Robust Adaptive Neurocontrol of SISO Nonlinear Systems Preceded by Unknown Deadzone

Abstract

In this study, the problem of controlling an unknown SISO nonlinear system in Brunovsky canonical form with unknown deadzone input in such a way that the system output follows a specified bounded reference trajectory is considered. Based on universal approximation property of the neural networks, two schemes are proposed to handle this problem. The first scheme utilizes a smooth adaptive inverse of the deadzone. By means of Lyapunov analyses, the exponential convergence of the tracking error to a bounded zone is proven. The second scheme considers the deadzone as a combination of a linear term and a disturbance-like term. Thus, the estimation of the deadzone inverse is not required. By using a Lyapunov-like analyses, the asymptotic converge of the tracking error to a bounded zone is demonstrated. Since this control strategy requires the knowledge of a bound for an uncertainty/disturbance term, a procedure to find such bound is provided. In both schemes, the boundedness of all closed-loop signals is guaranteed. A numerical experiment shows that a satisfactory performance can be obtained by using any of the two proposed controllers.

1. Introduction

The deadzone is a nonsmooth nonlinearity commonly found in many practical systems such as hydraulic positioning systems [1], pneumatic servo systems [2], DC servo motors, among others. When the deadzone is not considered explicitly during the design process, the performance of the control system could be degraded by an increase of the steady-state error, the presence of limit cycles, or inclusive instability [3–6]. A direct way of compensating the deleterious effect of the deadzone is by calculating its inverse. However, this is not an easy question because in many practical situations, both the parameters and the output of the deadzone are unknown. To overcome this problem, in a pioneer work [3], Tao and Kokotović proposed to employ an adaptive inverse of the deadzone. This scheme was applied to linear systems in transfer function form. Cho and Bai [7] extended this work and achieved a perfect asymptotic adaptive cancellation of the deadzone. However, their work assumed that the deadzone output was measurable. In [8], the work of Tao and Kokotović was extended to linear systems in a state space form with nonmeasurable deadzone output. In [9], a new smooth parameterization of the deadzone was proposed and a class of SISO systems with completely known nonlinear functions and with linearly parameterized unknown constants was controlled by using backstepping technique. In order to avoid the construction of the adaptive inverse, in [10], the same class of nonlinear systems as in [9] was controlled by means of a robust adaptive approach and by modeling the deadzone as a combination of a linear term and a disturbance-like term. The controller design in [10] is based on the assumption that maximum and minimum values for the deadzone parameters are a priori known. However, a specific procedure to find such bounds is not provided. In order to extend the class of systems previously considered in [9, 10], in this paper, we propose the development of two controllers based on universal approximation property of the neural networks. The first scheme utilizes a smooth adaptive inverse of the deadzone as in [9]. By means of Lyapunov analyses, the exponential convergence of the tracking error to a bounded zone is proven. The second scheme considers the deadzone as a combination of a linear term and a disturbance-like term as in [10]. Thus, the estimation of the deadzone inverse is not required. By using the Lyapunov-like analyses, the asymptotic converge of the tracking error to a bounded zone is demonstrated. Since this control strategy requires the knowledge of a bound for an uncertainty/disturbance term, a procedure to find such bound is provided. In both schemes, the boundedness of all closed-loop signals is guaranteed. A numerical experiment with a second-order nonlinear system shows that a satisfactory performance can be obtained by using any of the two proposed controllers.

2. Preliminaries and Problem Statement

In this study, the system which will be controlled is composed of an unknown nonlinear plant preceded by an actuator with an unknown deadzone in such a way that the deadzone output is the input to the plant. Consider that the n-order dynamics of the nonlinear plant can be represented as follows: 𝑦^(𝑛)(𝑡)=𝑓𝑦(𝑡),𝑦̇(𝑡),…,𝑦^(𝑛−1)(𝑡)+𝑏𝑢(𝑡)+𝜉(𝑡), (2.1) where the scalar 𝑦(𝑡) is the output of interest, 𝑦^(𝑖)(𝑡) for 𝑖=1,…,𝑛−1 represents the 𝑖th derivative of 𝑦(𝑡)—each one of these derivatives are assumed measurable, 𝑢(𝑡)∈ℜ is the deadzone output (and the input to the plant), 𝑓∶ℜ^𝑛→ℜ is an unknown but continuous nonlinear function, 𝑏 is an unknown constant, and 𝜉(𝑡)∈ℜ is an unknown but bounded disturbance. Defining the state variables as 𝑥₁(𝑡)∶=𝑦(𝑡),𝑥₂(𝑡)∶=𝑦(̇𝑡),…,𝑥_𝑛(𝑡)∶=𝑦^(𝑛−1)(𝑡), (2.1) can be expressed as follows: 𝑥̇PLANT:^𝑖(𝑡)=𝑥_𝑖+1(𝑡)𝑥̇,𝑖=1,…,𝑛−1,_𝑛(𝑡)(𝑥(𝑡(𝑡)(𝑡),𝑦(𝑡)=𝑓))+𝑏𝑢+𝜉=𝑥₁(𝑡), (2.2) where 𝑥(𝑡)∈ℜ^𝑛 is the measurable state vector for 𝑡∈ℜ⁺∶={𝑡∶𝑡≥0}, which is defined as 𝑥(𝑡)∶=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇=[𝑥₁(𝑡),𝑥̇₁(𝑡),…,𝑥₁^(𝑛−1)(𝑡)]^𝑇. The nonsymmetric deadzone can be represented by (𝑡)(𝑣(𝑡=𝑚DEADZONE:𝑢=DZ))^𝑟𝑣(𝑡)−𝑏_𝑟𝑣(𝑡)≥𝑏_𝑟,0𝑏_𝑙<𝑣(𝑡)<𝑏_𝑟,𝑚_𝑙𝑣(𝑡)−𝑏_𝑙𝑣(𝑡)≤𝑏_𝑙, (2.3) where 𝑚_𝑟 and 𝑚_𝑙 are the right and left constant slopes for the deadzone characteristic, 𝑏_𝑟 and 𝑏_𝑙 represent the right and left constant breakpoints, 𝑢(𝑡)∈ℜ and 𝑣(𝑡)∈ℜ are the output and the input of the deadzone, respectively. Note that 𝑣(𝑡) is the actual control input to the global system formed by the actuator and the plant. In accordance with [3, 4], the deadzone model (2.3) is a static simplification of diverse physical phenomena with negligible fast dynamics. Hereafter, it is considered that the following assumptions are valid.

Assumption 2.1. Without loss of generality, the unknown constant 𝑏 is assumed positive.

Assumption 2.2. The deadzone output 𝑢(𝑡) is not available for measurement.

Assumption 2.3. Although the deadzone parameters 𝑏_𝑟,𝑏_𝑙,𝑚_𝑟, and 𝑚_𝑙 are unknown constants, we can assure that 𝑏_𝑟>0,𝑏_𝑙<0,𝑚_𝑟>0, and 𝑚_𝑙>0.

The objective that we are trying to achieve is to determine a control signal 𝑣(𝑡) such that the output of the plant (2.2), 𝑦(𝑡)=𝑥₁(𝑡), follows a specified reference trajectory 𝑦_𝑟(𝑡), and, at the same time, all closed-loop signals stay bounded.

