Observer-Based Robust Adaptive Fuzzy Control for MIMO Nonlinear Uncertain Systems with Delayed Output

Abstract

An observer-based robust adaptive fuzzy control scheme is presented to tackle the problem of the robust stability and the tracking control for a class of multiinput multioutput (MIMO) nonlinear uncertain systems with delayed output. Because the nonlinear system functions and the uncertainties of the controlled system including structural uncertainties are supposed to be unknown, fuzzy logic systems are utilized to approximate these nonlinear system functions and the upper bounded functions of the uncertainties. Moreover, the upper bound of uncertainties caused by these fuzzy modeling errors is also estimated. In addition, the state observer based on state variable filters is designed to estimate all states which are not available for measurement in the controlled system. By constructing an appropriate Lyapunov function and using strictly positive-real (SPR) stability theorem, the proposed robust adaptive fuzzy controller not only guarantees the robust stability of a class of multivariable nonlinear uncertain systems with delayed output but also maintains a good tracking performance. Finally, some simulation results are illustrated to verify the effectiveness of the proposed control approach.

1. Introduction

Controller design for nonlinear systems has been given a lot of attention in the control community during the last two decades. Recently, the methods of feedback linearization have been successfully applied in the development of controllers for a class of nonlinear systems [1, 2]. However, the above techniques can be only applied to nonlinear systems whose dynamics are exactly known. Actually, it is difficult to obtain the exact construction of the systems in most of industrial and engineering systems because of the complexity of systems. In addition, the existence of uncertainties such as parameter uncertainties, modeling errors, and external disturbances may lead to poor performance and instability [3–8]. Hence, the study of robust stability of nonlinear uncertain systems in the presence of uncertainties is an important topic for the control design engineer.

Recently, fuzzy control technique has been considered extensively in the control problems of complex and ill-defined nonlinear systems in the presence of incomplete knowledge of the plant [3, 9–15]. However, the above adaptive fuzzy controllers are only limited to the systems under the conservative assumption that system states are available for measurement. In order to relax this restriction, an observer-based fuzzy adaptive output feedback control for the SISO nonlinear systems is presented to tackle the nonlinear systems whose states are not available [16–23]. So far, few research results have been extended to MIMO nonlinear systems [24, 25]. The common feature of most previous results [16, 19, 24, 25] is the assumption that the controlled systems are free of uncertainties. Although the above restrictive assumption can be relaxed in [17, 18, 20–23], the uncertainties are assumed to be a bounded external disturbance. Therefore, the motivation of this paper is to synthesize an observer-based robust adaptive fuzzy control scheme to deal with the tracking control problem for a class of MIMO nonlinear uncertain systems with delayed output in the presence of uncertainties including the structural uncertainty.

On the other hand, the design problem of nonlinear time-delay systems has received considerable attention in [26–31] because time-delay characteristic often encountered in various engineering systems may not only cause instability but also lead to serious deterioration in the performance of the plants. 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 [26]. The robust 𝐻_∞ control problem for stochastic systems with a delay in the state is investigated in [28], and the results are further extended to the stochastic time-delay systems with parameter uncertainties. The problems of stability analysis and robust control for uncertain systems with input delay were examined in [29, 31]. In [27], the design scheme of output feedback controller for a class of SISO nonlinear systems with delayed output was proposed to construct a delay-dependent controller making the closed-loop system globally asymptotically stable. In [30], a new approach for the construction of a state observer for SISO nonlinear systems with delayed output was presented to ensure the global exponential convergence to zero of the observation error. It should be pointed out that the uncertainty was not taken into account in [27, 30]. Therefore, the robust control problem of MIMO nonlinear systems with delayed output in the presence of uncertainties will be considered in this paper.

In this paper, the problem of controller design for a class of MIMO nonlinear uncertain output-delay systems whose states are not available is considered. The main features of the proposed observer-based robust adaptive fuzzy controller are summarized as follows. (i) Fuzzy logic systems with some appropriate learning laws are applied to approximate the nonlinear system functions and the upper bounded functions of the uncertainties including the structural uncertainty. (ii) The unknown upper bound of the uncertainties caused by approximation (or fuzzy modeling) error is estimated by a simple adaptive law. (iii) The state observer based on state variable filters is designed to estimate all states which are not available for measurement in the controlled system. (iv) By constructing an appropriate Lyapunov function and using strictly positive-real (SPR) stability theorem, the proposed robust adaptive fuzzy output feedback controller can not only guarantee the robust stability of the whole closed-loop system but also obtain the good tracking performance.

This paper is organized as follows: the description of the system and the concept of fuzzy logic systems are given together in Section 2. Section 3 proposes an observer-based robust adaptive fuzzy output feedback controller to achieve the purpose of asymptotic stabilization and output tracking performance of the whole closed-loop MIMO nonlinear uncertain systems with delayed output. In Section 4, a series of simulation results are illustrated to show the effectiveness of the proposed control scheme. Finally, a conclusion is given in Section 5.

2. Problem Formulation and Preliminaries

2.1. Problem Formulation

Consider a class of MIMO nonlinear uncertain systems with delayed output in the following form: 𝑥̇₁₁(𝑡)=𝑥₁₂(𝑡),𝑥̇₁₂(𝑡)=𝑥₁₃(𝑡),⋮𝑥̇_1(𝑛1−1)(𝑡)=𝑥_1𝑛1(𝑡),𝑥̇_1𝑛1(𝑡)=𝑓₁(𝐱(𝑡))+^𝑝_𝑗=1𝑔_1𝑗(𝐱(𝑡))𝑢_𝑗(𝑡)+Δ𝜙₁(𝐱(𝑡,⋮𝑥̇))_𝑝1(𝑡)=𝑥_𝑝2(𝑡),𝑥̇_𝑝2(𝑡)=𝑥_𝑝3(𝑡),⋮𝑥̇_{𝑝(𝑛𝑝−1)}(𝑡)=𝑥_𝑝𝑛𝑝(𝑡),𝑥̇_𝑝𝑛𝑝(𝑡)=𝑓_𝑝(𝐱(𝑡))+^𝑃_𝑗=1𝑔_𝑝𝑗(𝐱(𝑡))𝑢_𝑗(𝑡)+Δ𝜙_𝑝(𝐱(𝑡,𝑦))₁(𝑡)=𝑥₁₁(),⋮𝑦𝑡−𝜏_𝑝(𝑡)=𝑥_𝑝1(𝑡−𝜏), (1) or equivalently 𝐱̇(𝑡)(𝑡)[𝐅(𝐱(𝑡(𝐱(𝑡𝐮(𝑡)(𝐱(𝑡,𝐲(𝑡)(),=𝐀𝐱+𝐁))+𝐆))+ΔΦ))]=𝐂𝐱𝑡−𝜏 (2) where 𝐅(𝐱)=𝑓₁(𝐱),…,𝑓_𝑝(𝐱)^𝑇∈𝑅^𝑝,𝐆(𝐱)=𝐆₁(𝐱),…,𝐆_𝑝(𝐱)∈𝑅^𝑝×𝑝,𝐆_𝑖(𝐱)=𝑔_1𝑖(𝐱),…,𝑔_𝑝𝑖(𝐱)^𝑇∈𝑅^𝑝𝐀,𝑖=1,2,…,𝑝,𝐀=diag₁,…,𝐀_𝑝∈𝑅^𝑛×𝑛,𝐁=diag𝐁₁,…,𝐁_𝑝∈𝑅^𝑛×𝑝,𝐂=diag𝐂₁,…,𝐂_𝑝∈𝑅^𝑝×𝑛,𝐀_𝑖=010⋯0001⋯0⋮⋮⋮⋯1000⋯0∈𝑅^{𝑛𝑖×𝑛𝑖},𝐁_𝑖=00⋮1∈𝑅^𝑛𝑖,𝐂_𝑖^𝑇=10⋮0∈𝑅^𝑛𝑖, (3) where 𝐱=[𝑥₁₁,𝑥₁₂,…,𝑥_1𝑛1,…,𝑥_𝑝1,𝑥_𝑝2,…,𝑥_𝑝𝑛𝑝]^𝑇∈𝑅^𝑛 is the system state vector which is assumed to be unavailable for measurement, 𝐮=[𝑢₁,…,𝑢_𝑝]^𝑇∈𝑅^𝑝 is the control input, 𝐲=[𝑦₁,…,𝑦_𝑝]^𝑇∈𝑅^𝑝 is the output vector, 𝑛₁+𝑛₂+⋯+𝑛_𝑝=𝑛, 𝜏>0 is a known time delay of the system, 𝑓_𝑖(𝐱(𝑡)) and 𝑔_𝑖𝑗(𝐱(𝑡)) are the unknown nonlinear system functions, and ΔΦ(𝐱(𝑡))  =  [Δ𝜙₁(𝐱(𝑡),Δ𝜙₂(𝐱(𝑡),…,Δ𝜙_𝑝(𝐱(𝑡)]^𝑇∈𝑅^𝑝 is the vector of uncertainties.

