Fuzzy adaptive control for SISO nonlinear uncertain systems based on backstepping and small-gain approach

Accepted Manuscript

Fuzzy adaptive control for SISO nonlinear uncertain systems based on backstepping and small-gain approach Hang Su, Tianliang Zhang, Weihai Zhang PII: DOI: Reference:

S0925-2312(17)30165-0 10.1016/j.neucom.2017.01.057 NEUCOM 17977

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Neurocomputing

Received date: Revised date: Accepted date:

18 November 2016 14 January 2017 22 January 2017

Please cite this article as: Hang Su, Tianliang Zhang, Weihai Zhang, Fuzzy adaptive control for SISO nonlinear uncertain systems based on backstepping and small-gain approach, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.01.057

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ACCEPTED MANUSCRIPT

Fuzzy adaptive control for SISO nonlinear uncertain systems based on backstepping and small-gain approach Hang Su 1 , Tianliang Zhang 2 , and Weihai Zhang 1,∗ College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China 2

College of Information and Control Engineering, China University of Petroleum (East China), Qingdao 266580, China

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Abstract: With the combination of the adaptive backstepping technique and small-gain approach, the adaptive fuzzy tracking control for a class of uncertain single-input and single-output (SISO) nonlinear systems with unmodeled dynamics, dynamic disturbances, unknown dead zone output and unmeasurable states is studied. The designed controller guarantees that all the signals of the closed-loop system are semi-globally uniformly ultimately bounded, and the output of the closed-loop system converges to a small neighborhood of the desired output. In addition, only three adaptive parameters need to be updated online, which simplifies the structure of the controller and promotes the implementation of the proposed scheme in engineering. A Nussbaum-type function is introduced to overcome the difficulty caused by the non-sensitivity of the output in the dead band. A simulation example is given to show the effectiveness of the proposed algorithm. Keywords: Backstepping; Small-gain approach; Adaptive fuzzy tracking control; Fuzzy dead zone

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1 Introduction

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Adaptive control of strict-feedback nonlinear systems has attracted a great deal of attention over the past few decades. The early study assumed that the uncertain nonlinearities can be linearly parameterized or have a prior knowledge of the bound. However, because of some inevitable elements in practice, it is impossible to guarantee that these assumptions can always be satisfied. In order to remove these restrictions, the approximation-based adaptive control has been adopted for nonlinear systems with triangular structure. In recent years, with the combination of neural networks and backstepping technique, many significant results have been obtained; see [1–8], where not only all the signals contained in the closed-loop systems could be uniformly ultimately bounded, but also the designed controllers achieved the desired control performances. Meanwhile, many researchers have paid much attention to analyzing the triangular nonlinear systems by combining fuzzy logic systems with the backstepping technique such as [9–13]. In [14, 15], Takagi-Sugeno (T-S [16, 17]) fuzzy logic systems have been utilized to approximate the unknown nonlinear functions to develop the robust adaptive fuzzy control. The adaptive controllers that have been constructed in [1–15] were adapted to the nonlinear systems without satisfying the accordant conditions, and the unknown nonlinear functions in [1–15] were not required to be linearly parameterized. However, the unmodeled dynamics or dynamic disturbances haven’t been considered in these nonlinear systems. The unmodeled dynamics and dynamic disturbances often destroy the stability and performance of the systems [18– 20]. Therefore, the controller design for the unmodeled dynamics and dynamical disturbances is necessary. To this end, adaptive output feedback controllers have been discussed in [18–20], where a class of systems that contained unstructured uncertainties and dynamic disturbances have been studied. To cancel the influence of dynamic disturbances, the references [19, 20] adopted a nonlinear damping technique, and to treat with the unmodeled dynamics, a dynamic bounded signal was utilized. Recently, the authors of [21–23] have extended some results of [19] and [20] to the systems with dynamic ∗ Corresponding

author. E-mail:w [email protected].

