Error-driven nonlinear feedback-based fuzzy adaptive output dynamic surface control for nonlinear systems with partially constrained tracking errors

Accepted Manuscript

Error-driven nonlinear feedback-based fuzzy adaptive output dynamic surface control for nonlinear systems with partially constrained tracking errors Shigen Gao, Hairong Dong, Bin Ning, Hongwei Wang PII: DOI: Reference:

S0016-0032(18)30370-3 10.1016/j.jfranklin.2018.05.042 FI 3477

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

1 July 2017 23 November 2017 22 May 2018

Please cite this article as: Shigen Gao, Hairong Dong, Bin Ning, Hongwei Wang, Error-driven nonlinear feedback-based fuzzy adaptive output dynamic surface control for nonlinear systems with partially constrained tracking errors, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2018.05.042

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Error-driven nonlinear feedback-based fuzzy adaptive output dynamic surface control for nonlinear systems with partially constrained tracking errors1 Shigen Gao2a , Hairong Donga , Bin Ninga , Hongwei Wangb a

State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China National Research Center of Railway Safety Assessment, Beijing Jiaotong University, Beijing 100044, China

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Abstract

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In this paper, a novel error-driven nonlinear feedback technique is designed for partially constrained errors fuzzy adaptive observer-based dynamic surface control of a class of multiple-input-multiple-output nonlinear systems in the presence of uncertainties and interconnections. There is no requirements that the states are available for the controller design by constructing fuzzy adaptive observer, which can online identify the unmeasurable states using available output information only. By transforming partial tracking errors into new error variables, partially constrained tracking errors can be guaranteed to be confined in pre-specified performance regions. The feature of the error-driven nonlinear feedback technique is that the feedback gain self-adjusts with varying tracking errors, which prevents high-gain chattering with large errors and guarantees disturbance attenuation with small errors. Based on a new non-quadratic Lyapunov function, it is proved that the signals in the resulted closed-loop system are kept bounded. Simulation and comparative results are given to demonstrate the effectiveness of the proposed method.

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Keywords: Fuzzy adaptive observer, Dynamic surface control, Nonlinear feedback, Partially constrained error, Prescribed performance, Uncertain MIMO nonlinear system.

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1. Introduction

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Nonlinear control design has been a research hotspot in the past decades due to its general better performance than linear control [1, 2, 3, 4, 5, 6, 7]. Backstepping method was originally proposed to design stabilizing control for nonlinear dynamical systems [8, 9, 10, 11, 12], which laid the research foundation for some subsequent theoretical results in particular for the control of high-order systems with various nonlinearities, including uncertain dynamics, input saturation, unknown control directions, backlash-like hysteresis, and so on [13, 14, 15, 16]. Nevertheless, the problem of “explosion of complexity” is an inherent imperfection of the backstepping method raised by the differentiations of virtual controllers in the recursive design procedures, which worsens particularly in the backstepping-based control of high-order nonlinear systems. Fortunately, a pioneering technique named after dynamic surface control (DSC) was proposed to circumvent such a problem by filtering the intermediate controllers, and as a result, thus the differentiations of virtual 1 This work is supported jointly by the National Natural Science Foundation of China (61703033), the Fundamental Research Funds for Central Universities (2016RC054) and the State Key Laboratory of Rail Traffic Control and Safety (RCS2017ZQ001). 2 Corresponding author. E-mail address: [email protected] (S. Gao).

Preprint submitted to Journal of the Franklin Institute

June 14, 2018

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controllers are not required in the recursive design [17, 18]. In the meantime, the DSC method relaxes the requirements of smoothness systematic functions and target trajectories. Inspired by the DSC technique, some subsequent varietal and improved DSC-based methods have been reported in [19, 20, 21, 22] on some theoretical researches and practical applications. In these literatures, neural networks (NNs) or fuzzy logic systems (FLSs) are employed to online approximate the uncertainties, thus the matching conditions or assumptions of linearly parameterize to unknown nonlinear uncertainties are removed, these methods can guarantee the closed-loop stability with rigourous proof by Lyapunov theorem, nevertheless, the transient and ultimate tracking performance are guaranteed to be confined to residual sets with bounds characterized by design parameters and some unknown bounded terms. In some control systems with high-precision requirements on the performance [23, 24, 25, 26, 27], the previous methods can guarantee prescribed transient and steady-state performance only after several trial and error tests before implementations for some specific plants to be controlled. This motivates the developments of control design under prescribed performance constraints, such as the barrier Lyapunov function (BLF) based methods [28, 29, 30] and the error transformation (ET) based methods [31, 32, 33]. The BLF-based methods constrain the states by the prescribed time-invariant or time-variant constraints defined in the logarithm-based BLF, yet, extra efforts needs to be made in the BLF-based methods on the continuity and differentiability of stabilizing control laws since piecewise smooth BLF is used. ET-based methods transform the tracking errors into new errors variables associated with prescribed performance functions to achieve prescribed performance control, however, such ET-based method suffers from a potential singularity problem for some prescribed performance conditions, which may result in the violation of prescribed regions and uncontrollability. Following up, Han and Lee proposed a new ET-based DSC method with partial states tracking errors kept within the prescribed regions at all times [34], which also circumvents the restrictions in [28, 29, 30, 31, 32, 33], nevertheless, these methods are developed based on the assumptions that all states are available for control design, which cannot be applied to systems with immeasurable states. Until recently, the conservations of DSC method has been founded as follows using singular perturbation analysis [35]: i) the stability of resulted closed-loop system relies on both high control gains and small filter coefficients; ii) the consequent bound of output tracking error is large; iii) the convergent time of steady-state tracking accuracy greatly relies on large control gains. As everyone knows, high control gains can guarantee the quick response, disturbance attenuation, and insensitivity to parameter variations properties [36, 37, 38] of resulting closed-loop systems, the high gain feedback can be developed using some available prior knowledge of system uncertainties, thus high gain coefficients can be chosen large enough to easily guaranteed the closed-loop stability. However, high-gain feedback scheme faces the risk of modeling errors and unknown or neglected nonlinearities in practical applications, it is also well known that high gains lead to overshoot problem with system order larger than 1, meanwhile, it has been reported in [39] that the stability regions may vanish with high-gain feedback in nonlinear systems since neglected nonlinearities give rise to an unstable limit cycle around an asymptotically stable equilibrium. It is necessary to point out that [35] presents the features for understanding the DSC and helps establishing an efficient strategy of parameter selection and system debugging, no results on how to improve the control design were given. Motivated by these observations, a fuzzy adaptive observer-based DSC for nonlinear systems with uncertainties is developed using error-driven nonlinear output and virtual tracking errors feedback, which can achieved the partially constrained tracking errors control. The main merits and contributions of this paper can be summarized as follows. 1). With comparison to the pioneering DSC [17, 20] and recent advances [40, 41, 42, 43], where the constant gain tracking errors feedback are utilized in the closed-loop control, this paper presents a novel continuous nonlinear output and virtual tracking errors feedback method implemented in the intermedi2

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ate and final controllers. In the nonlinear feedback control scheme, the feedback gains self-adjust versus errors with different amplitudes, which lightens the conservations of high-gain dependent DSC [35]. To be specific, if small tracking error shows up, high-gain feedback is implemented to guarantee disturbance attenuation properties, and if large tracking error shows up, low-gain feedback is implemented to avoid the amplifications of disturbance-like terms and the overshoot problems. 2). The proposed fuzzy adaptive observer-based DSC has solved the unmeasured states and partial tracking errors constrained problems by constructing fuzzy systems based observer and utilizing the error transformation. Some recent works in [44, 45, 46] also presented some significant results on prescribed performance control, they require the direct feedback of full states using linear-gain feedback, which hold the similar conservations of high-gain dependent DSC [35]. 3). Different from the widely-used quadratic Lyapunov function, a non-quadratic one is constructed in this paper to analyse the closed-loop stability, the resulted closed-loop system is much more complex raised by the supererogatory compound function of error dynamics. Furthermore, systematic parameterselection criterions are provided detailedly with rigorous mathematical proof.

