# Fuzzy Adaptive DSC Design for an Extended Class of MIMO Pure-Feedback Non-Affine Nonlinear Systems in the Presence of Input Constraints

Authors: Ning Wang; Ying Wang; Maolong Lv
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

A novel adaptive fuzzy dynamic surface control (DSC) scheme is for the first time constructed for a larger class of (multi-input multi-output) MIMO non-affine pure-feedback systems in the presence of input saturation nonlinearity. First of all, the restrictive differentiability assumption on non-affine functions has been canceled after using the piecewise functions to reconstruct the model for non-affine nonlinear functions. Then, a novel auxiliary system with bounded compensation term is firstly introduced to deal with input saturation, and the dynamic system employed in this work designs a bounded compensation term of tangent function. Thus, we successfully relax the strictly bounded assumption of the dynamic system. Additionally, the fuzzy logic systems (FLSs) are used to approximate unknown continuous systems functions, and the minimal learning parameter (MLP) technique is exploited to simplify control design and reduce the number of adaptive parameters. Finally, two simulation examples with input saturation are given to validate the effectiveness of the developed method.


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## Body

## 1. Introduction

In the past several decades, approximation-based adaptive control of nonlinear systems has been attracting much attention, and many significant results have been achieved [1–11]. Among them, the fuzzy logic systems (FLSs) and neural networks (NNs) have been successfully employed to approximate the unknown nonlinear functions. In addition, as a breakthrough in nonlinear control, approximation-based adaptive backstepping control has been extensively introduced to achieve global stability for many classes of nonlinear systems [12–17]. For example, in [12], an adaptive fuzzy control scheme was proposed for a class of nonlinear pure-feedback systems under the framework of backstepping, which requires nopriori knowledge of the systems dynamic. In [14], an adaptive fuzzy control scheme is presented for a class of pure-feedback nonlinear systems with immeasurable states by utilizing backstepping methodology. Recently, for a class of stochastic nonlinear systems with unknown control direction and unknown dead-zones, an adaptive fuzzy backstepping control method is presented in [17]. However, the problem of “explosion of complexity” caused by repeated differentiations of the virtual control law seriously limits the application of conventional backstepping technique. Thus, the dynamic surface control (DSC) technique has been creatively proposed to avoid this problem effectively by introducing a first-order low-pass filter at each step. Furthermore, compared with strict-feedback systems, pure-feedback systems have a non-affine fashion that the control inputs or variables appear nonlinearly in uncertain systems functions, which leads to the design being more difficult [18, 19]. Moreover, in contrast with SISO pure-feedback nonlinear systems, the control design of MIMO pure-feedback nonlinear systems is, as well known, more complicated due to the couplings among various inputs and outputs [20].On the other hand, input saturation nonlinearity, as one of the most important input constraints, usually appears in many industrial control systems [21]. In many applications, the input saturation nonlinearity may severely cause degradation of system performance, instability, or even damage. Consequently, the adaptive control of nonlinear systems in the presence of input saturation nonlinearity has been an active topic and attracted increasing attention in recent years [22–30]. For example, in [22], an adaptive fuzzy controller is constructed for pure-feedback stochastic nonlinear systems to deal with input constraints based on the adjustment of commanded input signal. In [25], an adaptive neural controller is investigated for a class of pure-feedback nonlinear time-varying systems with asymmetric input saturation nonlinearity in combination with the Gaussian error function. Recently, for a class of uncertain nonlinear systems with input saturation constraint and external disturbances, a tracking control scheme is proposed by introducing an auxiliary system in [27]. However, it should be pointed out that, for all above the state-of-the-art schemes [22–30] to work for pure-feedback uncertain nonlinear systems subject to input saturation, the non-affine function is always assumed to be differentiable with respect to control variables or inputs, which is restrictive arising from the fact that non-smooth nonlinearities such as dead zone, backlash, and saturation widely exist in various kinds of practical systems [22–25], which makes the non-affine functions non-differentiable and motives us to explore new methods to overcome this limitation [22].As a matter of fact, overcoming this limitation is challenging. This is because FLSs approximation errors will inevitably occur while adopting FLSs to approximate unknown systems functions within a compact set, this, in combination with external disturbances, may seriously degrade control performance or even give rise to closed-loop system instability. Additionally, there also exist a large number of fuzzy weights that need to be tuned online, which drastically increases the computational burden [28]. Therefore, a design technique needs to be developed that is able to guarantee that all system trajectories stay in the appropriate compact sets all the time, and the MLP technique needs to be employed to solve the explosion of learning parameters. Based on the aforementioned observations, this paper addresses the control problem for a more general class of MIMO pure-feedback nonlinear systems in the presence of input saturation nonlinearity. What is more, to the best of authors’ knowledge, the control design of this huger class of nonlinear systems has not been reported, which is still an open problem with theoretical and applicable significance. The main contributions of this paper are highlighted as follows: (1) it seems that this is the first work that considers both the MIMO non-affine nonlinear systems and input saturation even though some existing works focused on the same topic; (2) to handle input saturation, compared with the auxiliary system ℘˙j=-κj℘j+satoj-oj presented in [27, 30], the dynamic system employed in this work designs a bounded compensation term ξjtanh℘j, and, thus, the assumption that ℘j is bounded is cancelled; (3) in contrast to the existing strategies [22–30], we allow the non-affine functions of MIMO input-saturated nonlinear systems to be non-differentiable via the reconstruction of non-affine functions using appropriate piecewise functions, which removes the restrictive differentiability assumption on non-affine functions.The rest of this paper is organized as follows. Section2 presents the problem statement and preliminaries. The adaptive controller design is given in Section 3. Section 4 is devoted to stability analysis. In Section 5 simulation results are presented to show the effectiveness of the proposed scheme, followed by the conclusion in Section 6.