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uncertainties, dynamic disturbances and unstructured uncertainties. With the combination of backstepping technique and stochastic small-gain approach, a direct adaptive fuzzy output feedback control has been developed for a class of stochastic nonlinear systems with unmodeled dynamics preceded by hysteretic quantized input in [24]. In engineering practice, the presence of non-smooth nonlinear characteristics such as dead zone, backlash and hysteresis in actuators and sensors is unavoidable. It is known that dead zone can not only degrade the performance of the systems but also be the source of instability. In recent years, many scholars have focused their attention on improving the performances of the control systems with dead zones in actuators. The problem of dead zone input nonlinearity has been dealt with mainly in two ways. One way was to study adaptive fuzzy output feedback control via inverse approach for SISO, MIMO and large-scale nonlinear systems with unknown dead zones [25–27]. The other way, without constructing the dead zone inverse, was to design robust adaptive controllers for nonlinear systems with symmetric or nonsymmetric dead zone inputs; see [28, 29], where the dead zone model was a combination of a line and a disturbance-like term. Based on fuzzy neural networks, an adaptive controller has been constructed for the nonlinear system in [30], where the form of the dead zone outside the dead-band was nonlinear. It can be seen that most previous works were limited to the dead zone in the input, while the non-smooth nonlinearities in sensors were not investigated in [25–30]. Up to now, few results have been reported on adaptive control algorithm for dead zone output, hence, the work on the stability analysis and controller design for strict-feedback system with dead zone output is challenging. Based on backstepping technique and small-gain approach, this paper studies the input-to-state practical stability (ISpS) problem via adaptive fuzzy control for a class of strict-feedback systems with unmodeled dynamics, dynamic disturbances, unmeasurable states and unknown dead zone output. All the states are not available in this scheme, therefore, the variables x j ( j = 1, 2, . . . , n) could not be contained in the virtual control signal α j and the output u. In addition, a Nussbaum function is introduced to cope with the nonlinearity caused by the unknown dead zone output. The main contributions of this paper are as follows: (i) Compared with [31–34], unmodeled dynamics, unknown dead zone output, dynamic disturbances and unmeasurable states have been all taken into account to improve the adaptability of the system. The external disturbances ∆ j (z, x) ( j = 1, 2, . . . , n) of the system depend on the measured output and unmeasurable states of the system, which is different from the dynamic disturbances ∆ j (z, y) ( j = 1, 2, . . . , n) that have been constructed in [34]. So, this paper has extended the results of [31–34]. (ii) Computation explosion phenomenon is avoided by applying the estimates of unknown parameters. No matter how many states and fuzzy rules there are in the system, there are only three adaptive parameters to be updated online in our proposed algorithm, which greatly alleviates the computation complex and accordingly makes the algorithm more easily applicable in practice. The paper is organized as follows. Section 2 is concerned with the control problem of nonlinear strict-feedback systems with unmodeled dynamics, unknown dead zone output, immeasurable states and dynamic uncertainties. Combining the adaptive backstepping technique with small-gain approach, an adaptive fuzzy control algorithm is presented in Section 3. In order to demonstrate the effectiveness of the proposed method, we give a simulation example in Section 4. Finally, Section 5 is a conclusion of the whole paper.

Problem formulation and preliminaries

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In this section, we formulate our concerned problems and make some preliminaries.

2.1

System description and control problem

Consider a class of uncertain SISO nonlinear systems with unknown dead zone output and dynamic uncertainties in the following form:

z˙ = q(z, x) x˙ j = x j+1 + f j ( x¯ j ) + ∆ j (z, x), j = 1, 2, . . . , n − 1, x˙n = u + fn ( x¯n ) + ∆n (z, x), y = H(x1 ),

(1)

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where x¯ j = [x1 , x2 , . . . , x j ]T ∈ R j is a part of the system state and x = x¯n = [x1 , x2 , . . . , xn ]T ∈ Rn . f j ( x¯ j ) ( j = 1, 2, . . . , n) are the unknown smooth functions. z ∈ Rn0 is the unmeasured portion of the system state, and the z-dynamic shown in (1) represents the unmodeled dynamics. ∆ j ( j = 1, 2, . . . , n) are the dynamic disturbances. It is supposed that ∆ j and q are uncertain Lipschitz continuous functions. u and y are the control input and the measurable output of the system, respectively. In this paper, only the output y can be measured. According to [31], y is the unknown dead zone output which can be defined as follows:   hr (x1 ), x1 ≥ br ,      (2) y = H(x1 ) =  0, bl < x1 < br ,      h (x ), x ≤ b . l 1 1 l

where the parameters bl < 0 and br > 0 are the breakpoints of the output nonlinearity. hr (·) and hl (·) represent the unknown smooth nonlinear functions. The control objective is to design a controller which makes not only the tracking error as small as possible but also the closed-loop trajectory bounded.

