s2,∀l∈1…v2,U¯ijl=C¯jlK¯i⊗In+wpj,Y¯ihjl=Q¯hjlK¯i⊗In+wph+j,(51)∀j∈1…s1,∀l∈1…v1,∀i∈1…q2,∀h∈1…s2,∀c∈1…v2N¯jli=B¯iL¯jl⊗In+wpi,Z¯jlihc=Q¯ihcL¯jl⊗In+wpi+h.By substituting the constrained inputs controlsatut by the unconstrained inputs control ut in state equation (39) and by taking into account the relations (41) and (47)–(51), it comes out:(52)X˙t=∑i=1q1MiXit+∑j=1s1∑l=1v1HjlXjt−τl+∑i=1q1∑j=1s1∑l=1v1G¯ijlXit⊗Xjt−τl+E¯yct−∑i=1q1∑j=1q2L¯ijXit⊗Xjt−∑j=1s1∑l=1v1∑i=1q2N¯jliXjt−τl⊗Xit−∑i=1q1∑j=1s2∑l=1v2U¯ijlXit⊗Xjt−τl−∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihXjt−τl⊗Xit−τh−∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlXit⊗Xht⊗Xjt−τl−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcXjt−τl⊗Xit⊗Xht−τc.With respect tot over the interval 0,T, the integration of equation (52) leads to(53)Xt−X0=∑i=1q1Mi∫0tXiσdσ+∑j=1s1∑l=1v1Hjl∫0tXjσ−τldσ+∑i=1q1∑j=1s1∑l=1v1G¯ijl∫0tXiσ⊗Xjσ−τldσ+E¯∫0tycσdσ−∑i=1q1∑j=1q2L¯ij∫0tXiσ⊗Xjσdσ−∑j=1s1∑l=1v1∑i=1q2N¯jli∫0tXjσ−τl⊗Xiσdσ−∑i=1q1∑j=1s2∑l=1v2U¯ijl∫0tXiσ⊗Xjσ−τldσ−∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlih∫0tXjσ−τl⊗Xiσ−τhdσ−∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjl∫0tXiσ⊗Xhσ⊗Xjσ−τldσ−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihc∫0tXjσ−τl⊗Xiσ⊗Xhσ−τcdσ.Over the basis of the block-pulse functions truncated to an orderN, the expansion of the fixed reference input vector yct can be done as follows:(54)yct=Y1,cNSNt,where Y1,cN denote reference input coefficients, computing from the scalar product (15).We indicate that the essential key is based on the equalization of the undelayed state vector of the controlled augmented system and the state vector of the reference model. By way of explanation∀t∈0,T:(55)Xt=Xrt=X1,rNSNt,where X1,rN denote state coefficients, computing from the scalar product (15).As a result, the delayed state vector of the controlled augmented systemXt−τl, for each l∈1…v, can be expressed as follows:(56)Xt−τl=ζ¯tfor0≤t≤τlXt−τl=Xrt−τlforτl<t≤T,and then, its expansion over the considered basis leads to(57)Xt−τl=X1,rN∗τlSNt,where X1,rN∗τl denote the delayed state coefficients, computing from the scalar product (20).Using the operational matrix of product, theith Kronecker power of Xt could be expressed as follows:(58)X2t=Xt⊗Xt=X1,rNSNt⊗X1,rNSNt=X1,rN⊗X1,rNSNt⊗SNt=X1,rN⊗X1,rNMNSNt=X2,rNSNt⋮Xq1t=Xq1−1t⊗Xt=Xq1−1,rNSNt⊗X1,rNSNt=Xq1−1,rN⊗X1,rNSNt⊗SNt=Xq1−1,rN⊗X1,rNMNSNt=Xq1,rNSNt.In the same way, thejth Kronecker power of Xt−τl, for each l∈1…v1, could be expressed as follows:(59)X2t−τl=Xt−τl⊗Xt−τl=X1,rN∗τlSNt⊗X1,rN∗τlSNt=X1,rN∗τl⊗X1,rN∗τlSNt⊗SNt=X1,rN∗τl⊗X1,rN∗τlMNSNt=X2,rN∗τlSNt⋮Xs1t−τl=Xs1−1t−τl⊗Xt−τl=Xs1−1,rN∗τlSNt⊗X1,rN∗τlSNt=Xs1−1,rN∗τl⊗X1,rN∗τlSNt⊗SNt=Xs1−1,rN∗τl⊗X1,rN∗τlMNSNt=Xs1,rN∗τlSNt.In addition to this, for eachi∈1…q1, j∈1…s1, and l∈1…v1, the Kronecker product terms Xit⊗Xjt−τl can be expressed as follows:(60)Xit⊗Xjt−τl=Xi,rNSNt⊗Xj,rN∗τlSNt=Xi,rN⊗Xj,rN∗τlSNt⊗SNt=Xi,rN⊗Xj,rN∗τlMNSNt.As well, for eachi∈1…q1 and j∈1…q2, the Kronecker product terms Xit⊗Xjt can be also expressed as follows:(61)Xit⊗Xjt=Xi,rNSNt⊗Xj,rNSNt=Xi,rN⊗Xj,rNSNt⊗SNt=Xi,rN⊗Xj,rNMNSNt.As well, for eachi∈1…q2, j∈1…s1, and l∈1…v1, the Kronecker product terms Xjt−τl⊗Xit can be also expressed as follows:(62)Xjt−τl⊗Xit=Xj,rN∗τlSNt⊗Xi,rNSNt=Xj,rN∗τl⊗Xi,rNSNt⊗SNt=Xj,rN∗τl⊗Xi,rNMNSNt.As well, for eachi∈1…q1, j∈1…s2, and l∈1…v2, the Kronecker product terms Xit⊗Xjt−τl can be also expressed as follows:(63)Xit⊗Xjt−τl=Xi,rNSNt⊗Xj,rN∗τlSNt=Xi,rN⊗Xj,rN∗τlSNt⊗SNt=Xi,rN⊗Xj,rN∗τlMNSNt.As well, for eachi∈1…s2, j∈1…s1, l∈1…v1, and h∈1…v2, the Kronecker product terms Xjt−τl⊗Xit−τh can be also expressed as follows:(64)Xjt−τl⊗Xit−τh=Xj,rN∗τlSNt⊗Xi,rN∗τhSNt=Xj,rN∗τl⊗Xi,rN∗τhSNt⊗SNt=Xj,rN∗τl⊗Xi,rN∗τhMNSNt.As well, for eachi∈1…q1, h∈1…q2, j∈1…s2, and l∈1…v2, the Kronecker product terms Xit⊗Xht⊗Xjt−τl can be also expressed as follows:(65)Xit⊗Xht⊗Xjt−τl=Xi,rNSNt⊗Xh,rNSNt⊗Xj,rN∗τlSNt=Xi,rN⊗Xh,rN⊗Xj,rN∗τlSNt⊗SNt⊗SNt=Xi,rN⊗Xh,rN⊗Xj,rN∗τlIN⊗MNMNSNt.As well, for eachi∈1…q2, h∈1…s2, j∈1…s1, l∈1…v1, and c∈1…v2, the Kronecker product terms Xjt−τl⊗Xit⊗Xht−τc can be also expressed as follows:(66)Xjt−τl⊗Xit⊗Xht−τc=Xj,rN∗τlSNt⊗Xi,rNSNt⊗Xh,rN∗τcSNt=Xj,rN∗τl⊗Xi,rN⊗Xh,rN∗τcSNt⊗SNt⊗SNt=Xj,rN∗τl⊗Xi,rN⊗Xh,rN∗τcIN⊗MNMNSNt.For simplicity, let us denote from relations (60)–(66) the following constants matrices:(67)∀i∈1…q1,∀j∈1…s1,∀l∈1…v1,Gijl=Xi,rN⊗Xj,rN∗τlMN,(68)∀i∈1…q1,∀j∈1…q2,ℒij=Xi,rN⊗Xj,rNMN,(69)∀j∈1…s1,∀l∈1…v1,∀i∈1…s2,∀h∈1…v2,Vjlih=Xj,rN∗τl⊗Xi,rN∗τhMN,(70)∀i∈1…q1,∀h∈1…q2,∀j∈1…s2,∀l∈1…v2,Uijl=Xi,rN⊗Xj,rN∗τlMN,Yihjl=Xi,rN⊗Xh,rN⊗Xj,rN∗τlIN⊗MNMN,(71)∀j∈1…s1,∀l∈1…v1,∀i∈1…q2,∀h∈1…s2,∀c∈1…v2Njli=Xj,rN∗τl⊗Xi,rNMN,Zjlihc=Xj,rN∗τl⊗Xi,rN⊗Xh,rN∗τcIN⊗MNMN.Then, it results from the expansion of the whole state model (53) over the considered orthogonal functions basis:(72)X1,rNSNt−X0NSNt=∑i=1q1MiXi,rN∫0tSNσdσ+∑j=1s1∑l=1v1HjlXj,rN∗τl∫0tSNσdσ+∑i=1q1∑j=1s1∑l=1v1G¯ijlGijl∫0tSNσdσ+E¯Y1,cN∫0tSNσdσ−∑i=1q1∑j=1q2L¯ijℒij∫0tSNσdσ−∑j=1s1∑l=1v1∑i=1q2N¯jliNjli∫0tSNσdσ−∑i=1q1∑j=1s2∑l=1v2u¯ijlUijl∫0tSNσdσ−∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihVjlih∫0tSNσdσ−∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlYihjl∫0tSNσdσ−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcZjlihc∫0tSNσdσ.where X0N=x00n,1…0n,10wp,10wp,1…0wp,1∈Rn+wp×N.The application of the integration operational matrixPN, defined by relation (22), leads to(73)X1,rNSNt−X0NSNt=∑i=1q1MiXi,rNPNSNt+∑j=1s1∑l=1v1HjlXj,rN∗τlPNSNt+∑i=1q1∑j=1s1∑l=1v1G¯ijlGijlPNSNt+E¯Y1,cNPNSNt−∑i=1q1∑j=1q2L¯ijLijPNSNt−∑j=1s1∑l=1v1∑i=1q2N¯jliNjliPNSNt−∑i=1q1∑j=1s2∑l=1v2U¯ijlUijlPNSNt−∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihVjlihPNSNt−∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlYihjlPNSNt−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcZjlihcPNSNt.By removing the block-pulse vector basisSNt in both sides of the last relation (73), we attain the following algebraic equation:(74)X1,rN−X0N=∑i=1q1MiXi,rNPN+∑j=1s1∑l=1v1HjlXj,rN∗τlPN+∑i=1q1∑j=1s1∑l=1v1G¯ijlGijlPN+E¯Y1,cNPN−∑i=1q1∑j=1q2L¯ijℒijPN−∑j=1s1∑l=1v1∑i=1q2N¯jliNjliPN−∑i=1q1∑j=1s2∑l=1v2U¯ijlUijlPN−∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihVjlihPN−∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlYihjlPN−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcZjlihcPN.By taking into account the relations (47)–(51), it results the following algebraic relation, where the only unknowns are the control law parameters:(75)X1,rN−X0N=∑i=1q1A¯i−B¯K¯iXi,rNPN+∑j=1s1∑l=1v1D¯jl−B¯L¯jlXj,rN∗τlPN+∑i=1q1∑j=1s1∑l=1v1G¯ijlGijlPN+E¯Y1,cNPN−∑i=1q1∑j=1q2B¯jK¯i⊗In+wpjℒijPN−∑j=1s1∑l=1v1∑i=1q2B¯iL¯jl⊗In+wpiNjliPN−∑i=1q1∑j=1s2∑l=1v2C¯jlK¯i⊗In+wpjUijlPN−∑j=1s1∑l=1v1∑i=1s2∑h=1v2C¯ihL¯jl⊗In+wpiVjlihPN−∑i=1q1∑h=1q2∑j=1s2∑l=1v2Q¯hjlK¯i⊗In+wph+jYihjlPN−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Q¯ihcL¯jl⊗In+wpi+hZjlihcPN.Using the vectorization operatorvec (see Appendix B in [34]) allows us to develop the last relation (75) in the following form:(76)β=−∑i=1q1χivecK¯i−∑j=1s1∑l=1v1ηjlvecL¯jl−∑i=1q1∑j=1q2ξijvecK¯i⊗In+wpj−∑j=1s1∑l=1v1∑i=1q2σjlivecL¯jl⊗In+wpi−∑i=1q1∑j=1s2∑l=1v2ζijlvecK¯i⊗In+wpj−∑j=1s1∑l=1v1∑i=1s2∑h=1v2μjlihvecL¯jl⊗In+wpi−∑i=1q1∑h=1q2∑j=1s2∑l=1v2ϖihjlvecK¯i⊗In+wph+j−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2κjlihcvecL¯jl⊗In+wpi+h,with β=vecX1,rN−vecX0N−∑i=1q1PNT⊗A¯ivecXi,rN−PNT⊗E¯vecY1,cN−∑j=1s1∑l=1v1PNT⊗D¯jlvecXj,rN∗τl−∑i=1q1∑j=1s1∑l=1v1PNT⊗G¯ijlvecGijl, where for each i∈1…q1, j∈1…s1, l∈1…v1: χi=Xi,rNPNT⊗B¯,ηjl=Xj,rN∗τlPNT⊗B¯ and for each