Now let the output of the system and its derivatives be expressed as follows: 𝑦_𝑖(𝑡)=𝑥_𝑖1(),𝑦̇𝑡−𝜏_𝑖(𝑡)=𝑥̇_𝑖1(𝑡−𝜏)=𝑥_𝑖2(),⋮𝑦𝑡−𝜏_{𝑖(𝑛𝑖−1)}(𝑡)=𝑥̇_{𝑖(𝑛𝑖−1)}(𝑡−𝜏)=𝑥_𝑖𝑛𝑖(),𝑦𝑡−𝜏_{𝑖(𝑛𝑖)}(𝑡)=𝑥̇_𝑖𝑛𝑖(𝑡−𝜏)=𝑓_𝑖(𝐱(𝑡−𝜏))+^𝑝_𝑗=1𝑔_𝑖𝑗(𝐱(𝑡−𝜏))𝑢_𝑗(𝑡−𝜏)+Δ𝜙_𝑖(𝐱(𝑡−𝜏)). (4) Then, let 𝐲_𝑑=[𝑦_1𝑑,…,𝑦_𝑝𝑑]^𝑇∈𝑅^𝑝 be the given bounded output desired signal and contain finite derivatives up to the 𝑛th order. The tracking errors of the system can be defined as 𝑒_𝑖𝑗(𝑡)=𝑦_𝑖^(𝑗−1)(𝑡)−𝑦^(𝑗−1)_𝑖𝑑(𝑡), for 𝑖=1,2,…,𝑝 and 𝑗=1,2,…,𝑛_𝑖, and its derivatives can be obtained as 𝑒̇_𝑖1(𝑡)=𝑦̇_𝑖(𝑡)−𝑦̇_𝑖𝑑(𝑡)=𝑒_𝑖2(𝑡),𝑒̇_𝑖2(𝑡)=𝑦_𝑖⁽²⁾(𝑡)−𝑦_𝑖𝑑⁽²⁾(𝑡)=𝑒_𝑖3(𝑡),⋮𝑒̇_{𝑖(𝑛𝑖−1)}(𝑡)=𝑦_𝑖^{(𝑛𝑖−1)}(𝑡)−𝑦_𝑖𝑑^{(𝑛𝑖−1)}(𝑡)=𝑒_1𝑛𝑖(𝑡),𝑒̇_𝑖𝑛𝑖(𝑡)=𝑦_𝑖^(𝑛𝑖)(𝑡)−𝑦_𝑖𝑑^(𝑛𝑖)(𝑡)=𝑓_𝑖(𝐱(𝑡−𝜏))+^𝑝_𝑗=1𝑔_𝑖𝑗(𝐱(𝑡−𝜏))𝑢_𝑗(𝑡−𝜏)+Δ𝜙_𝑖(𝐱(𝑡−𝜏))−𝑦_𝑖𝑑^(𝑛𝑖)(𝑡)=𝑓_𝑖(𝐱(𝑡−𝜏))+^𝑝_𝑗=1𝑔_𝑖𝑗(𝐱(𝑡−𝜏))𝑢_𝑗(𝑡)−𝑦_𝑖𝑑^(𝑛𝑖)(𝑡)+Δ𝜓_𝑖(𝐱,𝐮), (5) where the uncertainty Δ𝜓_𝑖(𝐱,𝐮) including the structural uncertainty is defined as Δ𝜓_𝑖(𝐱,𝐮)=^𝑝_𝑗=1𝑔_𝑖𝑗(𝐱(𝑡−𝜏))𝑢_𝑗(𝑡−𝜏)−𝑢_𝑗(𝑡)+Δ𝜙_𝑖(𝐱(𝑡−𝜏)). (6) Thus, (5) can be expressed as 𝐞̇_𝑖(𝑡)=𝐀_𝑖𝐞_𝑖(𝑡)+𝐁_𝑖𝑓_𝑖(𝐱(𝑡−𝜏))+^𝑝_𝑗=1𝑔_𝑖𝑗(𝐱(𝑡−𝜏))𝑢_𝑗(𝑡)−𝑦^(𝑛𝑖)_𝑖𝑑(𝑡)+Δ𝜓_𝑖(𝐱,𝐮). (7) From (7), it is easy to verify that the tracking error dynamics of the system (2) can be represented in the following form: 𝐞̇(𝑡)=𝐀𝐞(𝑡)+𝐁𝐅(𝐱(𝑡−𝜏))+𝐆(𝐱(𝑡−𝜏))𝐮(𝑡)−𝐘_𝑑(𝑡)(.𝐞+ΔΨ𝐱,𝐮)₁(𝑡)=𝐂𝐞(𝑡), (8) where 𝐞=𝐞^𝑇₁,𝐞^𝑇₂,…,𝐞^𝑇_𝑝^𝑇=𝑒₁₁,…,𝑒_1𝑛1,…,𝑒_𝑝1,…,𝑒_𝑝𝑛𝑝^𝑇∈𝑅^𝑛,𝐞₁=𝑒₁₁,𝑒₂₁,…,𝑒_𝑝1^𝑇∈𝑅^𝑝,ΔΨ=Δ𝜓₁,Δ𝜓₂,…,Δ𝜓_𝑝^𝑇∈𝑅^𝑝,𝐞_𝑖=𝑒_𝑖1,𝑒_𝑖2,…,𝑒_𝑖𝑛𝑖^𝑇∈𝑅^𝑛𝑖,𝐘_𝑑=𝑦^(𝑛1)_1𝑑,𝑦^(𝑛2)_2𝑑,…,𝑦^(𝑛𝑝)_𝑝𝑑^𝑇∈𝑅^𝑝,𝐘_𝑑=𝑦_1𝑑,…,𝑦_1𝑑^(𝑛1−1),…,𝑦_𝑝𝑑,…,𝑦_𝑝𝑑^{(𝑛𝑝−1)𝑇}∈𝑅^𝑛. (9)

Based on the above discussion, the following assumptions should be made for the controller design.

Assumption 1. The matrix 𝐆(𝐱) is nonsingular (i.e., 𝐆(𝐱)⁻¹ exists) for all 𝐱∈𝑈_𝑥 where 𝑈_𝑥 is a compact set in 𝑅^𝑛.