2.1. Smooth Parameterization of the Deadzone

A direct way of compensating the deleterious effect of the deadzone is by calculating its inverse. From (2.3), the deadzone inverse can be obtained as 𝑣(𝑡)=DZ⁻¹(𝑢(𝑡=𝑢(𝑡)))+𝑚_𝑟𝑏_𝑟𝑚_𝑟𝑢(𝑡)𝑏≥0,_𝑙,𝑏_𝑟𝑢(𝑡)𝑢(𝑡)=0,+𝑚_𝑙𝑏_𝑙𝑚_𝑙𝑢(𝑡)≤0. (2.4) Notwithstanding, clearly this is a discontinuous function. A smooth approximation of (2.4) was presented in [9] as 𝑣(𝑡)≈𝑢(𝑡)+𝑚_𝑟𝑏_𝑟𝑚_𝑟𝜙_𝑟(𝑢(𝑡+𝑢(𝑡)))+𝑚_𝑙𝑏_𝑙𝑚_𝑙𝜙_𝑙(𝑢(𝑡)), (2.5) where 𝜙_𝑟(𝑢(𝑡𝑢(𝑡)))∶=exp/𝜀_𝑜exp𝑢(𝑡)/𝜀_𝑜+exp−𝑢(𝑡)/𝜀_𝑜,𝜙_𝑙(𝑢(𝑡(𝑡)))∶=exp−𝑢/𝜀_𝑜exp𝑢(𝑡)/𝜀_𝑜+exp−𝑢(𝑡)/𝜀_𝑜 (2.6) and 𝜀_𝑜 is a positive constant chosen by the designer. Since both the parameters and the output of the deadzone are unknown, approximation (2.5) cannot be utilized directly. To overcome this problem, a smooth parameterization of the deadzone was proposed in [9]. For completeness, this parameterization is explained here. Note that (2.3) can be expressed alternatively as 𝑢(𝑡)=−𝜃^𝑇𝜔(𝑡), (2.7) where 𝜃=[𝑚_𝑟,𝑚_𝑟𝑏_𝑟,𝑚_𝑙,𝑚_𝑙𝑏_𝑙]^𝑇,𝜔(𝑡)=[−𝜎_𝑟(𝑡)𝑣(𝑡),𝜎_𝑟(𝑡),−𝜎_𝑙(𝑡)𝑣(𝑡),𝜎_𝑙(𝑡)]^𝑇∶𝜎_𝑟(𝑡)=1if𝑣(𝑡)≥𝑏_𝑟,𝜎0otherwise,^𝑙(𝑡)=1if𝑣(𝑡)≤𝑏_𝑙,0otherwise. (2.8) Given that 𝑢(𝑡), 𝜃, and 𝜔(𝑡) are unknown, the deadzone output 𝑢(𝑡) is approximated by 𝑢_𝑑(𝑡)∶=−𝜃^𝑇(𝑡)𝜔(𝑡), (2.9) where 𝜃(𝑡)∶=[𝑚_𝑟,𝑚_𝑟𝑏_𝑟,𝑚_𝑙,𝑚_𝑙𝑏_𝑙]^𝑇 is an estimation of 𝜃 and 𝜔(𝑡)∶=[−𝜙_𝑟(𝑣(𝑡))𝑣(𝑡),𝜙_𝑟(𝑣(𝑡)),−𝜙_𝑙(𝑣(𝑡))𝑣(𝑡),𝜙_𝑙(𝑣(𝑡))]^𝑇. From (2.7) and (2.9), 𝑢(𝑡) can be expressed as 𝑢(𝑡)=𝑢_𝑑(𝑡)+𝜃(𝑡)−𝜃^𝑇𝜔(𝑡)+𝑑_𝑁(𝑡), (2.10) where 𝑑_𝑁(𝑡)∶=𝜃^𝑇(𝜔(𝑡)−𝜔(𝑡)). Although 𝑑_𝑁(𝑡) is an unknown term, its boundedness can be guaranteed [11]. Consider that the positive constant 𝑑_𝑁 is an upper bound for 𝑑_𝑁(𝑡), that is, |𝑑_𝑁(𝑡)|≤𝑑_𝑁. From (2.9), 𝑣(𝑡) can be expressed in terms of 𝑢_𝑑(𝑡) as 𝑣(𝑡)=𝑢_𝑑(𝑡)+𝑚_𝑟𝑏_𝑟𝑚_𝑟𝜙_𝑟𝑢_𝑑(𝑡)+𝑢_𝑑(𝑡)+𝑚_𝑙𝑏_𝑙𝑚_𝑙𝜙_𝑙𝑢_𝑑(𝑡). (2.11) This expression can be utilized only if vector 𝜃(𝑡) is properly determined. To avoid the singularity problem, note that 𝑚_𝑟 and 𝑚_𝑙 must always be different from zero.

2.2. Deadzone Representation as a Linear Term and a Disturbance-Like Term

For the particular case when 𝑚∶=𝑚_𝑟=𝑚_𝑙, the deadzone model (2.3) can alternatively be described as [10, 12] 𝑢(𝑡)=𝑚𝑣(𝑡)+𝑑(𝑡), (2.12) where 𝑑(𝑡) is given by 𝑑(𝑡)=−𝑚𝑏_𝑟𝑣(𝑡)≥𝑏_𝑟,−𝑚𝑣(𝑡)𝑏_𝑙<𝑣(𝑡)<𝑏_𝑟,−𝑚𝑏_𝑙𝑣(𝑡)≤𝑏_𝑙. (2.13) Note that (2.13) is the negative of a saturation function. Thus, although 𝑑(𝑡) could not be exactly known, its boundedness can be assured. Consider that the positive constant 𝑑 is an upper bound for 𝑑(𝑡), that is, |𝑑(𝑡)|≤𝑑.

3. Neurocontroller Design and Stability Analyses

Based on the universal approximation property of the neural networks, two control schemes are presented in this section to handle the trajectory tracking problem.

Assumption 3.1. The reference trajectory 𝑦_𝑟(𝑡) and their first n-derivatives are continuous and bounded. Besides, all these variables are available for the design.

Given the reference trajectory 𝑦_𝑟(𝑡) and their first (𝑛−1)-derivatives, the vector 𝑥_𝑟(𝑡) can be defined as 𝑥_𝑟(𝑡)∶=[𝑥_𝑟,1(𝑡),𝑥_𝑟,2(𝑡),…,𝑥_𝑟,𝑛(𝑡)]^𝑇=[𝑦_𝑟(𝑡),𝑦̇_𝑟(𝑡),…,𝑦_𝑟^(𝑛−1)(𝑡)]^𝑇. Let us define the filtered tracking error 𝑟(𝑡) as 𝑟(𝑡)𝑑∶=𝑑𝑡+𝜆_𝑟^𝑛−1𝑒₁(𝑡), (3.1) where 𝑒₁(𝑡) is the first element of the tracking error vector 𝑒(𝑡) which is defined simply as 𝑒(𝑡)∶=𝑥(𝑡)−𝑥_𝑟(𝑡)=[𝑒₁(𝑡),𝑒₂(𝑡),…,𝑒_𝑛(𝑡)]^𝑇=[𝑒₁(𝑡),𝑒₁̇(𝑡),…,𝑒₁^(𝑛−1)(𝑡)]^𝑇 and 𝜆_𝑟 is a positive constant chosen by the designer. Note that 𝑟(𝑡) can also be expressed as 𝑟(𝑡)=Λ^𝑇_𝑟𝑒(𝑡), (3.2) where Λ_𝑟∈ℜ^𝑛 is a constant vector given by Λ_𝑟∶=[𝜆_𝑟^𝑛−1,(𝑛−1)𝜆_𝑟^𝑛−2,…,1]^𝑇.