i∈1…q1 and j∈1…q2ξij=ℒijPNT⊗B¯j and for each j∈1…s1, l∈1…v1, i∈1…s2, and h∈1…v2μjlih=VjlihPNT⊗C¯ih and for each i∈1…q1, h∈1…q2, j∈1…s2, l∈1…v2: ζijl=UijlPNT⊗C¯jl,ϖihjl=YihjlPNT⊗Q¯hjl and for each j∈1…s1, l∈1…v1, i∈1…q2, h∈1…s2, c∈1…v2: σjli=NjliPNT⊗B¯i,κjlihc=ZjlihcPNT⊗Q¯ihc.With the famous property of the vectorization operator, given for each matrixA∈ℝm×n by(77)vecA⊗Ip=vecE11m×n⊗Ip⋮…⋮vecEm1m×n⊗Ip⋮vecE22m×n⊗Ip⋮…⋮vecEm2m×n⊗Ip⋮…⋮vecE1nm×n⊗Ip⋮…⋮vecEmnm×n⊗Ip=Πm,nIpvecA,where Eijm×n is the elementary matrix defined in relation (23), it results the following outcome from the previous relation (76):(78)β=−∑i=1q1χivecK¯i−∑j=1s1∑l=1v1ηjlvecL¯jl−∑i=1q1∑j=1q2ξijΠm,n+wpiIn+wpjvecK¯i−∑j=1s1∑l=1v1∑i=1q2σjliΠm,n+wpjIn+wpivecL¯jl−∑i=1q1∑j=1s2∑l=1v2ζijlΠm,n+wpiIn+wpjvecK¯i−∑j=1s1∑l=1v1∑i=1s2∑h=1v2μjlihΠm,n+wpjIn+wpivecL¯jl−∑i=1q1∑h=1q2∑j=1s2∑l=1v2ϖihjlΠm,n+wpiIn+wph+jvecK¯i−∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2κjlihcΠm,n+wpjIn+wpi+hvecL¯jl.It should be borne in mind that our goal is the gain determination of the adopted control law. Thus, from relations (42) and (43), we can infer that(79)vecK¯1=Imn0mwp,mnvecK1−0mn,mwpImwpvecN¯,and for each i∈2…q1, j∈1…s1, l∈1…v1:(80)vecK¯i=ΨiT⊗ImvecKi,vecL¯jl=ΨjT⊗ImvecLjl.Consequently, relation (78) can be transformed into the following form:(81)β=α1vecN¯−∑i=1q1χ¯ivecKi−∑j=1s1∑l=1v1η¯jlvecLjl,with α1=α1∗0mn,mwpImwp,χ¯1=χ¯1∗Imn0mwp,mn where(82)α1∗=χ1+∑j=1q2ξ1jΠm,n+wpIn+wpj+∑j=1s2∑l=1v2ζ1jlΠm,n+wpIn+wpj+∑h=1q2∑j=1s2∑l=1v2ϖ1hjlΠm,n+wpIn+wph+j,χ¯1∗=χ1+∑j=1q2ξ1jΠm,n+wpIn+wpj+∑j=1s2∑l=1v2ζ1jlΠm,n+wpIn+wpj+∑h=1q2∑j=1s2∑l=1v2ϖ1hjlΠm,n+wpIn+wph+j,and for each i∈2…q1, j∈1…s1, l∈1…v1: χ¯i=χ¯i∗ΨiT⊗Im,η¯jl=η¯jl∗ΨjT⊗Im where χ¯i∗=χi+∑j=1q2ξijΠm,n+wpiIn+wpj+∑j=1s2∑l=1v2ζijlΠm,n+wpiIn+wpj+∑h=1q2∑j=1s2∑l=1v2ϖihjlΠm,n+wpiIn+wph+jη¯jl∗=ηjl+∑i=1q2σjliΠm,n+wpjIn+wpi+∑i=1s2∑h=1v2μjlihΠm,n+wpjIn+wpi+∑i=1q2∑h=1s2∑c=1v2κjlihcΠm,n+wpjIn+wpi+h.Lettingα2=−χ¯1χ¯2…χ¯q1 and ∀j∈1…s1,γj=−η¯j1η¯j2…η¯jv1then, from equation (81), it comes out(83)β=α1vecN¯+α2θK+∑j=1s1γjθLj,with θK=vecK1⋮vecKq1 and ∀j∈1…s1,θLj=vecLj1⋮vecLjv1 which is equivalent to(84)β=α1vecN¯+α2θK+α3θL,where α3=γ1γ2…γs1andθL=θL1⋮θLs1.It is appreciable to systematize such problem as follows:(85)Aθ=B,where constant matrix A, constant vector B, and vector of searched parameters θ are given by(86)A=α1α2α3,B=β,θ=vecN¯θKθL.Now, by taking into account that the control lawut has to respect upper and lower bounds (30), and then from relation (41), we get the two following inequalities:(87)−∑i=1q1K¯iXit−∑j=1s1∑l=1v1L¯jlXjt−τl≤umax∑i=1q1K¯iXit+∑j=1s1∑l=1v1L¯jlXjt−τl≤vmax.Undertaken the equality (55), the expansion of the latter inequalities over the considered block-pulse functions basis and the use of the vec operator yield to the following inequality system:(88)Cθ≤d,where constant matrix C and constant vector d are given as follows:(89)C=−D1−D2−D3D1D2D3,d=vecUmax,NvecVmax,N.with(90)D1=−X1,rNT⊗Im0mn,mwpImwp,D2=X1,rNT⊗ImImn0mwp,mnD¯2,D3=d1…ds1,where(91)D¯2=Ψ2X2,rNT⊗Im…Ψq1Xq1,rNT⊗Im,dj=ΨjXj,rN∗τ1T⊗Im…ΨjXj,rN∗τv1T⊗Im,and Umax,N and Vmax,N denote the upper bound coefficients, which are computed from the scalar product (15).In the end, the posed control problem has been effectively reduced to an overdetermined linear system subject to an inequality constraint, which can be formulated from equality (85) and inequality (88) as follows:(92a)Aθ=B,subject to(92b)Cθ≤d.The above constrained linear system is ofn+wpN algebraic equations subject to 2mN algebraic constraints with respect to mwp+∑i=1q1ni+v1∑j=1s1nj unknowns.This kind of system can be easily solved in the constrained least square sense, by the formulation as a quadratic programming problem as follows:(93a)minθAθ−B,subject to(93b)Cθ≤d.