Assumption 2. ‖ΔΨ(𝐱,𝐮)‖≤ℎ₁(𝐱)+ℎ₂(𝐱(𝑡−𝜏)), where ℎ₁(𝐱) and ℎ₂(𝐱(𝑡−𝜏)) are the unknown positive smooth continuous functions.

Remark 3. Compared with the previous results [27, 30], the uncertainties including the structural uncertainty are taken into account in this paper. In addition, the uncertainties considered in this paper are not supposed to be a bounded external disturbance unlike the previous results [17, 18, 20–23].

The control objective of this paper is to design a control law 𝐮(𝑡) such that 𝐲(𝑡) can follow a given desired reference signal 𝐲_𝑚(𝑡) 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 [12, 16]. 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 fuzzifier maps a crisp point in 𝑈 into a fuzzy set in 𝑈. The fuzzy rule base consists of a collection of fuzzy IF-THEN rules: 𝑅^(𝑙)∶IF𝑥₁is𝐹^𝑙₁,and𝑥₂is𝐹^𝑙₂,…,𝑥_𝑛is𝐹^𝑙_𝑛THEN𝑦_𝑗is𝐺^𝑙_𝑗,𝑗=1,…,𝑝,𝑙=1,…,𝑁, (10) in which 𝐱=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇∈𝑈 and 𝑦∈𝑉⊂𝑅 are the input and output of the fuzzy logic system, 𝐹^𝑙_𝑖 and 𝐺^𝑙_𝑗 are fuzzy sets in 𝑈_𝑖 and 𝑉_𝑗, respectively. 𝑁 is the number of rules. 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. Through singleton fuzzification, center average defuzzification, and product inference, the output of the fuzzy logic system can be expressed as 𝑦_𝑗=𝜃^𝑇_𝑗𝜉(𝑥),𝑗=1,…,𝑝, (11) where 𝜃^𝑇_𝑗=[𝜃¹_𝑗,…,𝜃^𝑁_𝑗] with each variable 𝜃^𝑙_𝑗 as the point at which the fuzzy membership function of 𝐺^𝑙_𝑗 achieves the maximum value, and 𝜉(𝑥)=[𝜉₁(𝑥),…,𝜉_𝑁(𝑥)]^𝑇 with each variable 𝜉_l as the fuzzy basis function defined as 𝜉_𝑙(𝑥)=∏^𝑛_𝑖=1𝜇_𝐹𝑙𝑖𝑥_𝑖∑^𝑁_𝑙=1∏^𝑛_𝑖=1𝜇_𝐹𝑙𝑖𝑥_𝑖, (12) where 𝜇_𝐹𝑙𝑖(𝑥_𝑖) is the membership function of the fuzzy set.

Now let MIMO fuzzy logic systems be expressed as follows: 𝐲=Φ(𝐱)𝜽, (13) where 𝜽^𝑇=𝜃₁^𝑇,…,𝜃_𝑝^𝑇,𝜃^𝑇_𝑖=𝜃¹_𝑖,…,𝜃^𝑁_𝑖Φ(𝐱)𝜉for𝑖=1,2,…,𝑝,=diag^𝑇(𝐱),…,𝜉^𝑇(𝐱). (14)

3. Observer-Based Robust Adaptive Fuzzy Controller Design and Stability Analysis

According to the description of the fuzzy logic systems presented in Section 2.2, we can construct the following fuzzy logic systems, over a compact set Ω_𝐱, to approximate the unknown nonlinear functions 𝑓_𝑖(𝐱(𝑡)) and 𝑔_𝑖𝑗(𝐱(𝑡)) and the unknown upper bounded functions ℎ₁(𝐱), ℎ₂(𝐱(𝑡−𝜏)) as follows: 𝑓_𝑖𝐱∣𝜃_𝑓1=𝜃^𝑇_𝑓𝑖𝜉(𝐱)𝑔,𝑖=1,…,𝑝,_𝑖𝑗𝐱∣𝜃_𝑔𝑖𝑗=𝜃^𝑇_𝑔𝑖𝑗𝜉(𝐱)ℎ,𝑖,𝑗=1,…,𝑝,₁(𝐱)=𝜃^𝑇_ℎ1𝜉(𝐱),ℎ₂(𝐱(𝑡−𝜏))=𝜃^𝑇_ℎ2𝜉(𝐱(𝑡−𝜏)). (15) Based on the above statements, it can be easily shown that 𝐅𝐱∣𝜽₁=𝑓₁𝐱∣𝜃_𝑓1,…,𝑓_𝑝𝐱∣𝜃_𝑓𝑝^𝑇=Φ(𝐱)𝜽₁,𝐆𝐱∣𝜽₂=𝑔₁₁𝐱∣𝜃_𝑔11,…,𝑔_1𝑝𝐱∣𝜃_𝑔1𝑝𝑔⋮⋱⋮_𝑝1𝐱∣𝜃_𝑔𝑝1,…,𝑔_𝑝𝑝𝐱∣𝜃_𝑔𝑝𝑝=Φ(𝐱)𝜽₂, (16) where 𝜽^𝑇₁=𝜃^𝑇_𝑓1,…,𝜃^𝑇_𝑓𝑝,𝜃^𝑇_𝑓𝑖=𝜃¹_𝑓𝑖,…,𝜃^𝑁_𝑓𝑖,𝜽₂=𝜃_𝑔11,…,𝜃_𝑔1𝑝𝜃⋮⋱⋮_𝑔𝑝1,…,𝜃_𝑔𝑝𝑝,𝜃^𝑇_𝑔𝑖𝑗=𝜃¹_𝑔𝑖𝑗,…,𝜃^𝑁_𝑔𝑖𝑗,𝜃^𝑇_ℎ1=𝜃¹_ℎ1,…,𝜃^𝑁_ℎ1,𝜃^𝑇_ℎ2=𝜃¹_ℎ2,…,𝜃^𝑁_ℎ2,Φ(𝐱)=diag𝜉^𝑇(𝐱),…,𝜉^𝑇(𝐱). (17) Suppose the state variables of the controlled system (1) are available for measurement, then the following robust adaptive fuzzy controller 𝐮(𝑡) can be adopted to let the system achieve the above control objective and can be defined as follows: 𝐮(𝑡)=𝐆(𝐱(𝑡−𝜏))⁻¹−𝐅(𝐱(𝑡−𝜏))−𝐊_𝑐𝐞(𝑡)+𝐘_𝑑(𝑡)+𝐮_𝑠(𝑡), (18) where 𝐊_𝑐=diag[𝐊_𝑐1,…,𝐊_𝑐𝑝]∈𝑅^𝑝×𝑛, where 𝐊_𝑐𝑖=[𝑘^𝑐_𝑖1,…,𝑘^𝑐_𝑖𝑛𝑖]∈𝑅^1×𝑛𝑖 for 𝑖=1,2,…,𝑝, is the control gain matrix such that the characteristic polynomial of 𝐀−𝐁𝐊_𝑐 is Hurwitz. The robust compensator 𝐮_𝑠(𝑡) will be designed to compensate the fuzzy approximation errors and the uncertainties.