Remark 3.2. Note that, from (3.1), 𝑟(𝑡) can be considered as the input to a stable linear system whose output is 𝑒₁(𝑡). Consequently, if 𝑟(𝑡)∈𝐿_∞, then 𝑒₁(𝑡),𝑒(𝑡)∈𝐿_∞. Specifically, 𝑒(𝑡) has the following properties proven in [13]: (i) 𝑒(𝑡) converges exponentially to zero, if 𝑟(𝑡)=0, (ii) if 𝑒(0)=0 and |𝑟(𝑡)|≤𝜍 where 𝜍 is a positive constant, then 𝑒(𝑡) belongs to a compact set 𝑅 given by 𝑅={𝑒(𝑡)∈ℜ^𝑛suchthat|𝑒_𝑖(𝑡)|≤2^𝑖−1𝜆_𝑟^𝑖−𝑛𝜍,𝑖=1,…,𝑛} for ∀𝑡≥0, and (iii) if 𝑒(0)≠0 and |𝑟(𝑡)|≤𝜍, then 𝑒(𝑡) will converge to 𝑅 within a time-constant (𝑛−1)/𝜆_𝑟.

The first derivative of 𝑟(𝑡) can be calculated as 𝑟̇(𝑡)=𝑑𝑑𝑡+𝜆_𝑟^𝑛𝑒₁(𝑡)=𝑥̇_𝑛(𝑡)−𝑥̇_𝑟,𝑛(𝑡)+Λ^𝑇_𝑟𝑒(𝑡), (3.3) where Λ_𝑟∶=[0,𝜆_𝑟^𝑛−1,(𝑛−1)𝜆_𝑟^𝑛−2,…,(𝑛−1)𝜆_𝑟]^𝑇.

Now then, it is well known that any unknown continuous function can be approximated on a compact set Ω by a neural network as follows [14–16]: 𝑓(𝑥)=𝑊^∗𝜎(𝑥)+𝜂(𝑥),∀𝑥∈Ω⊂ℜ^𝑛, (3.4) where 𝜎(⋅) is the activation vector function with sigmoidal components, that is, 𝜎(⋅)∶=[𝜎₁(⋅),…,𝜎_𝑠(⋅)]^𝑇: 𝜎_𝑗(𝑥(𝑡𝑎))∶=^𝜎𝑗1+exp−∑^𝑛_𝑖=1𝑐_{𝜎𝑗,𝑖}𝑥_𝑖(𝑡)−𝑑_𝜎𝑗for𝑗=1,…,𝑠, (3.5) where 𝑎_𝜎𝑗,𝑐_{𝜎𝑗,𝑖}, and 𝑑_𝜎𝑗 are positive constants which can be specified by the designer, 𝜂∶ℜ^𝑛→ℜ is the approximation error which is bounded by |𝜂(𝑥)|<𝜂 for all 𝑥∈Ω, 𝜂 is a positive constant, and 𝑊^∗∈ℜ^1×𝑠 is an unknown constant vector. Typically, 𝑊^∗ is considered as the optimal weight such that 𝑊^∗=argmin_{𝑊∈ℜ1×𝑠}sup_𝑥∈Ω𝑓(𝑥)−𝑊𝜎(𝑥). (3.6) By substituting (3.4) into (2.2), the following alternative representation for the plant dynamics can be obtained: 𝑥̇_𝑛(𝑡)=𝑊^∗𝜎(𝑥)+𝑏𝑢(𝑡)+𝜂(𝑥)+𝜉(𝑡). (3.7) By substituting (3.7) into (3.3), we get 𝑟̇(𝑡)=𝑊^∗𝜎(𝑥)+𝑏𝑢(𝑡)−𝑥̇_𝑟,𝑛(𝑡)+Λ^𝑇_𝑟𝑒(𝑡)+𝜂(𝑥)+𝜉(𝑡). (3.8)

3.1. Scheme I

A control scheme which does not require the specific knowledge of the upper bound for the term 𝜂(𝑥)+𝜉(𝑡)+𝑏𝑑_𝑁(𝑡) is developed below.

In order to take into account the effect of the deadzone, the adaptive parameterization model (2.10) is substituted into (3.8): 𝑟̇(𝑡)=𝑊^∗𝜎(𝑥)+𝑏𝑢_𝑑(𝑡)+𝑏𝜃(𝑡)−𝜃^𝑇𝜔(𝑡)−𝑥̇_𝑟,𝑛(𝑡)+Λ^𝑇_𝑟𝑒(𝑡)+𝜂(𝑥)+𝜉(𝑡)+𝑏𝑑_𝑁(𝑡). (3.9) Consider that 𝑢_𝑑(𝑡) is chosen as 𝑢_𝑑(𝑡)=𝑝(𝑡)𝑢_𝑑(𝑡), (3.10) where 𝑝(𝑡) is an online estimation of 𝑝∶=1/𝑏 and 𝑢_𝑑(𝑡) is selected as 𝑢_𝑑(𝑡)=−𝑊(𝑡)𝜎(𝑥(𝑡))−𝑘𝑟(𝑡)+𝑥̇_𝑟,𝑛(𝑡)−Λ^𝑇_𝑟𝑒(𝑡), (3.11) where 𝑊(𝑡) is an online estimation of 𝑊^∗ and 𝑘 is a positive constant. Note that 𝑢_𝑑(𝑡)=𝑝(𝑡)𝑢_𝑑(𝑡)=𝑝(𝑡)+𝑝−𝑝𝑢_𝑑(𝑡)=𝑝−𝑝−𝑝(𝑡)𝑢_𝑑(𝑡)=𝑝𝑢_𝑑(𝑡)−𝑝(𝑡)𝑢_𝑑(𝑡), (3.12) where 𝑝(𝑡)∶=𝑝−𝑝(𝑡). If (3.12) is substituted into (3.9), the following is obtained: 𝑟̇(𝑡)=𝑊^∗𝜎(𝑥)+𝑏𝑝𝑢_𝑑(𝑡)−𝑝(𝑡)𝑢_𝑑(𝑡)+𝑏𝜃(𝑡)−𝜃^𝑇𝜔(𝑡)−𝑥̇_𝑟,𝑛(𝑡)+Λ^𝑇_𝑟𝑒(𝑡)+𝜂(𝑥)+𝜉(𝑡)+𝑏𝑑_𝑁(𝑡)=𝑊^∗𝜎(𝑥)+𝑢_𝑑(𝑡)−𝑏𝑝(𝑡)𝑢_𝑑(𝑡)+𝑏𝜃(𝑡)−𝜃^𝑇𝜔(𝑡)−𝑥̇_𝑟,𝑛(𝑡)+Λ^𝑇_𝑟𝑒(𝑡)+𝜂(𝑥)+𝜉(𝑡)+𝑏𝑑_𝑁(𝑡). (3.13) Now, by substituting (3.11) into (3.13) and reducing like terms, the filtered tracking error dynamics can be expressed as 𝑟̇(𝑡)=𝑊^∗𝜎(𝑥(𝑡))−𝑊(𝑡)𝜎(𝑥(𝑡))−𝑘𝑟(𝑡)+𝑥̇_𝑟,𝑛(𝑡)−Λ^𝑇_𝑟𝑒(𝑡)−𝑏𝑝(𝑡)𝑢_𝑑(𝑡)+𝑏𝜃(𝑡)−𝜃^𝑇𝜔(𝑡)−𝑥̇_𝑟,𝑛(𝑡)+Λ^𝑇_𝑟𝑒(𝑡)+𝜂(𝑥)+𝜉(𝑡)+𝑏𝑑_𝑁(𝑡)=𝑊(𝑡)𝜎(𝑥(𝑡))−𝑘𝑟(𝑡)−𝑏𝑝(𝑡)𝑢_𝑑(𝑡)+𝑏𝜃(𝑡)−𝜃^𝑇𝜔(𝑡)+𝛿(𝑡), (3.14) where 𝑊(𝑡)∶=𝑊^∗−𝑊(𝑡) and 𝛿(𝑡)∶=𝜂(𝑥)+𝜉(𝑡)+𝑏𝑑_𝑁(𝑡) is an unknown term but bounded by the positive constant 𝛿, that is, |𝛿(𝑡)|≤𝛿.

Once the filtered tracking error dynamics has been determined, the following Lyapunov function candidate is suggested: 𝑉(𝑡)=12𝑟²(𝑡)+12𝑘₁𝑊(𝑡)𝑊^𝑇(𝑡)+12𝑘₂𝑝²(𝑡)+12𝑘₃𝜃^𝑇(𝑡)𝜃(𝑡), (3.15) where 𝜃(𝑡)∶=𝜃−𝜃(𝑡), and 𝑘₁, 𝑘₂, and 𝑘₃ are positive constants. The first derivative of 𝑉(𝑡) is 𝑉̇(𝑡)=𝑟(𝑡)𝑟̇(𝑡)+𝑑1𝑑𝑡2𝑘₁𝑊(𝑡)𝑊^𝑇(𝑡)+𝑑1𝑑𝑡2𝑘₂𝑝²(𝑡)+𝑑1𝑑𝑡2𝑘₃𝜃^𝑇(𝑡)𝜃(𝑡). (3.16) The first derivative of (1/2𝑘₁)𝑊(𝑡)𝑊^𝑇(𝑡) can be calculated as follows. 𝑑1𝑑𝑡2𝑘₁𝑊(𝑡)𝑊^𝑇(𝑡)=12𝑘₁𝑊̇(𝑡)𝑊^𝑇(𝑡)+12𝑘₁𝑊(𝑡)𝑊̇^𝑇(𝑡). (3.17) Since 𝑊̇(𝑡)𝑊^𝑇(𝑡) is a scalar and the transpose of a scalar is the same scalar, then 𝑑1𝑑𝑡2𝑘₁𝑊(𝑡)𝑊^𝑇(𝑡)=12𝑘₁𝑊(𝑡)𝑊̇^𝑇(𝑡)+12𝑘₁𝑊(𝑡)𝑊̇^𝑇(𝑡)=1𝑘₁𝑊(𝑡)𝑊̇^𝑇(𝑡). (3.18) Proceeding in a similar way for (𝑑/𝑑𝑡)((1/2𝑘₃)𝜃^𝑇(𝑡)𝜃(𝑡)), it can be determined that 𝑑1𝑑𝑡2𝑘₃𝜃^𝑇(𝑡)𝜃(𝑡)=1𝑘₃𝜃^𝑇(𝑡)𝜃̇(𝑡). (3.19) Substituting (3.14), (3.18), and (3.19) into (3.16) yields 𝑉̇(𝑡)=𝑟(𝑡)𝑊(𝑡)𝜎(𝑥(𝑡))−𝑘𝑟²(𝑡)−𝑏𝑝(𝑡)𝑢_𝑑(𝑡)𝑟(𝑡)+𝑏𝜃(𝑡)−𝜃^𝑇𝜔(𝑡)𝑟(𝑡)(𝑡)𝑟(𝑡)+1+𝛿𝑘₁𝑊(𝑡)𝑊̇^𝑇(𝑡)+𝑏𝑘₂𝑝(𝑡)𝑝̇(𝑡)+𝑏𝑘₃𝜃^𝑇(𝑡)𝜃̇(𝑡). (3.20) Consider that the learning laws for 𝑊(𝑡), 𝑝(𝑡), and 𝜃(𝑡) are chosen as 𝑊̇(𝑡)=𝑘₁𝑟(𝑡)𝜎^𝑇(𝑥(𝑡))−ℓ₁𝑊(𝑡)−𝑊⁰,𝑝̇(𝑡)=−𝑘₂𝑢_𝑑(𝑡)𝑟(𝑡)−ℓ₂𝑝(𝑡)−𝑝⁰̇𝜃(𝑡),=Proj−𝑘₃𝑟(𝑡)𝜔(𝑡)−ℓ₃𝜃(𝑡)−𝜃⁰, (3.21)(3.22)(3.23) where ℓ₁, ℓ₂, and ℓ₃ are positive constants, 𝑊⁰, 𝑝⁰, and 𝜃⁰ are ideally (but not necessarily) good estimations of 𝑊^∗, 𝑝, and 𝜃, respectively, and Proj represents a smooth projection operation as in [17] or [18] in order to avoid that 𝑚_𝑟 and 𝑚_𝑙 can be equal to zero. In accordance with (3.21), (3.22), and (3.23) and taking into account that 𝑊̇(𝑡)=−𝑊̇(𝑡), 𝑝̇(𝑡)=−𝑝̇(𝑡), and 𝜃̇(𝑡)=−𝜃̇(𝑡), (3.20) can be expressed as 𝑉̇(𝑡)=−𝑘𝑟²(𝑡)+𝛿(𝑡)𝑟(𝑡)+𝑏𝜃(𝑡)−𝜃^𝑇𝜔(𝑡)𝑟(𝑡)+ℓ₁𝑘₁𝑊(𝑡)𝑊(𝑡)−𝑊^0𝑇+ℓ₂𝑏𝑘₂𝑝(𝑡)𝑝(𝑡)−𝑝⁰−𝑏𝑘₃𝜃^𝑇(𝑡)Proj−𝑘₃𝑟(𝑡)𝜔(𝑡)−ℓ₃𝜃(𝑡)−𝜃⁰. (3.24) In [11], it is mentioned that the projection operation has the following property: −𝜃−𝜃(𝑡)^𝑇Proj𝜓(𝑡)≤−𝜃−𝜃(𝑡)^𝑇𝜓(𝑡). (3.25) On the other hand, the truthfulness of the following inequalities was proven in [9] ℓ₂𝑏𝑘₂𝑝(𝑡)𝑝(𝑡)−𝑝⁰ℓ≤−²2𝑘₂𝑏𝑝²(𝑡)+ℓ₂2𝑘₂𝑏𝑝−𝑝⁰²,ℓ₃𝑘₃𝜃−𝜃(𝑡)^𝑇𝜃(𝑡)−𝜃₀ℓ≤−³2𝑘₃𝜃^𝑇(𝑡)𝜃(𝑡)+ℓ₃2𝑘₃𝜃−𝜃^0𝑇𝜃−𝜃⁰. (3.26) Likewise, it can be demonstrated that ℓ₁𝑘₁𝑊(𝑡)𝑊_𝑡−𝑊^0𝑇ℓ≤−¹2𝑘₁𝑊(𝑡)𝑊^𝑇(𝑡)+ℓ₁2𝑘₁𝑊^∗−𝑊⁰𝑊^∗−𝑊^0𝑇,𝛿(𝑡)𝑟(𝑡)≤12𝑟²(𝑡)+12𝛿²(𝑡)≤12𝑟²(𝑡)+12𝛿². (3.27) If the inequalities (3.25), (3.26), (3.27) are substituted into (3.24), we obtain 𝑉̇(𝑡)≤−(2𝑘−1)2𝑟²(𝑡)−ℓ₁2𝑘₁𝑊(𝑡)𝑊^𝑇(𝑡)−ℓ₂2𝑘₂𝑏𝑝²(𝑡)−ℓ₃2𝑘₃𝜃^𝑇(𝑡)𝜃(𝑡)+ℓ₁2𝑘₁𝑊^∗−𝑊⁰𝑊^∗−𝑊^0𝑇+ℓ₂2𝑘₂𝑏𝑝−𝑝⁰²+ℓ₃2𝑘₃𝜃−𝜃^0𝑇𝜃−𝜃⁰+12𝛿². (3.28) If 𝑘>0.5 and defining 𝛼∶=min{(2𝑘−1),ℓ₁,ℓ₂,ℓ₃} and ℓ𝛽∶=¹2𝑘₁𝑊^∗−𝑊⁰𝑊^∗−𝑊^0𝑇+ℓ₂2𝑘₂𝑏𝑝−𝑝⁰²+ℓ₃2𝑘₃𝜃−𝜃^0𝑇𝜃−𝜃⁰+12𝛿², (3.29) the following bound as a function of 𝑉(𝑡) can finally be determined for 𝑉̇(𝑡): 𝑉̇(𝑡)≤−𝛼𝑉(𝑡)+𝛽 (3.30) (3.30) can be rewritten in the following form: 𝑉̇(𝑡)+𝛼𝑉(𝑡)≤𝛽. (3.31) Multiplying both sides of the last inequality by exp(𝛼𝑡), it is possible to obtain exp(𝛼𝑡)𝑉̇(𝑡)+𝛼exp(𝛼𝑡)𝑉(𝑡)≤𝛽exp(𝛼𝑡). (3.32) The left-hand side of (3.32) can be rewritten as 𝑑()𝑉(𝑡)()𝑑𝑡exp𝛼𝑡≤𝛽exp𝛼𝑡 (3.33) or, equivalently, as 𝑑exp(𝛼𝑡)𝑉(𝑡)≤𝛽exp(𝛼𝑡)𝑑𝑡. (3.34) Integrating both sides of the last inequality from 0 to 𝑡 yields exp(𝛼𝑡)𝑉(𝑡)−𝑉(0)≤^𝑡₀𝛽exp(𝛼𝜏)𝑑𝜏. (3.35) Adding 𝑉(0) to both sides of the last inequality, we obtain exp(𝛼𝑡)𝑉(𝑡)≤𝑉(0)+^𝑡₀𝛽exp(𝛼𝜏)𝑑𝜏. (3.36) Multiplying both sides of the inequality (3.36) by exp(−𝛼𝑡) yields 𝑉(𝑡)≤exp(−𝛼𝑡)𝑉(0)+exp(−𝛼𝑡)^𝑡₀𝛽exp(𝛼𝜏)𝑑𝜏, (3.37) and, consequently 𝑉(𝑡)(0)()+𝛽≤𝑉exp−𝛼𝑡𝛼1−exp(−𝛼𝑡). (3.38) As 𝛼 and 𝛽 are positive constants, the right-hand side of the last inequality can be bounded by 𝑉(0)+𝛽/𝛼. Thus, 𝑉(𝑡)∈𝐿_∞ and since by construction 𝑉(𝑡) is a nonnegative function, the boundedness of 𝑟(𝑡), 𝑊(𝑡), 𝑝(𝑡), and 𝜃(𝑡) can be guaranteed. Because 𝑊^∗, 𝑝, and 𝜃 are bounded, 𝑊(𝑡)=𝑊^∗−𝑊(𝑡), 𝑝(𝑡)=𝑝−𝑝(𝑡), and 𝜃(𝑡)=𝜃−𝜃(𝑡) must be bounded too. If 𝑟(𝑡)∈𝐿_∞, from Remark 3.2, we can assure that 𝑒(𝑡)∈𝐿_∞ and converges to 𝑅. From (3.11), it can be seen that 𝑢_𝑑(𝑡) is formed by bounded terms and consequently 𝑢_𝑑(𝑡)∈𝐿_∞. From (3.10), it can be seen that 𝑢_𝑑(𝑡) is the product of two bounded variables. Therefore, 𝑢_𝑑(𝑡)∈𝐿_∞. As 𝑢_𝑑(𝑡),𝜃(𝑡)∈𝐿_∞, and 𝑚_𝑟 and 𝑚_𝑙 are always different from zero, from (2.11), the boundedness of 𝑣(𝑡) can be concluded. Now, note that the following is true: (1/2)𝑟²(𝑡)≤𝑉(𝑡). Taking into account this fact and from (3.38), we get |𝑟(𝑡)|≤2𝑉(0)exp(−𝛼𝑡)+2𝛽𝛼1−exp(−𝛼𝑡). (3.39) By taking the limit as 𝑡→∞ of the last inequality, we can guarantee that |𝑟(𝑡)| converges exponentially fast to a zone bounded by the term 2𝛽/𝛼. Based on this fact together with Remark 3.2, we can conclude that 𝑒₁(𝑡)=𝑦(𝑡)−𝑦_𝑟(𝑡) converges exponentially fast to a region around zero bounded by the term (1/𝜆_𝑟^𝑛−1)2𝛽/𝛼. Thus, the following theorem has been proven.