## 2. Problem Statement and Preliminaries

Consider the following MIMO pure-feedback systems [23]:(1)χ˙j,ij=φj,ijχ-j,ij,χj,ij+1+Dj,ijχ,t,1≤ij≤ρj-1χ˙j,ρj=φj,ρjχ-j,ρj,ujoj+Dj,ρjχ,tyj=χj,1,j=1,2,…,mwhere χj,ij∈R is the state of the jth subsystem, χ=χ-1,ρ1T,…,χ-j,ρjT,…,χ-m,ρmTT∈RN is the state vector of the whole system N=ρ1+⋯+ρm, where χ-j,ρj=χj,1,χj,2,…,χj,ρjT∈Rρj and ρj is the order of the jth subsystem. χ-j,ij=χj,1,…,χj,ijT∈Rij, uj∈R and yj∈R are the input and output of the jth subsystem, respectively. φj,ij· are unknown non-affine continuous functions, and Dj,ijχ,t,ij=1,…,ρj,j=1,…,m are the unknown external disturbances. ujoj is the plant input subject to saturation and satisfying [30](2)ujoj=satoj=signojuj,M,oj≥uj,Moj,oj<uj,Mwhere uj,M is the bound of ujoj, oj∈R is the input saturation, and uj=ujoj.The design objective of this work is to construct a novel dynamic surface controlleruj such that (1) the output tracking error zj,1=χj,1-yj,d achieves preselected transient and steady bounds; (2) all signals of system (1) are semiglobally uniformly ultimately bounded (SGUUB); (3) the control input constraint is not violated.Assumption 1.
Define the functionsψj,ijχ-j,ij,χj,ij+1=φj,ijχ-j,ij,χj,ij+1-φj,ijχ-j,ij,0. We assume that the functions ψj,ijχ-j,ij,χj,ij+1 satisfy(3)ψ_j,ijχj,ij+1+ħj,1ij≤ψj,ijχ-j,ij,χj,ij+1≤ψ-j,ijχj,ij+1+ħj,2ij,χj,ij+1≥0ψ_j,ij′χj,ij+1+ħj,3ij≤ψj,ijχ-j,ij,χj,ij+1≤ψ-j,ij′χj,ij+1+ħj,4ij,χj,ij+1<0where ψ_j,ij, ψ-j.ij, ψ_j,ij′, and ψ-j,ij′ are unknown positive constants; ħj,1ij, ħj,2ij, ħj,3ij, and ħj,4ij are unknown constants. And denote χj,ρj+1=uj, χ-j,ρj+1=χ-j,ρjT,ujT for notation conciseness.Remark 2.
In [22–30], the non-affine functions are always assumed to satisfy q_j,ij+1≤∂φj,ijχ-j,ij,χj,ij+1/∂χj,ij+1≤q-j,ij+1 and q_j,ρj≤∂φj,ijχ-j,ρj,uj/∂uj≤q-j,ρj with q_j,ij+1>0, q-j,ij+1>0, q_j,ρj>0, and q-j,ρj>0 being unknown constants. In fact, this assumption is used to ensure the controllability of system (1). However, the assumption q_j,ρj≤∂φj,ijχ-j,ρj,uj/∂uj≤q-j,ρj is too restrictive due to the fact that many kinds of non-smooth nonlinearities (e.g., dead-zone, backlash, or saturation, and so on) extensively exist in control input, leading to the non-differentiability of non-affine functions, even instability of closed-loop systems [10]. Even though some existing works like [16, 19] focus on the same topic, none of them addresses the control problem for both MIMO non-affine systems and input saturation problem. In other words, in this paper, we for the first time investigate a larger class of MIMO nonlinear systems considering both non-differentiable non-affine functions and input saturation.Remark 3.
From (3), there exist functions lj,1ijχ-j,ij+1 and lj,2ijχ-j,ij+1 taking values in 0,1 and satisfying(4)ψj,ijχ-j,ij,χj,ij+1=1-lj,1ijχ-j,ij+1ψ_j,ijχj,ij+1+ħj,1ij+lj,1ijχ-j,ij+1ψ-j,ijχj,ij+1+ħj,2ij,χj,ij+1≥0ψj,ijχ-j,ij,χj,ij+1=1-lj,2ijχ-j,ij+1ψ_j,ij′χj,ij+1+ħj,3ij+lj,2ijχ-j,ij+1ψ-j,ij′χj,ij+1+ħj,4ij,χj,ij+1≥0
To make the control design succinct, define the functionsQj,ijχ-j,ij+1 and Δj,ijχ-j,ij+1 as(5)Qj,ijχ-j,ij+1=1-lj,1ijχ-j,ij+1ψ_j,ij+lj,1ijχ-j,ij+1ψ-j,ij,χj,ij+1≥01-lj,2ijχ-j,ij+1ψ_j,ij′+lj,2ijχ-j,ij+1ψ-j,ij′,χj,ij+1<0Δj,ijχ-j,ij+1=1-lj,1ijχ-j,ij+1ħj,1ij+lj,1ijχ-j,ij+1ħj,2ij,χj,ij+1≥01-lj,2ijχ-j,ij+1ħj,3ij+lj,2ijχ-j,ij+1ħj,4ij,χj,ij+1<0
Using (5), we can model the non-affine terms ψj,ijχ-j,ij,χj,ij+1 as(6)ψj,ijχ-j,ij,χj,ij+1=Qj,ijχ-j,ij+1χj,ij+1+Δj,ijχ-j,ij+1
In view of (5), it can be known that(7)0<Q_j,ij≤Qj,ijχ-j,ij+1≤Q-j,ij,0≤Δj,ijχ-j,ij+1≤ħj,ij∗where Q_j,ij=minij=1,2,…,ρjψ_j,ij,ψ-j,ij,ψ_j,ij′,ψ-j,ij′, Q-j,ij=maxij=1,2,…,ρjψ_j,ij,ψ-j,ij,ψ_j,ij′,ψ-j,ij′ and ħj,ij∗=maxħj,1ij+ħj,2ij+ħj,3ij+ħj,4ij. According to (6) and the definition of ψj,ijχ-j,ij,χj,ij+1, system (1) can be rewritten as(8)χ˙j,ij=φj,ijχ-j,ij,0+Qj,ijχ-j,ij+1χj,ij+1+Δj,ijχ-j,ij+1+Dj,ijχ,t,ij=1,2,…,ρj-1χ˙j,ρj=φj,ρjχ-j,ρj,0+Qj,ρjχ-j,ρj+1uj+Δj,ρjχ-j,ρj+1+Dj,ρjχ,tyj=χj,1Assumption 4.
The reference signalyj,dt is continuous and available, and there exists a positive constant Pj0 such that Ωj0≔yj,d,y˙j,d,y¨j,d:yj,d2+y˙j,d2+y¨j,d2≤Pj0.Assumption 5.
Forij=1,…,ρj,j=1,…,m, there exist unknown positive constants Dj,ij∗ satisfying Dj,ijχ,t≤Dj,ij∗.Lemma 6 (see [8]).
Consider the first-order dynamical system:(9)
Υ
˙
t
=
-
ƛ
Υ
t
+
b
τ
twith ƛ>0, b>0 and τt a positive function. Then, for any given bounded initial condition Υt0≥0, the inequality Υt≥0,∀t≥0 holds.Lemma 7 (see [17]).
For anyγ∈R and ∀ℵ>0, the hyperbolic tangent function tanh· fulfills(10)0≤γ-γtanhγℵ≤0.2785ℵ0≤γtanhγℵ
The fuzzy logic systems (FLSs) are employed as function approximator. Construct FLSs with the following IF-THEN rules:(11)Rl:Ifχ1isF1land…andχnisFnlTHENyisBl,l=1,2,…,N.where χ=χ1,χ2,…,χnT∈Rn and y are input and output of the FLSs. Based on the singleton fuzzifier, product inference, and center average defuzzifier, the FLSs can be formulated as(12)yχ=∑i=1Nmaxy∈R⁡uBiy∏j=1nuFjiχj∑i=1N∏j=1nuFjiχjwhere uFjiχj and uBiy are the membership of Fji and Bi, respectively. Let(13)ϕ-iχ=∏j=1nuFjiχj∑i=1N∏j=1nuFjiχjwhere Θ-i=maxy∈R⁡uBiy, Θ=Θ-1,Θ-2,…,Θ-NT, and ϕχ=ϕ-1χ,ϕ-2χ,…,ϕ-NχT. Then, the FLSs can be expressed as follows:(14)yχ=ΘTϕχLemma 8 (see [23]).
On a compact setΩχ, if fχ is a continuous function, for any given constant ϖ∗>0, then there exist FLSs yχ such that(15)supχ∈Ωχ⁡fχ-yχ≤ϖ∗