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2. Plant Descriptions, Problem Formulation and Some Preliminaries

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Consider a class of uncertain MIMO strict feedback nonlinear systems composed of N subsystems:  x˙ j,ij =xj,ij +1 + fj,ij (¯ xj,ij ) + dj,ij (t, x)     ij = 1, · · · , nj − 1 (1)  x˙ j,nj =uj + fj,nj (x) + dj,nj (t, x)    yj =xj,1

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where j = 1, · · · , N represents the subscript of the jth subsystem, xj,ij is the ij th state variable in the jth subsystem, ij = 1, · · · , nj with nj being the order of theP jth subsystem, x¯j,ij = [xj,1 , · · · , xj,ij ]T ∈ Rij , x = [¯ xT1,n1 , · · · , x¯TN,nN ]T ∈ Rn is the full state vector, n = nnNj =n1 nj , fj,ij (·) is unknown function of its arguments, dj,ij (t, x) denotes the composite time-varying external disturbances and coupled dynamic terms among full states. uj and yj are the control input to be designed and the output signal of jth subsystem, respectively. The target is to design output feedback control with immeasurable states such that i) the output yj can track the pre-specified trajectory yj,r with sufficiently small error, ii) all the signals in the resulted closed-loop system are guaranteed bounded, and iii) partial tracking error constrains are not violated. To this ends, the following widely adopted assumptions and lemmas are introduced before designing a feasible controller.

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Assumption 1. [47] The target trajectories yj,r , j = 1, · · · , N are sufficiently smooth, that is, there 2 2 2 2 exists a positive constant Lj,0 such that yj,r + y˙ j,r + y¨j,r ≤ Lj,0 . Assumption 2. [48] dj,ij (t, x) is bounded by unknown constant with j = 1, · · · , N and ij = 1, · · · , nj , that is, there exists unknown constant d∗j,ij satisfying |dj,ij (t, x)|≤ d∗j,ij . Assumption 3. [49] The functions fj,ij (·) with j = 1, · · · , N and ij = 1, · · · , nj satisfy the global Lipschitz conditions, i.e., one can find some constants hj,ij , then the following inequality is true for any X1 ∈ R and X2 ∈ R: |fj,ij (X1 ) − fj,ij (X2 )|≤ hj,ij kX1 − X2 k with k·k being the 2-norm of corresponding input arguments.

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Lemma 1. By choosing some proper knowledge base, fuzzifier, fuzzy inference engine, and defuzzifier, for some unknown continuous function f (x), x ∈ Rn defined on compact set Ω, there exists fuzzy logic systems fˆ(x) = θ∗T ξ(x) + δ(x), which can approximate the function f (x) with arbitrarily small reconstruction error δ(x) if enough fuzzy rules are set. θ∗ is the optimal weights of fuzzy logic systems that minimizes the approximation error δ(x), which is bounded by some unknown constant δ ∗ , i.e., |δ(x)|≤ δ ∗ . ξ(x) = [ξ1 (x), · · · , ξK (x)]T is the fuzzy basis function vector with K > 1 being the number of fuzzy rules.

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Lemma 2. [34] The smooth monotone exponentially decreasing function ρj,il (t) : R+ → R+ will facilitate the descriptions of partially constrained tracking errors: ρj,il (t) = (ρ0,j,il − ρ∞,j,il ) exp(−aj,il t) + ρ∞,j,il , where ρ0,j,il and ρ∞,j,il are the initial and ultimate values of ρj,il (t), respectively, aj,il is the damped coefficient. It is easily known that lim ρj,il (t) → ρ∞,j,il as t → ∞. The bounds of partially constrained t→∞ transient and steady-state tracking errors can be described as follows: (2) (3)

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− λj,il ρj,il (t) < zj,il (t) < ρj,il (t), if zj,il (0) ≥ 0 − ρj,il (t) < zj,il (t) < λj,il ρj,il (t), if zj,il (0) < 0

for all t ≥ 0, where 0 ≤ λj,il ≤ 1. Then, the error transformation methods are given as follows: z l , ηj,il (t) = b¯ ηj,il (t) + (1 − b)η j,i (t) with b equals to 1 if zj,il (t) ≥ 0 and 0 otherwise. η¯j,il (t) ξj,il (t) = ηj,ij,i(t) l l and η j,i (t) are defined as follows: if zj,il (0) ≥ 0, η¯j,il (t) = ρj,il (t), η j,i (t) = −λj,il ρj,il (t), and if zj,il (0) < 0, l l η¯j,il (t) = λj,il ρj,il (t), η j,i (t) = −ρj,il (t). The introduced ξj,il (t) satisfies the inequality 0 < ξj,il (t) < 1 if l and only if ρ0,j,il , ρ∞,j,il , aj,il and λj,il satisfy (2) and (3).

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3. Main Results

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3.1. Observer Design Using Fuzzy Systems Since the intermediate states in (1) are not measurable and thus unavailable for control design, an observer is needed before the design procedures. To this ends, an observer by virtue of the function approximation properties of fuzzy logic systems (FLSs) is designed in this subsection. Define Fj,ij = fj,ij (¯ xj,ij ) − fj,ij (xˆ¯j,ij ), and Fj,nj = fj,nj (x) − fj,nj (ˆ x). Based on the function approximation properties ∗T of FLSs, fj,ij (·) can be reconstructed as fj,ij (·) = θj,i φ (·) + δj,ij , then, one can design the following j j,ij FLSs-based observer:  T  xˆ˙ =ˆ xj,2 + kj,1 (yj − xˆj,1 ) + θˆj,1 φj,1 (¯ xj,1 )   j,1 T xˆ˙ j,ij =ˆ xj,ij +1 + kj,ij (yj − xˆj,1 ) + θˆj,ij φj,ij (xˆ¯j,ij ), ij = 2, · · · , nj − 1 (4)    xˆ˙ T ˆj,1 ) + θˆj,nj φj,nj (xˆ¯j,nj ), j = 1, · · · , N j,nj =uj + kj,nj (yj − x

∗ where kj,ij , ij = 1, · · · , nj are positive constants, θˆj,ij is the estimation of optimal vector θj,i with the j ∗ ˜ ˆ estimation error vector θj,ij := θj,ij − θj,ij . Define the state observation error ej,ij = xj,ij − xˆj,ij , the dynamics of observation error can obtained as follows:  T  e˙ =ej,2 − kj,1 (ej,1 ) − θ˜j,1 φj,1 (¯ xj,1 ) + δj,1 + dj,1   j,1 T e˙ j,ij =ej,ij +1 − kj,ij (ej,1 ) − θ˜j,i φ (xˆ¯j,ij ) + δj,ij + Fj,ij + dj,ij (5) j j,ij    e˙ ˜T φj,n (xˆ¯j,n ) + δj,n + Fj,n + dj,n j,n = − kj,n (ej,1 ) − θ j

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which can be rewritten as the following compact form ˜ j + F j + δj + d j e˙ j = Aj ej + Θ

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with  ej = 

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ej,1  ..  , A =    j .  ej,nj

−kj,1 −kj,2 .. .

1 0 .. .

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−kj,nj 0 · · ·   0 δj,1  ˆ Fj,1 (¯ xj,ij , x¯j,ij )   ..  , δj =  . ..  . δj,nj Fj,nj (¯ xj,nj , xˆ¯j,nj )

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T φj,1 (¯ xj,1 ) −θ˜j,1 T ˜ −θj,ij φj,1 (xˆ¯j,ij ) .. . T −θ˜j,n φ (x¯ˆj,nj ) j j,nj

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 dj,1 (t, x)    ..  , dj =  . . dj,nj (t, x)

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Invoking the Young’s inequality, one gets the inequality

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  ˜ j k2 +γj kej k2 +kδ ∗ k2 +k∆∗ k2 ˜ j + Fj + δj + ∆j ) ≤4ζj kPj k2 kej k2 +Cζ,j kΘ 2eTj Pj (Θ j j n o ζj =1/Cζ,j , γj = (nj − 1)max h2j,ij ij

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note that φj,ij (xˆ¯j,ij )T φj,ij (xˆ¯j,ij ) ≤ 1, it follows

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2 2 ∗ 2 ∗ 2 T ˜ ˙ Ve,j ≤ − λj,min (Qj ) − 4ζj kPj k −γj kej k +Cζ,j kδj k +k∆j k + Cζ,j θ˜j,i θ j j,ij

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where λj,min (Qj ) is the minimum eigenvalue Qj . Define qj := λj,min (Qj ) − 4ζj kPj k2 −γj , one
of matrix P n j T ˜ 2 ∗ 2 ∗ 2 θ . obtains V˙ e,j ≤ −qj kej k +Cζ,j kδj k +kdj k + Cζ,j ij =1 θ˜j,i j j,ij

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3.2. Novel Error-driven Nonlinear Feedback Function In order to improve the closed-loop robustness, improve the dynamic performance of controllers, and facilitate the stability analysis by Lyapunov theorem, the following continuous differentiable nonlinear gain function is proposed:  s, if |s|≤ κ N (s) = (10) [loga (1 − ln a · κ + ln a · |s|) + κ] sign(s), else where a > 1, κ > 0. This designed nonlinear gain function holds the following properties. Property 1: The function N (s) is a continuous differentiable and strictly-monotone increasing function versus its argument s, and its derivative is  1, if |s|≤ κ Nd (s) = (11) −1 (1 − ln a · κ + ln a · |s|) , if |s|> κ 5

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Figure 1: Plots of Nf (s) versus s (solid line, a = 10, κ = 0.1, dot-dash line, a = 10, κ = 0.05) and linear feedback (dot line).

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Property 2: Define Nf (s) := Nd (s)·s+N (s), then Nf (s) is a monotone increasing function versus s, and Nf (s) · s ≥ N (s) · s is true. N (s) Property 3: Let Nh (s) := fs , it is known that Nh (s) > 0 is true for any s 6= 0. Define the  Nf (s) , if s 6= 0 s nonlinear function Nh+ (s) := . As a result, if s 6= 0, Nf (s)/Nh+ (s) = s, and if s = 0, 2, if s = 0 Nf (s)/Nh+ (s) = Nf (s)/2 = 0 = s.

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Remark 1. As can be seen from Fig. 1, trajectory of Nf (s) is given with s defined on [−1, 1], the main difference between the idea of the following nonlinear feedback scheme and the one using traditional linear feedback is clearly demonstrated. The empirical control design principle “small error vs high gain, large error vs small gain” is well reflected. Meanwhile, it is actually not trivial work to simply replace the linear feedback by any nonlinear feedback, because the resulted closed-loop system by nonlinear feedback is more complicated than the one by linear feedback. Particularly, the nonlinear feedback will introduce compound functions in the closed-loop system, as a result, and it is a hard work to analyze the stability via general quadratic Lyapunov function. To overcome this problem, the nonlinear gain function Nf (s) is used in the following control design procedures, and the above-mentioned properties can facilitate the stability analysis using novel Lyapunov function. 3.3. Control Design The following transformation of coordinates will facilitate the controller design for the jth subsystem for j = 1, · · · , N :  zj,1 =yj − yj,r      xj,il − αj,il −1 , il = 2, · · · , pj  zj,il =ˆ (12) ξj,il zj,i  , il = 1, · · · , pj ξj,il = l , Sj,il =   ηj,il 1 − ξj,il    Sj,ik =ˆ xj,ik − αj,ik −1 , ik = pj + 1, · · · , nj 6

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where pj is the number of partially constrained tracking errors, αj,ij , ij = 1, · · · , nj − 1 are the virtual controllers to be designed later. Step (j, 1): The derivative of Sj,1 defined in (12) can be obtained as z˙j,1 − ξj,1 η˙ j,1 S˙ j,1 = (1 − ξj,1 )2 ηj,1 =ϑj,1 [ej,2 + ξj,2 ηj,2 + αj,1 + fj,1 (¯ xj,1 )] + ϑj,1 [dj,1 (t, x) − y˙ j,r − ξj,1 η˙ j,1 ]

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where ϑj,1 := 1/[(1 − ξj,1 )2 ηj,1 ]. According to the function approximation capacity of FLSs, fj,1 (¯ xj,1 ) ∗ ∗T xj,1 ) + δj,1 with θj,1 being an unknown optimal weight vector can be reconstructed as fj,1 (¯ xj,1 ) = θj,1 φj,1 (¯ of FLSs that minimizes the approximation error.

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Remark 2. The system (13) maintains the accordant controllability of the original system (1) due to the fact that the term |1/[(1 − ξj,1 )2 ηj,1 ] | are guaranteed bounded away from zero and infinity known from Lemma 1. Similar arguments are applicable to the following |1/[(1 − ξj,il )2 ηj,il ] | with il = 2, · · · , pj , that is, 0 < |1/[(1 − ξj,il )2 ηj,il ] |< ξ¯j,il for il = 1, · · · , pj with ξ¯j,il being some unknown constant. It is noticeable that (13) is used only for theoretical analysis, it cannot be implemented directly since it contains unknown state observation error ej,2 , unknown nonlinear function fj,1 and external disturbance dj,1 .

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Select the virtual control αj,1 as follows:   Sj,1 T αj,1 = − Kj,1 Nf (Sj,1 )ηj,1 + − ξj,2 ηj,2 + y˙ j,r + ξj,1 η˙ j,1 − θˆj,1 φj,1 (¯ xj,1 ) mj,1 ϑj,1

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∗ , which where Kj,1 and mj,1 are positive design parameters, θˆj,1 is the estimation of optimal vector θj,1 is updated by: h i ˙ θˆj,1 = Γj,1 φj,1 (¯ xj,1 )Nf (Sj,1 )ϑj,1 − σj,1 θˆj,1 (15)

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where Γj,1 > 0 is a diagonal matrix composed of adaptation rate coefficients, σj,1 > 0 is design parameter, Nf (Sj,1 ) is the nonlinear function defined in Property 2 with Sj,1 being the function input. According to the standard dynamic surface control design procedure, the following first-order is introduced to generate a new signal ϕj,1 : τj,1 ϕ˙ j,1 + ϕj,1 = αj,1 , ϕj,1 (0) = αj,1 (0) (16) with τj,1 being the constant coefficient of filter.