Theorem 3.3. If Assumptions 2.1–3.1 are satisfied, 𝑘>0.5, and the control law (2.11), (3.10), (3.11) with the learning laws (3.21), (3.22), and (3.23) are applied to the system formed by (2.2)-(2.3), then the following hold:(a)the filtered tracking error, tracking error, the weights, the estimations of the control gain reciprocal and the deadzone parameters, and the control signal are bounded: 𝑟(𝑡),𝑒(𝑡),𝑊(𝑡),𝑝(𝑡),𝜃(𝑡),𝑣(𝑡)∈𝐿_∞, (3.40)(b)the actual tracking error 𝑦(𝑡)−𝑦_𝑟(𝑡) converges exponentially fast to a region around zero bounded by the term: 1𝜆_𝑟^𝑛−12𝛽𝛼. (3.41)

3.2. Scheme II

A simpler scheme can be obtained by using the model (2.12) and the tuning error 𝑟_𝜀(𝑡) instead of the model (2.10) and the filtered tracking error 𝑟(𝑡), respectively. However, the implementation of this scheme requires necessarily the a priori knowledge of a good bound for the term 𝜂(𝑥)+𝜉(𝑡)+𝑏𝑑(𝑡). The development of this scheme is explained below.

If the deadzone model (2.12) is substituted into (3.8), we have 𝑟̇(𝑡)=𝑊^∗𝜎(𝑥(𝑡))+𝑏𝑚𝑣(𝑡)−𝑥̇_𝑟,𝑛(𝑡)+Λ^𝑇_𝑟𝑒(𝑡)+𝜂(𝑥)+𝜉(𝑡)+𝑏𝑑(𝑡). (3.42) Note that, by using the model (2.12), the actual control input 𝑣(𝑡) appears now directly into the error dynamics (3.42) only multiplied by a constant gain 𝑏𝑚.

Consider that 𝑣(𝑡) is chosen as 𝑣(𝑡)=𝑞(𝑡)𝑣(𝑡), (3.43) where 𝑞(𝑡) is an online estimation of 𝑞∶=1/𝑏𝑚 and 𝑣(𝑡) is selected as 𝑣(𝑡)=−𝑊(𝑡)𝜎(𝑥(𝑡))−𝑘𝑟(𝑡)−𝑘^∗𝑟(𝑡)sat𝜀+𝑥̇_𝑟,𝑛(𝑡)−Λ^𝑇_𝑟𝑒(𝑡), (3.44) where 𝑊(𝑡) is an online estimation of 𝑊^∗; 𝑘, 𝑘^∗, and 𝜀 are positive constants, and sat(⋅) represents a saturation function given by sat(𝑧)=1for𝑧≥1,𝑧for−1<𝑧<1,−1for𝑧≤−1. (3.45) Note that 𝑣(𝑡)=𝑞(𝑡)𝑣(𝑡)=𝑞(𝑡)+𝑞−𝑞𝑣(𝑡)=𝑞𝑣(𝑡)−𝑞(𝑡)𝑣(𝑡), (3.46) where 𝑞(𝑡)∶=𝑞−𝑞(𝑡).