## 3. Fuzzy Adaptive Controller Design

In this section, an adaptive fuzzy controller is proposed for a larger class of MIMO pure-feedback nonlinear systems (1) utilizing the DSC technique. To start, consider the following change of coordinates:(16)zj,1=χj,1-yj,d,zj,ij=χj,ij-νj,ij,ij=2,3,…,ρj-1,zj,ρj=χj,ρj-νj,ρj-ξjtanh℘jwhere zj,1 is the output tracking error, νj,ij is the output of the first-order filter with sj,ij-1 as the input, ξj is a positive design parameter, and ℘j is a dynamic system defined as(17)℘˙j=cosh2℘jξj-κjtanh℘j+satoj-oj,℘j0=0where κj>0 is a design parameter.Remark 9.
It has to be noted that, compared with the existing work [27, 30], a novel auxiliary system is proposed, and the dynamic system employed in this brief designed a bounded compensation term ξjtanh℘j to cope with input saturation problem. Therefore, the restrictive bounded assumption of the dynamic system has been deleted.
Sinceφj,ijχ-j,ij,0,ij=1,…,ρj are unknown continuous functions, we use fuzzy logic systems (FLSs) to approximate them as follows:(18)φj,ijχ-j,ij,0=Θj,ijTϕχ-j,ij+ϖj,ij,ij=1,…,ρjwhere ϖj,ij is the approximation error and satisfies ϖj,ij≤ϖj,ij∗ with ϖj,ij∗>0 being an unknown constant.
Define(19)Φj,ij=Q_j,ij-1Θj,ij2,ij=1,…,ρjwhere Φj,ij are unknown constants and Φ^j,ij is the estimate of Φj,ij with Φ^j,ij=Φj,ij-Φ~j,ij.Step j , 1. Differentiating zj,1 along with (16) yields(20)z˙j,1=φj,1χ-j,1,0+Qj,1χ-j,2χj,2+Δj,1χ-j,2+Dj,1χ,t-y˙j,dConsider the following quadratic Lyapunov function candidate:(21)Vzj,1=12zj,12Invoking (7), (20), and Assumption 5, we have(22)V˙zj,1≤zj,1φj,1χ-j,1,0+Qj,1χ-j,2χj,2zj,1-zj,1y˙j,d+zj,1ħj,1∗+zj,1Dj,1∗Substituting (18) into (22) gives(23)V˙zj,1≤zj,1Θj,1Tϕχ-j,1+Qj,1χ-j,2χj,2zj,1-zj,1y˙j,d+zj,1Q_j,1ηj,1∗where ηj,1∗=Q_j,1-1ϖj,1∗+ħj,1∗+Dj,1∗. In view of Young’s inequality, we can further have(24)V˙zj,1≤Qj,1χ-j,2χj,2zj,1+zj,12Θj,122aj,12ϕTχ-j,1ϕχ-j,1+aj,122+zj,1Q_j,1ηj,1∗-zj,1y˙j,dwhere aj,1 is positive constant.Then, construct the virtual control lawsj,1 and parameters adaptation laws η^j,1 and Φ^j,1 as(25)sj,1=-cj,1zj,1-Φ^j,1zj,12aj,12ϕTχ-j,1ϕχ-j,1-η^j,1tanhzj,1υj,1-αj,1y˙j,dtanhzj,1y˙j,dυj,1(26)η^˙j,1=ξj,1zj,1tanhzj,1υj,1-σj,1ξj,1η^j,1(27)Φ^˙j,1=βj,1zj,122aj,12ϕTχ-j,1ϕχ-j,1-σj,1βj,1Φ^j,1where cj,1>0, βj,1>0, ξj,1>0, σj,1>0, aj,1>0, υj,1>0 and αj,1≥Q_j,1-1 are design parameters, and η^j,1 is the estimate of ηj,1∗.Remark 10.
Note that (26) and (27) satisfy Lemma 6. Thus, by choosing η^j,10≥0 and Φ^j,10≥0, one has η^j,1t≥0 and Φ^j,1t≥0 for ∀t≥0. Furthermore, since the initial conditions η^j,10 and Φ^j,10 are selected by control law designer, we choose η^j,10≥0 and Φ^j,10≥0.