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Remark 3. With comparison to some recent advances of constrained control problem, for example, Ref. [50], the proposed virtual control αj,1 is quite different from the existing ones. The error variable Sj,1 is utilized in a nonlinear manner in this paper, which is the key point to achieve the “error-driven thought, that is, the feedback gain self-adjusts with varying virtual errors. Similar arguments are also applicable to the following design steps. Step (j, il ), (il = 2, · · · , pj ): The differentiation of Sj,il can be obtained as follows: h i T S˙ j,il =ϑj,il [zj,il +1 + αj,il + kj,il (yj − xˆj,1 )] + ϑj,il θˆj,i φ − α ˙ − ξ η ˙ j,i j,i −1 j,i j,i l l l l l

where ϑj,il := 1/[(1 − ξj,il )2 ηj,il ].

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Select the virtual control αj,il and update law of θˆj,il as follows:   Sj,il αj,il = − Kj,il Nf (Sj,il )ηj,il + − kj,il ej,1 − ξj,il +1 ηj,il +1 + ϕ˙ j,il −1 + ξj,il η˙ j,il mj,il ϑj,il i h ˙ xj,il )Nf (Sj,il )ϑj,il + σj,il θˆj,il θˆj,il = − Γj,il φj,il (¯

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where Kj,il and mj,il are positive design parameters, Γj,il > 0 is a diagonal matrix composed of adaptation rate coefficients, σj,il > 0 is design parameter, Nf (Sj,il ) is the nonlinear function defined in Property 2 with Sj,il being the function input. Similar to above step, the following first-order is introduced to generate a new signal ϕj,il : τj,il ϕ˙ j,il + ϕj,il = αj,il , ϕj,il (0) = αj,il (0)

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with τj,il being the constant coefficient of filter. Step (j, pj + 1): The differentiation of Sj,pj +1 can be obtained as

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T S˙ j,pj +1 =Sj,pj +2 + αj,pj +1 + kj,pj +1 ej,1 + θˆj,p φ (¯ xj,pj +1 ) − α˙ j,pj j +1 j,pj +1

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Select the virtual control αj,pj +1 and update law of θˆj,pj +1 as follows:   Sj,pj +1 αj,pj +1 = − Kj,pj +1 Nf (Sj,pj +1 ) + − kj,pj +1 ej,1 + ϕ˙ j,pj mj,pj +1   ˙ θˆj,pj +1 = − Γj,pj +1 φj,pj +1 (¯ xj,pj +1 )Nf (Sj,pj +1 ) − Γj,pj +1 σj,pj +1 θˆj,pj +1

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where Kj,pj +1 and mj,pj +1 are positive design parameters, Γj,pj +1 > 0 is a diagonal matrix composed of adaptation rate coefficients, σj,pj +1 > 0 is design parameter, Nf (Sj,pj +1 ) is the nonlinear function defined in Property 2 with Sj,pj +1 being the function input. The following first-order is introduced to generate a new signal ϕj,pj +1 : (22)

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τj,pj +1 ϕ˙ j,pj +1 + ϕj,pj +1 = αj,pj +1 , ϕj,pj +1 (0) = αj,pj +1 (0) with τj,pj +1 being the constant coefficient of filter. Step (j, ik ), (ik = pj + 2, · · · , nj − 1): The differentiation of Sj,ik is

CE

T S˙ j,ik =Sj,ik +1 + αj,ik + kj,ik ej,1 + θˆj,i φ (¯ xj,ik ) − α˙ j,ik −1 k j,ik

AC

Select the virtual control αj,ik and update law of θˆj,ik as follows:   Nf (Sj,ik −1 ) Sj,ik αj,ik = − Kj,ik Nf (Sj,ik ) + − kj,ik ej,1 + ϕ˙ j,ik −1 − mj,ik Nh+ (Sj,ik ) h i ˙ θˆj,ik = − Γj,ik φj,ik (¯ xj,ik )Nf (Sj,ik ) + σj,ik θˆj,ik

(23)

(24a) (24b)

where Kj,ik and mj,ik are positive design parameters, Γj,ik > 0 is a diagonal matrix composed of adaptation rate coefficients, σj,ik > 0 is design parameter, Nf and Nh are the nonlinear functions defined in the above Property 2 and Property 3 with corresponding arguments. The following first-order is introduced to generate a new signal ϕj,ik : τj,ik ϕ˙ j,ik + ϕj,ik = αj,ik , ϕj,ik (0) = αj,ik (0) 8

(25)

ACCEPTED MANUSCRIPT

with τj,pj +1 being the constant coefficient of filter. Step (j, nj ): The differentiation of Sj,nj can be obtained as: T φ (xˆ¯) − α˙ j,nj −1 S˙ j,nj = uj + kj,nj ej,1 + θˆj,n j j,nj

(26)

Choose the control law uj and update law of θˆj,nj as follows:  Nf (Sj,nj −1 ) Sj,nj − kj,nj ej,1 + ϕ˙ j,nj −1 − uj = − Kj,nj Nf (Sj,nj ) + mj,nj Nh+ (Sj,nj ) h i ˙ θˆj,nj = − Γj,nj φj,nj (¯ xj,nj )Nf (Sj,nj ) + σj,nj θˆj,nj

CR IP T



(27a) (27b)

AN US

where Kj,nj and mj,nj are positive design parameters, Γj,nj > 0 is a diagonal matrix composed of adaptation rate coefficients, σj,nj > 0 is design parameter, Nf and Nh are the nonlinear functions defined in the above Property 2 and Property 3 with corresponding arguments. The structure of the above-designed controller can be shown schematically in the Fig. 2, and the above design procedures can be summarized as the following theorem. Theorem 1. For a class of uncertain MIMO strict nonlinear systems described as (1), under Assumptions 1-3 and initial conditions satisfying nj −1 nj h i X X −1 ˜ T T 2 ˜ 2N (Sj,ij (0))Sj,ij (0) + θj,ij (0)Γj,ij θj,ij (0) + 2ej (0)Pj ej (0) + rj,i (0) ≤ 2µj j

(28)

ij =1

M

ij =1

PT

ED

with µj being any positive constant, then, the actual controller (27a), intermediate virtual controllers (14), (18a),(21a), (24a), and the adaptation laws (15), (18b),(21b), (24b), (27b) guarantee that i) all the signals in the resulted closed-loop system are kept semi-globally uniformly ultimately bounded (SGUUB), ii) the output tracking and partially constrained errors are straightly smaller than the prescribed error bounds, meanwhile, the transient and ultimate errors can be adjusted to arbitrarily small by choosing proper design parameters in an explicit manner.

