#### 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.

#### 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], 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 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 dimensional Euclidean space is denoted by , while is referred to the set of all real matrices with rows and columns, whereas stands for the set of all real matrices with rows and columns, where and are two natural numbers. denotes the identity matrix of size , whereas denotes the zero matrix of size . corresponds to the transpose of the matrix . denotes the dimensional unit vector which has 1 in the 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 interval , an set of block-pulse functions (BPFs) can be defined as follows [38]:where for each , 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 interval :

###### 2.1.2. Orthogonality

The block-pulse functions are orthogonal with each other in the interval :

The orthogonal property of block-pulse functions is the basis of expanding functions into their block-pulse series. Thus, any absolutely integrable function of the Lebesgue measure in the interval may be expanded over a BPFs basis as follows:where the block-pulse coefficient vector is given byand the block-pulse vector basis of order is given bywith is the block-pulse coefficient computing as follows:

###### 2.1.3. Completeness

The block-pulse function set is complete when the order approaches infinity. This means that for any square-integrable function of the Lebesgue measure in the interval , we have

##### 2.2. Error in BPFs Approximation

Assuming that is a differentiable function, where his first derivative on the time interval is bounded, that is to say

Over every subinterval , the residual error is given by

Furthermore, the total error can be defined as follows:which can be bounded as follows [43]:allowing to conclude that

Consequently, over a BPFs basis with a high order , we can represent exactly the function .

##### 2.3. BPFs Expansion of a Vector Function

A vector function of dimensional components which are square-integrable with respect to Lebesgue measure in the interval can be expressed practically by a finite series of block-pulse functions as follows:with where are the block-pulse coefficients of vector function , as computed through the orthogonality property of the block-pulse functions as follows:

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

Assume that the vector has its initial aswith . Then, for , the time-delayed vector is presented by

As a result, the expansion of time-delayed vector over the BPFs basis is given by [44]withwhere is the number of BPFs considered over , and

##### 2.5. Operational Matrix of Integration

For the BPFs basis, an operational matrix of integration is defined in [44] bywhere

##### 2.6. Operational Matrix of Product

The elementary matrix can be defined as follows: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]

#### 3. Problem Formulation and System Description

Consider a continuous-time nonlinear polynomial systems with known and fixed multiple time delays in states , which are described by the following state equation:where is the input vector, is the nondelayed state vector, is the delayed state vector with denoted time delay, and is the output vector. The continuous vector-valued function denotes the initial data, where for each , with for each .

In (25), , , , , , , , , and is the Kronecker power of the state vector , defined by

##### 3.1. Assumption

The consider system (25) is locally controllable around where its 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 vector . For this purpose, we intend to design a memory polynomial state feedback controller accompanied by compensator gains:withwherewith is a positive integer which depends on the type of reference input vector. In (26), , , , and are the derivative of .

Each element of the actuator vector should satisfy the following constraint:which is equivalent towhere and are given positive reals vectors.

The constrained inputs, denoted , are saturating functions and defined for each as follows:with

##### 3.3. Control Objective

The undelayed augmented state vector as well as the delayed augmented state vector can be expressed using the augmented error modeling technique, as follows:

When taking into consideration that

Then, the augmented system subject to the constrained input control can be described by the following state equation:with , and for each , and and for each , and .

Assuming that the obtained augmented system is locally controllable around , then the dynamical behavior of the closed-loop augmented system must exactly match the behavior of a linear reference model , which is represented also by its state equations:where and .

The output vector of the above reference model, which should be adequately designed, must track perfectly the reference input vector .

*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:with

*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 gain is designed to eliminate the steady-state error for a time-varying set point reference input vector:(ii)The control gains and for each are designed to ensure the stability performance of the following closed-loop nonlinear polynomial time delays system under input control saturation : with

#### 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:

By substituting the constrained inputs control by the unconstrained inputs control in state equation (39) and by taking into account the relations (41) and (47)–(51), it comes out:

With respect to over the interval , the integration of equation (52) leads to

Over the basis of the block-pulse functions truncated to an order , the expansion of the fixed reference input vector can be done as follows:where 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 :where denote state coefficients, computing from the scalar product (15).

As a result, the delayed state vector of the controlled augmented system , for each , can be expressed as follows:and then, its expansion over the considered basis leads towhere denote the delayed state coefficients, computing from the scalar product (20).

Using the operational matrix of product, the Kronecker power of could be expressed as follows:

In the same way, the Kronecker power of , for each , could be expressed as follows:

In addition to this, for each , , and , the Kronecker product terms can be expressed as follows:

As well, for each and , the Kronecker product terms can be also expressed as follows:

As well, for each , , and , the Kronecker product terms can be also expressed as follows:

As well, for each , , and , the Kronecker product terms can be also expressed as follows:

As well, for each , , , and , the Kronecker product terms can be also expressed as follows:

As well, for each , , , and , the Kronecker product terms can be also expressed as follows:

As well, for each , , , , and , the Kronecker product terms can be also expressed as follows:

For simplicity, let us denote from relations (60)–(66) the following constants matrices:

Then, it results from the expansion of the whole state model (53) over the considered orthogonal functions basis:where

The application of the integration operational matrix , defined by relation (22), leads to

By removing the block-pulse vector basis in both sides of the last relation (73), we attain the following algebraic equation: