Adaptive Fuzzy Tracking Control for Uncertain Nonlinear Time-Delay Systems with Unknown Dead-Zone Input

Abstract

The tracking control problem of uncertain nonlinear time-delay systems with unknown dead-zone input is tackled by a robust adaptive fuzzy control scheme. Because the nonlinear gain function and the uncertainties of the controlled system including matched and unmatched uncertainties are supposed to be unknown, fuzzy logic systems are employed to approximate the nonlinear gain function and the upper bounded functions of these uncertainties. Moreover, the upper bound of the uncertainty caused by the fuzzy modeling error is also estimated. According to these learning fuzzy models and some feasible adaptive laws, a robust adaptive fuzzy tracking controller is developed in this paper without constructing the dead-zone inverse. Based on the Lyapunov stability theorem, the proposed controller not only guarantees that the robust stability of the whole closed-loop system in the presence of uncertainties and unknown dead-zone input can be achieved, but it also obtains that the output tracking error can converge to a neighborhood of zero exponentially. Some simulation results are provided to demonstrate the effectiveness and performance of the proposed approach.

1. Introduction

In general systems, there exist some nonsmooth nonlinearities in the actuators, such as dead-zone, saturation, and backlash [1–7]. The information of the dead-zone is usually poorly known and time variant. Recently, high accuracy position control is required, such as DC servosystems, pressure control systems, power systems, chemical reactor systems, and machine tools [1–3, 8]. However, the dead-zone characteristics in actuators may severely limit the performance of the systems and let the output of the systems not reach our requirements. The robust adaptive control was proposed to deal with nonlinear systems with unknown dead-zone [2]. In Corradini and Orlando [3], the sliding mode controller was presented to robustly stabilize a nonlinear uncertain input. Robust adaptive dead-zone compensation method was used in a DC servo-motor control system [4]. Variable structure control laws were proposed for uncertain large-scale system with dead-zone input [5]. In [8, 9], adaptive control approach was used to cope with nonlinear systems with nonsymmetric dead-zone input. The proposed controllers in [10, 11] tackled the plants with unknown dead-zone via dead-zone inverse. However, the common feature of most previous results [1, 2, 4–6, 8, 9, 12] is the nonlinear gain function which is assumed to be a constant. Although the Previous restrictive assumption can be relaxed in [3, 7, 10, 11], the unmatched uncertainty is not taken into account. Therefore, the motivation of this paper is to synthesize a controller to handle the tracking control problem for a class of uncertain nonlinear state time-delay systems in the presence of an unknown dead-zone input and unmatched uncertainties without constructing the dead-zone inverse.

It is well known that a real system is difficult to be described by the exact mathematical model, owing to the existence of uncertain elements, such as parameter variation, modeling errors, unmodeled dynamics, and external disturbances. These uncertainties may affect the stability of the systems. Robust stabilization of the nonlinear uncertain system has widely been investigated [13–16]. In [13], the purpose of this direct robust adaptive fuzzy controller was to deal with a class of nonlinear systems containing both unconstructed state-dependent unknown nonlinear uncertain and gain functions. Bartolini et al. [14] suggested the second-order sliding mode controller to cope with the uncertain system nonaffine in the control law and the presence of the unmodeled dynamic actuator. The methods of robust adaptive control [15, 16] were utilized to solve the nonlinear uncertain problem. In [15], the robust adaptive controller for SISO nonlinear uncertain system was presented by the input/output linearization approach. In the case where the nonlinear uncertain systems include constant linearly parameterized uncertainty and nonlinear state-dependent parametric uncertainty, the direct robust adaptive control framework was developed in [16].

In recent years, the design problem of nonlinear time-delay systems has received considerable attention in [17–23] because time-delay characteristic usually confronted in engineering systems may degrade the control performance and make the systems unstable. By employing the input-output approach and the scaled small gain theorem, the filtering problem for discrete-time T-S fuzzy systems with time-varying delay has been studied [17]. In [18], the stabilization of LTI systems with time delay was considered by using a low-order controller. The stability analysis and robust control for time-delay systems attracted a large number of researchers over the past years [19–21]. Recently, the problem of stability analysis for stochastic neural networks with discrete interval and distributed time-varying was investigated by applying the idea of delay partitioning method [23].

On the other hand, the fuzzy control techniques have been widely used in many control problems in recent years [24–26]. The fuzzy logic system is constructed from a collection of fuzzy IF-THEN rules. It becomes a useful way to approximate the unknown nonlinear functions and uncertainties in the nonlinear systems. An adaptive interval type-2 fuzzy sliding mode controller for a class of unknown nonlinear discrete-time systems corrupted by external disturbances was presented [24]. In [25], an adaptive neural-fuzzy control design was examined for tracking of nonlinear affine in the control dynamic systems with unknown nonlinearities. Based on a novel fuzzy Lyapunov-Krasovskii functional, a delay partitioning method has been developed for the delay-dependent stability analysis of fuzzy time-varying state delay systems [26].

In this paper, the problem of output tracking control is investigated for a class of uncertain nonlinear state time-delay systems containing unknown dead-zone input and unmatched uncertainties. The main features of the proposed robust adaptive fuzzy controller are summarized as follows. (i) By utilizing a description of a dead-zone feature, an adaptive law is used to estimate the properties of the dead-zone model intuitively and mathematically, without constructing a dead-zone inverse. (ii) Fuzzy logic systems with some appropriate learning laws are applied to approximate the nonlinear gain function and the upper bounded functions of matched and unmatched uncertainties. (iii) The unknown upper bound of the uncertainties caused by approximation (or fuzzy modeling) error is estimated by a simple adaptive law. (iv) By means of Lyapunov stability theorem, the proposed controller cannot only guarantee the robust stability of the whole closed-loop system but also obtain the good tracking performance.

This paper is organized as follows. In Section 2, the form of the uncertain nonlinear state time-delay system with unknown dead-zone input is described. The fuzzy logic systems and fuzzy basis functions are also reviewed. Section 3 presents the robust adaptive fuzzy tracking controller to deal with a class of nonlinear uncertain state time-delay systems containing unknown dead-zone input. By Lyapunov stability theorem, the presented controller can ensure the stability of the controlled systems. Simulation results are demonstrated along with the effectiveness and performance of the proposed controller in Section 4. Finally, a conclusion is given in Section 5.

2. Problem Statement and Preliminaries

2.1. Problem Statement

Consider a class of uncertain nonlinear state time-delay systems containing an unknown dead-zone in the following form: 𝑥̇₁=𝑥₂+Δ𝜙₁(𝐱),𝑥̇₂=𝑥₃+Δ𝜙₂(𝐱),𝑥̇₃=𝑥₄+Δ𝜙₃(𝐱),⋮𝑥̇_𝑛−1=𝑥_𝑛+Δ𝜙_𝑛−1(𝐱),𝑥̇_𝑛=^𝑀_𝑖=1𝜃_1𝑖𝑓_1𝑖(𝐱(𝑡))+Δ𝑓₁(𝐱(𝑡+))_𝑁𝑗=1𝜃_2𝑗𝑓_2𝑗(𝐱(𝑡−𝜏))+Δ𝑓₂(𝐱((𝐱)𝑍(𝑣(𝑡𝑡−𝜏))+𝑔))+Δ𝜙_𝑛(𝐱),𝑦=𝑥₁, (1) or equivalently, 𝐱̇=𝐀𝐱+𝐁^𝑀_𝑖=1𝜃_1𝑖𝑓_1𝑖(𝐱(𝑡))+Δ𝑓₁(𝐱(𝑡+))_𝑁𝑗=1𝜃_2𝑗𝑓_2𝑗(𝐱(𝑡−𝜏))+Δ𝑓₂(𝐱(𝑡−𝜏))+𝑔(𝐱)𝑍(𝑣(𝑡))+Θ(𝐱),𝑦=𝐂𝐱, (2) where 𝐀=010⋯0001⋱⋮⋮⋮⋱⋱000⋯0100⋯00∈𝑅^𝑛×𝑛00⋮01,𝐁=∈𝑅^𝑛×1,𝐂^𝑇=10⋮00∈𝑅^𝑛×1, (3) where 𝐱(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇∈𝑅^𝑛 is the system state vector which is assumed to be available for measurement, and 𝑣(𝑡)∈𝑅 and 𝑦(𝑡)∈𝑅 are the input and output of the system, respectively. 𝜏 is the value of time delay. The unknown nonlinear system functions are assumed to be in the linearly parameterized form and consist of two parts: (i) the sum of 𝜃_1𝑖𝑓_1𝑖(𝐱(𝑡)) for 𝑖=1,2,…,𝑀; (ii) the sum of 𝜃_2𝑗𝑓_2𝑗(𝐱(𝑡−𝜏)) for 𝑗=1,2,…,𝑁. The parameters 𝜃_1i and 𝜃_2j are unknown but constant. 𝑓_1𝑖(𝐱(𝑡)) and 𝑓_2𝑗(𝐱(𝑡−𝜏)) are known continuous, linear or nonlinear functions. Δ𝑓₁(𝐱(𝑡)) and Δ𝑓₂(𝐱(𝑡−𝜏)) are the unknown matched uncertainties. 𝑔(𝐱(𝑡)) is the unknown nonlinear gain function, and Θ(𝐱)= [Δ𝜙₁(𝐱),Δ𝜙₂(𝐱),…,Δ𝜙_𝑛(𝐱)]^𝑇∈𝑅^𝑛×1 is the vector of unknown unmatched uncertainties. Without loss of generality, it is assumed that the sign of 𝑔(𝐱(𝑡)) is positive. 𝑍(𝑣(𝑡))∶𝑅→𝑅 is the nonlinear input function containing a dead-zone.