AC

CE

3.4. Stability Analysis and Parameter-Selection Criterions This subsection presents the rigorous stability using Lyapunov and Invariant-Set theorems, and parameter-selection criterions are explained in details since the proposed control scheme involves multiple control parameters. Define the filter error of introduced first-order filters as rj,ij = ϕj,ij − αj,ij , ij = 1, · · · , nj − 1, then rj,i r˙j,ij = − τj,ij + Rj,ij (·), where j

Rj,1

Rj,il



 dNf (Sj,1 ) ˙ Kj,1 (S˙ j,1 ϑj,1 − Sj,1 ϑ˙ j,1 ) =Kj,1 Sj,1 ηj,1 + Nf (Sj,1 )η˙ j,1 − y¨j,r + + ξ˙j,2 ηj,2 + ξj,2 η˙ j,2 Sj,1 mj,1 ϑ2j,1 ∂φj,1 ˙T − ξ˙j,1 η˙ j,1 − ξj,1 η¨j,1 + θˆj,1 φj,1 (¯ xj,1 ) + x¯˙ j,1 , (29a) ∂ x¯j,1   dNf (Sj,il ) ˙ =Kj,il Sj,il ηj,il + Nf (Sj,il )η˙ j,il + kj,il e˙ j,1 + ξ˙j,il +1 η˙ j,il +1 + ξj,il +1 η¨j,il +1 Sj,il − ϕ¨j,i −1 − ξ˙j,i η˙ j,i − ξj,i η¨j,i , il = 2, · · · , pj , (29b) l

l

l

l

l

9

ACCEPTED MANUSCRIPT

y j  x j ,1

jth subsystem

Neural networks-based observer (4)

y j ,r

z j ,1 Coordinate transformation in (12)

S j ,1 y j ,r

Nonlinear function (NF) f ( S j ,1 )

xˆ j ,2

Virtual control (14)

 j ,1

z j ,2

First-order filter (16)

 j ,1 z j, p j

 j , p 1 e j ,1 j

AN US

Coordinate transformation & NF f

Neural adaptive law (18b)

xˆ j ,n j

CR IP T

Neural adaptive law (15)

(S j , p j )

Virtual control (18a)  j, p j

First-order filter (19)

 j, p

j

M

 j ,n 1 j

First-order filter (25) e j ,1 

Nonlinear function

j , n j 1

Control law (27a)

ED

S j ,n j

f

( S j ,n j )

"

# dNf (Sj,pj +1 ) ˙ S˙ j,pj +1 =Kj,pj +1 Sj,pj +1 + + kj,pj +1 e˙ j,1 − ϕ¨j,pj , Sj,pj +1 mj,pj +1 dNf (zj,ij −1 ) " # z˙j,ij −1 Nh+ (zj,ij ) dzj,ij −1 S˙ j,ik dNf (Sj,ik ) ˙ Sj,ik + =Kj,ik + kj,ik e˙ j,1 + Sj,ik mj,ik Nh+2 (zj,ij )

AC

Rj,ik

Figure 2: Control diagram.

CE

Rj,pj +1

PT

Neural adaptive law (27b)

− ϕ¨j,ik −1 −

Nf (zj,ij −1 )

dNh+ (zj,ij ) dzj,ij

Nh+2 (zj,ij )

z˙j,ij

, ik = pj + 2, · · · , nj − 1

(29c)

(29d)

T T ∗T Using the facts that θˆj,i φ (¯ xj,ij ) = θ˜j,i φ (¯ xj,ij ) + θj,i φ (¯ xj,ij ), ij = 2, · · · , nj − 1, the resulted j j,ij j j,ij j j,ij closed-loop system using Theorem 1 is obtained as follows   S j,1 2 T + ϑj,1 ej,2 + ϑj,1 θ˜j,1 φj,1 + ϑj,1 (δj,1 + dj,1 ), (30a) S˙ j,1 = − Kj,1 Nf (Sj,1 )χj,1 + mj,1

10

ACCEPTED MANUSCRIPT

  Sj,il rj,i −1 2 T ˙ + θ˜j,i φ ϑ + θ∗ Tj,il φj,il ϑj,il − ϑj,il l + ϑj,il Rj,ij −1 (·), Sj,il = − Kj,il Nf (Sj,il )χj,il + l j,il j,il mj,il τj,ij −1

S˙ j,ik

S˙ j,nj

AN US

αj,ij =τj,ij ϕ˙ j,ij + ϕj,ij , ϕj,ij (0) = αj,ij (0), ij = 1, · · · , nj − 1, ˜ j + Fj + δj + dj . e˙ j =Aj ej + Θ

CR IP T

S˙ j,pj +1

il = 2, · · · , pj , (30b)   Sj,pj +1 rj,pj T ∗T = − Kj,pj +1 Nf (Sj,pj +1 ) + + θ˜j,p φ + θ φ + S − + Rj,pj (·), j,p +1 j,p +1 j,p +2 +1 j,p +1 j j j j j mj,pj +1 τj,pj (30c)   Sj,ik T + Sj,ik +1 + θ˜j,i φ (¯ xj,ik ) + θ∗ Tj,ik φj,ik (¯ xj,ik ) = − Kj,ik Nf (Sj,ik ) + k j,ik mj,ik rj,i −1 Nf (Sj,ik −1 ) − k + Rj,ik −1 (·) − , ik = pj + 2, · · · , nj − 1, (30d) τj,ik −1 Nh+ (Sj,ik )   Sj,nj T = − Kj,nj Nf (Sj,nj ) + + θ˜j,n φ (¯ xj,nj ) + θ∗ Tj,nj φj,nj (¯ xj,nj ) j j,nj mj,nj rj,nj −1 Nf (Sj,nj −1 ) − + Rj,nj −1 (·) − , (30e) τj,nj −1 Nh+ (Sj,nj ) (30f) (30g)

 2 2 2 2 ≤ Lj,0 is com+ y¨j,r From Assumption 1, one knows that the set Πy,j := (yj,r , y˙ j,r , y¨j,r ) : yj,r + y˙ j,r P ij N +2i +1 ( ) pact on R3 , meanwhile, Πv,j := {Πv,1 + Πv,2 + Πv,3 } is compact on R kj =1 khj ,ij j where Nkj ,ij is the i P P n n −1 j j T 2 Γ−1 θ˜ . ≤ 2µj . Πv,3 := ij =1 2N (Sj,ij )Sj,ij + θ˜j,i dimension of θ˜kj ,ij , Πv,1 := 2eTj Pj ej , Πv,2 := ij =1 rj,i j j,ij j,ij j Pi

CE

PT

ED

M

j (N +2i +4) As a result, Πy,j × Πv,j is compact on R kj =1 kj ,ij j , consequently, there exists positive constant + + Rj,i satisfying |Rj,ij |≤ Rj,i on Πy,j × Πv,j with ij = 1, · · · , nj − 1. j j PN T The Lyapunov function can be chosen as V = j=1 Vj with Vj described as Vj = ej Pj ej + h i P P nj −1 2 nj −1 ˜ 1 1 ˜T ij =1 N (Sj,ij )Sj,ij + 2 θj,ij Γj,ij θj,ij ij =1 rj,ij + 2 . Taking the derivative of Vj along (30) yields   nj X T ˜  V˙ j ≤ − qj kej k2 +Cζ,j kδj∗ k2 +kd∗j k2 + θ˜j,i θ + Nf (Sj,1 )ϑj,1 (ej,2 + δj,1 + dj,1 ) j j,ij

ij =1

 nj  X X   X  Kj,ij Nf (Sj,ij )Sj,ij 2 2 2 T ˆ ˜ − Kj,il Nf (Sj,il )χj,il − Kj,ik Nf (Sj,ik ) − + σj,ij θj,ij θj,ij mj,ij ij =1 ik =pj +1 il =1  pj nj pj  X X     X Nf (Sj,il )ϑj,il rj,il −1 ∗T ∗T Nf (Sj,il )θj,il φj,il ϑj,il + Nf (Sj,ik )θj,ik φj,ik − + τj,il −1 i =2 i =p +1 i =2 nj