## 4.2. Practical Stability Problem Analysis

Presently, the control parametersN¯, Ki, and Ljl can be found by solving the resulted quadratic programming problem (93a)-(93b) for fixed reference input yct. Thereafter, we intend to analyze the practical stability of the closed-loop augmented system (39).For this purpose, let us consider the following constants matrices:(94)∀i∈1…q1,∀j∈1…s1,∀l∈1…v1,ℳi=A¯i−12B¯K¯i,ℋjl=D¯jl−12B¯L¯jl.Definition 1.
The controlled system (39) is called practically stable if every bounded input vector produces a bounded states vector. In other words, there exist 0<R0<r, such that (see [45]):(95)X0<R0⇒Xt<r,∀t≥0.Theorem 1.
The controlled system (39) is practical stable if all eigenvalues of matrix ℳ1 have a strictly negative real part and if(96)λ1∑l=1v1ℋ1l+ω1<0,(97)X0<R0,where R0>0, λ1>0 and ω1<0 are given scalars satisfying:(98)eℳ1t≤λ1eω1t,∀t≥0,(99)R0=R1λ1,with R1 is the unique positive solution of the following equation:(100)fR1+λ1∑l=1v1ℋ1l+ω1=0,where(101)fR1=λ1∑i=2q1MiR1i−1+∑j=2s1∑l=1v1HjlR1j−1+∑i=1q1∑j=1s1∑l=1v1G¯ijlR1i+j−1+λ12∑i=1q1∑j=1q2L¯ijR1i+j−1+∑j=1s1∑l=1v1∑i=1q2N¯jliR1j+i−1+∑i=1q1∑j=1s2∑l=1v2U¯ijlR1i+j−1+λ12∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihR1j+i−1+∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlR1i+h+j−1+λ12∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcR1j+i+h−1+∑i=1q2∑j=1q1B¯iK¯jR1i+j−1+λ12∑i=1q2∑j=1s1∑l=1v1B¯iL¯jlR1i+j−1+∑j=1s2∑l=1v2∑i=1q1C¯jlK¯iR1i+j−1+λ12∑j=1s2∑l=1v2∑i=1s1∑h=1v1C¯jlL¯ihR1i+j−1+∑i=1q2∑j=1s2∑l=1v2∑h=1q1Q¯ijlK¯hR1i+j+h−1+λ12∑i=1q2∑j=1s2∑l=1v2∑h=1s1∑c=1v1Q¯ijlL¯hcR1i+j+h−1.Proof 1.
By taking into account thatsatut=u¯t+1/2ut with u¯t=satut−1/2ut, the augmented closed-loop system can be written from relations (39), (41), (94), and (48)–(51) as follows:(102)X˙t−M1Xt=∑i=2q1MiXit+∑l=1v1H1lXt−τl+∑j=2s1∑l=1v1HjlXjt−τl+B¯u¯t+∑i=1q1∑j=1s1∑l=1v1G¯ijlXitXjt−τl+E¯yct−12∑i=1q1∑j=1q2L¯ijXit⊗Xjt−12∑j=1s1∑l=1v1∑i=1q2N¯jliXjt−τl⊗Xit−12∑i=1q1∑j=1s2∑l=1v2U¯ijlXit⊗Xjt−τl−12∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihXjt−τl⊗Xit−τh−12∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlXit⊗Xht⊗Xjt−τl−12∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcXjt−τl⊗Xit⊗Xht−τc+∑i=1q2B¯iu¯t⊗Xit+∑j=1s2∑l=1v2C¯jlu¯t⊗Xjt−τl+∑i=1q2∑j=1s2∑l=1v2Q¯ijlu¯t⊗Xit⊗Xjt−τl.
By respecting thatℳ1e−ℳ1t=e−ℳ1tℳ1 which can be deduced from the series definition, we get the following:(103)ddte−ℳ1tXt=−ℳ1e−ℳ1tXt+e−ℳ1tX˙t=e−ℳ1tX˙t−ℳ1Xt.Thus, by integrating the last differential equation from 0 to t, it comes out:(104)e−ℳ1tXt−X0=∫0te−ℳ1σX˙σ−ℳ1Xσdσ,or equivalently(105)Xt=eℳ1tX0+∫0teℳ1t−σX˙σ−ℳ1Xσdσ.
Since all eigenvalues of matrixℳ1 must have the negative real part, then there exist two reals λ1>0 and ω1<0 such that relation (98) holds on. Consequently, the state vector Xt can be bounded as:(106)Xt≤λ1eω1tX0+λ1eω1t∫0te−ω1σX˙σ−ℳ1Xσdσ,which is similar to the following:(107)e−ω1tXt≤λ1X0+λ1∫0te−ω1σX˙σ−ℳ1Xσdσ.
Let us suppose that(108)Xt<R.
Through the following matrix norm properties:(109)Xit<Ri−1Xt,Xjt−τl<Rj−1Xt−τlXit⊗Xjt−τl<Ri+j−1Xt−τl,Xit⊗Xjt<Ri+j−1XtXjt−τl⊗Xit<Rj+i−1XtXiσ⊗Xhσ⊗Xjσ−τl<Ri+h+j−1Xt−τlXjσ−τl⊗Xiσ⊗Xhσ−τc<Rj+i+h−1Xt−τc.then we can deduce, from inequality (107) and relation (102), the following expression:(110)e−ω1tXt<λ1X0+λ1∑i=2q1MiRi−1∫0te−ω1σXσdσ+λ1∑l=1v1H1l∫0te−ω1σXσ−τldσ+λ1∑j=2s1∑l=1v1HjlRj−1∫0te−ω1σXσ−τldσ+λ1∑i=1q1∑j=1s1∑l=1v1G¯ijlRi+j−1∫0te−ω1σXσ−τldσ+λ1E¯δ∫0te−ω1σdσ+λ1B¯δ¯∫0te−ω1σdσ+λ12∑i=1q1∑j=1q2L¯ijRi+j−1∫0te−ω1σXσdσ+λ12∑j=1s1∑l=1v1∑i=1q2N¯jliRj+i−1∫0te−ω1σXσdσ+λ12∑i=1q1∑j=1s2∑l=1v2U¯ijlRi+j−1∫0te−ω1σXσ−τldσ+λ12∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihRj+i−1∫0te−ω1σXσ−τhdσ+λ12∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlRi+h+j−1∫0te−ω1σXσ−τldσ+λ12∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcRj+i+h−1∫0te−ω1σXσ−τcdσ+λ1∑i=1q2B¯iRi∫0te−ω1σu¯σdσ+λ1∑j=1s2∑l=1v2C¯jlRj∫0te−ω1σu¯σdσ+λ1∑i=1q2∑j=1s2∑l=1v2Q¯ijlRi+j∫0te−ω1σu¯σdσ.where δ>0 satisfying yct<δ, and δ¯=maxvmax,umax.
Now by taking into account the following special property of function saturation (see [46]):(111)u¯t=satut−12ut≤12ut,which can be written from relation (41) and condition (108) as follows:(112)u¯t<12∑i=1q1K¯iXit+12∑j=1s1∑l=1v1L¯jlXjt−τl<12∑i=1q1K¯iRi−1Xt+12∑j=1s1∑l=1v1L¯jlRj−1Xt−τl.
It results the following inequality:(113)e−ω1tXt<λ1X0+λ1∑i=2q1MiRi−1∫0te−ω1σXσdσ+λ1∑l=1v1H1l∫0te−ω1σXσ−τhdσ+λ1∑j=2s1∑l=1v1HjlRj−1∫0te−ω1σXσ−τhdσ+λ1∑i=1q1∑j=1s1∑l=1v1G¯ijlRi+j−1∫0te−ω1σXσ−τhdσ+λ1E¯δ∫0te−ω1σdσ+λ1B¯δ¯∫0te−ω1σdσ+λ12∑i=1q1∑j=1q2L¯ijRi+j−1∫0te−ω1σXσdσ+λ12∑j=1s1∑l=1v1∑i=1q2N¯jliRj+i−1∫0te−ω1σXσdσ+λ12∑i=1q1∑j=1s2∑l=1v2U¯ijlRi+j−1∫0te−ω1σXσ−τldσ+λ12∑j=1s1∑l=1v1∑i=1s2∑h=1v2V¯jlihRj+i−1∫0te−ω1σXσ−τhdσ+λ12∑i=1q1∑h=1q2∑j=1s2∑l=1v2Y¯ihjlRi+h+j−1∫0te−ω1σXσ−τldσ+λ12∑j=1s1∑l=1v1∑i=1q2∑h=1s2∑c=1v2Z¯jlihcRj+i+h−1∫0te−ω1σXσ−τcdσ+λ12∑i=1q2∑j=1q1B¯iK¯jRi+j−1∫0te−ω1σXσdσ+λ12∑i=1q2∑j=1s1∑l=1v1B¯iL¯jlRi+j−1∫0te−ω1σXσ−τldσ+λ12∑j=1s2∑l=1v2∑i=1q1C¯jlK¯iRi+j−1∫0te−ω1σXσdσ+λ12∑j=1s2∑l=1v2∑i=1s1∑h=1v1C¯jlL¯ihRi+j−1∫0te−ω1σXσ−τhdσ+λ12∑i=1q2∑j=1s2∑l=1v2∑h=1q1Q¯ijlK¯hRi+j+h−1∫0te−ω1σXσdσ+λ12∑i=1q2∑j=1s2∑l=1v2∑h=1s1∑c=1v1Q¯ijlL¯hcRi+j+h−1∫0te−ω1σXσ−τcdσ.
Let us introduce the following function:(114)zt=supρ∈t−τ¯,tXρe−ω1t∀t≥0.then under considering that ∀t≥0 and l∈1…v: Xt≤supρ∈t−τ¯,tXρ and Xt−τl≤supρ∈t−τ¯,tXρ it turns out, from inequality (113), that(115)e−ω1tXt<λ1X0+fR+λ1∑l=1v1ℋ1l∫0tzσdσ+Ω∫0te−ω1σdσ,where function f. is given by relation (101) and Ω=λ1E¯δ+λ1B¯δ¯.
Now, by introducing a new functiongt as the right-hand member of inequality (115):(116)gt=λ1X0+fR+λ1∑l=1v1ℋ1l∫0tzσdσ+Ω∫0te−ω1σdσ.we can show that(117)∀t≥0,zt<gt.
By deriving the functiongt:(118)dgtdt=fR+λ1∑l=1v1ℋ1lzt+Ωe−ω1t,it comes out from inequality (117) that(119)dgtdt<fR+λ1∑l=1v1ℋ1lgt+Ωe−ω1t.
By integrating the last inequality from 0 to t, it follows, based on the use of the Bellman-Gronwall inequality (see appendix D in [34]) that(120)gt<efR+λ1∑l=1v1ℋ1ltg0+ΩfR+λ1∑l=1v1ℋ1l+ω1efR+λ1∑l=1v1ℋ1lt−e−ω1t,where g0=λ1X0.
Inequalities (117) and (120) imply(121)zt<λ1X0efR+λ1∑l=1v1ℋ1lt+ΩfR+λ1∑l=1v1ℋ1l+ω1efR+λ1∑l=1v1ℋ1lt−e−ω1t.
From the definition of functionzt, we can derived that(122)Xt<λ1X0efR+λ1∑l=1v1ℋ1l+ω1t+ΩfR+λ1∑l=1v1ℋ1l+ω1efR+λ1∑l=1v1ℋ1l+ω1t−1.
If condition (96) holds on, then for(123)fR+λ1∑l=1v1ℋ1l+ω1<0.which is verified for each radius R satisfying(124)R<R1,where R1 is the unique positive solution of the equality (100), we can make out the following result:(125)Xt<r=λ1X0−ΩfR+λ1∑l=1v1ℋ1l+ω1.
At this point, in order to make sure the hypothesis (108) for all t, it is enough to have(126)λ1X0<R1.
Consequent, from inequality (125) and condition (126), it allows that(127)X0<R0=R1λ1⇒Xt<r=λ1R0−ΩfR+λ1∑l=1v1ℋ1l+ω1.
Hence, the controlled system (39) is practically stable.