Now, let the output of the system and its derivatives be expressed as follows: 𝑦=𝑥₁,𝑦⁽¹⁾=𝑥̇₁=𝑥₂+Δ𝜙₁,𝑦⁽²⁾=𝑥̇₂+(Δ𝜙₁)⁽¹⁾=𝑥₃+Δ𝜙₂+(Δ𝜙₁)⁽¹⁾,𝑦⁽³⁾=𝑥̇₃+(Δ𝜙₂)⁽¹⁾+(Δ𝜙₁)⁽²⁾=𝑥₄+Δ𝜙₃+(Δ𝜙₂)⁽¹⁾+(Δ𝜙₁)⁽²⁾,⋮𝑦^(𝑛−1)=𝑥̇_(𝑛−1)+Δ𝜙_(𝑛−2)⁽¹⁾+Δ𝜙_(𝑛−3)⁽²⁾+⋯+Δ𝜙₂^(𝑛−3)+Δ𝜙₁^(𝑛−2)=𝑥_𝑛+Δ𝜙_(𝑛−1)+Δ𝜙_(𝑛−2)⁽¹⁾+Δ𝜙_(𝑛−3)⁽²⁾+⋯+Δ𝜙₂^(𝑛−3)+Δ𝜙₁^(𝑛−2),𝑦^(𝑛)=𝑥̇_𝑛+Δ𝜙_(𝑛−1)⁽¹⁾+Δ𝜙_(𝑛−2)⁽²⁾+Δ𝜙_(𝑛−3)⁽³⁾+⋯+Δ𝜙₂^(𝑛−2)+Δ𝜙₁^(𝑛−1)=^𝑀_𝑖=1𝜃_1𝑖𝑓_1𝑖(𝐱(𝑡))+Δ𝑓₁(𝐱(𝑡+))_𝑁𝑗=1𝜃_2𝑗𝑓_2𝑗(𝐱(𝑡−𝜏))+Δ𝑓₂(𝐱((𝐱)𝑍(𝑣(𝑡𝑡−𝜏))+𝑔))+Δ𝜙_𝑛+(Δ𝜙_(𝑛−1))⁽¹⁾+(Δ𝜙_(𝑛−2))⁽²⁾+(Δ𝜙_(𝑛−3))⁽³⁾+⋯+(Δ𝜙₂)^(𝑛−2)+(Δ𝜙₁)^(𝑛−1)=^𝑀_𝑖=1𝜃_1𝑖𝑓_1𝑖(𝐱(𝑡))+Δ𝑓₁(𝐱(𝑡+))_𝑁𝑗=1𝜃_2𝑗𝑓_2𝑗(𝐱(𝑡−𝜏))+Δ𝑓₂(𝐱((𝐱)𝑍(𝑣(𝑡𝑡−𝜏))+𝑔))+ΔΦ, (4) where ΔΦ=Δ𝜙₁^(𝑛−1)+Δ𝜙₂^(𝑛−2)+⋯+Δ𝜙⁽¹⁾_𝑛−1+Δ𝜙_𝑛. (5)

The dead-zone with input 𝑣(𝑡) and output as shown in Figure 1 is described by 𝑍(𝑣(𝑡=𝑚))^𝑟𝑣(𝑡)−𝑐_𝑎for𝑣(𝑡)≥𝑐_𝑎,0for𝑐_𝑏<𝑣(𝑡)<𝑐_𝑎,𝑚_𝑙𝑣(𝑡)−𝑐_𝑏for𝑣(𝑡)≤𝑐_𝑏, (6) where 𝑐_𝑎>0,𝑐_𝑏<0 and 𝑚_𝑟>0,𝑚_𝑙>0 are parameters and slopes of the dead-zone, respectively. In order to investigate the key features of the dead-zone in the control problems, the following assumptions should be made.

Figure 1 

Dead-zone model.

Assumption 1. The dead-zone output 𝑍(𝑣(𝑡)) is not available to obtain.

Assumption 2. The dead-zone slopes are of the same value; that is, 𝑚_𝑟=𝑚_𝑙=𝑚.

Assumption 3. There exist known constants 𝑐_𝑎min, 𝑐_𝑎max, 𝑐_𝑏min, 𝑐_𝑏max, 𝑚_min, and 𝑚_max such that the unknown dead-zone parameters 𝑐_𝑎,𝑐_𝑏, and 𝑚 are bounded; that is, 𝑐_𝑎∈[𝑐_𝑎min, 𝑐_𝑎max], 𝑐_𝑏∈[𝑐_𝑏min,  𝑐_𝑏max], and 𝑚∈[𝑚_min, 𝑚_max].

Based on the previous assumptions, the expression (6) can be represented as 𝑍(𝑣(𝑡))=𝑚𝑣(𝑡)+𝑧(𝑣(𝑡)), (7) where 𝑧(𝑣(𝑡)) can be calculated from (6) and (7) as 𝑧(𝑣(𝑡))=−𝑚𝑐_𝑎for𝑣(𝑡)≥𝑐_𝑎,−𝑚𝑣(𝑡)for𝑐_𝑏<𝑣(𝑡)<𝑐_𝑎,−𝑚𝑐_𝑏for𝑣(𝑡)≤𝑐_𝑏. (8) From Assumptions 2 and 3, we can conclude that 𝑧(𝑣(𝑡)) is bounded and satisfies |𝑧(𝑣(𝑡))|≤𝜌, where 𝜌 is the upper bound which can be chosen as 𝜌=max𝑚_max𝑐_𝑎max,−𝑚_max𝑐_𝑏max, (9) where 𝑐_𝑏min is a negative value.

Then, let 𝑦_𝑚 be a given bounded reference signal and contain finite derivatives up to the 𝑛th order, define the tracking error as 𝑒_𝑖=𝑦_𝑚^(𝑖−1)−𝑦^(𝑖−1),for𝑖=1,2,…,𝑛, (10) and denote 𝐞=[𝑒₁,𝑒₂,…,𝑒_𝑛]^𝑇,  𝐲=[𝑦,𝑦,̇…,𝑦^(𝑛−1)]^𝑇, and 𝐲_𝑚=[𝑦_𝑚,𝑦̇_𝑚,…,𝑦_𝑚^(𝑛−1)]^𝑇.

The control objective of this paper is to design a control law 𝑣(𝑡) such that 𝑦 can follow a given desired reference signal 𝑦_𝑚 and guarantee that all the signals involved in the whole closed-loop system are bounded.

2.2. Description of Fuzzy Logic Systems

The basic configuration of the fuzzy logic system consists of four main components: fuzzy rule base, fuzzy inference engine, fuzzifier, and defuzzifier [27]. The fuzzy logic system performs a mapping from 𝑈⊂𝑅^𝑛 to 𝑉⊂𝑅. Let 𝑈=𝑈₁×⋯×𝑈_𝑛, where 𝑈_𝑖⊂𝑅,𝑖=1,2,…,𝑛. The fuzzy rule base consists of a collection of fuzzy IF-THEN rules as follows: 𝑅^(𝑙)∶IF𝑥₁is𝐹^𝑙₁and…and𝑥_𝑛is𝐹^𝑙_𝑛,THEN𝑦is𝐺^𝑙, (11) where 𝐱=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇∈𝑈 and 𝑦∈𝑉⊂𝑅 are the input and output of the fuzzy logic system, and 𝐹^𝑙_𝑖 and 𝐺^𝑙 are fuzzy sets in 𝑈_𝑖 and 𝑉, respectively. The fuzzifier maps a crisp point 𝐱=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇 into a fuzzy set in 𝑈. The fuzzy inference engine performs a mapping from fuzzy sets in 𝑈 to fuzzy sets in 𝑉, based upon the fuzzy IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The defuzzifier maps a fuzzy set in 𝑉 to a crisp point in 𝑉.