AC

pj

l

j

k

l

!  nX j −1 2 X X  Nf (Sj,i )rj,i −1 r j,i k k + [Nf (Sj,il )ϑj,il Rj,il −1 ] − − Nf (Sj,ik )Rj,ik −1 + − j + |rj,ij Rj,ij | τ τj,ij j,i −1 k i =2 i =p +1 i =1 pj

l

nj

k

j

j

(31)

2 Remark 4. The term N (Sj,ij )Sj,ij in the Lyapunov function is quite different from the general Sj,i in j existing literatures, which needs to be guaranteed as positive definite before further analysis. From the

11

ACCEPTED MANUSCRIPT

2 definition of N (·), it is easy to know the following facts: i), if |Sj,ij |≤ κj,ij , N (Sj,ij )Sj,ij = Sj,i ≥ 0, j 
 and ii), if |Sj,ij |≥ κj,ij , N (Sj,ij )Sj,ij = loga 1 − ln a · κj,ij + ln a · |Sj,ij | + κ |Sj,ij |, since a >
1 and − ln a · κj,ij + ln a · |Sj,ij | is easily known as positive definitive, loga 1 − ln a · κj,ij + ln a · |Sj,ij | + κ is positive consequently. The positive definite property of N (Sj,ij )Sj,ij is thus guaranteed.

By using Assumption 2 and completing the squares, one obtains the following inequalities: σj,ij σj,ij ∗ 2 ˜T −1 ˜ kθj,ij k −1 θj,ij Γj,ij θj,ij + 2 2λj,ij ,min (Γj,ij ) Nf2 (Sj,1 )ξ¯j,1 e2j,2 ξ¯j,1 Nf (Sj,1 )ϑj,1 ej,2 ≤ + 2 2 2ηj,1 2ηj,1 Nf2 (Sj,1 )ξ¯j,1 2j,1 ξ¯j,1 Nf (Sj,1 )ϑj,1 (δj,1 + dj,1 ) ≤ + 2 2 2ηj,1 2ηj,1 ∗ Nf2 (Sj,il )ξ¯j,il kθj,i k2 ξ¯j,il ∗T l + Nf (Sj,il )θj,i φ ϑ ≤ 2 2 l j,il j,il 2ηj,i 2ηj,i l l

CR IP T

T ˆ θ ≤− −σj,ij θ˜j,i j j,ij

−

Nf (Sj,il )ϑj,il rj,il −1 τj,il −1

AN US

∗ εj,ik kθj,i k2 1 k Nf2 (Sj,ik ) + 2εj,ik 2 2 2 ¯ ¯ Nf (Sj,il )ξj,il rj,il −1 ξj,il ≤ + 2 2 2τj,il −1 ηj,il 2τj,il −1 ηj,i l

∗T φ ≤ Nf (Sj,ik )θj,i k j,ik

2 rj,i Nf (Sj,ik )rj,ik −1 Nf2 (Sj,ik ) k −1 ≤ + τj,ik −1 2τj,ik −1 2τj,ik −1 2 Nf2 (Sj,il )ξ¯j,il Rj,i ξ¯ l −1 j,il + Nf (Sj,il )ϑj,il Rj,il −1 ≤ 2 2 2ηj,i 2ηj,i l l

M

−

(32a) (32b) (32c) (32d) (32e) (32f) (32g) (32h)

ED

−1 where λj,ij ,min (Γ−1 j,ij ) is the minimum eigenvalue of matrix Γj,ij , and j,1 is the unknown bound value of composite δj,1 + dj,1 . As a result, (31) becomes

AC

CE

PT

nj
Nf2 (Sj,1 )ξ¯j,1 (e2j,2 + 2j,1 )ξ¯j,1 X   2 2 ∗ 2 ∗ 2 ˙ Vj ≤ − qj kej k +Cζ,j kδj k +kdj k + N (S ) + − K j,i j,i f j j 2 2 ηj,1 2ηj,1 ij =1 # " " #  X nj  nj pj ∗ 2¯ X X Nf2 (Sj,il )ξ¯j,il kθj,i k ξ Kj,ij Nf (Sj,ij )Sj,ij σj,ij − 2Cζ,j ˜T −1 ˜ j,i l l − − + −1 θj,ij Γj,ij θj,ij + 2 2 m 2η 2η 2λ (Γ ) j,ij j,ij ,min j,il j,il j,ij ij =1 il =2 ij =1 " # " #   p nj pj j 2 2 2 ¯ ¯j,i ¯ X X Nf2 (Sj,il )ξ¯j,il X N (S ) ξ r ξ R ξ 1 j,il j,il j,il −1 f j,il −1 j,il l + Nf2 (Sj,ik ) + + + + 2 2 2 2 2ε 2τ η 2τ η 2η 2ηj,i j,ik j,il −1 j,il j,il −1 j,il j,il l ik =pj +1 il =2 il =2 !  2  nj nj nj −1 2 2 X X X rj,i Nf (Sj,ik ) rj,i k −1 − j + |rj,ij Rj,ij | + + + [Nf (Sj,ik )Rj,ik −1 ] + 2τj,ik −1 2τj,ik −1 τj,ij ik =pj +1 ik =pj +1 ij =1   nj nj X 1 X ∗ 2 ∗ + σj,i kθ k + εj,il kθj,i k2  k 2 i =1 j j,ij i =p +1 j

k

j

(33)

12

ACCEPTED MANUSCRIPT

It is known that

e2j,2 2 2ηj,1

≤

kej k2 . 2λ2j,1 ρ2∞,j,1

Choosing Kj,1 and mj,1 accord with the following criterions:

kj,0 ≤ min

ij =1,···,nj

Kj,1 ≥

(

σj,ij − 2Cζ,j 2λj,ij ,max (Γ−1 j,ij ) ξ¯j,1

)

>0

(34a) (34b)

λ2j,1 ρ2∞,j,1

Kj,1 ≥ kj,0 mj,1

CR IP T

(34c)

(33) becomes  V˙ j ≤ − qj −

j

j

j

ED

k

M

AN US


2j,1 ξ¯j,1 ξ¯j,1 2 ∗ 2 ∗ 2 kej k +Cζ,j kδj k +kdj k + 2 2 − kj,0 Nf (Sj,1 )Sj,1 2λ2j,1 ρ2∞,j,1 2λj,1 ρ∞,j,1  nj  nj nj h i X X  X  Kj,ij Nf (Sj,ij )Sj,ij −1 ˜ 2 T θ − − kj,0 Γ Kj,ij Nf (Sj,ij ) − θ˜j,i j,i j j j,ij mj,ij ij =2 ij =2 ij =1 " # " # pj pj 2¯ 2 2 ∗ ¯j,i ¯j,i X X Nf2 (Sj,il )ξ¯j,il Nf2 (Sj,il )ξ¯j,il ξ ξ k ξ rj,i R kθ j,i j,il −1 j,il l l l l −1 + + + + + 2 2 2 2 2 η 2τ η 2τ η 2η 2η j,i −1 j,i −1 j,il j,il j,il j,il j,il l l il =2 il =2     nj nj X X Nf2 (Sj,ik ) 1 + + Nf (Sj,ik )Rj,ik −1 Nf2 (Sj,ik ) + 2εj,ik 2τj,ik −1 ik =pj +1 ik =pj +1   !  2 nj nj nj nj −1 2 ∗ 2 X X X X rj,i rj,ik −1 εj,ik kθj,ik k 1 ∗ ∗ k2 + kθj,i k2  σj,ij kθj,i + + + − j + |rj,ij Rj,ij | +  j l 2τ 2 τ 2 j,ik −1 j,ij i =2 i =1 i =p +1 i =1

(35)

For any positive constant ωj , select the filter time constant τj,ij satisfying +2 Rj,i l −1

+2 Rj,i k −1

l

1 τj,il −1

−

ξ¯j,il 2τj,il −1 λ2j,i ρ2∞,j,i l

k

l

≥

PT

+ kj,0 with il = 2, · · · , pj , and τj,i1 ≥ ωj + j,0 with ik = pj + 1, · · · , nj , the following inequality 2 k is always true: ! !  2  nX pj nj j −1 2 2 X X rj,i rj,ik −1 rj,i ξ¯ j l −1 j,il + + − + |rj,ij Rj,ij | 2 2τj,il −1 ηj,i 2τj,ik −1 τj,ij l il =2 ik =pj +1 ij =1 ! !  2  nX pj nj j −1 2 2 ¯j,i X X r rj,i ξ r j,i j,ik −1 l l −1 ≤ + + − j + |rj,ij Rj,ij | 2 2 2τ λ ρ 2τ τ j,il −1 j,il ∞,j,il j,ik −1 j,ij il =2 ik =pj +1 ij =1 ! !  2  nX pj nj j −1 2 2 2 2 ¯j,i X X r r R r rj,i ξ ω j,i j,i j,i j j,i −1 −1 l j j l k ≤ + + − + j + 2 2 2τ λ ρ 2τ 2τ 2ω 2 j,i −1 j,i −1 j,i j j j,il ∞,j,il l k i =2 i =p +1 i =1

AC

CE

ωj

l

≤ nj

X

ik =pj +1

nj −1 h

X

ij =1

k

2 −kj,0 rj,i j

ωj i + 2

[Nf (Sj,ik )Rj,ik −1 ] ≤

nj X

ik =pj +1

j

j

|Nf (Sj,ik )Rj,ik −1 | ≤ 13

nj X

ik =pj +1

"

2 Nf2 (Sj,ik )Rj,i j −1

2ωj

ωj + 2

#

ACCEPTED MANUSCRIPT

Remark 5. The time-constant selection criterions of first-order filters +2 Rj,i k −1

ξ¯

1

− 2τj,i −1 λj,i2 l ρ2 τj,i −1 l

l

j,il ∞,j,il

≥

+2 Rj,i −1 l

ωj

+

k

kj,0 with il = 2, · · · , pj and τj,i1 ≥ ωj + j,0 with ik = pj + 1, · · · , nj actually require sufficiently small 2 k time constants for the filters, which accords with the analysis results in [35]. Choosing Kj,ij and mj,ij accord with the following criterions: Kj,ij ≥kj,0 mj,ij

Kj,ik

ξ¯j,il ξ¯j,il + , il = 2, · · · , pj 2λ2j,il ρ2∞,j,il 2τj,il −1 λ2j,il ρ2∞,j,il

CR IP T

Kj,il ≥

(36a)

+2 Rj,i 1 1 k −1 + + , ik = pj + 1, · · · , nj ≥ 2εj,ik 2τj,ik −1 2ωj

After some manipulations, (35) becomes 1 2 2λj,1 ρ2∞,j,1



2

kej k −kj,0

nj −1

X

2 rj,i j

(36c)

nj h i X −1 ˜ T θ + Dj Nf (Sj,ij )Sj,ij + θ˜j,i Γ − kj,0 j j,ij j,ij

AN US

 ˙ Vj ≤ − qj −

(36b)

ij =1

ij =1



Pp +2 2 ∗ 2¯ 2 ¯j,i /2λ2 ρ2 + ρ k ξ /2λ + kθ ξ where Dj := Cζ,j kδj∗ k2 +kd∗j k2 +2j,1 ξ¯j,1 /2λ2j,1 ρ2∞,j,1 + ilj=2 Rj,i j,i ∞,j,i j,i j,i ∞,j,1 j,1 −1 l l l l l l i hP P P nj n nj 1 2 ∗ 2 ∗ ∗ + (2nj − pj − 1)ωj /2 + ikj=pj +1 εj,ik kθj,i k2 . il =2 kθj,il k ij =1 σj,ij kθj,ij k + 2 k

M

Remark 6. From (34b) and (36b), it is known that sufficiently large control gains Kj,il for il = 1, · · · , pj are prerequisite to guarantee partial prescribed performance under small ultimate permissible regions. Invoking the Property 2, one knows Nf (Sj,ij )Sj,ij ≥ N (Sj,ij )Sj,ij is always true, then  ˙ Vj ≤ − qj −

j

j

CE

PT

ED

 nj h nj −1 i X X ξ¯j,il −1 ˜ 2 2 T ˜ N (Sj,ij )Sj,ij + θj,ij Γj,ij θj,ij + Dj kej k −kj,0 rj,ij − kj,0 2λ2j,1 ρ2∞,j,1 ij =1 ij =1   ξ¯ l nj −1 nj h i qj − 2λ2 j,i X X 2 j,1 ρ∞,j,1 −1 ˜ T T 2  ej Pj ej − kj,0 θ ≤− N (Sj,ij )Sj,ij + θ˜j,i Γ rj,ij + Dj − kj,0 j j,ij j,ij λj,min (Pj ) i =1 i =1

≤ − Cj Vj + Dj

AC

where λj,min (Pj ) is the minimum eigenvalue of matrix Pj , Cj := min

 ¯ l  qj − 2λ2 ξj,i ρ2 

j,1 ∞,j,1

λj,min (Pj )

 

(37)

, kj,0 . Consequently, 

P V˙ ≤ −C V + D is true with C := min {Cj } and D := N j=1 Dj . j=1,···,N P ˙ 1) Choose proper C such that C > UD with U := N j=1 µj , one knows V ≤ 0 on V = U , that is to say, V ≤ U is an invariant set, this implies that if V (0) ≤ U , V (t) ≤ U is true for all t ≥ 0. From Lemma 1, one knows that the function approximation property of FLSs are true on some properly compact set, the main results are thus semi-globally stable. 2) From the definition of V and inequality V˙ ≤ −C V + D, one gets   D D 0 ≤ V (t) ≤ V (0) − exp(−C t) + (38) C C 14

ACCEPTED MANUSCRIPT

q then, the bound values of ej,ij , θ˜j,ij and rj,ij remain the compact sets: |ej,ij |≤ V (0) − r  q  exp(−C t)   kθ˜j,ij k≤ 2 V (0) − D + C kΓ2D−1 k and |rj,ij |≤ 2 V (0) − D exp(−C t) + 2D . C C C kΓ−1 k j,ij

D C

 exp(−C t) kPj k

+

D , C kPj k

j,ij

j

CR IP T

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Remark 7. From these analysis, one knows that the transient and ultimate performance bounds are guaranteed by several parameters. Similar to the constant-gain feedback controllers, theoretically speaking, increasing the C guarantees better performance, and the constants σj,· and filter coefficient τj,· should be chosen as small as possible. While too-small sets of σj,· may invalidate the σ-modification, which cannot circumvent the adaptation values from drifting to very large, and the filter coefficient τj,· should chosen possibly-small with hardware sampling limitation. Other parameters should satisfy the conditions in (34) and (36), which are the sufficient conditions of the proposed controller.

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4. Simulation and Comparative Results

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This section presents simulation and comparative results to demonstrate the effectiveness and advantages of the proposed method versus existing methods. In order to execute unprejudiced comparisons, fuzzy adaptive observed-based DSC (Labelled as “NAO-DSC” for short in the results) and traditional output feedback DSC control without constrained errors (Label as “T-DSC” for short in the results) can be obtained trivially using the similar design procedures, which are given in details in the appendix section. The system to be controlled is composed of two second-order subsystems with f1,1 = x31,1 sin2 (x1,1 ), f1,2 = x31,1 sin(x1,1 x1,2 x2,1 x2,2 ) tanh(x1,1 x2,1 ), d1,1 = 0.01 sin(t), d1,2 = 0.01 sin(x1,1 x2,2 t), f2,1 = x32,1 sin2 (x2,1 ), f2,2 = x32,1 sin(x1,1 x1,2 x2,1 x2,2 ), d2,1 = 0.01 sin(x2,1 t), d2,2 = 0.01 sin(x1,1 x1,2 x2,2 t). The initial values are set as ϕ1,1 = 7.0815, ϕ2,1 = 8.3516, and others are zeros. The fuzzy membership functions and fuzzy basis functions hold the same structure as [51]. The fuzzy membership functions are all Gaussian functions with widthes all being 2, and the centers are {−2, −1, 0, 1, 2}. The target is to guarantee that the partial states x1,1 and x2,1 tracks predefined trajectories 0.5 sin(2πt) with specified performance. The parameters of performance functions are chosen as ρ0,1,1 = 1, ρ0,2,1 = 0.8, λ1,1 = λ2,1 = 0.8. The parameters in the observers are set as kj,1 = 2 and kj,2 = 20 with j = 1, 2. The control parameters are selected as Γj,ij = 0.001diag(I), σj,ij = 0.005 with j = 1, 2 and ij = 1, 2. The parameters in nonlinear functions Nj,ij (·) are set as κj,ij = 0.01, aj,ij = 9 × 1030 . The following three sets of remaining control parameters and initial values with the same abovementioned conditions are chosen to show the effectiveness. Case 1 : a = 5, ρ∞,j,1 = 0.02, Kj,1 = 15, 15

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Kj,2 = 18, xj,1 (0) = 0.1 with j = 1, 2; Case 2 : a = 5, ρ∞,j,1 = 0.02, Kj,1 = 25, Kj,2 = 28, xj,1 (0) = 0.15 with j = 1, 2; Case 3 : a = 5, ρ∞,j,1 = 0.002, Kj,1 = 25, Kj,2 = 28, xj,1 (0) = 0.15 with j = 1, 2. Note that these initial tracking errors are chosen to satisfy z1,1 (0) < ρ0,1,1 and z2,1 (0) < ρ0,2,1 , and the initial states are exactly the same among three comparative algorithms. Case 1). In this case, the decay coefficient of performance function is 5, and the ultimate target performance is set as non-restrictive 0.02. The comparative results of this case are shown in Fig. 3 (a) and (b), it is observed that the Theorem 1 and NAO-DSC guarantee satisfactory tracking performance, while the T-DSC ruins the closed-loop stability. Case 2). In this case, the initial output tracking errors and the control gains are enlarged purposely. The results of this case are shown in Fig. 3 (c) and (d), it can be observed that the Theorem 1 performance better than the NAO-DSC, while T-DSC cannot guarantee the closed-loop stability. Case 3). In this case, the ultimate target performance is quite stringent with allowed maximum tracking error as 0.002, it is observed in Fig. 3 (e) and (f) that the Theorem 1 performs better than the NAO-DSC, while the T-DSC cannot guarantee pre-specified performance even with high-gain control. The state variables x1,2 and x2,2 and their estimations of Case 1) are given in Fig. 4, it is observed that the observers guarantee satisfactory state estimations. The implemented control signals of Case 1) are shown in Fig. 5. All the remaining closed-loop signals are guaranteed bounded, which are not given due to the page limit. Apparently, the proposed controller achieves better prescribed steady error bounds with existing constant-feedback controllers by the nonlinear feedback, in this sense, the advantage and effectiveness of the proposed method are well demonstrated by these comparative results.

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Remark 8. The recent works in [34] addresses the homologous issue with the requirements of all states are available for measurement, which was extended to output measurement case in [50] similar to the algorithm given in the following Appendix A. While, this paper further extends these results to the nonlinear virtual and output tracking errors feedback in order to improve the closed-loop robustness and the dynamic performance of controllers. 5. Conclusions

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In this paper, a new fuzzy adaptive observer-based DSC method using error-driven nonlinear output and virtual tracking errors feedback has been proposed for a class of uncertain multiple-input-multipleoutput nonlinear systems. By virtue of FLSs approximating the uncertain functions, an observer using FLSs has been developed to online estimate the unmeasurable states using available output information only. Nonlinear feedback technique is implemented in the intermediate and final controllers based on a continuous differentiable feedback nonlinear function, which gives the controller a self-regulation ability against pre-defined periodical tracking errors appeared in the recursive backstepping at different amplitude ranges. At the same time, the nonlinear feedback function enables the stability analysis using a novel Lyapunov function. Besides circumventing the complexity problem in backstepping design, the proposed controller ensures better dynamic performance, and guarantees the bound properties of all the closed-loop signals. Furthermore, the output tracking errors can be tuned to sufficiently small sets around zero and partial constrained errors can be guaranteed in the prescribed performance regions, which is characterized by design parameters in an explicit manner. Simulation and comparative results are given to demonstrate the effectiveness of the proposed method. Future work is underway to enlarge the determination regions of filters time constants, since the results in this paper and previous works [17, 35] all require sufficiently small filters time constant, which may limits the practical applications caused by the constraints on hardware sampling frequency in control systems. Meanwhile, it is a remaining and challenging problem to study the prescribed performance 16

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control without requirements on initial states and input saturations issues, since the performance bounds are defined related as the signs and values of initial tracking errors and assumption that unlimited control input signal is available, which are certain conservative conditions. Prescribed performance control without initial requirements and input saturation is particularly expected for convenient applications in controlling various plants. Appendix A. Existing Output Feedback DSC Control with Partially Constrained Tracking Errors

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By removing the nonlinear feedback function in Theorem 1, the following output feedback DSC control with partially constrained tracking errors can be easily obtained using the observer (4) and coordinates changes (12): T αj,1 = − Kj,1 Sj,1 ηj,1 − ξj,2 ηj,2 + y˙ j,r + ξj,1 η˙ j,1 − θˆj,1 φj,1 (¯ xj,1 ), h i ˙ θˆj,1 =Γj,1 φj,1 (¯ xj,1 )Sj,1 ϑj,1 − σj,1 θˆj,1 ,

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αj,ik = − Kj,ik Sj,ik − kj,ik ej,1 + ϕ˙ j,ik −1 , ik = pj + 1, · · · , nj − 1, uj = − Kj,nj Sj,nj − kj,nj ej,1 + ϕ˙ j,nj −1 , h i ˙ θˆj,ik = − Γj,ik φj,ik (¯ xj,ik )Sj,ik + σj,ik θˆj,ik , ik = pj + 1, · · · , nj ,

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where Pnj the notations are defined as the same as the ones in the previous design procedure. The term ij =1 N (Sj,ij )Sj,ij in the previous Lyapunov function can be replaced by general quadratic form one P nj 1 2 ij =1 Sj,ij to guarantee the closed-loop stability trivially, which is not discussed in details here. 2

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