In line with the DSC technique, introduce variableνj,2. Let sj,1 pass through a first-order filter with time constant ιj,2 to obtain νj,2 as(28)ιj,2ν˙j,2+νj,2=sj,1,νj,20=sj,10
Define the filter errorej,2=νj,2-sj,1, which yields ν˙j,2=-ej,2/ιj,2 and(29)e˙j,2=-ej,2ιj,2+Ξj,2zj,1,zj,2,ej,2,Φ^-j,1,η^-j,1,yj,d,y˙j,d,y¨j,dwhere Ξj,2· is the introduced continuous function.
Byχj,2=zj,2+νj,2 and ej,2=νj,2-sj,1, we have(30)χj,2=zj,2+sj,1+ej,2
Noting thatΦj,1=Q_j,1-1Θj,12 and αj,1≥Q_j,1-1, and substituting (25) and (30) into (24), we can further obtain(31)V˙zj,1≤Qj,1χ-j,2zj,2+ej,2zj,1-cj,1Q_j,1zj,12+zj,1y˙j,d+aj,122-Q_j,1η^j,1zj,1tanhzj,1υj,1-zj,1y˙j,dtanhzj,1y˙j,dυj,1+Q_j,1Φ~j,1zj,122aj,12ϕTχ-j,1ϕχ-j,1+zj,1Q_j,1ηj,1∗
Take the following Lyapunov function candidate:(32)Vj,1=Vzj,1+Q_j,1η~j,122ξj,1+Q_j,1Φ~j,122βj,1where η~j,1=ηj,1∗-η^j,1 and Φ~j,1=Φj,1-Φ^j,1 are the estimates of ηj,1 and Φj,1, respectively.
It follows from (31) that the time derivative of Vj,1 is(33)V˙j,1≤Qj,1χ-j,2zj,2+ej,2zj,1-cj,1Q_j,1zj,12+aj,122-Q_j,1Φ~j,1βj,1Φ^˙j,1-βj,1zj,122aj,12ϕTχ-j,1ϕχ-j,1+zj,1y˙j,d-zj,1y˙j,dtanhzj,1y˙j,dυj,1-Q_j,1η~j,1ξj,1η^˙j,1-ξj,1zj,1tanhzj,1υj,1+Q_j,1ηj,1∗zj,1-zj,1tanhzj,1υj,1
Applying (26), (27), and Lemma 7, one has(34)V˙j,1≤Qj,1χ-j,2zj,2+ej,2zj,1-cj,1Q_j,1zj,12+Q_j,1σj,1η~j,1η^j,1+Φ~j,1Φ^j,1+aj,122+0.2785Q_j,1ηj,1∗υj,1+0.2785υj,1Step j , i j   2 ≤ i j ≤ ρ j - 1 ,   j = 1 , … , m. Similar to the design method in Step j,1 differentiating zj,ij along with (16) yields(35)z˙j,ij=φj,ijχ-j,ij,0+Qj,ijχ-j,ij+1χj,ij+1+Δj,ijχ-j,ij+1+Dj,ijχ,t-ν˙j,ijConsider the following quadratic Lyapunov function candidate:(36)Vzj,ij=12zj,ij2In view of Young’s inequality and using (35), we can obtain the time derivative of (36) as(37)V˙zj,ij≤Qj,ijχ-j,ij+1χj,ij+1zj,ij+zj,ij2Θj,ij22aj,ij2ϕTχ-j,ijϕχ-j,ij+aj,ij22+zj,ijQ_j,ijηj,ij∗-zj,ijν˙j,ijwhere ηj,ij∗=Q_j,ij-1ϖj,ij∗+ħj,ij∗+Dj,ij∗, and aj,ij is positive constant.Take the virtual control lawsj,ij and parameters adaptation laws η^j,ij and Φ^j,ij as(38)sj,ij=-cj,ijzj,ij-Φ^j,ijzj,ij2aj,ij2ϕTχ-j,ijϕχ-j,ij-η^j,ijtanhzj,ijυj,ij+αj,ijν˙j,ijtanhzj,ijν˙j,ijυj,ij(39)η^˙j,ij=ξj,ijzj,ijtanhzj,ijυj,ij-σj,ijξj,ijη^j,ij(40)Φ^˙j,ij=βj,ijzj,ij22aj,ij2ϕTχ-j,ijϕχ-j,ij-σj,ijβj,ijΦ^j,ijThe design process of parameters is similar to Stepj,1. Then, let sj,ij pass through a first-order filter with time constant ιj,ij+1 as follows:(41)ιj,ij+1ν˙j,ij+1+νj,ij+1=sj,ij,νj,ij+10=sj,ij0Defineej,ij+1=νj,ij+1-sj,ij, it yields ν˙j,ij+1=-ej,ij+1/ιj,ij+1 and(42)e˙j,ij+1=-ej,ij+1ιj,ij+1+Ξj,ij+1z-j,ij+1,e-j,ij+1,Φ^-j,ij,η^-j,ij,yj,d,y˙j,d,y¨j,dwhere z-j,ij+1=zj,1,…,zj,ij+1T, e-j,ij+1=ej,2,…,ej,ij+1T, Φ^-j,ij=Φ^j,1,…,Φ^j,ijT, η^-j,ij=η^j,1,…,η^j,ijT and Ξj,ij+1· is a continuous function.According toχj,ij+1=zj,ij+1+νj,ij+1 and ej,ij+1=νj,ij+1-sj,ij, one reaches(43)χj,ij+1=zj,ij+1+sj,ij+ej,ij+1Substituting (38) and (43) into (37) results in(44)V˙zj,ij≤Qj,ijχ-j,ij+1zj,ij+1+ej,ij+1zj,ij-cj,ijQ_j,ijzj,ij2+aj,ij22-Q_j,ijη^j,ijzj,ijtanhzj,ijυj,ij-zj,ijν˙j,ijtanhzj,ijν˙j,ijυj,ij+Q_j,ijΦ~j,ijzj,ij22aj,ij2ϕTχ-j,ijϕχ-j,ij+zj,ijν˙j,ij+zj,ijQ_j,ijηj,ij∗Consider the following Lyapunov function candidate:(45)Vj,ij=Vzj,ij+Q_j,ijη~j,ij22ξj,ij+Q_j,ijΦ~j,ij22βj,ijwhere η~j,ij=ηj,ij∗-η^j,ij and Φ~j,ij=Φj,ij-Φ^j,ij.Notingη~j,ij=ηj,ij∗-η^j,ij and following the same way as Step j,1 give rise to(46)V˙j,ij≤Qj,ijχ-j,ij+1zj,ij+1+ej,ij+1zj,ij-cj,ijQ_j,ijzj,ij2+aj,ij22-Q_j,ijΦ~j,ijβj,ijΦ^˙j,ij-βj,ijzj,ij22aj,ij2ϕTχ-j,ijϕχ-j,ij-Q_j,ijη~j,ijξj,ijη^˙j,ij-ξj,ijzj,ijtanhzj,ijυj,ij+zj,ijν˙j,ij-zj,ijν˙j,ijtanhzj,ijν˙j,ijυj,ij+Q_j,ijηj,ij∗zj,ij-zj,ijtanhzj,ijυj,ijApplying (39), (40), and Lemma 7 yields(47)V˙j,ij≤Qj,ijχ-j,ij+1zj,ij+1+ej,ij+1zj,ij+0.2785υj,ij+0.2785Q_j,ijηj,ij∗υj,ij+Q_j,ijσj,ijη~j,ijη^j,ij+Φ~j,ijΦ^j,ij-cj,ijQ_j,ijzj,ij2+aj,ij22Step j , ρ j   j = 1 , … , m. Similar to the former design process, we can obtain(48)z˙j,ρj=φj,ρjχ-j,ρj,0+Qj,ρjχ-j,ρj+1uj+Δj,ρjχ-j,ρj+1+Dj,ρjχ,t-ν˙j,ρj+κjtanh℘j-satoj+ojForuj=satoj≤uj,M, there exists a continuous function Qj,ρjχ-j,ρj such that(49)Qj,ρjχ-j,ρj+1uj=Qj,ρjχ-j,ρj,satojsatoj≤Qj,ρjχ-j,ρjConsider a compact setΩj,ρj≔{∑l=1ρjQ_j,lη~j,l2/ξj,l+Q_j,lΦ~j,l2/βj,l + zj,12+∑l=2ρjej,l2+zj,l2 ≤ 2ωj,ρj}. It can be seen from (43) that all the variables of χ-j,ρj are included in the compact set Ωj,ρj×Ωj0. Thus, Qj,ρjχ-j,ρj=Qj,ρjχ-j,ρj-1,zj,ρj+sj,ρj-1+ej,ρj have maximums on Ωj,ρj×Ωj0. There exist unknown positive constants Qj,ρj∗ such that Qj,ρjχ-j,ρj≤Qj,ρj∗.Choosing the quadratic functionVzj,ρj as Vzj,ρj=zj,ρj2/2, it gives(50)V˙zj,ρj≤Θj,ρjTϕχ-j,ρjzj,ρj+zj,ρjηj,ρj∗-zj,ρjν˙j,ρj+ojzj,ρjwhere ηj,ρj∗=ϖj,ρj∗+ħj,ρj∗+Dj,ρj∗+Qn∗+κj+uj,M.According to Young’s inequality, one has(51)V˙zj,ρj≤zj,ρj2Θj,ρj22aj,ρj2ϕTχ-j,ρjϕχ-j,ρj+aj,ρj22+ojzj,ρj+zj,ρjηj,ρj∗-zj,ρjν˙j,ρjwhere aj,ρj is positive constant.Similarly, construct the actual control lawoj and the adaptation laws η^j,ρj and Φ^j,ρj as(52)oj=-cj,ρjzj,ρj-Φ^j,ρjzj,ρj2aj,ρj2ϕTχ-j,ρjϕχ-j,ρj-η^j,ρjtanhzj,ρjυj,ρj-αj,ρjν˙j,ρjtanhzj,ρjν˙j,ρjυj,ρj(53)η^˙j,ρj=ξj,ρjzj,ρjtanhzj,ρjυj,ρj-σj,ρjξj,ρjη^j,ρj(54)Φ^˙j,ρj=cj,ρjzj,ρj22aj,ρj2ϕTχ-j,ρjϕχ-j,ρj-σj,ρjβj,ρjΦ^j,ρjThe design process of parameters is also similar to Stepj,ij and Step j,1. Take the following Lyapunov function candidate:(55)Vj,ρj=Vzj,ρj+η~j,ρj22ξj,ρj+Φ~j,ρj22βj,ρjwhere η~j,ρj=ηj,ρj∗-η^j,ρj and Φ~j,ρj=Φj,ρj-Φ^j,ρj.Following the same way as the former steps gives(56)V˙j,ρj≤σj,ρjη~j,ρjη^j,ρj+Φ~j,ρjΦ^j,ρj-cj,ρjzj,ρj2+0.2785υj,ρj+0.2785ηj,ρj∗υj,ρj+aj,ρj22


## 4. Stability Analysis

The main stability results of the MIMO pure-feedback nonlinear systems (1) are presented.Theorem 11.
Supposing that Assumptions1, 4, and 5 hold and the above proposed design procedure is employed to MIMO pure-feedback nonlinear systems described by (1), for Φ^j,ij0≥0, η^j,ij0≥0, ∀ω>0 and Vj0≤ω, there exist design parameters cj,ij, βj,ij, ξj,ij, σj,ij, aj,ij, υj,ij, αj,ij and ιj,ij such that
(1)Vjt≤ω for ∀t>0, and hence all of the signals in the closed-loop systems remain semiglobally uniformly ultimately bounded;
(2) the output tracking errorzj,1 is such that limt→∞zj,1t≤Δj,1, where Δj,1 is a positive constant depending on the design parameters. Furthermore, the whole system output tracking error z1=z1,1,…,zm,1T satisfies limt→∞z1t≤Δ1 with Δ1 a positive constant that relies on the design parameters;
(3) the dynamic system℘j is bounded, and the control input constraint is not violated.Proof.
Choose the following Lyapunov function candidate for the whole systems:(57)V=∑j=1mV-jwhere V-j = 1/2∑ij=1ρj-1zj,ij2+Q_j,ijη~j,ij2/ξj,ij+Q_j,ijΦ~j,ij2/βj,ij+ej,ij+12 + 1/2zj,ρj2+η~j,ρj2/ξj,ρj+Φ~j,ρj2/βj,ρj.
According to (34), (47), and (56), we can obtain the time derivative of V-j as(58)V-˙j≤∑ij=1ρj-1Q-j,ijzj,ij+1+ej,ij+1zj,ij-∑ij=1ρj-1cj,ijQ_j,ijzj,ij2+aj,ρj22+σj,ρjη~j,ρjη^j,ρj+Φ~j,ρjΦ^j,ρj+0.2785ηj,ρj∗υj,ρj+∑ij=1ρj-1aj,ij22+0.2785Q_j,ijηj,ij∗υj,ij+0.2785υj,ij+∑ij=1ρj-1Q_j,ijσj,ijη~j,ijη^j,ij+Φ~j,ijΦ^j,ij-cj,ρjzj,ρj2+∑ij=1ρj-1-ej,ij+12ιj,ij+1+ej,ij+1Ξj,ij+1·+0.2785υj,ρj
Using the following inequalities(59)η~j,ijη^j,ij≤ηj,ij∗22-η~j,ij22,Φ~j,ijΦ^j,ij≤Φj,ij22-Φ~j,ij22
we can arrive at(60)V-˙j≤-∑ij=1ρj-1cj,ijQ_j,ijzj,ij2-12∑ij=1ρj-1Q_j,ijσj,ijη~j,ij2+Φ~j,ij2+∑ij=1ρj-1Q-j,ijzj,ij+1+ej,ij+1zj,ij-cj,ρjzj,ρj2+∑ij=1ρj-1-yj,ij+12ιj,ij+1+ej,ij+1Ξj,ij+1·+Cj,0-12σj,ρjη~j,ρj2+Φ~j,ρj2where Cj,0 = ∑ij=1ρj-1(aj,ij2/2+0.2785Q_j,ijηj,ij∗υj,ij+0.2785υj,ij+Q_j,ijσj,ijηj,ij∗2+Φj,ij2/2) + 0.2785ηj,ρj∗υj,ρj + aj,ρj2/2+σj,ρjηj,ρj∗2+Φj,ρj2/2+0.2785υj,ρj.
By completion of squares, one has(61)ej,ij+1Ξj,ij+1·≤ej,ij+12Ξj,ij+12·2kj,1+kj,12Q-j,ijej,ij+1zj,ij≤Q-j,ij2ej,ij+12kj,22+zj,ij22kj,2Q-j,ijzj,ij+1zj,ij≤Q-j,ijzj,ij22+Q-j,ijzj,ij+122with kj,1 and kj,2 being positive constants. Then, we can further rewrite (60) as(62)V-˙j≤-∑ij=1ρj-1cj,ijQ_j,ijzj,ij2-12∑ij=1ρj-1Q_j,ijσj,ijη~j,ij2+Φ~j,ij2-cj,ρjzj,ρj2+∑ij=1ρj-1Q-j,ijzj,ij22+zj,ij+122+∑ij=1ρj-1zj,ij22kj,2+ρj-1kj,12+∑ij=1ρj-1-ej,ij+12ιj,ij+1+ej,ij+12Ξj,ij+12·2kj,1+Q-j,ij2ej,ij+12kj,22+Cj,0-12σj,ρjη~j,ρj2+Φ~j,ρj2
Then, it can be known from [16] that Ξj,ij+1· has a maximum Mj,ij+1 on the compact set Ωj,ij+1×Ωj0. Let 1/ιj,ij+1=Mj,ij+12/2kj,1+Q-j,ij2kj,2/2+ωj with ωj being a positive constant. Setting cj,ij=Q_j,ij-1Q-j,ij+1/2kj,2+ϑj, cj,ρj=Q-j,ρj+1/2kj,2+ϑj with ϑj being any positive constant, one has(63)V-˙j≤-∑ij=1ρjϑjzj,ij2-12∑ij=1ρj-1Q_j,ijσj,ijη~j,ij2+Φ~j,ij2+Cj-12σj,ρjη~j,ρj2+Φ~j,ρj2-∑ij=1ρj-1ωjej,ij+12where Cj=Cj,0+ρj-1kj,1/2. Noting (32), (45), (55), and (57), it yields(64)V-˙j≤-λjV-j+Cjwhere λj=minij=1,…,ρj2ωj,2ϑj,σj,ijξj,ij,σj,ijβj,ij. Note that Cj/λj can be made arbitrarily small by decreasing cj,ij, βj,ij, and ξj,ij and meanwhile increasing σj,ij, aj,ij, υj,ij, αj,ij and ιj,ij. Hence we can have Cj/λj≤ω by appropriately choosing the design parameters. It follows from Cj/λj≤ω and (64) that V-˙j≤0 on the level set V-j=ω. Therefore, all the signals of the closed-loop systems are SGUUB. The property (1) of Theorem 11 is proved.
Solving (64) shows(65)V-jt≤V-j0+Σwith Σ=Cj/λj a positive constant. According to (21), (36), and (57), we have ∑ij=1ρjzj,ij2/2≤V-j. Using the first inequality in (65), the following inequality holds:(66)limt→∞zj,1≤limt→∞2V-j≤2Σ=Δj,1
Now let us consider the Lyapunov function candidate for the whole systems asV=∑j=1mV-j. From (65), it can be derived that(67)V˙≤∑j=1m-λjV-j+Cj≤-RV+Πwhere R=minλ1,…,λm and Π=∑j=1mCj. Then, we further have(68)Vt≤V0-Γz-Rt+Γwhere Γ=Π/R is a positive constant.
Similarly, we havelimt→∞Vt≤Γ,which leads to(69)limt→∞z1t≤limt→∞2Vt≤2Γ=Δ1
Noting that the size ofΔ1 depends on the design parameters cj,ij, βj,ij, ξj,ij, σj,ij, aj,ij, υj,ij, αj,ij and ιj,ij. Thus, by appropriately online-tuning the design parameters, the tracking error z1 can be regulated to a neighborhood of the origin as small as desired and property (2) of Theorem 11 is proved.
Furthermore, for input saturation, there exists a nonnegative scalarIj to satisfy Λj≤Ij with Λj=satoj-oj and κj>Ij. Choosing the Lyapunov function candidate quadratic function V℘j as V℘j=C℘j2/2, we can obtain(70)V˙℘j=cosh2℘j-κj℘jtanh℘j+℘jsatoj-oj≤cosh2℘jκj℘j-κj℘jtanh℘j-κj℘j+Ij℘j≤cosh2℘j0.2785κj-κj-Ij℘j
If℘j>0.2785κj/κj-Ij, we have V˙℘j<0. Therefore, ℘j will lie in the compact set ℘j|℘j≤0.2785κj/κj-Ij∀t≥0 and property (3) of Theorem 11 is proved. This completes the proof.