However, the state variables of the system are unavailable for measurement in many engineering systems. In addition, although the tracking error vector 𝐞 can be obtained ideally by successive differentiation of 𝑒_𝑖1(𝑖=1,2,…,𝑝), ideal differentiators are physically unrealizable. Obviously, we must employ a state observer to estimate 𝐱 and 𝐞 and let 𝐱 and 𝐞̂ be the estimates of 𝐱 and 𝐞, respectively. Then replacing 𝐅(𝐱∣𝜽₁), 𝐆(𝐱∣𝜽₂), ℎ₁(𝐱∣𝜽_ℎ), ℎ₂(𝐱(𝑡−𝜏)), 𝐱, and 𝐞 by 𝐅(𝐱∣𝜽₁), 𝐆(𝐱∣𝜽₂), ℎ₁(𝐱∣𝜽_ℎ), ℎ₂(𝐱(𝑡−𝜏)), 𝐱, and 𝐞̂, respectively, the robust adaptive fuzzy controller (18) cannot be used to control the nonlinear system (2) and will be modified as the following form: 𝐮(𝑡)=𝐆(𝐱(𝑡−𝜏))⁻¹−𝐅(𝐱(𝑡−𝜏))−𝐊_𝑐𝐞̂(𝑡)+𝐘_𝑑(𝑡)+𝐮_𝑠(𝑡), (19) where 𝐅𝐱∣𝜽₁=𝑓₁𝐱∣𝜃_𝑓1,…,𝑓_𝑝𝐱∣𝜃_𝑓𝑝^𝑇=Φ(𝐱)𝜽₁,𝐆𝐱∣𝜽₂=𝑔₁₁𝐱∣𝜃_𝑔11,…,𝑔_1𝑝𝐱∣𝜃_𝑔1𝑝𝑔⋮⋱⋮_𝑝1𝐱∣𝜃_𝑔𝑝1,…,𝑔_𝑝𝑝𝐱∣𝜃_𝑔𝑝𝑝=Φ(𝐱)𝜽₂,ℎ₁(𝐱)=𝜃^𝑇_ℎ1𝜉(𝐱);ℎ₂(𝐱(𝑡−𝜏))=𝜃^𝑇_ℎ2𝜉(𝐱(𝑡−𝜏)). (20)(21)(22) Substituting (19) into (8) yields 𝐞̇(𝑡)=𝐀𝐞(𝑡)−𝐁𝐊_𝑐𝐞̂(𝑡)+𝐁𝐅(𝐱(𝑡−𝜏))−𝐅𝐱(𝑡−𝜏)∣𝜽₁+𝐆(𝐱(𝑡−𝜏))−𝐆𝐱(𝑡−𝜏)∣𝜽₂𝐮(𝑡)+𝐮_𝑠(𝑡)(,𝐞+ΔΨ𝐱,𝐮)₁(𝑡)=𝐂𝐞(𝑡). (23) In order to estimate the output tracking error vector, we design the observer as follows: 𝐞̇̂(𝑡)=𝐀−𝐁𝐊_𝑐𝐞̂(𝑡)+𝐊_𝑜𝐞₁(𝑡)−𝐞̂₁(𝑡,𝐞̂₁)(𝑡)=𝐂𝐞̂(𝑡), (24) where 𝐊_𝑜=diag[𝐊_𝑜1,…,𝐊_𝑜𝑝]∈𝑅^𝑛×𝑝, where 𝐊_𝑜𝑖=[𝑘^𝑜_1𝑖,…,𝑘^𝑜_𝑛𝑖𝑖]^𝑇 for 𝑖=1,2,…,𝑝, is the observer gain matrix such that the characteristic polynomial of 𝐀−𝐊_𝑜𝐂 is Hurwitz. Let us define the observation error vector as 𝐞^∼(𝑡)=𝐞(𝑡)−𝐞̂(𝑡). (25) Then by (23) and (24), we obtain 𝐞̇‌^∼(𝑡)=𝐀−𝐊_𝑜𝐂𝐞^∼(𝑡)+𝐁𝐅(𝐱(𝑡−𝜏))−𝐅𝐱(𝑡−𝜏)∣𝜽₁+𝐆(𝐱(𝑡−𝜏))−𝐆𝐱(𝑡−𝜏)∣𝜽₂𝐮(𝑡)+𝐮_𝐬(𝑡)+ΔΨ(𝐱,𝐮)_∼,𝐞₁(𝑡)=𝐂𝐞^∼(𝑡). (26) It is assumed that 𝜽₁,𝜽₂,𝜃_ℎ1, and 𝜃_ℎ2belong to compact sets Ω_𝜽1,  Ω_𝜽2,  Ω_𝜃ℎ1, and Ω_𝜃ℎ2, respectively, which are defined as Ω_𝜽1=𝜽₁∈𝑅^𝑁×𝑝∶𝜽₁≤𝑀_𝜽1,Ω<∞_𝜽2=𝜽₂∈𝑅^{𝑝𝑁×𝑝}∶𝜽₂≤𝑀_𝜽2,Ω<∞_𝜃ℎ1=𝜃_ℎ1∈𝑅^𝑁∶𝜃_ℎ1≤𝑀_𝜃ℎ1,Ω<∞_𝜃ℎ2=𝜃_ℎ2∈𝑅^𝑁∶𝜃_ℎ2≤𝑀_𝜃ℎ2<∞, (27) where 𝑀_𝜽1, 𝑀_𝜽2, 𝑀_𝜃ℎ1, and 𝑀_𝜃ℎ2 are the designed parameters by the designer, and 𝑁 is the number of fuzzy inference rules.