The fuzzy systems with center-average defuzzifier, product inference, and singleton fuzzifier are of the following form: 𝑦(𝐱)=∑^𝑀_𝑙=1𝜃^𝑙∏^𝑛_𝑖=1𝜇_𝐹𝑙𝑖𝑥_𝑖∑^𝑀_𝑙=1∏^𝑛_𝑖=1𝜇_𝐹𝑙𝑖𝑥_𝑖, (12) where 𝑀 is the number of rules, 𝜃^𝑙 is the point at which the fuzzy membership function 𝜇_𝐺𝑙(𝜃^𝑙) of fuzzy sets 𝐺^𝑙 achieves its maximum value, and it is assumed that 𝜇_𝐺𝑙(𝜃^𝑙)=1. Equation (12) can be rewritten as 𝑦(𝐱)=𝜽^𝑇𝝃(𝐱), (13) where 𝜽=[𝜃¹,𝜃²,…,𝜃^𝑀]^𝑇 is a parameter vector, and 𝝃(𝐱)=[𝜉¹(𝐱),…,𝜉^𝑀(𝐱)]^𝑇 is a regressive vector with the regressor 𝜉^l(𝐱), which is defined as fuzzy basis function 𝜉^𝑙(𝐱)=∏^𝑛_𝑖=1𝜇_𝐹𝑙𝑖𝑥_𝑖∑^𝑛_𝑙=1∏^𝑛_𝑖=1𝜇_𝐹𝑙𝑖𝑥_𝑖. (14)

3. Adaptive Fuzzy Tracking Controller Design and Stability Analysis

According to (4), (7), and (10), the tracking error dynamic equation can be expressed as 𝐞̇=𝐀𝐞+𝐁𝑦_𝑚^(𝑛)−^𝑀_𝑖=1𝜃_1𝑖𝑓_1𝑖(𝐱(𝑡))−Δ𝑓₁(𝐱(𝑡−))_𝑁𝑗=1𝜃_2𝑗𝑓_2𝑗(𝐱(𝑡−𝜏))−Δ𝑓₂(𝐱((𝐱)(𝑡)(𝐱)𝑧(𝑣(𝑡.𝑡−𝜏))−𝑔𝑚𝑣−𝑔))−ΔΦ (15)

Now, let us choose a vector 𝐊=[𝑘₁,𝑘₂,…,𝑘_𝑛]∈𝑅^1×𝑛 such that 𝐀_𝑚=𝐀−𝐁𝐊 is Hurwitz; then, the tracking error dynamic equation (15) can be rewritten as 𝐞̇=𝐀_𝑚𝐞+𝐁𝐊𝐞+𝑦_𝑚^(𝑛)−^𝑀_𝑖=1𝜃_1𝑖𝑓_1𝑖(𝐱(𝑡))−Δ𝑓₁(𝐱(𝑡−))_𝑁𝑗=1𝜃_2𝑗𝑓_2𝑗(𝐱(𝑡−𝜏))−Δ𝑓₂(𝐱((𝐱)(𝑡)(𝐱)𝑧(𝑣(𝑡.𝑡−𝜏))−𝑔𝑚𝑣−𝑔))−ΔΦ (16)

It is worth noting that Δ𝑓₁(𝐱(𝑡)), 𝑓_2𝑗(𝐱(𝑡−𝜏)), and ΔΦ are unknown uncertainties and satisfy the following assumption.

Assumption 4. |ΔΦ|≤ℎ₁(𝐱),|Δ𝑓₁(𝐱(𝑡))|≤ℎ₂(𝐱(𝑡)), and |Δ𝑓₂(𝐱(𝑡−𝜏))|≤ℎ₃(𝐱(𝑡−𝜏)), where ℎ₁(𝐱),  ℎ₂(𝐱(𝑡)), and ℎ₃(𝐱(𝑡−𝜏)) are unknown smooth positive functions and can be estimated by fuzzy logic systems with some adaptive laws which will be determined later.
First, the nonlinear gain function 𝑔(𝐱) and the upper bounded functions ℎ₁(𝐱),ℎ₂(𝐱(𝑡)), and ℎ₃(𝐱(𝑡−𝜏)) of unmatched and matched uncertainties can be approximated, over a compact set Ω_𝐱, by the fuzzy logic systems as follows: 𝑔𝐱∣𝜽_𝑔=𝜽^𝑇_𝑔𝝃(𝐱),ℎ₁𝐱∣𝜽_ℎ1=𝜽^𝑇_ℎ1𝝃(𝐱),ℎ₂𝐱∣𝜽_ℎ2=𝜽^𝑇_ℎ2𝝃(𝐱),ℎ₃𝐱(𝑡−𝜏)∣𝜽_ℎ3=𝜽^𝑇_ℎ3𝝃(𝐱(𝑡−𝜏)), (17) where 𝝃(𝐱) and 𝝃(𝐱(𝑡−𝜏)) are the fuzzy basis vectors, and 𝜽_𝑔,𝜽_ℎ1,𝜽_ℎ2, and 𝜽_ℎ3 are the corresponding adjustable parameter vectors of each fuzzy logic system. It is assumed that 𝜽_𝑔,𝜽_ℎ1,𝜽_ℎ2, and 𝜽_ℎ3 belong to compact sets Ω_𝜽𝑔,Ω_𝜽ℎ1,Ω_𝜽ℎ2, and Ω_𝜽ℎ3, respectively, which are defined as Ω_𝜽𝑔=𝜽_𝑔∈𝑅^𝑀∶𝜽_𝑔≤𝑁₁,Ω<∞_𝜽ℎ1=𝜽_ℎ1∈𝑅^𝑀∶𝜽_ℎ1≤𝑁₂,Ω<∞_𝜽ℎ2=𝜽_ℎ2∈𝑅^𝑀∶𝜽_ℎ2≤𝑁₃,Ω<∞_𝜽ℎ3=𝜽_ℎ3∈𝑅^𝑀∶𝜽_ℎ3≤𝑁₄<∞, (18) where 𝑁₁,𝑁₂,𝑁₃, and 𝑁₄ are the designed parameters, and 𝑀 is the number of fuzzy inference rules. Let us define the optimal parameter vectors 𝜽^∗_𝑔,𝜽^∗_ℎ1,𝜽^∗_ℎ2, and 𝜽^∗_ℎ3 as follows: 𝜽^∗_𝑔=argmin_{𝜽𝑔∈Ω𝜽𝑔}sup_{𝐱∈Ω𝐱}𝑔(𝐱)−𝑔𝐱∣𝜽_𝑔,𝜽^∗_ℎ1=argmin_{𝜽ℎ1∈Ω𝜽ℎ1}sup_{𝐱∈Ω𝐱}ℎ₁(𝐱)−ℎ₁𝐱∣𝜽_ℎ1,𝜽^∗_ℎ2=argmin_{𝜽ℎ2∈Ω𝜽ℎ2}sup_{𝐱∈Ω𝐱}ℎ₂(𝐱)−ℎ₂𝐱∣𝜽_ℎ2,𝜽^∗_ℎ3=argmin_{𝜽ℎ3∈Ω𝜽ℎ3}sup_{𝐱∈Ω𝐱}ℎ₃(𝐱(−ℎ𝑡−𝜏))₁𝐱(𝑡−𝜏)∣𝜽_ℎ1, (19) where 𝜽^∗_𝑔,𝜽^∗_ℎ1,𝜽^∗_ℎ2, and 𝜽^∗_ℎ3 are bounded in the suitable closed sets Ω_𝜽𝑔,Ω_𝜽ℎ1,Ω_𝜽ℎ2, and Ω_𝜽ℎ3, respectively. The parameter estimation errors can be defined as 𝜽_𝑔=𝜽_𝑔−𝜽^∗_𝑔,𝜽_ℎ1=𝜽_ℎ1−𝜽^∗_ℎ1,𝜽_ℎ2=𝜽_ℎ2−𝜽^∗_ℎ2,𝜽_ℎ3=𝜽_ℎ3−𝜽^∗_ℎ3,𝜔₁+𝜔₂≤𝜔, (20) where 𝜔 is an unknown positive constant, and 𝜔₁=ℎ₁(𝐱)−ℎ₁𝐱∣𝜽^∗_ℎ1+ℎ₂(𝐱(𝑡))−ℎ₂𝐱∣𝜽^∗_ℎ2+ℎ₃𝐱(𝐱(𝑡−𝜏))−ℎ₃𝐱∣𝜽^∗_ℎ3,𝜔₂=𝑔(𝐱)−𝑔𝐱∣𝜽^∗_𝑔(𝑚𝑣(𝑡)+𝑧(𝑣(𝑡))) (21) as the minimum approximation errors, which correspond to approximation errors obtained when optimal parameters are used.
Secondly, we define 𝜽𝜙=𝜙−𝜙,₁=𝜽₁−𝜽₁,𝜽₂=𝜽₂−𝜽₂,𝜔=𝜔−𝜔, (22) where 𝜙 is an estimate of 𝜙, which is defined as 𝜙=(𝑚)⁻¹. 𝜽₁ and 𝜽₂ are the estimates of 𝜽₁ and 𝜽₂, respectively, which are defined as 𝜽₁=(𝑚)⁻¹𝜃₁₁,(𝑚)⁻¹𝜃₁₂,…,(𝑚)⁻¹𝜃_1𝑀^𝑇∈𝑅^𝑀,𝜽₂=(𝑚)⁻¹𝜃₂₁,(𝑚)⁻¹𝜃₂₂,…,(𝑚)⁻¹𝜃_2𝑁^𝑇∈𝑅^𝑁, (23) and 𝜔 is an estimate of 𝜔.

Based on the previous discussion and under Assumptions 1–4, we are in a position to propose the robust adaptive fuzzy controller in the following form: 𝑣=𝑣₁+𝑣₂+𝑣₃+𝑣₄+𝑣₅, (24) where 𝑣₁=1𝑔𝐱∣𝜽_𝑔𝜙𝐊𝐞+𝑦_𝑚^(𝑛)+𝐞^𝑇𝐏𝐁^𝑇𝐞^𝑇×ℎ𝐏𝐁₁𝐱∣𝜽_ℎ1+ℎ₂𝐱∣𝜽_ℎ2+ℎ₃𝐱(𝑡−𝜏)∣𝜽_ℎ3,𝑣₂1=−𝑔𝐱∣𝜽_𝑔𝐟^𝑇₁(𝐱(𝑡))𝜽₁,𝑣₃1=−𝑔𝐱∣𝜽_𝑔𝐟^𝑇₂(𝐱(𝑡−𝜏))𝜽₂,𝑣₄=1𝑚_min1𝑔𝐱∣𝜽_𝑔(𝐞^𝑇𝐏𝐁)^𝑇𝐞^𝑇𝑣𝐏𝐁𝜔,₅=𝜌𝑚_min𝐞tanh^𝑇𝐏𝐁𝜀, (25)(26) where 𝐟₁(𝐱(𝑡))=[𝑓₁₁,𝑓₁₂,…,𝑓_1𝑀]^𝑇∈𝑅^𝑀 and 𝐟₂(𝐱(𝑡−𝜏))=[𝑓₂₁,𝑓₂₂,…,𝑓_2𝑁]^𝑇∈𝑅^𝑁, 𝜌 is defined in (9), and 𝐏 is a symmetric positive definite matrix, which is a solution of the following Lyapunov equation: 𝐀^𝑇_𝑚𝐏+𝐏𝐀_𝑚=−𝐐, (27) where 𝐐 is a positive definite matrix, and the parameter update laws are as follows: 𝜽̇_𝑔=−𝛾_𝑔𝐞^𝑇𝐏𝐁𝝃(𝐱)𝑣(𝑡)+𝑧₁(𝑣(𝑡,𝜽̇))_ℎ1=𝛾_ℎ1𝐞^𝑇𝝃(𝐱(𝑡,𝜽̇𝐏𝐁))_ℎ2=𝛾_ℎ2𝐞^𝑇𝝃(𝐱(𝑡,𝜽̇𝐏𝐁))_ℎ3=𝛾_ℎ3𝐞^𝑇𝝃(𝐱(,𝜽̇𝐏𝐁𝑡−𝜏))₁=−𝛾₁𝐞^𝑇𝐏𝐁𝐟₁(𝐱(𝑡,𝜽̇))₂=−𝛾₂𝐞^𝑇𝐏𝐁𝐟₂(𝐱(,̇𝑡−𝜏))𝜔=𝛾_𝜔𝐞^𝑇,̇𝐞𝐏𝐁𝜙=𝜂^𝑇𝐏𝐁𝐊𝐞+𝑦_𝑚^(𝑛)+𝐞^𝑇𝐏𝐁𝐞^𝑇ℎ𝐏𝐁¹𝐱∣𝜽_ℎ1+𝐞^𝑇𝐏𝐁𝐞^𝑇ℎ𝐏𝐁²𝐱(𝑡)∣𝜽_ℎ2+𝐞^𝑇𝐏𝐁𝐞^𝑇ℎ𝐏𝐁³𝐱(𝑡−𝜏)∣𝜽_ℎ3, (28)(29)(30)(31)(32)(33)(34) where the scalars 𝛾_ℎ1,𝛾_ℎ2,𝛾_ℎ3,𝛾_𝑔,𝛾₁,𝛾₂,𝛾_𝜔, and 𝜂 are positive constants, determining the rates of adaptations, and 𝑧₁(𝑣(𝑡))=−𝑐_𝑎for𝑣(𝑡)≥𝑐_𝑎,−𝑣(𝑡)for𝑐_𝑏<𝑣(𝑡)<𝑐_𝑎,−𝑐_𝑏for𝑣(𝑡)≤𝑐_𝑏. (35)