## 5. Study of an Illustrative Benchmark System

In this section, simulations will be performed on the model of double-inverted pendulums, which can be described concisely in Figure1.Figure 1
Double-inverted pendulums connected by a spring.The parametersm1=2kg, J1=5kg.m2 and m2=2.5kg, J2=6.5kg.m2 are, respectively, the masses and the moments of inertia of the two pendulums, r=0.5m is the pendulum height, k=100N/m is the constant of the connecting torsional spring, b=0.5m is the distance between the pendulum hinges, l=0.5m is the spring natural length, and g=9.81m/s2 is the gravitational acceleration.


### 5.1. System Modeling

According to [47], the considered benchmark system is described as follows:(128)x˙1t=x2tx˙2t=m1grJ1−kr24J1sinx1t+kr2J1l−b+kr24J1sinx3t+f1x1t−τ1,x2t+g1x1t,x3tsatu1tx˙3t=x4tx˙4t=m2grJ2−kr24J2sinx3t+kr2J2l−b+kr24J2sinx1t+f2x3t−τ2,x4t+g2x1t,x3tsatu2t,where x1t=θ1t and x3t=θ2t denote the angular displacements of the pendulums from vertical, f1. and f2. are nonlinear time delays functions associated, respectively, with each pendulum, and g1. and g2. are nonlinear functions associated, respectively, with each input channel u1t and u2t. The output systems are θ1t and θ2t.In addition, we assumed that(129)f1x1t−τ1,x2t=x1t−τ11+x12t−τ1+1J1x1t−τ1x2tcosx1t−τ1,f2x3t−τ2,x4t=x3t−τ21+x32t−τ2+1J2x3t−τ2x4tcosx3t−τ2,g1x1t,x3t=1J1−0.25x12t+0.25x32t,g2x1t,x3t=1J2−0.25x12t+0.25x32t.For simulation, we also give the values of the time delays, the initial conditions, and the two saturating actuators as follows:(130)τ1=2s,τ2=4s,(131)ζt=12sint,1−cost,sint,1−costTfort∈−4,0,(132)satu1t=50ifu1t≻50,u1tif−50≤u1t≤50,−50ifu1t≺−50,satu2t=80ifu2t≻80,u2tif−60≤u2t≤80,−60ifu2t≺−60.


### 5.2. Polynomial Model for Studied Benchmark System

Using Taylor series expansions, the nonlinear benchmark model (128), without actuator saturation, can be developed on a third-order polynomial system (25) which is described by the following state equations:(133)x˙t=∑i=13Aixit+∑j=13∑l=12Djlxjt−τl+∑i=13∑j=13∑l=12Gijlxit⊗xjt−τl+∑i=12Biut⊗xit+Butyt=Cxtxt=ζtfort∈−4,0,with(134)xt=x1tx2tx3tx4t,A1=0100m1grJ1−kr24J10kr24J100001kr24J20m2grJ2−kr24J20,D11=0000100000000000,D12=0000000000000010,B=01J1000001J2,C=10000010,G1112,5=1J1,G111i,j=0elsewherei=1…4;j=1…42;G121i,j=0fori=1…4;j=1…43;G1312,65=−12J1,G131i,j=0elsewherei=1…4;j=1…44;G1124,15=1J2,G112i,j=0elsewherei=1…4;j=1…42;G122i,j=0fori=1…4;j=1…43;G1324,235=−12J2,G132i,j=0elsewherei=1…4;j=1…44;B1i,j=0fori=1…4;j=1…8;B22,1=−0.25,B22,11=0.25,B24,17=−0.25,B22,27=0.25,B2i,j=0elsewherei=1…4;j=1…32;A2i,j=D21i,j=D22i,j=0fori=1…4;j=1…42;A32,1=−16m1grJ1−kr24J1,A32,43=−kr224J1,A34,1=−kr224J2;A34,43=−16m2grJ2−kr24J2,A3i,j=0elsewherei=1…4;j=1…43;D312,1=−1,D31i,j=0elsewherei=1…4;j=1…43;D324,43=−1,D32i,j=0elsewherei=1…4;j=1…43;G1112,5=1J1,G111i,j=0elsewherei=1…4;j=1…42;G121i,j=0fori=1…4;j=1…43;G1312,65=−12J1,G131i,j=0elsewherei=1…4;j=1…44;G1124,15=1J2,G112i,j=0elsewherei=1…4;j=1…42;G122i,j=0fori=1…4;j=1…43;G1324,235=−12J2,G132i,j=0elsewherei=1…4;j=1…44;B1i,j=0fori=1…4;j=1…8;B22,1=−0.25,B22,11=0.25,B24,17=−0.25,B22,27=0.25,B2i,j=0elsewherei=1…4;j=1…32;and for each i∈2,3, j∈1…3, and l∈1,2, Gijl are null matrices with appropriate dimensions.


### 5.3. Fidelity of the Adopted Polynomial Model

Given the following inputs control:(135)u1t=−36x1t−16x2t−20x1t−τ1−10x2t−τ1−16x1t−τ1−12x2t−τ2,u2t=−36x3t−12x4t−20x3t−τ1−10x4t−τ1−16x3t−τ1−12x4t−τ2.Then, the evolution of the state variables in closed loop for both nonlinear (128) and polynomial (133) systems is shown in Figure 2.Figure 2
Evolution of state variables for both real model and polynomial model .From the simulation results, it is proved that the dynamical behavior of the real system is adequately described through the adopted polynomial model affected by multiple time delays in states. This is confirmed by the fact that the behavior of the considered polynomial model can replicate with a enough accuracy the real system one.