Now let us define the optimal parameter vectors 𝜽^∗₁, 𝜽^∗₂, 𝜃^∗_ℎ1, and 𝜃^∗_ℎ2 as follows: 𝜽^∗₁=argmin_{𝜽1∈Ω𝜽1}sup_{𝐱∈Ω𝐱,}𝐱_∈Ω𝐱𝐅(𝐱)−𝐅𝐱∣𝜽₁,𝜽^∗₂=argmin_{𝜽2∈Ω𝜽2}sup_{𝐱∈Ω𝐱,}𝐱^∈Ω𝐱𝐆(𝐱)−𝐆𝐱∣𝜽₂,𝜃^∗_ℎ1=argmin_{𝜽ℎ1∈Ω𝜃ℎ1}sup_{𝐱∈Ω𝐱,}𝐱_∈Ω𝐱ℎ₁(𝐱)−ℎ₁𝐱∣𝜃_ℎ1,𝜃^∗_ℎ2=argmin_{𝜽ℎ2∈Ω𝜃ℎ2}sup_{𝐱∈Ω𝐱,}𝐱_∈Ω𝐱ℎ₂(𝐱(−ℎ𝑡−𝜏))₂𝐱(𝑡−𝜏))∣𝜃_ℎ2, (28) where 𝜽^∗₁, 𝜽^∗₂, 𝜃^∗_ℎ1, and 𝜃^∗_ℎ2 are bounded in the suitable closed sets Ω_𝜽1, Ω_𝜽2, Ω_𝜃ℎ1, and Ω_𝜃ℎ2, respectively. Also the parameter estimation errors are defined as 𝜽^∼₁=𝜽^∗₁−𝜽₁,_∼𝜽₂=𝜽^∗₂−𝜽₂,_∼𝜃_ℎ1=𝜃^∗_ℎ1−𝜃_ℎ1;𝜃^∼_ℎ2=𝜃^∗_ℎ2−𝜃_ℎ2. (29)(30)(31) Then, the minimum approximation errors which correspond to the optimal parameter vectors are defined as 𝝎₁=𝐅(𝐱(𝑡−𝜏))−𝐅𝐱(𝑡−𝜏)∣𝜽^∗₁+𝐆(𝐱(𝑡−𝜏))−𝐆𝐱(𝑡−𝜏)∣𝜽^∗₂𝜔𝐮,₂=ℎ₁(𝐱)−ℎ₁𝐱∣𝜃^∗_ℎ1+ℎ₂(𝐱(𝑡−𝜏))−ℎ₂𝐱(𝑡−𝜏)∣𝜃^∗_ℎ2. (32)(33) Applying (29), (30), and (32), (26) can be rewritten as 𝐞̇‌^∼(𝑡)=𝐀−𝐊_𝑜𝐂𝐞^∼(𝑡)+𝐁Φ(𝐱)𝜽^∼₁+Φ(𝐱)𝜽^∼₂𝐮(𝑡)+𝐅(𝐱(𝑡−𝜏))−𝐅𝐱(𝑡−𝜏)∣𝜽^∗₁+𝐆(𝐱(𝑡−𝜏))−𝐆𝐱(𝑡−𝜏)∣𝜽^∗₂𝐮(𝑡)+𝐮_𝑠(𝑡)(=+ΔΨ𝐱,𝐮)𝐀−𝐊_𝑜𝐂𝐞^∼(𝑡)+𝐁Φ(𝐱)𝜽^∼₁+Φ(𝐱)𝜽^∼₂𝐮(𝑡)+𝝎₁+𝐮_𝑠(𝑡)+ΔΨ(𝐱,𝐮)_∼,𝐞₁(𝑡)=𝐂𝐞^∼(𝑡). (34) Then, the output error dynamics in (34) can be expressed as follows: 𝐞^∼₁=𝐇(𝐬)Φ(𝐱)𝜽^∼₁+Φ(𝐱)𝜽^∼₂𝐮(𝑡)+𝝎₁+𝐮_𝑠(𝑡)+ΔΨ(𝐱,𝐮), (35) where 𝐇(𝐬)=𝐂𝐬𝐈−𝐀−𝐊_𝑜𝐂⁻¹𝐁. (36) Obviously, the transfer function 𝐇(𝐬) is a known stable transfer function matrix. In order to utilize the SPR-Lyapunov design approach, (35) can be represented as 𝐞^∼₁=𝐇(𝐬)𝐋(𝐬)𝐋(𝐬)⁻¹Φ(𝐱)𝜽^∼₁+𝐋(𝐬)⁻¹Φ(𝐱)𝜽^∼₂𝐮(𝑡)+𝐋(𝐬)⁻¹𝝎₁+𝐮_𝑠(𝑡)+ΔΨ(𝐱,𝐮), (37) with 𝐋(𝑠)=diag𝐿_𝑖(𝑠),…,𝐿_𝑝(𝑠,𝐿_𝑖)(𝑠)=𝑠^𝑚𝑖+𝑏_𝑖1𝑠^𝑚𝑖−1+⋯+𝑏_𝑖𝑚𝑖𝑚_𝑖=𝑛_𝑖−1,𝑖=1,2,…,𝑝. (38) It is worth noting that 𝐋(𝑠) is chosen to be a proper stable transfer function matrix and to make 𝐇(𝐬)𝐋(𝐬) to be a proper SPR transfer function matrix. Then the state-space realization of (37) can be written as 𝐞̇‌^∼_𝑐(𝑡)=𝐀−𝐊_𝑜𝐂𝐞^∼_𝑐+𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₁+Φ_𝑓(𝐱)𝜽^∼₂𝐮(𝑡)+𝝎_1𝑓+𝐮_𝑠𝑓(𝑡)+ΔΨ_𝑓(𝐱,𝐮)_∼,𝐞₁(𝑡)=𝐂𝐞^∼_𝑐(𝑡), (39) with 𝐁_𝑐=diag𝐁_𝑐1,…,𝐁_𝑐𝑝∈𝑅^𝑛×𝑝,𝐁_𝑐𝑖=1,𝑏_𝑖1,𝑏_𝑖2,…,𝑏_𝑖𝑚𝑖^𝑇∈𝑅^𝑛𝑖×1,𝑖=1,2,…,𝑝, (40)(41)Φ_𝑓(𝐱)=𝐋(𝐬)⁻¹Φ(𝐱),𝝎_1𝑓=𝐋(𝐬)⁻¹𝝎₁,𝐮_𝑠𝑓=𝐋(𝐬)⁻¹𝐮_𝑠,ΔΨ_𝑓(𝐱,𝐮)=𝐋(𝐬)⁻¹ΔΨ(𝐱,𝐮). (42a)(42b)(42c)(42d)We define 𝝎_1𝑓+𝜔₂≤𝜔_𝑓,_∼𝜔_𝑓=𝜔_𝑓−𝜔_𝑓, (43) where 𝜔_𝑓 is the estimate of an unknown positive constant 𝜔_𝑓.

Remark 4. Since 𝐋(𝐬) is chosen as a proper stable transfer function matrix, it is obvious that the norm of ΔΨ_𝑓(𝐱,𝐮) also satisfies the inequality in Assumption 2. Moreover, the upper bound functions will be replaced by ℎ₁(𝐱∣𝜽_ℎ) and ℎ₂(𝐱(𝑡−𝜏)) because the state variables are unavailable for measurement.
Based on the Lyapunov stability theorem, we can obtain the robust compensator 𝐮_𝑠𝑓(𝑡) as follows: 𝐮_𝑠𝑓(𝑡)=𝐮₁(𝑡)+𝐮₂(𝑡), (44) where 𝐮₁(𝑡)𝐁=−^𝑇𝑐𝐏𝐞₁^∼_𝑐∼𝐞^𝑇_𝑐𝐏₁𝐁_𝑐×𝜔_𝑓+ℎ₁𝐱∣𝜃_ℎ1+ℎ₂𝐱(𝑡−𝜏)∣𝜃_ℎ2,𝐮₂(𝑡)𝐁=−^𝑇𝑐𝐏𝐞₁^∼_𝑐∼𝐞^𝑇_𝑐𝐏₁𝐁_𝑐²𝐞̂^𝑇𝐏₂𝐊𝐞_𝑜^∼₁, (45) where 𝐊_𝑐 is the control gain matrix, 𝐊_𝑜 is the observer gain matrix, and 𝐏₁ and 𝐏₂ are the symmetric positive definite matrices and will be solved later.
Then, the parameter adaptive learning laws are chosen as 𝜽̇₁=𝛾₁Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐,𝜽̇₂=𝛾₂Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐𝐮^𝑇, (46)(47)𝜃̇_ℎ1=𝛾𝐞₃^∼𝑇_𝑐𝐏₁𝐁_𝑐𝝃(𝐱),𝜃̇_ℎ2=𝛾𝐞₄^∼𝑇_𝑐𝐏₁𝐁_𝑐𝝃(𝐱(𝑡−𝜏)), (48a)(48b)𝜔̇_𝑓=𝛾𝐞₅^∼𝑇_𝑐𝐏₁𝐁_𝑐, (49) where 𝛾₁>0,  𝛾₂>0,  𝛾₃>0,  𝛾₄>0, and 𝛾₅>0 are the positive adaptive gain constants and can be chosen by the designer.