Remark 1. Without loss of generality, the adaptive laws used in this paper are assumed that the parameter vectors are within the constraint sets or on the boundaries of the constraint sets but moving toward the inside of the constraint sets. If the parameter vectors are on the boundaries of the constraint sets but moving toward the outside of the constraint sets, we have to use the projection algorithm [27] to modify the adaptive laws such that the parameter vectors will remain inside of the constraint sets. The proposed adaptive law (28)–(30) can be modified as the following form: 𝜽̇_𝑔=−𝛾_𝑔𝐞^𝑇𝐏𝐁𝝃(𝐱(𝑡))𝑣(𝑡)+𝑧₁(𝑣(𝑡,𝜽))if_𝑔<𝑁₁𝜽or_𝑔=𝑁₁𝐞and_𝑇𝐏𝐁𝜽^𝑇_𝑔𝝃(𝐱(𝑡))𝑣(𝑡)+𝑧₁(𝑣(𝑡,𝑃))≥0−𝛾_𝑔𝐞^𝑇𝐏𝐁𝝃(𝐱(𝑡))𝑣(𝑡)+𝑧₁(𝑣(𝑡,𝜽))if^𝑔=𝑁₁𝐞and_𝑇𝐏𝐁𝜽^𝑇_𝑔𝝃(𝐱(𝑡))𝑣(𝑡)+𝑧₁(𝑣(𝑡,))<0 (36) where 𝑃{−𝛾_𝑔‖𝐞^𝑇𝐏𝐁‖𝝃(𝐱(𝑡))(𝑣(𝑡)+𝑧₁(𝑣(𝑡)))} is defined as 𝑃−𝛾_𝑔𝐞^𝑇𝐏𝐁𝝃(𝐱(𝑡))𝑣(𝑡)+𝑧₁(𝑣(𝑡))=−𝛾_𝑔𝐞^𝑇𝐏𝐁𝝃(𝐱(𝑡))𝑣(𝑡)+𝑧₁(𝑣(𝑡))+𝛾_𝑔𝐞^𝑇𝜽𝐏𝐁^𝑔𝜽^𝑇_𝑔𝜽_𝑔²𝝃(𝐱(𝑡×𝑣(𝑡)))+𝑧₁(𝑣(𝑡,𝜽̇))_ℎ1=𝛾_ℎ1𝐞^𝑇𝝃(𝐱(𝑡,𝜽𝐏𝐁))if_ℎ1<𝑁₂𝜽or_ℎ1=𝑁₂and𝐞^𝑇𝐏𝐁𝜽^𝑇_ℎ1𝝃(𝐱(𝑡,𝑃𝛾))≤0_ℎ1𝐞^𝑇𝝃(𝐱(𝑡,𝜽𝐏𝐁))if_ℎ1=𝑁₂and𝐞^𝑇𝐏𝐁𝜽^𝑇_ℎ1𝝃(𝐱(𝑡))>0, (37) where 𝑃{𝛾_ℎ1‖𝐞^𝑇𝐏𝐁‖𝝃(𝐱(𝑡))} is defined as 𝑃𝛾_ℎ1𝐞^𝑇𝐏𝐁𝝃(𝐱(𝑡))=𝛾_ℎ1𝐞^𝑇𝐏𝐁𝝃(𝐱(𝑡))−𝛾_ℎ1𝐞^𝑇𝜽𝐏𝐁^ℎ1𝜽^𝑇_ℎ1𝜽_ℎ1²𝝃(𝐱(𝑡,𝜽̇))_ℎ2=𝛾_ℎ2𝐞^𝑇𝝃(𝐱(𝑡,𝜽𝐏𝐁))if_ℎ2<𝑁₃𝜽or_ℎ2=𝑁₃and𝐞^𝑇𝐏𝐁𝜽^𝑇_ℎ2𝝃(𝐱(𝑡,𝑃𝛾))≤0_ℎ2𝐞^𝑇𝝃(𝐱(𝑡,𝜽𝐏𝐁))if_ℎ2=𝑁₃and𝐞^𝑇𝐏𝐁𝜽^𝑇_ℎ2𝝃(𝐱(𝑡))>0, (38) where 𝑃{𝛾_ℎ2‖𝐞^𝑇𝐏𝐁‖𝝃(𝐱(𝑡))} is defined as 𝑃𝛾_ℎ2𝐞^𝑇𝐏𝐁𝝃(𝐱(𝑡))=𝛾_ℎ2𝐞^𝑇𝐏𝐁𝝃(𝐱(𝑡))−𝛾_ℎ2𝐞^𝑇𝜽𝐏𝐁^ℎ2𝜽^𝑇_ℎ2𝜽_ℎ2²𝝃(𝐱(𝑡,𝜽̇))_ℎ3=𝛾_ℎ3𝐞^𝑇𝝃(𝐱(,𝜽𝐏𝐁𝑡−𝜏))if_ℎ3<𝑁₄𝜽or_ℎ3=𝑁₄and𝐞^𝑇𝐏𝐁𝜽^𝑇_ℎ3𝝃(𝐱(,𝑃𝛾𝑡−𝜏))≤0_ℎ3𝐞^𝑇𝝃(𝐱(,𝜽𝐏𝐁𝑡−𝜏))if_ℎ3=𝑁₄𝐞and_𝑇𝐏𝐁𝜽^𝑇_ℎ3𝝃(𝐱(𝑡−𝜏))>0, (39) where 𝑃{𝛾_ℎ3‖𝐞^𝑇𝐏𝐁‖𝝃(𝐱(𝑡−𝜏))} is defined as 𝑃𝛾_ℎ3𝐞^𝑇𝐏𝐁𝝃(𝐱(𝑡−𝜏))=𝛾_ℎ3𝐞^𝑇𝐏𝐁𝝃(𝐱(𝑡−𝜏))−𝛾_ℎ3𝐞^𝑇𝜽𝐏𝐁^ℎ3𝜽^𝑇_ℎ3𝜽_ℎ3²𝝃(𝐱(𝑡−𝜏)). (40)

The main result of the proposed robust adaptive fuzzy tracking control scheme is summarized in the following theorem.

Theorem 2. Consider the uncertain nonlinear state time-delay system (1) with unknown dead-zone input (7). If Assumptions 1–4 are satisfied, then the proposed robust adaptive fuzzy tracking controller defined by (24)–(26) with some adaptation laws (28)–(34) ensures that all the signals of the whole closed-loop system are bounded, and the output tracking errors converge to a neighborhood of zero exponentially.