Substituting (3.46) into (3.42) yields 𝑟̇(𝑡)=𝑊^∗𝜎(𝑥)+𝑏𝑚𝑞𝑣(𝑡)−𝑞(𝑡)𝑣(𝑡)−𝑥̇_𝑟,𝑛(𝑡)+Λ^𝑇_𝑟𝑒(𝑡)+𝜂(𝑥)+𝜉(𝑡)+𝑏𝑑(𝑡)=𝑊^∗𝜎(𝑥)+𝑣(𝑡)−𝑏𝑚𝑞(𝑡)𝑣(𝑡)−𝑥̇_𝑟,𝑛(𝑡)+Λ^𝑇_𝑟𝑒(𝑡)+𝜂(𝑥)+𝜉(𝑡)+𝑏𝑑(𝑡). (3.47) If (3.44) is substituted into (3.47) and reducing like terms, the filtered tracking error dynamics can be expressed as 𝑟̇(𝑡)=𝑊^∗𝜎(𝑥(𝑡))−𝑊(𝑡)𝜎(𝑥(𝑡))−𝑘𝑟(𝑡)−𝑘^∗𝑟(𝑡)sat𝜀+𝑥̇_𝑟,𝑛(𝑡)−Λ^𝑇_𝑟𝑒(𝑡)−𝑏𝑚𝑞(𝑡)𝑣(𝑡)−𝑥̇_𝑟,𝑛(𝑡)+Λ^𝑇_𝑟𝑒(𝑡)(𝑥)(𝑡)(𝑡)=𝑊(𝑡)𝜎(𝑥(𝑡(𝑡)+𝜂+𝜉+𝑏𝑑))−𝑘𝑟−𝑘^∗𝑟(𝑡)sat𝜀−𝑏𝑚𝑞(𝑡)𝑣(𝑡)+𝜁(𝑡), (3.48) where 𝑊(𝑡)∶=𝑊^∗−𝑊(𝑡) and 𝜁(𝑡)∶=𝜂(𝑥)+𝜉(𝑡)+𝑏𝑑(𝑡) is an unknown term bounded by the positive constant 𝜁, that is, |𝜁(𝑡)|≤𝜁. Consider the following Lyapunov function candidate: 𝑉(𝑡)=12𝑟²_𝜀(𝑡)+12𝑘₁𝑊(𝑡)𝑊^𝑇(𝑡)+𝑏𝑚2𝑘₂𝑞²(𝑡), (3.49) where 𝑘₁ and 𝑘₂ are positive constants and 𝑟_𝜀(𝑡) is the tuning error defined as [10] 𝑟_𝜀(𝑡)(𝑡)𝑟(𝑡)=𝑟−𝜀sat𝜀. (3.50) Taking into account (3.18), the first derivative of 𝑉(𝑡) can be calculated as 𝑉̇(𝑡)=𝑟_𝜀(𝑡)𝑟̇_𝜀(𝑡)+1𝑘₁𝑊(𝑡)𝑊̇^𝑇(𝑡)+𝑏𝑚𝑘₂𝑞(𝑡)𝑞̇(𝑡). (3.51) It can be demonstrated that 𝑟_𝜀(𝑡)𝑟̇_𝜀(𝑡)=𝑟_𝜀(𝑡)𝑟(̇𝑡). By substituting (3.48) into the last equality and the resulting expression into (3.51), we get 𝑉̇(𝑡)=𝑊(𝑡)𝜎(𝑥(𝑡))𝑟_𝜀(𝑡)−𝑘𝑟(𝑡)𝑟_𝜀(𝑡)−𝑘^∗𝑟_𝜀(𝑡)sat𝑟(𝑡)𝜀−𝑏𝑚𝑞(𝑡)𝑣(𝑡)𝑟_𝜀(𝑡)+𝑟_𝜀(𝑡)𝜁(𝑡)+1𝑘₁𝑊(𝑡)𝑊̇^𝑇(𝑡)+𝑏𝑚𝑘₂𝑞(𝑡)𝑞̇(𝑡). (3.52) Consider that 𝑊̇(𝑡) and 𝑞̇(𝑡) are chosen as 𝑊̇(𝑡)=−𝑘₁𝜎^𝑇(𝑥(𝑡))𝑟_𝜀(𝑡),𝑞̇(𝑡)=𝑘₂𝑣(𝑡)𝑟_𝜀(𝑡). (3.53)(3.54) If (3.53) and (3.54) are substituted into (3.52), we obtain 𝑉̇(𝑡)=−𝑘𝑟(𝑡)𝑟_𝜀(𝑡)−𝑘^∗𝑟_𝜀(𝑡)sat𝑟(𝑡)𝜀+𝑟_𝜀(𝑡)𝜁(𝑡). (3.55) Now, from (3.50), 𝑟(𝑡)=𝑟_𝜀(𝑡)𝑟(𝑡)+𝜀sat𝜀. (3.56) Besides, 𝑟_𝜀(𝑡)𝜁(𝑡)≤𝑟_𝜀(𝑡)𝜁(𝑡)=𝑟_𝜀(𝑡)𝜁(𝑡)≤𝑟_𝜀(𝑡)𝜁. (3.57) Substituting (3.56) and (3.57) into (3.55) yields 𝑉̇(𝑡)=−𝑘𝑟_𝜀(𝑡)+𝜀sat𝑟(𝑡)𝜀𝑟_𝜀(𝑡)−𝑘^∗𝑟_𝜀(𝑡)𝑟(𝑡)sat𝜀+𝑟_𝜀(𝑡)𝜁(𝑡)≤−𝑘𝑟²_𝜀(𝑡)−𝑘𝜀𝑟_𝜀(𝑡)𝑟(𝑡)sat𝜀−𝑘^∗𝑟_𝜀(𝑡)𝑟(𝑡)sat𝜀+𝑟_𝜀(𝑡)𝜁. (3.58) Considering that 𝑟_𝜀(𝑡)sat(𝑟(𝑡)/𝜀)=|𝑟_𝜀(𝑡)|, (3.58) can be expressed as 𝑉̇(𝑡)≤−𝑘𝑟²_𝜀(𝑡)−𝑘𝜀𝑟_𝜀(𝑡)−𝑘^∗𝑟_𝜀(𝑡)+𝑟_𝜀(𝑡)𝜁. (3.59) Note that, if 𝑘^∗ and 𝑘 are selected in such a way that 𝑘^∗+𝑘𝜀>𝜁, then 𝑉̇(𝑡)≤−𝑘𝑟²_𝜀(𝑡)−𝑘^∗+𝑘𝜀−𝜁𝑟_𝜀(𝑡)≤−𝑘𝑟²_𝜀(𝑡). (3.60) Because 𝑘 is a positive constant, the last inequality implies that 𝑉̇(𝑡)≤0. Based on this fact, it is possible to establish that 𝑉(𝑡)≤𝑉(0) and, therefore, 𝑉(𝑡)∈𝐿_∞. Since 𝑉(𝑡) is a nonnegative function, 𝑟_𝜀(𝑡), 𝑊(𝑡) and 𝑞(𝑡) belongs to 𝐿_∞. Because 𝑊(𝑡)=𝑊^∗−𝑊(𝑡), and 𝑞(𝑡)=𝑞−𝑞(𝑡), and as 𝑊^∗ and 𝑞 are constants, the boundedness of 𝑊(𝑡) and 𝑞(𝑡) can be guaranteed. From the definition of tuning error (3.50) and as 𝜀sat(𝑟(𝑡)/𝜀) is a bounded term, if 𝑟_𝜀(𝑡)∈𝐿_∞, then 𝑟(𝑡)∈𝐿_∞. Keeping in view the above fact, and on the basis of Remark 3.2, the boundedness of 𝑒(𝑡) can be assured. Now, it can be seen from (3.44) that 𝑣(𝑡) is formed by bounded terms. Thus, 𝑣(𝑡)∈𝐿_∞. Likewise, it can be seen from (3.43) that 𝑣(𝑡) is the product of two bounded variables and consequently 𝑣(𝑡) is also bounded. On the other hand, an inspection of (3.48) reveals that 𝑟(̇𝑡)∈𝐿_∞. From (3.50), this means that 𝑟̇_𝜀(𝑡) is bounded too. Integrating both sides of (3.60) from 0 to 𝑡 yields 𝑉(𝑡)−𝑉(0)≤−𝑘^𝑡₀𝑟²_𝜀(𝜏)𝑑𝜏. (3.61) Note that the last inequality can be expressed as ^𝑡₀𝑟²_𝜀(𝜏)𝑉(0)(𝑡)𝑑𝜏≤−𝑉𝑘. (3.62) Since 𝑉(𝑡) is a nonnegative function, the following is true: 𝑉(𝑡)≥0⇒𝑉(𝑡)−𝑉(0)≥−𝑉(0)⇒𝑉(0)−𝑉(𝑡)≤𝑉(0). (3.63) Substituting (3.63) into (3.62) yields ^𝑡₀𝑟²_𝜀(𝜏)𝑉(0)𝑑𝜏≤𝑘 (3.64) taking the limit as 𝑡→∞ of both sides of the last inequality, finally, we obtain ^∞₀𝑟²_𝜀(𝜏)𝑉(0)𝑑𝜏≤𝑘. (3.65) This means that 𝑟_𝜀(𝑡)∈𝐿₂. As 𝑟_𝜀(𝑡)∈𝐿₂∩𝐿_∞ and 𝑟̇_𝜀(𝑡)∈𝐿_∞, from Barbalat’s Lemma, we can conclude that 𝑟_𝜀(𝑡) converges asymptotically to zero. From definition (3.50), this implies that 𝑟(𝑡) asymptotically converges to a region around zero bounded by 𝜀. In view of the above and from Remark 3.2, we can conclude that 𝑒₁(𝑡)=𝑦(𝑡)−𝑦_𝑟(𝑡) converges asymptotically to a region around zero bounded by the term 𝜀/𝜆_𝑟^𝑛−1. Thus, the following theorem has been proven.