Remark 5. Without loss of generality, the adaptive laws used in this paper are assumed such 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 [12] to modify the adaptive laws such that the parameter vectors will remain inside of the constraint sets. The proposed adaptive laws (46)–(49) can be modified as the following form: 𝜽̇₁=𝛾₁Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐,if𝜽₁<𝑀_𝜃1or𝜽₁=𝑀_𝜃1,𝜽^𝑇₁Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐,𝑃𝛾≤0₁Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐,if𝜽₁=𝑀_𝜃1,𝜽^𝑇₁Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐>0, (50) where 𝑃{𝛾₁Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐} is defined as𝑃𝛾₁Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐=𝛾₁Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐−𝛾₁𝜽₁𝜽^𝑇₁𝜽₁²Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐,𝜽̇₂=𝛾₂Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐𝐮^𝑇,if𝜽₂<𝑀_𝜃2or𝜽₂=𝑀_𝜃2,𝜽^𝑇₂Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐𝐮^𝑇,𝑃𝛾≤0₂Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐𝐮^𝑇,if𝜽₂=𝑀_𝜃2,𝜽^𝑇₂Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐𝐮^𝑇>0, (51)(52)where 𝑃{𝛾₂Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐𝐮^𝑇} is defined as 𝑃𝛾₂Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐𝐮^𝑇=𝛾₂Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐𝐮^𝑇−𝛾₂𝜽₂𝜽^𝑇₂𝜽₂²Φ^𝑇_𝑓(𝐱)𝐁^𝑇_𝑐𝐏𝐞₁^∼_𝑐𝐮^𝑇, (53)𝜃̇_ℎ1=𝛾𝐞₃^∼𝑇_𝑐𝐏₁𝐁_𝑐𝝃(𝐱),if𝜃_ℎ1<𝑀_𝜃ℎ1or𝜃_ℎ1=𝑀_𝜃ℎ1,_∼𝐞^𝑇_𝑐𝐏₁𝐁_𝑐⋅𝜃^𝑇_ℎ1𝝃(𝐱),𝑃𝛾≤0_3∼𝐞^𝑇_𝑐𝐏₁𝐁_𝑐𝝃(𝐱),if𝜃_ℎ1=𝑀_𝜃ℎ1,_∼𝐞^𝑇_𝑐𝐏₁𝐁_𝑐⋅𝜃^𝑇_ℎ1𝝃(𝐱)>0, (54a)where 𝑃{𝛾₃‖𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐‖𝝃(𝐱)} is defined as 𝑃𝛾𝐞₃^∼𝑇_𝑐𝐏₁𝐁_𝑐𝝃(𝐱)=𝛾𝐞₃^∼𝑇_𝑐𝐏₁𝐁_𝑐𝝃(𝐱)−𝛾₃𝜃_ℎ1𝜃^𝑇_ℎ1𝜃_ℎ1^2∼𝐞^𝑇_𝑐𝐏₁𝐁_𝑐𝝃(𝐱), (54b)𝜃̇_ℎ2=𝛾𝐞₄^∼𝑇_𝑐𝐏₁𝐁_𝑐𝝃(𝐱(𝑡−𝜏)),if𝜃_ℎ2<𝑀_𝜃ℎ2or𝜃_ℎ2=𝑀_𝜃ℎ2,_∼𝐞^𝑇_𝑐𝐏₁𝐁_𝑐⋅𝜃^𝑇_ℎ2𝝃(𝐱(,𝑃𝛾𝑡−𝜏))≤0_4∼𝐞^𝑇_𝑐𝐏₁𝐁_𝑐𝝃(𝐱),if𝜃_ℎ2=𝑀_𝜃ℎ2,_∼𝐞^𝑇_𝑐𝐏₁𝐁_𝑐⋅𝜃^𝑇_ℎ2𝝃(𝐱(𝑡−𝜏))>0, (55a)where 𝑃{𝛾₄‖𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐‖𝝃(𝐱(𝑡−𝜏))} is defined as 𝑃𝛾𝐞₄^∼𝑇_𝑐𝐏₁𝐁_𝑐𝝃(𝐱(𝑡−𝜏))=𝛾𝐞₄^∼𝑇_𝑐𝐏₁𝐁_𝑐𝝃(𝐱(𝑡−𝜏))−𝛾₄𝜃_ℎ2𝜃^𝑇_ℎ2𝜃_ℎ2^2∼𝐞^𝑇_𝑐𝐏₁𝐁_𝑐𝝃(𝐱(𝑡−𝜏)). (55b)
The main result of the proposed observer-based robust adaptive fuzzy control scheme is summarized in the following theorem.

Theorem 6. Consider the MIMO nonlinear system (2) in the presence of output delay and the uncertainties subject to Assumptions 1-2. The observer-based robust adaptive fuzzy controller is defined by (19) and (44)-(45) with adaptation laws given by (46)–(49). For the given positive definite matrices 𝐐₁ and 𝐐₂, if there exist symmetric positive definite matrices 𝐏₁ and 𝐏₂ such that the following Lyapunov equations 𝐀−𝐊_𝑜𝐂^𝑇𝐏₁+𝐏₁𝐀−𝐊_𝑜𝐂=−𝐐₁,𝐀−𝐁𝐊_𝑐^𝑇𝐏₂+𝐏₂𝐀−𝐁𝐊_𝑐=−𝐐₂,𝐏₁𝐁_𝑐=𝐂^𝑇, (56)(57)(58) are satisfied, then all the closed-loop signals are bounded, and the tracking errors converge to a neighborhood of zero.