### 5.4. Control Qualification Statements

The aim of the present application is the control of the angular positionsθ1t and θ2t, in order to force them to track the following reference input vector:(136)yct=yc1tyc2t=sint+sin0.5tsint+sin0.5t.For this purpose, we adopt the following parameters matrices of the reference model (40):(137)E=0100000000−480.9−35.8−122.4−4.59518.96513.68825.34682.23019.31175.40001000000−110.8−4−477.8−35.75864.39597.14218.68821.21060.930040000001000000000010000000000100000000001−100000000000−10000000,F=00000000100000000001T,Z=10000000000010000000.It should be noted that the chosen number of integratorsw=3 allows for the output vector Yrt of the reference model to track adequately the considered reference input vector yct.


### 5.5. Parameters Determination of Block-Pulse Functions Basis

For the selected reference input vectoryct, the exact solution Xrt=x1,rt,x2,rt,…,xn+wp,rtT of the adopted reference model can be obtained from the following relation:(138)Xrt=eEtXr0+∫0teEt−τFycτdτ.In order to determine the parameters of block-pulse functions basis, namely the time interval0,T and the order N, it suffices to follow the following steps:(i)
The time interval0,T is sufficiently long to achieve reference model behavior.(ii)
By taking into consideration that each elementxi,rt of the state vector Xrt is of smooth functions, so from what has been indicated in Section 2.2, it can be approximated efficiently by a finite number of block-pulse functions under the condition that it is high as enough. To satisfy this condition, we compare the exact solution of the reference model Xrt with the approximate one X1,rNSNt, which leads to the suitable choice of the order N.From Figure3, we can deduce that the agreement is excellent for N=28 (number of BPFs) and T=32s.Figure 3
State variables of the reference model:, exact solutions; , BPFs approximations.In Table1, three numerical methods, namely, root mean square error (RMSE), mean squared error (MSE), and mean absolute error (MAE), are used to measure the errors between the exact solution Xrt and the approximate one X1,rNSNt for N=28 and T=32s. The errors are defined as follows:(139)Er=Xrt−X1,rNSNt=er,1,er,2,er,3,er,4,er,5,er,6,er,7,er,8,er,9,er,10T.Table 1
The errors between the exact solution and the approximate one.
MethodsErrorser,1er,2er,3er,4er,5er,6er,7er,8er,9er,10RMSE0.029 30.041 10.029 40.040 90.001 00.001 00.000 90.001 00.001 20.001 2MSE0.000 90.001 70.000 90.001 70.000 00.000 00.000 00.000 00.000 00.000 0MAE0.021 60.024 00.021 60.024 00.000 70.000 70.000 70.000 70.000 80.000 8It appears clear from Table1 that the errors can be judged to be very satisfactory with respect to the effectiveness expansivity of the exact solution over the considered block-pulse functions basis for N=28 and T=32s.


### 5.6. Results and Discussion

ForN=28 (number of BPFs) and T=32s, which permit us to represent adequately the exact solution over the considered BPFs basis of the adopted reference model Xrt under the fixed input reference vector yct, the execution of the proposed saturated tracking control approach leads to the following numerical results of the controller’s parameters:(140)N¯=3610343147330313394512799810026870687771890462039645518875,K1=2305.6174.7708.826.3968.838.52704208.5,K2=K2a,K2b,K2c,K2d,with:(141)K2a=−12.6331−159.64399.8312161.4933−15.2547−922.426212.6334921.7037,K2b=−159.643913.0846162.3417−13.1298−922.42628.9307923.6831−10.0853,K2c=9.8312162.3417−7.0379−164.183212.6334923.6831−10.0214−922.9536,K2d=161.4993−13.1298−164.183213.1800921.7037−10.0853−922.953611.2408,K3=K3a,K3b,K3c,K3d,K3e,K3f,K3g,K3h,K3i,K3j,K3k,with:(142)K3a=−72−883.2110.3401.6118.3124.4−167−832.7294.51225.1−621.240.3,K3b=272.8−102.871.1118.3−37.9434623.4−217.527.11205.2−126.9−861.4,K3c=−398−102.891.379.9118.3124.4589−150.5−241.1305.9−621.240.3,K3d=272.8−102.8124.459.2−3.4−23.4623.4−217.540.3−95.3152.343.9,K3e=395.2−3.4192.6−18.7−102.8−23.4−640.7152.3−994.1−20.8−150.543.9,K3f=−18.7−13.971.1118.3−37.9434−20.85.727.11205.2−126.9−861.4,K3g=395.2−3.4192.6−18.7−181.2−1312.6−640.7152.3−994.1−20.8112.61686.2,K3h=74.91436.2−271.8−18.7−773.742.2−42.51783.0−414.5−20.8−575.4−111.3,K3i=−398−102.891.379.9−102.8−23.4589−150.5−241.1305.9−150.543.9,K3j=−18.7−13.9−271.8−18.7−773.742.2−20.85.7−414.5−20.8−575.4−111.3,K3k=79.9−13.942.253305.95.7−111.3−53.6,(143)L111,1=10,L11i,j=0elsewherei=1…2;j=1…4;L122,3=12.5,L12i,j=0elsewherei=1…2;j=1…4;L21i,j=L22i,j=0foreachi=1…2;j=1…16;L311,1=−10,L31i,j=0elsewherei=1…2;j=1…64;L32i,j=0foreachi=1…2;j=1…64.Consequently, the following simulation results are obtained:In Figure4, the responses of controlled inverted pendulums system using the proposed saturated tracking control method and the considered reference model are plotted.Figure 4
Responses of the controlled inverted pendulums system using the proposed tracking control method under input saturation and the considered reference model .It can be seen that the time delay polynomial state feedback controllers with triple-action integral, applied to the inverted pendulums system, permit them to achieve their purpose. In fact, the performance of the controlled inverted pendulums system is judged to be very satisfactory with respect to the effectiveness trackability of the reference input vector.Figure5 displays the errors between the output vector of the controlled benchmark system Yt and the input reference vector yct, which are denoted as follows:(144)e˜t=Yt−yct=e1te2t=θ1t−yc1tθ2t−yc2t.Figure 5
Variation of the tracking errorse1t and e2t.It is clear that the error signals are not significant. In fact, we can see that they converge toward a small neighborhood of the origin, which confirm that the proposed saturated control law computed using the developed approach, permits to have a good tracking performance. This is confirmed by a close to zero steady-state error for the angular positionsθ1t and θ2t compared with its time-varying reference inputs yc1t and yc2t.According to the simulation result given in Figure6, it is seen that the control signals u1t and u2t are always inside, respectively, the interval −50,50 and −60,80. Due to the fact that the imposed limits on the control input by the actuator saturation have been incorporated into the procedure of computing tracking control law, the potential ability of the proposed saturated tracking control method to withstand these constraints can be confirmed from the obtained simulation results.Figure 6
Variation of the control signalsu1t and u2t.From what has been stated above, it can be deduced that the adopted strategy of control can guarantee for the closed-loop saturated nonlinear polynomial system subject to multiple time-delayed states an excellently trackability of the reference input vector.


### 5.7. Simulation Results Given in a Practical Environment

To demonstrate the effectiveness in a practical environment, some measurement noises and external disturbances are imposed on the system. The measurement noisesmn are assumed as mn=0.01rand4,1, and the external disturbances edt are set as(145)edt=0.2sint,0.2cost,0.2sint,0.2costT.The corresponding simulation results are shown in Figures7 and 8. We can deduce that the proposed strategy of control permits to conserve an acceptable tracking performance even in the presence of some measurement noises and external disturbances. Hence, this can be evaluated as a second advantage of the investigated approach in this study.Figure 7
Responses of the controlled inverted pendulums system under measurement noises and external disturbances and the considered reference model .Figure 8
Variation of the tracking errorse1t and e2t under measurement noises and external disturbances.