## 5. Simulation Analysis

In this section, two simulation examples are given to show validity of the proposed method in this paper.Example 1.
Consider the MIMO non-affine nonlinear uncertain systems as follows:(71)χ˙1,1=φ1,1χ1,1,χ1,2+D1,1χ,tχ˙1,2=φ1,2χ1,1,χ1,2,u1o1+D1,2χ,tχ˙2,1=φ2,1χ2,1,χ2,2+D2,1χ,tχ˙2,2=φ2,2χ2,1,χ2,2,u2o2+D2,2χ,ty1=χ1,1y2=χ2,1where φ1,1(χ1,1,χ1,2)=0.2χ1,1χ1,2/5+χ1,1+χ1,2, φ1,2(χ1,1,χ1,2,u1(o1))=0.3χ1,2χ1,1+0.25χ1,1+0.5u1(o1), φ2,1(χ2,1,χ2,2)=0.5χ2,1χ2,2+χ2,2+χ2,1 and φ2,2(χ2,1,χ2,2,u2o2)=0.8χ2,12χ2,2+χ2,2+0.5u2o2, D1,1χ,t=0.5sin⁡(t)cos⁡(0.2t3)2+χ1,12χ2,1χ2,2, D1,2χ,t = 0.5u1+0.3cos⁡(t)3sin⁡(t)+χ1,22+χ1,2χ2,1, D2,1χ,t=0.2sin⁡(t2)cos⁡(0.5t)2+χ1,1χ2,1χ1,22 and D2,2χ,t=0.5cos⁡(0.2t2)sin⁡(5t)+χ2,22+χ2,12. u1(o1) and u2(o2) are defined as follows:(72)u1o1=sato1signo15.5,o1≥5.5o1,o1<5.5u2o2=sato2signo24.5,o2≥4.5o2,o2<4.5
It can be known that the existence of input saturation nonlinearity implies that non-affine functionsφ1,2χ1,1,χ1,2,u1(o1) and φ2,2χ2.1,χ2.2,u2(o2) are non-differentiable. In this case, the existing approaches cannot be used. However, Assumption 1 in this paper is still satisfied which means that the scheme proposed here is able to deal with the control design difficulty in spite of the input saturation nonlinearity.
According to Theorem11, the virtual control laws and actual control laws are constructed as(73)s1,1=-8z1,1-Φ^1,1z1,12×0.252ϕTχ1,1ϕχ1,1-5y˙1,dtanhz1,1y˙1,d0.25-η^1,1tanhz1,10.25s2,1=-6z2,1-Φ^2,1z2,12×0.12ϕTχ2,1ϕχ2,1-2y˙2,dtanhz2,1y˙2,d0.1-η^2,1tanhz2,10.1o1=-8z1,2-Φ^1,2z1,22×0.252ϕTχ1,2ϕχ1,2-2ν˙1,2tanhz1,2ν˙1,20.5-η^2,1tanhz2,10.5o2=-12z2,2-Φ^2,2z2,22×0.12ϕTχ2,2ϕχ2,2-2ν˙2,2tanhz2,2ν˙2,20.2-η^2,2tanhz2,20.2with adaptive laws(74)η^˙1,1=2z1,1tanhz1,10.25-0.15×2η^1,1,Φ^˙1,1=2z1,122×0.252ϕTχ-1,1ϕχ-1,1-0.15×2Φ^1,1η^˙2,1=2z2,1tanhz2,10.1-0.1×2η^2,1,Φ^˙2,1=z2,122×0.12ϕTχ-2,1ϕχ-2,1-0.1Φ^2,1η^˙1,2=1.5z1,2tanhz1,20.5-0.15×1.5η^1,2,Φ^˙1,2=2z1,222×0.252ϕTχ-1,2ϕχ-1,2-0.15×2Φ^1,2η^˙2,2=1.5z2,2tanhz2,20.2-0.1×1.5η^2,2,Φ^˙2,2=z2,222×0.12ϕTχ-2,2ϕχ-2,2-0.1Φ^2,2where z1,1=χ1.1-y1,d, z1,2=χ1.2-ν1,2, z2,1=χ2,1-y2,d and z2,2=χ2,2-ν2,2. Let the initial conditions be χ1,10,χ1,20,χ2,10,χ2,20T=0.5,0.2,0.35,0.15T, Φ^1,10=Φ^1,20=Φ^2,10=Φ^2,20=0 and η^1,10=η^1,20=η^2,10=η^2,20=0. The simulation results are provided in Figures 1–5.Figure 1
Outputsy1 and y2 and desired trajectories y1,d and y2,d.Figure 2
System inputsu1 and u2.Figure 3
Adaptation parametersΦ^1,1, η^1,1, Φ^1,2, η^1,2, Φ^2,1, η^2,1, Φ^2,2, and η^2,2.Figure 4
Phase portrait ofz1,1, z1,2, and z2,2.Figure 5
Phase portrait of statesχ1,1, χ1,2, χ2,1 and χ2,2.From Figure1, we can see that the outputs y1 and y2 track the desired trajectories y1,d and y2,d with small tracking error. Figure 2 shows that the proposed scheme works well with bounded system inputs, and the response curves of adaptive parameters Φ^1,1, η^1,1, Φ^1,2, η^1,2, Φ^2,1, η^2,1, Φ^2,2, and η^2,2 are depicted in Figure 3. From Figure 4, it can be seen that the bounds for z1,1, z1,2, and z2,2 are not overstepped. Finally, Figure 5 is given to explain the boundedness of states χ1,1, χ1,2, χ2,1, and χ2,2.Example 2.