Proof. Consider the Lyapunov function candidate 1𝑉=2^∼𝐞^𝑇_𝑐𝐏𝐞₁^∼_𝑐+12𝐞̂^𝑇𝐏₂̂1𝐞+2𝛾₁^∼𝜽𝜽^𝑇₁^∼₁+12𝛾₂tr𝜽^∼𝜽^𝑇₂^∼₂+12𝛾₃^∼𝜃𝜃^𝑇_ℎ1^∼_ℎ1+12𝛾₄^∼𝜃𝜃^𝑇_ℎ2^∼_ℎ2+12𝛾₅^∼𝜔²_𝑓. (59) By the time derivative of 𝑉 and the facts that ̇‌^∼𝜽₁=−𝜽̇₁, ̇‌^∼𝜽₂=−𝜽̇₂, ̇‌^∼𝜃_ℎ1=−𝜃̇_ℎ1, ̇‌^∼𝜃_ℎ2=−𝜃̇_ℎ2  and ̇‌^∼𝜔_𝑓=−𝜔̇_𝑓, it can be easily shown from (24) and (39) that 𝑉̇=12𝐞̇^∼𝑇_𝑐𝐏𝐞₁^∼_𝑐+12^∼𝐞^𝑇_𝑐𝐏₁𝐞̇‌^∼_𝑐+12𝐞̇̂^𝑇𝐏₂𝐞̂+12𝐞̂^𝑇𝐏₂𝐞̇̂−1𝛾₁^∼𝜽^𝑇₁𝜽̇₁−1𝛾₂tr𝜽^∼𝑇₂𝜽̇₂−12𝛾₃^∼𝜃^𝑇_ℎ1𝜃̇_ℎ1−12𝛾₄^∼𝜃^𝑇_ℎ2𝜃̇_ℎ2−1𝛾₅^∼𝜔_𝑓𝜔̇_𝑓=12^∼𝐞^𝑇_𝑐𝐀−𝐊_𝑜𝐂^𝑇𝐏₁+𝐏₁𝐀−𝐊_𝑜𝐂𝐞^∼_𝑐+12𝐞̂^𝑇𝐀−𝐁𝐊_𝑐^𝑇𝐏₂+𝐏₂𝐀−𝐁𝐊_𝑐𝐞̂+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₁+Φ_𝑓(𝐱)𝜽^∼₂𝐮(𝑡)+𝜔_1𝑓+𝐮_𝑠𝑓(𝑡)+ΔΨ_𝑓(𝐱,𝐮)+𝐞̂^𝑇𝐏₂𝐊𝐞_𝑜^∼₁−1𝛾₁^∼𝜽^𝑇₁𝜽̇₁−1𝛾₂tr𝜽^∼𝑇₂𝜽̇₂−12𝛾₃^∼𝜃^𝑇_ℎ1𝜃̇_ℎ1−12𝛾₄^∼𝜃^𝑇_ℎ2𝜃̇_ℎ2−1𝛾₅^∼𝜔_𝑓𝜔̇_𝑓≤12^∼𝐞^𝑇_𝑐𝐀−𝐊_𝑜𝐂^𝑇𝐏₁+𝐏₁𝐀−𝐊_𝑜𝐂𝐞^∼_𝑐+12𝐞̂^𝑇𝐀−𝐁𝐊_𝑐^𝑇𝐏₂+𝐏₂𝐀−𝐁𝐊_𝑐𝐞̂+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₁+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₂𝐮+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐𝐮_𝑠𝑓+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐𝝎_1𝑓+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅ΔΨ_𝑓(𝐱,𝐮)+𝐞̂^𝑇𝐏₂𝐊𝐞_𝑜^∼₁−1𝛾₁^∼𝜽^𝑇₁𝜽̇₁−1𝛾₂tr𝜽^∼𝑇₂𝜽̇₂−12𝛾₃^∼𝜃^𝑇_ℎ1𝜃̇_ℎ1−12𝛾₄^∼𝜃^𝑇_ℎ2𝜃̇_ℎ2−1𝛾₅^∼𝜔_𝑓𝜔̇_𝑓. (60) Applying Assumption 2, (56), and (57), it yields 𝑉̇≤−12^∼𝐞^𝑇_𝑐𝐐𝐞₁^∼_𝑐−12𝐞̂^𝑇𝐐₂𝐞̂+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₁+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₂𝐮+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐𝐮_𝑠𝑓+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐𝝎_1𝑓+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅ℎ₁(𝐱)+ℎ₂(𝐱(𝑡−𝜏))+𝐞̂^𝑇𝐏₂𝐊𝐞_𝑜^∼₁−1𝛾₁^∼𝜽^𝑇₁𝜽̇₁−1𝛾₂tr𝜽^∼𝑇₂𝜽̇₂−12𝛾₃^∼𝜃^𝑇_ℎ1𝜃̇_ℎ1−12𝛾₄^∼𝜃^𝑇_ℎ2𝜃̇_ℎ2−1𝛾₅^∼𝜔_𝑓𝜔̇_𝑓1=−2^∼𝐞^𝑇_𝑐𝐐𝐞₁^∼_𝑐−12𝐞̂^𝑇𝐐₂𝐞̂+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₁+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₂𝐮+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐𝐮_𝑠𝑓+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐𝝎_1𝑓+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅ℎ₁(𝐱)−ℎ₁𝐱∣𝜃^∗_ℎ1+ℎ₁𝐱∣𝜃^∗_ℎ1−ℎ₁𝐱∣𝜃_ℎ1+ℎ₁𝐱∣𝜃_ℎ1+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅ℎ₂(𝐱(𝑡−𝜏))−ℎ₂𝐱(𝑡−𝜏)∣𝜃^∗_ℎ2+ℎ₂𝐱(𝑡−𝜏)∣𝜃^∗_ℎ2−ℎ₂𝐱(𝑡−𝜏)∣𝜃_ℎ2+ℎ₂(𝑡−𝜏)𝐱∣𝜃_ℎ2+𝐞̂^𝑇𝐏₂𝐊𝐞_𝑜^∼₁−1𝛾₁^∼𝜽^𝑇₁𝜽̇₁−1𝛾₂tr𝜽^∼𝑇₂𝜽̇₂−12𝛾₃^∼𝜃^𝑇_ℎ1𝜃̇_ℎ1−12𝛾₄^∼𝜃^𝑇_ℎ2𝜃̇_ℎ2−1𝛾₅^∼𝜔_𝑓𝜔̇_𝑓. (61) According to (33) and (43), we have 𝑉̇≤−12^∼𝐞^𝑇_𝑐𝐐𝐞₁^∼_𝑐−12𝐞̂^𝑇𝐐₂𝐞̂+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₁+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₂𝐮+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐𝐮_𝑠𝑓+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐𝝎_1𝑓+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅ℎ₁𝐱∣𝜃_ℎ1+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅𝜃^∼𝑇_ℎ1𝜉(𝐱)+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅ℎ₂𝐱(𝑡−𝜏)∣𝜃_ℎ2+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅𝜃^∼𝑇_ℎ2𝜉(𝐱(+𝑡−𝜏))_∼𝐞^𝑇_𝑐𝐏₁𝐁_𝑐⋅𝜔₂+𝐞̂^𝑇𝐏₂𝐊𝐞_𝑜^∼₁−1𝛾₁^∼𝜽^𝑇₁𝜽̇₁−1𝛾₂tr𝜽^∼𝑇₂𝜽̇₂−12𝛾₃^∼𝜃^𝑇_ℎ1𝜃̇_ℎ1−12𝛾₄^∼𝜃^𝑇_ℎ2𝜃̇_ℎ2−1𝛾₅^∼𝜔_𝑓𝜔̇_𝑓1≤−2^∼𝐞^𝑇_𝑐𝐐𝐞₁^∼_𝑐−12𝐞̂^𝑇𝐐₂𝐞̂+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₁+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₂𝐮+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐𝐮_𝑠𝑓+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐𝜔_𝑓+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅ℎ₁𝐱∣𝜃_ℎ1+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅𝜃^∼𝑇_ℎ1𝜉(𝐱)+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅ℎ₂𝐱(𝑡−𝜏)∣𝜃_ℎ2+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅𝜃^∼𝑇_ℎ2𝜉(𝐱(+𝑡−𝜏))_∼𝐞^𝑇_𝑐𝐏₁𝐁_𝑐⋅𝜔^∼_𝑓+𝐞̂^𝑇𝐏₂𝐊𝐞_𝑜^∼₁−1𝛾₁^∼𝜽^𝑇₁𝜽̇₁−1𝛾₂tr𝜽^∼𝑇₂𝜽̇₂−12𝛾₃^∼𝜃^𝑇_ℎ1𝜃̇_ℎ1−12𝛾₄^∼𝜃^𝑇_ℎ2𝜃̇_ℎ2−1𝛾₅^∼𝜔_𝑓𝜔̇_𝑓1=−2^∼𝐞^𝑇_𝑐𝐐𝐞₁^∼_𝑐−12𝐞̂^𝑇𝐐₂𝐞̂+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐𝐮_𝑠𝑓+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐𝜔_𝑓+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅ℎ₁𝐱∣𝜃_ℎ1+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅ℎ₂𝐱(𝑡−𝜏)∣𝜃_ℎ2+𝐞̂^𝑇𝐏₂𝐊𝐞_𝑜^∼₁+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₁−1𝛾₁^∼𝜽^𝑇₁𝜽̇₁+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐Φ_𝑓(𝐱)𝜽^∼₂1𝐮−𝛾₂tr𝜽^∼𝑇₂𝜽̇₂+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅𝜃^∼𝑇_ℎ1𝜉(𝐱)−1𝛾₃^∼𝜃^𝑇_ℎ1𝜃̇_ℎ1+𝐞^∼𝑇_𝑐𝐏₁𝐁_𝑐⋅𝜃^∼𝑇_ℎ2𝜉(𝐱(−1𝑡−𝜏))𝛾₄^∼𝜃^𝑇_ℎ2𝜃̇_ℎ2+𝐞^∼𝑇_𝑐𝐏₁𝐁𝜔_𝑐^∼_𝑓−1𝛾₅^∼𝜔_𝑓𝜔̇_𝑓. (62) By employing (46)–(49) and using the control laws (44)-(45), we can obtain 𝑉̇≤−12^∼𝐞^𝑇_𝑐𝐐𝐞₁^∼_𝑐−12𝐞̂^𝑇𝐐₂𝐞̂. (63) Therefore, it can be concluded that 𝑉̇≤0 from (63), and the output tracking error of the closed-loop system converges asymptotically to a neighborhood of zero based on the Lyapunov synthesis approach. This completes the proof.