### 5.8. Practical Stability Test

The obtained gains verified that all eigenvalues of matrixℳ1 have a strictly negative real part, and(146)eℳ1t<360e0.4t,which corresponds to inequality (98) with λ1=360 and ω1=−0.4. Then, Theorem 1 provides an estimation of the radius of practical stability of the studied system R0=0.00000000144.


## 5.1. System Modeling

According to [47], the considered benchmark system is described as follows:(128)x˙1t=x2tx˙2t=m1grJ1−kr24J1sinx1t+kr2J1l−b+kr24J1sinx3t+f1x1t−τ1,x2t+g1x1t,x3tsatu1tx˙3t=x4tx˙4t=m2grJ2−kr24J2sinx3t+kr2J2l−b+kr24J2sinx1t+f2x3t−τ2,x4t+g2x1t,x3tsatu2t,where x1t=θ1t and x3t=θ2t denote the angular displacements of the pendulums from vertical, f1. and f2. are nonlinear time delays functions associated, respectively, with each pendulum, and g1. and g2. are nonlinear functions associated, respectively, with each input channel u1t and u2t. The output systems are θ1t and θ2t.In addition, we assumed that(129)f1x1t−τ1,x2t=x1t−τ11+x12t−τ1+1J1x1t−τ1x2tcosx1t−τ1,f2x3t−τ2,x4t=x3t−τ21+x32t−τ2+1J2x3t−τ2x4tcosx3t−τ2,g1x1t,x3t=1J1−0.25x12t+0.25x32t,g2x1t,x3t=1J2−0.25x12t+0.25x32t.For simulation, we also give the values of the time delays, the initial conditions, and the two saturating actuators as follows:(130)τ1=2s,τ2=4s,(131)ζt=12sint,1−cost,sint,1−costTfort∈−4,0,(132)satu1t=50ifu1t≻50,u1tif−50≤u1t≤50,−50ifu1t≺−50,satu2t=80ifu2t≻80,u2tif−60≤u2t≤80,−60ifu2t≺−60.


## 5.2. Polynomial Model for Studied Benchmark System

Using Taylor series expansions, the nonlinear benchmark model (128), without actuator saturation, can be developed on a third-order polynomial system (25) which is described by the following state equations:(133)x˙t=∑i=13Aixit+∑j=13∑l=12Djlxjt−τl+∑i=13∑j=13∑l=12Gijlxit⊗xjt−τl+∑i=12Biut⊗xit+Butyt=Cxtxt=ζtfort∈−4,0,with(134)xt=x1tx2tx3tx4t,A1=0100m1grJ1−kr24J10kr24J100001kr24J20m2grJ2−kr24J20,D11=0000100000000000,D12=0000000000000010,B=01J1000001J2,C=10000010,G1112,5=1J1,G111i,j=0elsewherei=1…4;j=1…42;G121i,j=0fori=1…4;j=1…43;G1312,65=−12J1,G131i,j=0elsewherei=1…4;j=1…44;G1124,15=1J2,G112i,j=0elsewherei=1…4;j=1…42;G122i,j=0fori=1…4;j=1…43;G1324,235=−12J2,G132i,j=0elsewherei=1…4;j=1…44;B1i,j=0fori=1…4;j=1…8;B22,1=−0.25,B22,11=0.25,B24,17=−0.25,B22,27=0.25,B2i,j=0elsewherei=1…4;j=1…32;A2i,j=D21i,j=D22i,j=0fori=1…4;j=1…42;A32,1=−16m1grJ1−kr24J1,A32,43=−kr224J1,A34,1=−kr224J2;A34,43=−16m2grJ2−kr24J2,A3i,j=0elsewherei=1…4;j=1…43;D312,1=−1,D31i,j=0elsewherei=1…4;j=1…43;D324,43=−1,D32i,j=0elsewherei=1…4;j=1…43;G1112,5=1J1,G111i,j=0elsewherei=1…4;j=1…42;G121i,j=0fori=1…4;j=1…43;G1312,65=−12J1,G131i,j=0elsewherei=1…4;j=1…44;G1124,15=1J2,G112i,j=0elsewherei=1…4;j=1…42;G122i,j=0fori=1…4;j=1…43;G1324,235=−12J2,G132i,j=0elsewherei=1…4;j=1…44;B1i,j=0fori=1…4;j=1…8;B22,1=−0.25,B22,11=0.25,B24,17=−0.25,B22,27=0.25,B2i,j=0elsewherei=1…4;j=1…32;and for each i∈2,3, j∈1…3, and l∈1,2, Gijl are null matrices with appropriate dimensions.


## 5.3. Fidelity of the Adopted Polynomial Model

Given the following inputs control:(135)u1t=−36x1t−16x2t−20x1t−τ1−10x2t−τ1−16x1t−τ1−12x2t−τ2,u2t=−36x3t−12x4t−20x3t−τ1−10x4t−τ1−16x3t−τ1−12x4t−τ2.Then, the evolution of the state variables in closed loop for both nonlinear (128) and polynomial (133) systems is shown in Figure 2.Figure 2
Evolution of state variables for both real model and polynomial model .From the simulation results, it is proved that the dynamical behavior of the real system is adequately described through the adopted polynomial model affected by multiple time delays in states. This is confirmed by the fact that the behavior of the considered polynomial model can replicate with a enough accuracy the real system one.


## 5.4. Control Qualification Statements

The aim of the present application is the control of the angular positionsθ1t and θ2t, in order to force them to track the following reference input vector:(136)yct=yc1tyc2t=sint+sin0.5tsint+sin0.5t.For this purpose, we adopt the following parameters matrices of the reference model (40):(137)E=0100000000−480.9−35.8−122.4−4.59518.96513.68825.34682.23019.31175.40001000000−110.8−4−477.8−35.75864.39597.14218.68821.21060.930040000001000000000010000000000100000000001−100000000000−10000000,F=00000000100000000001T,Z=10000000000010000000.It should be noted that the chosen number of integratorsw=3 allows for the output vector Yrt of the reference model to track adequately the considered reference input vector yct.


## 5.5. Parameters Determination of Block-Pulse Functions Basis

For the selected reference input vectoryct, the exact solution Xrt=x1,rt,x2,rt,…,xn+wp,rtT of the adopted reference model can be obtained from the following relation:(138)Xrt=eEtXr0+∫0teEt−τFycτdτ.In order to determine the parameters of block-pulse functions basis, namely the time interval0,T and the order N, it suffices to follow the following steps:(i)
The time interval0,T is sufficiently long to achieve reference model behavior.(ii)
By taking into consideration that each elementxi,rt of the state vector Xrt is of smooth functions, so from what has been indicated in Section 2.2, it can be approximated efficiently by a finite number of block-pulse functions under the condition that it is high as enough. To satisfy this condition, we compare the exact solution of the reference model Xrt with the approximate one X1,rNSNt, which leads to the suitable choice of the order N.From Figure3, we can deduce that the agreement is excellent for N=28 (number of BPFs) and T=32s.Figure 3
State variables of the reference model:, exact solutions; , BPFs approximations.In Table1, three numerical methods, namely, root mean square error (RMSE), mean squared error (MSE), and mean absolute error (MAE), are used to measure the errors between the exact solution Xrt and the approximate one X1,rNSNt for N=28 and T=32s. The errors are defined as follows:(139)Er=Xrt−X1,rNSNt=er,1,er,2,er,3,er,4,er,5,er,6,er,7,er,8,er,9,er,10T.Table 1
The errors between the exact solution and the approximate one.
MethodsErrorser,1er,2er,3er,4er,5er,6er,7er,8er,9er,10RMSE0.029 30.041 10.029 40.040 90.001 00.001 00.000 90.001 00.001 20.001 2MSE0.000 90.001 70.000 90.001 70.000 00.000 00.000 00.000 00.000 00.000 0MAE0.021 60.024 00.021 60.024 00.000 70.000 70.000 70.000 70.000 80.000 8It appears clear from Table1 that the errors can be judged to be very satisfactory with respect to the effectiveness expansivity of the exact solution over the considered block-pulse functions basis for N=28 and T=32s.