Theorem 3.4. If Assumptions 2.1–3.1 are satisfied, 𝑘^∗+𝑘𝜀>𝜁, and the control input (3.43), (3.44) with the learning laws 𝑊̇(𝑡)=𝑘₁𝜎^𝑇(𝑥(𝑡))𝑟_𝜀(𝑡), 𝑞̇(𝑡)=−𝑘₂𝑣(𝑡)𝑟_𝜀(𝑡) are applied to the system (2.2)-(2.3), then (a)the tuning error, the filtered tracking error, tracking error, the weights, the 𝑏𝑚 reciprocal estimation, and the control signal are bounded: 𝑟_𝜀(𝑡),𝑟(𝑡),𝑒(𝑡),𝑊(𝑡),𝑞(𝑡),𝑣(𝑡)∈𝐿_∞, (3.66)(b)the actual tracking error 𝑦(𝑡)−𝑦_𝑟(𝑡) converges asymptotically to a region around zero bounded by the term 𝜀𝜆_𝑟^𝑛−1, (3.67) where 𝜀 and 𝜆_𝑟 are positive constants chosen by the designer.

3.2.1. Estimation of a Bound for 𝜁(𝑡)

Certainly, compared with the scheme I, the structure of the scheme II is simpler. Nevertheless, the implementation of this last scheme requires a good estimation of 𝜁 in order to guarantee the theoretical performance provided by Theorem 3.4. Here, we propose an offline practical procedure to achieve this goal.

Consider that some experimental data (𝑥_𝑡𝑖,𝑣(𝑡_𝑖))_{𝑖=1,…,𝑁} are available (see Remark 3.5). By substituting (2.12) into (3.7), we obtain 𝑥̇_𝑛(𝑡)=𝑊^∗𝜎(𝑥(𝑡))+𝑏𝑚𝑣(𝑡)+𝜁(𝑡). (3.68) Certainly, if 𝑥̇_𝑛(𝑡), 𝑊^∗, and 𝑏𝑚 could be known, 𝜁(𝑡) can be completely determined. By hypothesis, 𝑥_𝑛(𝑡) is known. However, because of the noise, 𝑥̇_𝑛(𝑡) must not be calculated directly from 𝑥_𝑛(𝑡). Instead, a robust differentiation strategy should be used such as sliding modes [19], high-order sliding modes [20, 21], or smoothing by least squares, among others. Hereafter, the estimation of 𝑥̇_𝑛(𝑡) is denoted by 𝑥̇‌_𝑛(𝑡). Once 𝑥̇‌_𝑛(𝑡) is obtained, let us consider the following regression model: 𝑊^∗𝜎𝑥𝑡_𝑖+𝑏𝑚𝑣𝑡_𝑖=𝑥̇_𝑛𝑡_𝑖+𝜒𝑡_𝑖, (3.69) where 𝜒(𝑡_𝑖) is simply an error term. By defining 𝑋_𝑡𝑖∶=[𝜎^𝑇(𝑥(𝑡_𝑖)),𝑣(𝑡_𝑖)], and Φ∶=[𝑊^∗,𝑏𝑚], (3.69) can be expressed as Φ𝑋^𝑇_𝑡𝑖=𝑥̇_𝑛𝑡_𝑖+𝜒𝑡_𝑖. (3.70) By using least-squares method, Φ can be estimated as Φ=𝑊^∗,𝑏𝑚=argmin_Φ^𝑁_𝑖=1𝜒²𝑡_𝑖=𝑋^𝑇𝑋⁻¹𝑋^𝑇𝑌^𝑇, (3.71) where 𝑋∶=[𝑋_𝑡1,…,𝑋_𝑡𝑁]^𝑇 and 𝑌=[𝑥̇_𝑛(𝑡₁),…,𝑥̇_𝑛(𝑡_𝑁)]^𝑇. Once Φ is determined, 𝜁(𝑡) can be approximated as 𝜁(𝑡)≈𝑥̇_𝑛(𝑡)−𝑊^∗𝜎(𝑥(𝑡))−𝑏𝑚𝑣(𝑡). (3.72) Next, 𝜁 can be estimated from (3.72).

Remark 3.5. These experimental data can be generated by trying to use the neurocontroller II with relatively large values for the constant parameters. A first attempt could be to let 𝑘=100 and 𝑘^∗=50. If the tracking is not satisfactory, then larger values could be tried.

Remark 3.6. Alternatively, in [22], the determination of an error term 𝛿(𝑡) is achieved by using of sliding modes.

4. Numerical Example

In this section, the proposed neurocontrollers are tested by simulation on the following second order nonlinear system: 𝑥̇₁(𝑡)=𝑥₂(𝑡),𝑥̇₂(𝑡)=−2.31−exp−𝑥₁(𝑡)1+exp−𝑥₂(𝑡)𝑥+3.7²(𝑡)sin𝑥₁(𝑡)𝑥₂(𝑡)cos𝑥₂(𝑡)𝑥²₁(𝑡)+𝑥²₂(𝑡)+1+1.5𝑥₁(𝑡)𝑥₂(𝑡)+0.7𝑥₁(𝑡)𝑥³₂(𝑡)sin2𝑥₁(𝑡)+0.4𝑥²₁(𝑡)𝑥₂(𝑡)+3.5𝑢(𝑡)+𝜉(𝑡). (4.1) The initial condition for system (4.1) is 𝑥₁(0)=1,𝑥₂(0)=−1;𝑢(𝑡) is the deadzone output whose parameters are 𝑚_𝑟=𝑚_𝑙=1.5,𝑏_𝑟=2.5,𝑏_𝑙=−2; 𝜉(𝑡), the disturbance term is selected as 𝜉(𝑡)=sin(10𝑡). The following reference trajectory is proposed 𝑦_𝑟(𝑡)=−cos(𝑡)+1.5cos(2𝑡)−0.5. The first and second derivative of 𝑦_𝑟(𝑡) can be calculated analytically. In another case, a robust differentiation method must be used. It is very important to mention that the nonlinear system (4.1), the exact values for deadzone parameters, and the disturbance term are assumed completely unknown for the neurocontrollers during the design and simulation process. That is, the system (4.1) is only used as a data generator.