4. An Example and Simulation Results

In this paper, a numerical example is illustrated to verify the performance of the proposed observer-based robust adaptive fuzzy controller. Consider the following MIMO nonlinear uncertain system with delayed output: 𝑥̇₁₁(𝑡)=𝑥₁₂(𝑡),𝑥̇₁₂(𝑡)=𝑓₁(𝐱)+𝑔₁₁(𝐱)𝑢₁(𝑡)+𝑔₁₂(𝐱)𝑢₂(𝑡)+Δ𝜙₁(𝐱),𝑥̇₂₁(𝑡)=𝑥₂₂(𝑡),𝑥̇₂₂(𝑡)=𝑓₂(𝐱)+𝑔₂₁(𝐱)𝑢₁(𝑡)+𝑔₂₂(𝐱)𝑢₂(𝑡)+Δ𝜙₂(𝐱),𝑦₁(𝑡)=𝑥₁₁(),𝑦𝑡−𝜏₂(𝑡)=𝑥₂₁(𝑡−𝜏). (64) According to (2), the above equation can be rewritten as the following compact form: 𝑥̇₁₁(𝑡)𝑥̇₁₂(𝑡)𝑥̇₂₁(𝑡)𝑥̇₂₂(𝑡)=0100𝑥000000010000¹¹(𝑡)𝑥₁₂(𝑡)𝑥₂₁(𝑡)𝑥₂₂(𝑡)+×𝑓00100001₁(𝐱)𝑓₂(𝐱)+𝑔₁₁(𝐱)𝑔₁₂(𝐱)𝑔₂₁(𝐱)𝑔₂₂(𝐱)𝑢₁(𝑡)𝑢₂(𝑡)+Δ𝜙₁(𝐱)Δ𝜙₂(𝐱),𝑦₁(𝑡)𝑦₂(𝑡)=𝑥10000010¹¹()𝑥𝑡−𝜏₁₂()𝑥𝑡−𝜏₂₁()𝑥𝑡−𝜏₂₂(),𝑡−𝜏 (65) where 𝐀₁=𝐀₂=⁰¹₀₀, 𝐁₁=𝐁₂=⁰₁, 𝐂₁=𝐂₂=[¹⁰], the nonlinear system functions 𝑓₁(𝐱)=(1−𝑒^−𝑥11)/(1+𝑒^−𝑥11), 𝑓₂(𝐱)=(1−𝑒^−𝑥21)/(1+𝑒^−𝑥21), 𝑔₁₂(𝐱)=𝑔₂₁(𝐱)=0, and 𝑔₁₁(𝐱)=𝑔₂₂(𝐱)=1, and the uncertainties ΔΦ=^{0.1𝑥21sin(𝑡)}_{0.1𝑥12cos(𝑡)} and the delay time is 𝜏=0.1. In the simulation, these nonlinear system functions and uncertainties are assumed to be unknown and are estimated by the fuzzy logic systems. The fuzzy membership functions are chosen as follows: 𝜇_𝐹1𝑖𝑥_𝑖=11+exp5𝑥_𝑖,𝜇+0.8_𝐹2𝑖𝑥_𝑖=exp−𝑥_𝑖+0.6²,𝜇_𝐹3𝑖𝑥_𝑖=exp−𝑥_𝑖+0.4²,𝜇_𝐹4𝑖𝑥_𝑖=exp−𝑥_𝑖²,𝜇_𝐹5𝑖𝑥_𝑖=exp−𝑥_𝑖−0.4²,𝜇_𝐹6𝑖𝑥_𝑖=exp−𝑥_𝑖−0.6²,𝜇_𝐹7𝑖𝑥_𝑖=11+exp−5𝑥_𝑖−0.8,𝑖=1,2,3,4. (66) Let 𝜉^𝑙(𝑥)=𝜇_𝐹𝑙1𝑥₁𝜇_𝐹𝑙2𝑥₂𝜇_𝐹𝑙3𝑥₃𝜇_𝐹𝑙4𝑥₄∑⁷_𝑙=1𝜇_𝐹𝑙1𝑥₁𝜇_𝐹𝑙2𝑥₂𝜇_𝐹𝑙3𝑥₃𝜇_𝐹𝑙4𝑥₄,𝜉(𝑥)=𝜉¹(𝑥),𝜉²(𝑥),𝜉³(𝑥),𝜉⁴(𝑥),𝜉⁵(𝑥),𝜉⁶(𝑥),𝜉⁷(𝑥)^𝑇. (67) In this section, we apply the observer-based robust adaptive fuzzy controller proposed in Section 3 to deal with the output tracking control problem of an MIMO nonlinear uncertain system with delayed output in (65).

The control objective is to force the system output to track the given desired trajectories 𝑦_1𝑑=𝑦_2𝑑=0.5sin(2𝜋/10). First, we select the control and observer gain matrices as 𝐊_𝑐=^1704000_0018030 and 𝐊_𝑜=^5010000_0075100^𝑇, using (20) and (21) to approximate the unknown 𝐅(𝐱) and 𝐆(𝐱), respectively. Given the positive definite matrices 𝐐₁=diag[1,1,1,1] and 𝐐₂=diag[1,1,1,1], and solving the Lyapunov equations (56), we can get 𝐏₁=,𝐏0.01010.005000.0051.2600000.00670.005000.0051.0483₂=.0.1302−0.500−0.52.137500000.1−0.500−0.53.0167 (68) The initial conditions are chosen as 𝐱(0)=[−0.5,−0.5,0.5,0.5]^𝑇, 𝐞̂(0)=[0.3,0.3,0.3,0.3]^𝑇, 𝜽_ℎ(0)=𝟎, 𝜃_𝑓1(0)=𝜃_𝑓2(0)=0, 𝛾₁=𝛾₂=0.13, 𝛾₃=0.1, 𝛾₄=0.01, 𝜃_𝑔11(0)=1, 𝜃_𝑔12(0)=0, 𝜃_𝑔21(0)=0, 𝜃_𝑔22(0)=1. The Simulation results are shown in Figures 1, 2, 3, 4, 5, 6, and 7. Figures 1 and 2 show the output tracking performance of the system output. Figures 3, 4, 5, and 6 show the trajectories of the system states and their estimation states. Figure 7 shows the control signal. Obviously, the proposed controller can not only ensure that all the signals of the resulting closed-loop system are bounded but also obtain that the good tracking performance of an MIMO nonlinear uncertain output-delay system is achieved.

Figure 1 

The trajectories of 𝑦1 and 𝑦1𝑑.

Figure 2 

The trajectories of 𝑦2 and 𝑦2𝑑.

Figure 3 

The trajectories of 𝑥11 and 𝑥11.

Figure 4 

The trajectories of 𝑥12 and 𝑥12.

Figure 5 

The trajectories of 𝑥21 and 𝑥21.

Figure 6 

The trajectories of 𝑥22 and 𝑥22.

Figure 7 

The control signals 𝑢1 and 𝑢2.

5. Conclusion

For a class of multiinput multioutput (MIMO) nonlinear systems with output delay and uncertainties, a robust adaptive fuzzy output feedback controller is proposed to deal with the problem of robust stability and output tracking control. Since the system states are unavailable for measurement, the elements of the observation error vector cannot be obtained completely. Hence, by means of the strictly positive-real (SPR) theory, all elements of the observation error vector will be obtained completely. Moreover, the fuzzy logic systems and some parameter adaptive laws are used to approximate the unknown nonlinear functions and the unknown upper bounds of the uncertainties including the structural uncertainty. By constructing an appropriate Lyapunov function, the proposed observer-based robust adaptive fuzzy controller can not only guarantee that all the signals in the whole closed-loop system are bounded but also obtain that the output tracking performance of the controlled system with delayed output and uncertainties can be achieved. Finally, some simulation results are provided to illustrate the validity of the proposed approach.