Proof. Consider the Lyapunov function candidate 1𝑉=21𝑚𝐞^𝑇1𝐏𝐞+𝛾₁𝜽^𝑇₁𝜽₁+1𝛾₂𝜽^𝑇₂𝜽₂+1𝑚⋅𝛾_ℎ1𝜽^𝑇_ℎ1𝜽_ℎ1+1𝑚⋅𝛾_ℎ2𝜽^𝑇_ℎ2𝜽_ℎ2+1𝑚⋅𝛾_ℎ3𝜽^𝑇_ℎ3𝜽_ℎ3+1𝛾_𝑔𝜽^𝑇_𝑔𝜽_𝑔+1𝜂𝜙²+1𝑚_min⋅𝛾_𝜔𝜔². (41) Differentiating the Lyapunov function 𝑉 with respect to time, we can obtain 𝑉̇=1𝐞̇2𝑚^𝑇1𝐏𝐞+𝐞2𝑚^𝑇𝐏̇1𝐞+𝛾₁𝜽^𝑇₁𝜽̇₁+1𝛾₂𝜽^𝑇₂𝜽̇₂+1𝑚𝛾_ℎ1𝜽^𝑇_ℎ1𝜽̇_ℎ1+1𝑚𝛾_ℎ2𝜽^𝑇_ℎ2𝜽̇_ℎ2+1𝑚𝛾_ℎ3𝜽^𝑇_ℎ3𝜽̇_ℎ3+1𝛾_𝑔𝜽^𝑇_𝑔𝜽̇_𝑔+1𝜂𝜙̇1𝜙+𝑚_min⋅𝛾_𝜔𝜔𝜔̇. (42) From (16) and by the fact that 𝜽̇₁=𝜽̇₁,𝜽̇₂=𝜽̇₂,𝜽̇_ℎ1=𝜽̇_ℎ1,𝜽̇_ℎ2=𝜽̇_ℎ2,𝜽̇_ℎ3=𝜽̇_ℎ3,𝜽̇_𝑔=𝜽̇_𝑔,𝜙̇=𝜙̇, and 𝜔̇=𝜔̇, the previous equation becomes 𝑉̇=1𝐞2𝑚^𝑇𝐀^𝑇_𝑚𝐏+𝐏𝐀_𝑚𝐞+1𝑚𝐞^𝑇𝐏𝐁𝐊𝐞+𝑦_𝑚^(𝑛)−^𝑀_𝑖=1𝜃_1𝑖𝑓_1𝑖(𝐱(𝑡))−Δ𝑓₁(𝐱(𝑡))−^𝑁_𝑗=1𝜃_2𝑗𝑓_2𝑗(𝐱(𝑡−𝜏))−Δ𝑓₂(𝐱((𝐱)(𝑡)(𝐱)𝑧(𝑣(𝑡+1𝑡−𝜏))−𝑔𝑚𝑣−𝑔))−ΔΦ𝛾₁𝜽^𝑇₁𝜽̇₁+1𝛾₂𝜽^𝑇₂𝜽̇₂+1𝑚𝛾_ℎ1𝜽^𝑇_ℎ1𝜽̇_ℎ1+1𝑚𝛾_ℎ2𝜽^𝑇_ℎ2𝜽̇_ℎ2+1𝑚𝛾_ℎ3𝜽^𝑇_ℎ3𝜽̇_ℎ3+1𝛾_𝑔𝜽^𝑇_𝑔𝜽̇_𝑔+1𝜂𝜙̇1𝜙+𝑚_min𝛾_𝜔𝜔𝜔̇. (43) Applying (27) and Assumption 4 to (43) yields 𝑉̇≤−1𝐞2𝑚^𝑇1𝐐𝐞+𝑚𝐞^𝑇×𝐏𝐁𝐊𝐞+𝑦_𝑚^(𝑛)−^𝑀_𝑖=1𝜃_1𝑖𝑓_1𝑖(𝐱(𝑡−))_𝑁𝑗=1𝜃_2𝑗𝑓_2𝑗(𝐱((𝐱)(𝑡)(𝐱)𝑧(𝑣(𝑡+1𝑡−𝜏))−𝑔𝑚𝑣−𝑔))𝑚𝐞^𝑇𝐏𝐁ℎ₁(𝐱)+1𝑚𝐞^𝑇𝐏𝐁ℎ₂(𝐱(𝑡+1))𝑚𝐞^𝑇𝐏𝐁ℎ₃(𝐱(+1𝑡−𝜏))𝛾₁𝜽^𝑇₁𝜽̇₁+1𝛾₂𝜽^𝑇₂𝜽̇₂+1𝑚𝛾_ℎ1𝜽^𝑇_ℎ1𝜽̇_ℎ1+1𝑚𝛾_ℎ2𝜽^𝑇_ℎ2𝜽̇_ℎ2+1𝑚𝛾_ℎ3𝜽^𝑇_ℎ3𝜽̇_ℎ3+1𝛾_𝑔𝜽^𝑇_𝑔𝜽̇_𝑔+1𝜂𝜙̇1𝜙+𝑚_min𝛾_𝜔𝜔𝜔̇. (44) Substituting (17) and (23) into (44), we obtain 𝑉̇≤−1𝐞2𝑚^𝑇𝐐𝐞+𝐞^𝑇×1𝐏𝐁𝑚𝐊𝐞+𝑦_𝑚^(𝑛)−𝐟^𝑇₁(𝐱(𝑡))𝜽₁−𝐟^𝑇₂(𝐱(𝑡−𝜏))𝜽₂+1𝑚𝐞^𝑇𝐏𝐁𝜔₁+𝜔₂−1𝑚𝐞^𝑇×𝑔𝐏𝐁𝐱∣𝜽_𝑔((𝑡)(𝑣(𝑡+1𝑚𝑣+𝑧)))𝑚𝐞^𝑇𝐏𝐁𝜽^𝑇_𝑔𝝃(𝐱(𝑡)(𝑣(𝑡+1)(𝑚𝑣+𝑧)))𝑚𝐞^𝑇𝐏𝐁ℎ₁𝐱∣𝜽_ℎ1−𝜽^𝑇_ℎ1𝝃(𝐱+1)𝑚𝐞^𝑇𝐏𝐁ℎ₂𝐱(𝑡)∣𝜽_ℎ2−𝜽^𝑇_ℎ2𝝃(𝐱(𝑡+1))𝑚𝐞^𝑇𝐏𝐁ℎ₃𝐱(𝑡−𝜏)∣𝜽_ℎ3−𝜽^𝑇_ℎ3𝝃(𝐱(+1𝑡−𝜏))𝛾₁𝜽^𝑇₁𝜽̇₁+1𝛾₂𝜽^𝑇₂𝜽̇₂+1𝑚⋅𝛾_ℎ1𝜽^𝑇_ℎ1𝜽̇_ℎ1+1𝑚⋅𝛾_ℎ2𝜽^𝑇_ℎ2𝜽̇_ℎ2+1𝑚⋅𝛾_ℎ3𝜽^𝑇_ℎ3𝜽̇_ℎ3+1𝛾_𝑔𝜽^𝑇_𝑔𝜽̇_𝑔+1𝜂𝜙̇1𝜙+𝑚_min⋅𝛾_𝜔𝜔𝜔̇1≤−𝐞2𝑚^𝑇𝐐𝐞+𝐞^𝑇×1𝐏𝐁𝑚𝐊𝐞+𝑦_𝑚^(𝑛)−𝐟^𝑇₁(𝐱(𝑡))𝜽₁−𝐟^𝑇₂(𝐱(𝑡−𝜏))𝜽₂+1𝑚𝐞^𝑇1𝐏𝐁𝜔−𝑚𝐞^𝑇×𝑔𝐏𝐁𝐱∣𝜽_𝑔((𝑡)(𝑣(𝑡+1𝑚𝑣+𝑧)))𝑚𝐞^𝑇𝐏𝐁𝜽^𝑇_𝑔𝝃(𝐱)𝑚𝑣(𝑡)+𝑚⋅𝑧₁(𝑣(𝑡+1))𝑚𝐞^𝑇𝐏𝐁ℎ₁𝐱∣𝜽_ℎ1−𝜽^𝑇_ℎ1𝝃(𝐱+1)𝑚𝐞^𝑇𝐏𝐁ℎ₂𝐱(𝑡)∣𝜽_ℎ2−𝜽^𝑇_ℎ2𝝃(𝐱(𝑡+1))𝑚𝐞^𝑇𝐏𝐁ℎ₃𝐱(𝑡−𝜏)∣𝜽_ℎ3−𝜽^𝑇_ℎ3𝝃(𝐱(+1𝑡−𝜏))𝛾₁𝜽^𝑇₁𝜽̇₁+1𝛾₂𝜽^𝑇₂𝜽̇₂+1𝑚⋅𝛾_ℎ1𝜽^𝑇_ℎ1𝜽̇_ℎ1+1𝑚⋅𝛾_ℎ2𝜽^𝑇_ℎ2𝜽̇_ℎ2+1𝑚⋅𝛾_ℎ3𝜽^𝑇_ℎ3𝜽̇_ℎ3+1𝛾_𝑔𝜽^𝑇_𝑔𝜽̇_𝑔+1𝜂𝜙̇1𝜙+𝑚_min⋅𝛾_𝜔𝜔𝜔̇. (45) According to adaptive laws (28)–(30), (45) can be rewritten as 𝑉̇≤−1𝐞2𝑚^𝑇1𝐐𝐞+𝑚𝐞^𝑇𝜔+1𝐏𝐁𝑚𝐞^𝑇𝐏𝐁𝐊𝐞+𝑦_𝑚^(𝑛)+𝐞^𝑇𝐏𝐁^𝑇𝐞^𝑇ℎ𝐏𝐁¹𝐱∣𝜽_ℎ1+𝐞^𝑇𝐏𝐁^𝑇𝐞^𝑇ℎ𝐏𝐁²𝐱(𝑡)∣𝜽_ℎ2+𝐞^𝑇𝐏𝐁^𝑇𝐞^𝑇ℎ𝐏𝐁³𝐱(𝑡−𝜏)∣𝜽_ℎ3+𝐞^𝑇𝐏𝐁−𝑔𝐱∣𝜽_𝑔𝑣(𝑡)−𝑔𝐱∣𝜽_𝑔𝑧(𝑣(𝑡))𝑚−𝐞^𝑇𝐏𝐁𝐟^𝑇₁(𝐱(𝑡))𝜽₁−𝐞^𝑇𝐏𝐁𝐟^𝑇₂(𝐱(𝑡−𝜏))𝜽₂+1𝛾₁𝜽^𝑇₁𝜽̇₁+1𝛾₂𝜽^𝑇₂𝜽̇₂+1𝜂𝜙𝜙̇+1𝑚_min⋅𝛾_𝜔𝜔𝜔̇1≤−𝐞2𝑚^𝑇1𝐐𝐞+𝑚𝐞^𝑇1𝐏𝐁𝜔+𝑚𝐞^𝑇×𝐏𝐁𝐊𝐞+𝑦_𝑚^(𝑛)+𝐞^𝑇𝐏𝐁^𝑇𝐞^𝑇ℎ𝐏𝐁¹𝐱∣𝜽_ℎ1+𝐞^𝑇𝐏𝐁^𝑇𝐞^𝑇ℎ𝐏𝐁²𝐱(𝑡)∣𝜽_ℎ2+𝐞^𝑇𝐏𝐁^𝑇𝐞^𝑇ℎ𝐏𝐁³𝐱(𝑡−𝜏)∣𝜽_ℎ3−𝐞^𝑇𝐏𝐁𝑔𝐱∣𝜽_𝑔𝑣(𝑡)+𝐞^𝑇𝐏𝐁𝑔𝐱∣𝜽_𝑔×|𝑧(𝑣(𝑡))||𝑚|−𝐞^𝑇𝐏𝐁𝐟^𝑇₁(𝐱(𝑡))𝜽₁−𝐞^𝑇𝐏𝐁𝐟^𝑇₂(𝐱(𝑡−𝜏))𝜽₂+1𝛾₁𝜽^𝑇₁𝜽̇₁+1𝛾₂𝜽^𝑇₂𝜽̇₂+1𝜂𝜙̇1𝜙+𝑚_min⋅𝛾_𝜔𝜔𝜔̇. (46) Using