With respect to the tuning process, we must recognize that, similarly to many techniques of adaptive control, we do not have a systematic procedure in order to find the proper values for the controller parameters. Instead, an approach by trial and error is used. Thus, through various iterative simulations, the following values for the parameters of the neurocontroller I were found: 𝑘=100,𝜀₀=0.2,𝑘₁=400,𝑙₁=2,𝑘₂=10,𝑙₂𝑘=1,₃=100,𝑙₃=1,𝜆_𝑟=1,𝑊⁰=[],𝑊(0)1,0,1,0.5=𝑊⁰,𝜃⁰=[1,1,1,−1]^𝑇,𝜃(0)=𝜃⁰,𝑝⁰=1,𝑝(0)=𝑝⁰,𝜎_𝑗(𝑥(𝑡=2))1+exp−∑²_𝑖=1𝑐_{𝜎𝑗,𝑖}𝑥_𝑖(𝑡𝑐)−1for𝑗=1,2,3,4,_𝜎1,1=1,𝑐_𝜎1,2=1,𝑐_𝜎2,1=0.5,𝑐_𝜎2,2𝑐=0.5,_𝜎3,1=2,𝑐_𝜎3,2=2,𝑐_𝜎4,1=0.4,𝑐_𝜎4,2=0.8. (4.2) Notice that a great freedom is allowed in order to select 𝑊⁰, 𝜃⁰, and 𝑝⁰. However, the designer should be aware of that as these parameters take values increasingly different from the optimal ones, the parameter 𝛽 in Theorem 3.3 becomes larger. On the other hand, we have seen that by setting 𝑊(0)=𝑊⁰, 𝜃(0)=𝜃⁰, and 𝑝(0)=𝑝⁰, a more regular behavior of the closed-loop system can be obtained [23, 24].

The simulation is carried out by means of Simulink with ode45 method, relative tolerance equal to 1𝑒−6, and absolute tolerance equal to 1𝑒−8. A Simulink block diagram for the neurocontroller based on scheme I is shown in Figure 1. The results of the tracking process are presented in Figures 2–4 for the first 20 seconds of the simulation. In Figure 2, the output of nonlinear system (4.1), 𝑦(𝑡)=𝑥₁(𝑡), is represented by dashed line whereas the reference trajectory 𝑦_𝑟(𝑡) is represented by solid line. In spite of the difference between 𝑦(0) and 𝑦_𝑟(0), the tracking process shows a satisfactory behavior. This can be verified more specifically in Figure 3 where the actual tracking error, 𝑦(𝑡)−𝑦_𝑟(𝑡), is illustrated. In order to appreciate better the quality of the tracking process, a zoom of Figure 3 is presented in Figure 4. From Figures 3 and 4, we can appreciate that the actual tracking error converges fast to a zone bounded by 0.02. Finally, the control signal 𝑣(𝑡) acting as the input of deadzone is shown in Figure 5.

Figure 1 

Simulink block diagram for the neurocontroller I.

Figure 2 

Tracking process for the neurocontroller I.

Figure 3 

Tracking error evolution for the neurocontroller I.

Figure 4 

Tracking error evolution for the neurocontroller I (zoom of Figure 3).

Figure 5 

Control signal of the scheme I.

For the case of the neurocontroller II, its parameters are selected very similarly to the neurocontroller I parameters in order to provide a proper reference for comparison. The parameters of the neurocontroller II are 𝑘=100,𝑘^∗=30,𝜀=0.1,𝑘₁=400,𝑘₂=10,𝜆_𝑟𝑊=1,₀=[1,0,1,0.5],𝑊(0)=𝑊⁰,𝑞⁰=1,𝑞(0)=𝑞⁰,𝜎_𝑗(𝑥(𝑡=2))1+exp−∑²_𝑖=1𝑐_{𝜎𝑗,𝑖}𝑥_𝑖(𝑡)𝑐−1for𝑗=1,2,3,4,_𝜎1,1=1,𝑐_𝜎1,2=1,𝑐_𝜎2,1=0.5,𝑐_𝜎2,2𝑐=0.5,_𝜎3,1=2,𝑐_𝜎3,2=2,𝑐_𝜎4,1=0.4,𝑐_𝜎4,2=0.8. (4.3) A Simulink block diagram for the neurocontroller based on scheme II is shown in Figure 6. Since a very similar performance is obtained, the signals of the actual tracking error for the two neurocontrollers are presented together in Figure 7. In Figure 8, a zoom of Figure 7 is presented. As can be seen from Figures 7 and 8, the actual tracking error for the controller II converges fast to a zone bounded by 0.02. This is a remarkable behavior since Theorem 3.4 only guarantees the asymptotical convergence to a zone bounded by 0.2. Finally, the control signals of the two neurocontrollers are displayed in Figure 9.

Figure 6 

Simulink block diagram for the neurocontroller II.

Figure 7 

Tracking error evolution for the two neurocontrollers.

Figure 8 

Tracking error evolution for the two neurocontrollers (zoom of Figure 7).

Figure 9 

Control signals of the two schemes.

So far, the neurocontroller II has been used in a nonrigorous way. In order to guarantee its stability, a bound for 𝜁(𝑡) must be first determined. By using the procedure presented in Section 3, 𝑊^∗,𝑏𝑚 can be estimated as 𝑊^∗=[10.7869,−29.0274,−7.4315,19.3033], 𝑏𝑚=0.2453. With these parameters, the maximum value for 𝜁(𝑡) is estimated as 7.5819 and a very conservative bound for 𝜁(𝑡) can be established as 𝜁=16. In view of the above and in accordance with the selected values for the neurocontroller II parameters and from Theorem 3.4, the stability of the closed-loop system can be guaranteed.

As the comparison between two empirically tuned techniques is necessarily limited, the main objective of this numerical example was rather to show that by using any of the two proposed controllers, a good performance can be obtained. Notwithstanding, it must be mentioned that the tuning process was easier for the case of Scheme II.

5. Conclusions

In this paper, two adaptive schemes based on universal approximation property of the neural networks are proposed in order to control an unknown SISO nonlinear system in Brunovsky canonical form with unknown deadzone input. The objective is to determine a control signal such that the output of the unknown plant follows a specified reference trajectory, and, at the same time, all closed-loop signals stay bounded in spite of the presence of unknown but bounded disturbances. The first scheme utilizes a smooth adaptive inverse of the deadzone while the second scheme considers the deadzone as a combination of a linear term and a disturbance-like term. For the first case, the exponential convergence of the tracking error is guaranteed by using a Lyapunov analyses. For the second case, only the asymptotical convergence can be guaranteed theoretically. However, a numerical example shows that a similarly satisfactory performance can be obtained by using any of the two proposed controllers. The designer should select between these two schemes in accordance with the particular features of his application and considering the following:(i)Scheme I does not need the specific knowledge of a bound for the unknown dynamics and/or the disturbance term.(ii)Scheme I can handle the case when 𝑚_𝑟≠𝑚_𝑙.(iii)The tuning and implementation process is easier for Scheme II.Finally, it must be mentioned that a compromise should be established between the accuracy of the tracking process and the smoothness of the control signal.

Acknowledgment

J. H. Pérez-Cruz would like to thank the financial support through a postdoctoral fellowship from the Mexican National Council for Science and Technology (CONACYT).