## 5.6. Results and Discussion

ForN=28 (number of BPFs) and T=32s, which permit us to represent adequately the exact solution over the considered BPFs basis of the adopted reference model Xrt under the fixed input reference vector yct, the execution of the proposed saturated tracking control approach leads to the following numerical results of the controller’s parameters:(140)N¯=3610343147330313394512799810026870687771890462039645518875,K1=2305.6174.7708.826.3968.838.52704208.5,K2=K2a,K2b,K2c,K2d,with:(141)K2a=−12.6331−159.64399.8312161.4933−15.2547−922.426212.6334921.7037,K2b=−159.643913.0846162.3417−13.1298−922.42628.9307923.6831−10.0853,K2c=9.8312162.3417−7.0379−164.183212.6334923.6831−10.0214−922.9536,K2d=161.4993−13.1298−164.183213.1800921.7037−10.0853−922.953611.2408,K3=K3a,K3b,K3c,K3d,K3e,K3f,K3g,K3h,K3i,K3j,K3k,with:(142)K3a=−72−883.2110.3401.6118.3124.4−167−832.7294.51225.1−621.240.3,K3b=272.8−102.871.1118.3−37.9434623.4−217.527.11205.2−126.9−861.4,K3c=−398−102.891.379.9118.3124.4589−150.5−241.1305.9−621.240.3,K3d=272.8−102.8124.459.2−3.4−23.4623.4−217.540.3−95.3152.343.9,K3e=395.2−3.4192.6−18.7−102.8−23.4−640.7152.3−994.1−20.8−150.543.9,K3f=−18.7−13.971.1118.3−37.9434−20.85.727.11205.2−126.9−861.4,K3g=395.2−3.4192.6−18.7−181.2−1312.6−640.7152.3−994.1−20.8112.61686.2,K3h=74.91436.2−271.8−18.7−773.742.2−42.51783.0−414.5−20.8−575.4−111.3,K3i=−398−102.891.379.9−102.8−23.4589−150.5−241.1305.9−150.543.9,K3j=−18.7−13.9−271.8−18.7−773.742.2−20.85.7−414.5−20.8−575.4−111.3,K3k=79.9−13.942.253305.95.7−111.3−53.6,(143)L111,1=10,L11i,j=0elsewherei=1…2;j=1…4;L122,3=12.5,L12i,j=0elsewherei=1…2;j=1…4;L21i,j=L22i,j=0foreachi=1…2;j=1…16;L311,1=−10,L31i,j=0elsewherei=1…2;j=1…64;L32i,j=0foreachi=1…2;j=1…64.Consequently, the following simulation results are obtained:In Figure4, the responses of controlled inverted pendulums system using the proposed saturated tracking control method and the considered reference model are plotted.Figure 4
Responses of the controlled inverted pendulums system using the proposed tracking control method under input saturation and the considered reference model .It can be seen that the time delay polynomial state feedback controllers with triple-action integral, applied to the inverted pendulums system, permit them to achieve their purpose. In fact, the performance of the controlled inverted pendulums system is judged to be very satisfactory with respect to the effectiveness trackability of the reference input vector.Figure5 displays the errors between the output vector of the controlled benchmark system Yt and the input reference vector yct, which are denoted as follows:(144)e˜t=Yt−yct=e1te2t=θ1t−yc1tθ2t−yc2t.Figure 5
Variation of the tracking errorse1t and e2t.It is clear that the error signals are not significant. In fact, we can see that they converge toward a small neighborhood of the origin, which confirm that the proposed saturated control law computed using the developed approach, permits to have a good tracking performance. This is confirmed by a close to zero steady-state error for the angular positionsθ1t and θ2t compared with its time-varying reference inputs yc1t and yc2t.According to the simulation result given in Figure6, it is seen that the control signals u1t and u2t are always inside, respectively, the interval −50,50 and −60,80. Due to the fact that the imposed limits on the control input by the actuator saturation have been incorporated into the procedure of computing tracking control law, the potential ability of the proposed saturated tracking control method to withstand these constraints can be confirmed from the obtained simulation results.Figure 6
Variation of the control signalsu1t and u2t.From what has been stated above, it can be deduced that the adopted strategy of control can guarantee for the closed-loop saturated nonlinear polynomial system subject to multiple time-delayed states an excellently trackability of the reference input vector.


## 5.7. Simulation Results Given in a Practical Environment

To demonstrate the effectiveness in a practical environment, some measurement noises and external disturbances are imposed on the system. The measurement noisesmn are assumed as mn=0.01rand4,1, and the external disturbances edt are set as(145)edt=0.2sint,0.2cost,0.2sint,0.2costT.The corresponding simulation results are shown in Figures7 and 8. We can deduce that the proposed strategy of control permits to conserve an acceptable tracking performance even in the presence of some measurement noises and external disturbances. Hence, this can be evaluated as a second advantage of the investigated approach in this study.Figure 7
Responses of the controlled inverted pendulums system under measurement noises and external disturbances and the considered reference model .Figure 8
Variation of the tracking errorse1t and e2t under measurement noises and external disturbances.


## 5.8. Practical Stability Test

The obtained gains verified that all eigenvalues of matrixℳ1 have a strictly negative real part, and(146)eℳ1t<360e0.4t,which corresponds to inequality (98) with λ1=360 and ω1=−0.4. Then, Theorem 1 provides an estimation of the radius of practical stability of the studied system R0=0.00000000144.


## 6. Conclusion

In this article, we have built up an original control scheme for a class of nonlinear polynomial systems subject to control input constraints. These systems are characterized by their complex and high nonlinearities as well as fixed and well-defined time delays. The design methodology makes full use of a memory nonlinear state feedback control formalism, accompanied by a compensator gain built through the integral actions technique. In the initial step, the closed-loop augmented form is determined using the augmented error modeling technique. In the subsequent step, an algebraic system with constraints of controller parameters is derived so as to solve the tracking control problem in the sense of constrained least squares optimization methodology. In the last step, the system’s closed-loop stability is assessed through providing new sufficient conditions for the model’s practical stability. As an original performance of the synthesized technique, it is straightforward to estimate the enlarged region of stability.The simulation analysis has demonstrated the viability of the developed control method in enhancing the performance of the studied benchmark model output tracking. Indeed, it is plainly observed throughout the simulation study that the suggested memory polynomial controllers with integrals action can quickly attenuate the steady-state error, as a result of a good tracking performance, even in the presence of actuator saturation.Future works will stretch out the developed technique to deal with saturated tracking control design for nonlinear polynomial time-varying systems affected with both unstructured uncertainties and time delays in states.



---

*Source: 6411693-2022-11-21.xml*