the control laws (24)–(26), the previous equation can be rewritten as 𝑉̇≤−1𝐞2𝑚^𝑇1𝐐𝐞+𝑚_min𝐞^𝑇𝐏𝐁⋅(𝜔−𝜔)+𝐞^𝑇𝜙×𝐏𝐁𝜙−𝐊𝐞+𝑦_𝑚^(𝑛)+𝐞^𝑇𝐏𝐁^𝑇𝐞^𝑇ℎ𝐏𝐁¹𝐱∣𝜽_ℎ1+𝐞^𝑇𝐏𝐁^𝑇𝐞^𝑇ℎ𝐏𝐁²𝐱(𝑡)∣𝜽_ℎ2+𝐞^𝑇𝐏𝐁^𝑇𝐞^𝑇ℎ𝐏𝐁³𝐱(𝑡−𝜏)∣𝜽_ℎ3+𝐞^𝑇𝐏𝐁𝑔𝐱∣𝜽_𝑔|𝑧(𝑣(𝑡))||𝑚|−𝐞^𝑇𝐏𝐁𝜽^𝑇₁−𝜽^𝑇₁⋅𝐟₁(𝐱(𝑡))−𝐞^𝑇𝐏𝐁𝜽^𝑇₂−𝜽^𝑇₂⋅𝐟₂(𝐱(𝑡−𝜏))−𝐞^𝑇𝐏𝐁⋅𝑔𝐱∣𝜽_𝑔⋅𝜌𝑚_min𝐞⋅tanh^𝑇𝐏𝐁𝜀+1𝛾₁𝜽^𝑇₁𝜽̇₁+1𝛾₂𝜽^𝑇₂𝜽̇₂+1𝜂𝜙̇1𝜙+𝑚_min⋅𝛾_𝜔𝜔𝜔̇1=−𝐞2𝑚^𝑇1𝐐𝐞−𝑚_min𝐞^𝑇𝐏𝐁⋅𝜔−𝐞^𝑇𝐏𝐁𝜙𝐊𝐞+𝑦_𝑚^(𝑛)+𝐞^𝑇𝐏𝐁^𝑇𝐞^𝑇ℎ𝐏𝐁¹𝐱∣𝜽_ℎ1+𝐞^𝑇𝐏𝐁^𝑇𝐞^𝑇ℎ𝐏𝐁²𝐱(𝑡)∣𝜽_ℎ2+𝐞^𝑇𝐏𝐁^𝑇𝐞^𝑇ℎ𝐏𝐁³𝐱(𝑡−𝜏)∣𝜽_ℎ3+𝐞^𝑇𝐏𝐁𝑔𝐱∣𝜃_𝑔|𝑧(𝑣(𝑡))||𝑚|+𝐞^𝑇𝐏𝐁⋅𝜽^𝑇₁⋅𝐟₁(𝐱(𝑡))+𝐞^𝑇𝐏𝐁⋅𝜽^𝑇₂⋅𝐟₂(𝐱(𝑡−𝜏))−𝐞^𝑇𝐏𝐁⋅𝑔𝐱∣𝜽_𝑔⋅𝜌𝑚_min𝐞⋅tanh^𝑇𝐏𝐁𝜀+1𝛾₁𝜽^𝑇₁𝜽̇₁+1𝛾₂𝜽^𝑇₂𝜽̇₂+1𝜂𝜙̇1𝜙+𝑚_min⋅𝛾_𝜔𝜔𝜔̇. (47) According to adaptive laws (31)–(33), we have 𝑉̇≤−1𝐞2𝑚^𝑇𝐐𝐞+𝐞^𝑇𝐏𝐁𝑔𝐱∣𝜽_𝑔|𝑧(𝑣(𝑡))||𝑚|−𝐞^𝑇𝐏𝐁⋅𝑔𝐱∣𝜽_𝑔⋅𝜌𝑚_min𝐞⋅tanh^𝑇𝐏𝐁𝜀1≤−𝐞2𝑚^𝑇𝐐𝐞+𝐞^𝑇𝐏𝐁𝑔𝐱∣𝜽_𝑔𝜌𝑚_min−𝐞^𝑇𝐏𝐁⋅𝑔𝐱∣𝜽_𝑔⋅𝜌𝑚_min𝐞⋅tanh^𝑇𝐏𝐁𝜀. (48) By considering the inequality |𝜙|−𝜙tanh(𝜙/𝜀)≤0.2785𝜀. We obtain 𝑉̇≤−1𝐞2𝑚^𝑇𝐐𝐞+0.2785𝑔𝐱∣𝜽_𝑔⋅𝜌𝑚_min1⋅𝜀=−𝐞2𝑚^𝑇1𝐐𝐞−2𝛾₁𝜽^𝑇₁𝜽₁−12𝛾₂𝜽^𝑇₂𝜽₂−12𝑚𝛾_ℎ1𝜽^𝑇_ℎ1𝜽_ℎ1−12𝑚𝛾_ℎ2𝜽^𝑇_ℎ2𝜽_ℎ2−12𝑚𝛾_ℎ3𝜽^𝑇_ℎ3𝜽_ℎ3−12𝛾_𝑔𝜽^𝑇_𝑔𝜽_𝑔−1𝜙2𝜂²−12𝑚_min⋅𝛾_𝜔𝜔²+12𝛾₁𝜽^𝑇₁𝜽₁+12𝛾₂𝜽^𝑇₂𝜽₂+12𝑚𝛾_ℎ1𝜽^𝑇_ℎ1𝜽_ℎ1+12𝑚𝛾_ℎ2𝜽^𝑇_ℎ2𝜽_ℎ2+12𝑚𝛾_ℎ3𝜽^𝑇_ℎ3𝜽_ℎ3+12𝛾_𝑔𝜽^𝑇_𝑔𝜽_𝑔+1𝜙2𝜂²+12𝑚_min⋅𝛾_𝜔𝜔²+0.2785𝑔𝐱∣𝜽_𝑔⋅𝜌𝑚_min⋅𝜀. (49) Let 1𝐿=2𝛾₁𝜽^𝑇₁𝜽₁+12𝛾₂𝜽^𝑇₂𝜽₂+12𝑚𝛾_ℎ1𝜽^𝑇_ℎ1𝜽_ℎ1+12𝑚𝛾_ℎ2𝜽^𝑇_ℎ2𝜽_ℎ2+12𝑚𝛾_ℎ3𝜽^𝑇_ℎ3𝜽_ℎ3+12𝛾_𝑔𝜽^𝑇_𝑔𝜽_𝑔+1𝜙2𝜂²+12𝑚_min⋅𝛾_𝜔𝜔²+0.2785𝑔𝐱∣𝜃_𝑔⋅𝜌𝑚_min⋅𝜀. (50) Then, 𝑉̇≤−1𝐞2𝑚^𝑇1𝐐𝐞−2𝛾₁𝜽^𝑇₁𝜽₁−12𝛾₂𝜽^𝑇₂𝜽₂−12𝑚𝛾_ℎ1𝜽^𝑇_ℎ1𝜽_ℎ1−12𝑚𝛾_ℎ2𝜽^𝑇_ℎ2𝜽_ℎ2−12𝑚𝛾_ℎ3𝜽^𝑇_ℎ3𝜽_ℎ3−12𝛾_𝑔𝜽^𝑇_𝑔𝜽_𝑔−1𝜙2𝜂²−12𝑚_min⋅𝛾_𝜔𝜔²1+𝐿≤−𝜆2𝑚^min(𝐐)𝐞^𝑇1𝐞−2𝛾₁𝜽^𝑇₁𝜽₁−12𝛾₂𝜽^𝑇₂𝜽₂−12𝑚𝛾_ℎ1𝜽^𝑇_ℎ1𝜽_ℎ1−12𝑚𝛾_ℎ2𝜽^𝑇_ℎ2𝜽_ℎ2−12𝑚𝛾_ℎ3𝜽^𝑇_ℎ3𝜽_ℎ3−12𝛾_𝑔𝜽^𝑇_𝑔𝜽_𝑔−1𝜙2𝜂²−12𝑚_min⋅𝛾_𝜔𝜔²1+𝐿≤−𝜆2𝑚^min(𝐐)𝜆_max(𝐏)𝐞^𝑇1𝐏𝐞−2𝛾₁𝜽^𝑇₁𝜽₁−12𝛾₂𝜽^𝑇₂𝜽₂−12𝑚𝛾_ℎ1𝜽^𝑇_ℎ1𝜽_ℎ1−12𝑚𝛾_ℎ2𝜽^𝑇_ℎ2𝜽_ℎ2−12𝑚𝛾_ℎ3𝜽^𝑇_ℎ3𝜽_ℎ3−12𝛾_𝑔𝜽^𝑇_𝑔𝜽_𝑔−1𝜙2𝜂²−12𝑚_min⋅𝛾_𝜔𝜔²+𝐿. (51) Let 𝜆_𝑣=𝜆_min(𝐐)/𝜆_max(𝐏). We obtain 𝑉̇≤−1𝜆2𝑚^𝑣𝐞^𝑇1𝐏𝐞−2𝛾₁𝜽^𝑇₁𝜽₁−12𝛾₂𝜽^𝑇₂𝜽₂−12𝑚𝛾_ℎ1𝜽^𝑇_ℎ1𝜽_ℎ1−12𝑚𝛾_ℎ2𝜽^𝑇_ℎ2𝜽_ℎ2−12𝑚𝛾_ℎ3𝜽^𝑇_ℎ3𝜽_ℎ3−12𝛾_𝑔𝜽^𝑇_𝑔𝜽_𝑔−1𝜙2𝜂²−12𝑚_min⋅𝛾_𝜔𝜔²𝜆+𝐿≤−min_𝑣,1𝛾₁,1𝛾₂,1𝑚⋅𝛾_ℎ1,1𝑚⋅𝛾_ℎ2,1𝑚⋅𝛾_ℎ3,1𝛾_𝑔,1𝜂,1𝑚⋅𝛾_𝜔×1𝐞2𝑚^𝑇1𝐏𝐞+2𝛾₁𝜽^𝑇₁𝜽₁+12𝛾₂𝜽^𝑇₂𝜽₂+12𝑚𝛾_ℎ1𝜽^𝑇_ℎ1𝜽_ℎ1+12𝑚𝛾_ℎ2𝜽^𝑇_ℎ2𝜽_ℎ2+12𝑚𝛾_ℎ3𝜽^𝑇_ℎ3𝜽_ℎ3+12𝛾_𝑔𝜽^𝑇_𝑔𝜽_𝑔+1𝜙2𝜂²+12𝑚_min⋅𝛾_𝜔𝜔²+𝐿. (52) Setting 𝑐=min{𝜆_𝑣,1/𝛾₁,1/𝛾₂,1/(𝑚⋅𝛾_ℎ1),1/(𝑚⋅𝛾_ℎ2),1/(𝑚⋅𝛾_ℎ3),1/𝛾_𝑔,1/𝜂,1/(𝑚⋅𝛾_𝜔)}, it yields that 𝑉̇≤−𝑐𝑉+𝐿. (53) Then, it is easy from (53) to show that 𝑉(𝑡)≤𝑒^−𝑐𝑡𝑉(0)+𝐿𝑐. (54) Therefore, the output tracking error converges to a neighborhood of zero exponentially.