Consider the following two inverted pendulums systems composed of spring and damper connections. The pendulum angle and angular velocity were controlled using the torque inputs generated by a servomotor at each base. The dynamic equations can be described as follows [6]:(75)J1θ¨1=m1grsin⁡θ1-0.5Frcos⁡θ1-θ-Tf1+u1J2θ¨2=m2grsin⁡θ2-0.5Frcos⁡θ2-θ-Tf2+u2where θ1 and θ2 are the angular positions, J1=1kgm2 and J2=1kgm2 are the moments of inertia, m1=2kg and m2=2.5kg are the masses, r=0.5m, F=kp-l+bp˙ denotes the force applied by the spring and damper at the connection points, and p is the distance between the connection points as follows:(76)p=d2+drsin⁡θ1-sin⁡θ2+r221-cos⁡θ2-θ1where d=0.5m, k=150N/m and b=1Nsec/m. The relative angular position θ can be defined as(77)θ=tan-1r/2cos⁡θ2-cos⁡θ1d+r/2sin⁡θ1-sin⁡θ2and Tfii=1,2 are assumed to be a LuGre friction model defined as(78)Tfi=s0ε˙i+s1ε˙i+s2θ˙iε˙i=θ˙i-s0θ˙iTc+Ts-Tcexp⁡-θ˙/θ˙22where s0=1Nm, s1=1Nmsec, s2=1Nmsec, θ˙s=0.1rad/sec, Ts=2Nm and Tc=1Nm.
Definingχ1,1=θ1, χ1,2=θ˙1, χ2,1=θ2 and χ2,2=θ˙2, system (75) can be rewritten in the following form:(79)χ˙1,1=φ1,1χ1,1,χ1,2+D1,1χ,tχ˙1,2=φ1,2χ1,1,χ1,2,u1o1+D1,2χ,tχ˙2,1=φ2,1χ2,1,χ2,2+D2,1χ,tχ˙2,2=φ2,2χ2,1,χ2,2,u2o2+D2,2χ,ty1=χ1,1y2=χ2,1where φ1,1(·)=1-e-1-χ1,1/1+e-1-χ2,1+χ1,2, φ1,2(·)=g1,2m1grsin⁡x1,1-0.5Frcos⁡χ1,1-θ-Tf1+g1,2u1(o1), g1,2=1/J1, φ2,2(·)=g2,2u2o2+g2,2m2grsin⁡x2,1-0.5Frcos⁡χ2,1-θ-Tf2, g2,2=1/J2, φ2,1(·)=χ2,1+χ2,2/1+χ2,1+χ2,2; D1,1χ,t=0.5sin⁡χ1,2χ2,1χ2,2cos⁡0.2t, D2,1χ,t=2×sin⁡χ1,1χ1,2χ2,12, D1,2χ,t=0.3cos⁡χ1,22+χ1,1χ2,1 and D2,2χ,t=cos⁡χ2,12+χ2,22+χ1,1sin⁡(t)2. Moreover, u1(o1) and u2(o2) are described by the following:(80)u1o1=sato1signo14,o1≥4o1,o1<4u2o2=sato2signo26.5,o2≥6.5o2,o2<6.5
It can be seen that the non-affine functions are non-differentiable with respect tou1 and u2. In simulation, choose the desired reference trajectories as y1,d=0.5sin⁡t+sin⁡0.5t and y2,d=sin⁡t; the virtual control laws, actual control laws, and adaption laws are provided by (25), (52) and (26)-(27), (53)-(54) with design parameters c1,1=c1,2=6, c2,1=c2,2=9, a1,1=a1,2=0.2, a2,1=a2,2=0.5, α1,1=α2.1=4, α1,2=α2,2=2, υ1,1=υ1,2=0.25, υ2,1=υ2,2=0.1, β1,1=β1,2=1.5, β2,1=β2,2=2, σ1,1=σ1,2=0.15, σ2,1=σ2,2=0.25, ξ1,1=2, ξ1,2=ξ2,2=2.5, ξ2,1=1.5. Let the initial conditions be χ1,10,χ1,20,χ2,10,χ2,20T=0.3,0.2,0.5,0.45T, Φ^1,10=Φ^1,20=Φ^2,10=Φ^2,20=0 and η^1,10=η^1,20=η^2,10=η^2,20=0. The simulation results are provided in Figures 6–8.Figure 6
Outputsy1 and y2 and reference signals y1,d and y2,d.Figure 7
Control inputsu1 and u2.Figure 8
Adaptive parametersΦ^1,1, η^1,1, Φ^1,2, η^1,2, Φ^2,1, η^2,1, Φ^2,2, and η^2,2.As can be seen in Figure6, the system outputs track the desired trajectories, perfectly. Figures 7-8 illustrate the system inputs and adaptive parameters, from which, we can see that the fairly good tracking performance is obtained.


## 6. Conclusion

This work for the first time proposes fuzzy adaptive dynamic surface control design for a larger class of MIMO non-affine nonlinear systems in the presence of input saturation. To overcome the design difficulty of input saturation, a novel auxiliary system with bounded compensation term has been proposed, and a bounded compensation term of tangent function is designed in this paper. Thanks to this design, we successfully relax the strictly bounded assumption of the dynamic system. SGUUB stability of the closed-loop systems is rigorously proved by combining Lyapunov theory and invariant set theory.



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