Remark 3. In the future work, the control problem of uncertain T-S fuzzy time-varying delay systems with unknown dead-zone input is an important topic and is worth to be studied. Based on a novel fuzzy Lyapunov-Krasovskii functional, a delay partitioning method has been developed for the delay-dependent stability analysis of fuzzy time-varying state delay systems [26]. Obviously, it provides a useful idea to deal with the aforementioned future research.

4. An Example and Simulation Results

Consider the second-order uncertain nonlinear time-delay system containing an unknown dead-zone that is modified from the simulation example in [7] as follows: 𝑥̇₁=𝑥₂+Δ𝜙₁(𝐱),𝑥̇₂=𝑥₁+𝑓₁₁(𝐱(𝑡))+𝑓₂₂(𝐱(𝑡−𝜏))+Δ𝑓₁(𝐱(𝑡))+Δ𝑓₂(𝐱((𝐱)𝑍(𝑣(𝑡𝑡−𝜏))+𝑔))+Δ𝜙₂(𝐱),𝑦=𝑥₁, (55) where the nonlinear functions 𝑓₁₁(𝐱(𝑡))=−0.3sin𝑥₁(𝑡),𝑓₁₂(𝐱(𝑡))=0,𝑓₂₁(𝐱(𝑡−𝜏))=0, and 𝑓₂₂(𝐱(𝑡−𝜏)) = 0.1𝑥²₁(𝑡−𝜏) are assumed to be known, and Δ𝑓₁(𝐱(𝑡))=−0.1𝑥₁sin(3𝑥₂(𝑡)),Δ𝑓₂(𝐱(𝑡−𝜏))=−0.1𝑥₁sin(3𝑥₂(𝑡−𝜏)) are unknown system uncertainties with unknown upper bound functions, where 𝜏 is the time delay. Δ𝜙₁(𝐱)=0.1𝑥₁sin(𝑡) and Δ𝜙₂(𝐱)=0.3𝑥₂sin(𝑡) are unknown external disturbances, and 𝑔(𝐱(𝑡))=2−sin²(𝑥₁(𝑡)). |Δ𝑓₁(𝐱(𝑡))|≤ ℎ₂(𝐱(𝑡)),|Δ𝑓₂(𝐱(𝑡−𝜏))|≤ℎ₃(𝐱(𝑡−𝜏)), and 𝑍(𝑣(𝑡)) is an output of a dead-zone. The goal of control is to maintain the system output 𝑦 to follow the reference signal 𝑦_𝑚=0.5[sin(𝑡)+sin(0.5𝑡)].

In the simulation, parameters of the dead-zone are 𝑚=1, 𝑐_𝑟=0.5, and 𝑐_𝑙=−0.5. And their bounds are chosen as 𝑚_max=1.5, 𝑚_min=0.6, 𝑐_𝑟max=0.9, 𝑐_𝑟min=0.1, 𝑐_𝑙max=−0.1, and 𝑐_𝑙min=−0.8. In the implementation, six fuzzy sets are defined over interval [−3,3] for both 𝑥₁ and 𝑥₂, with labels 𝐹1, 𝐹2, 𝐹3, 𝐹4, 𝐹5, and 𝐹6, and their membership functions are 𝜇_𝐹1𝑥_𝑖=11+exp5𝑥_𝑖,𝜇+2_𝐹2𝑥_𝑖=exp−𝑥_𝑖+1.5²,𝜇_𝐹3𝑥_𝑖=exp−𝑥_𝑖+0.5²,𝜇_𝐹4𝑥_𝑖=exp−𝑥_𝑖−0.5²,𝜇_𝐹5𝑥_𝑖=exp−𝑥_𝑖−1.5²,𝜇_𝐹6𝑥_𝑖=11+exp−5𝑥_𝑖−2,𝑖=1,2. (56)

In this section, we apply the proposed robust adaptive fuzzy tracking control approach in Section 3 to deal with the output tracking control problem of the second-order uncertain nonlinear time-delay system as shown in (55). Choose 𝐊=[10,10] and 𝐐=diag[5,5]; then, we solve the Lyapunov equation (27) to obtain 𝐏=5.250.250.250.275. (57) In this example, the sampling time is 0.01 sec. Initial values are chosen as 𝐱(0)=[−2,3]^T, 𝜽_𝑔(0)=1, 𝜽_ℎ1(0)=0, 𝜽_𝐡2(0)=0, and 𝜽_𝐡3(0)=0. The initial values of the parameters to be estimated are selected as 𝜙(0)=0.85, 𝜽₁(0)=[00]^𝑇, 𝜽₂(0)=[00]^𝑇.  𝛾_𝑔=2, 𝛾_ℎ1=1.5, 𝛾_ℎ2=1.5, 𝛾_ℎ3=1.5, 𝛾₁=1.5, 𝛾₂=1.5, 𝛾_𝜔=1.5, 𝜂=1.0, 𝜏=0.5s, and 𝜀=0.06. The simulation results are shown in Figures 2–5. Figures 2 and 3 show the trajectories of states 𝑥₁and𝑥₂ and the desired outputs 𝑦_𝑚1and𝑦_𝑚2, respectively. The phase plane of tracking errors of 𝑒₁ and 𝑒₂ is shown in Figure 4. Figure 5 shows the trajectory of the control signal. Obviously, the proposed robust adaptive fuzzy tracking control scheme can achieve the objective of good tracking performance and robust stability simultaneously in spite of the controlled system containing an unknown dead-zone and uncertainties.

Figure 2 

The trajectories of state 𝑥1 and desired output 𝑦𝑚1.

Figure 3 

The trajectories of state 𝑥2 and desired output 𝑦𝑚2.

Figure 4 

The phase plane of tracking errors 𝑒1 and 𝑒2.

Figure 5 

The trajectory of the control input 𝑣(𝑡).

5. Conclusion

The dead-zone input characteristics widely exist in the actuators of practical control systems, which are usually poorly known. The time-delay characteristics are usually confronted in engineering systems. The two characteristics may severely limit the performance of control. In this paper, the robust adaptive fuzzy tracking controller is designed to overcome the stabilization problem of a class of uncertain nonlinear state time-delay systems containing unknown dead-zone input and unmatched uncertainties. By utilizing a description of a dead-zone feature to estimate the properties of the dead-zone model intuitively and mathematically, the adaptive fuzzy tracking controller is proposed without constructing the dead-zone inverse. The nonlinear uncertainties are approximated by the fuzzy logic system according to the adaptive laws. Based on the Lyapunov stability theorem, the proposed robust adaptive tracking fuzzy controller can ensure that the output tracking error of the resulting closed-loop system converges to a neighborhood of zero exponentially. Finally, some simulations results are illustrated to verify the effectiveness and performance of the proposed approach.