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Definition 1 A class κ-function α is continuous, strictly increasing from R+ into R+ with α(0) = 0. A class κ∞ -function is of class κ and satisfies lim α(r) = ∞. A function β: R+ → R+ is said to belong to class κ` if for each fixed s ≥ 0, β(·, s) r→∞ is of class κ and, for each fixed r, β(r, ·) is a decreasing function and satisfies lim β(r, s) = 0. s→∞

Definition 2 A control system x˙ = f (x, u) is said to be input-to-state practically stable (ISpS) if there exist a class κ`-function β, a class κ-function α and a constant % ≥ 0 such that, for each measurable and bounded control u and the initial condition x(0), the solution x(t) satisfies the following inequality

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kx(t)k ≤ β(kx(0)k, t) + α(kut k) + %, ∀t ≥ 0,

when % = 0, the ISpS property turns into the input-to-state stability (ISS) property given in [19].

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Definition 3 A C 1 function V¯ is defined as an ISpS-Lyapunov function for the system x˙ = f (x, u), if there exist class κ∞ -functions α, α, α, class κ-function ψ and a nonnegative constant % such that

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¯ α(kxk) ≤ V(x) ≤ α(kxk), ∀x ∈ Rn , ∂V¯ ¯ f (x, u) ≤ −α(V(x)) + ψ(kuk) + %, ∂x

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when % = 0, V¯ is said to be an ISS-Lyapunov function given in [19].

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Definition 4 A C 1 function V¯ is defined as an exp-ISpS Lyapunov function for the system x˙ = f (x, u), if there exist ¯ and nonnegative constants ρ, % such that class κ∞ -functions α, α, ψ, ¯ α(kxk) ≤ V(x) ≤ α(kxk), ∀x ∈ Rn , ∂V¯ ¯ f (x, u) ≤ −ρV(x) + ψ(|u|) + %, ∂x

when % = 0, V¯ is said to be an exp-ISS Lyapunov function given in [19]. Assumption 1

[34, 35]. For 1 < j < n, there exist unknown positive constants q j1 and q j2 such that |∆ j | ≤ q j1 γ j1 (| x¯ j |) + q j2 γ j2 (kzk),

(3)

where γ j1 and γ j2 are known nonnegative smooth functions, and γ j2 (0) = 0. Assumption 2 [20]. For the system z˙ = q(z, x) in nonlinear system (1), there exist an exp-ISpS Lyapunov function

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¯ ¯ and a known nonnegative constant %0 such that V(z), class κ∞ -functions α, α, α and ψ, ¯ α(kzk) ≤ V(z) ≤ α(kzk), ¯ ∂V ¯ q(z, x) ≤ −α(kzk) + ψ(|y|) + %0 , ∂z

(4) (5)

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where ψ¯ is a known function, i.e., the unmodeled dynamics in system (1) are exponentially input-to-state practically stable (exp-ISpS). Remark 1 Compared with [31–34], the proposed nonlinear system (1) embraces more general expression, which in the meanwhile takes the unmodeled dynamics z, nonlinear uncertainties f j ( x¯ j ) ( j = 1, 2, . . . , n), the dynamical disturbances ∆ j ( j = 1, 2, . . . , n) and the unknown dead zone H into account. Therefore, the nonlinear system (1) has more general adaptability than the systems in [31–34]. Assumption 3 The reference signal yr as well as its time derivatives up to nth order is continuous and bounded. Assumption 4 A strictly increasing smooth function ϑ1 (·): R+ → R+ with ϑ1 (0) = 0 can be found to satisfy the following inequality: | f1 (x1 )| ≤ ϑ1 (|x1 |).

(6)

where h˙ l (x1 ) = 2.2

dhl (π) dπ |π=x1

min{|h˙ l (x1 )|, |h˙ r (x1 )|} ≥ h, ∀x1 ∈ (−∞, b]

and h˙ r (x1 ) =

dhr (π) dπ |π=x1 .

ISpS and small-gain theorem

(8)

[36]. Consider the following interconnected systems

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Lemma 1

[ [b, +∞),

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(2)

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Assumption 5 [30]. The functions hl (x1 ) and hr (x1 ) in (2) are smooth, and we can find the unknown positive constants h, h and b > max{|br |, |bl |} such that (1) [ max{|h˙ l (x1 )|, |h˙ r (x1 )|} ≤ h, ∀x1 ∈ (−∞, bl ] [br , +∞); (7)

x˙1 = f1 (x1 , x2 , u1 ), x˙2 = f2 (x1 , x2 , u2 ),

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where x j ∈ Rn j , u j ∈ Rm j , and f j : Rn1 × Rn2 × Rm j → Rn j ( j = 1, 2) are locally Lipschitz. For j=1, 2, it is assumed that an ISpS-Lyapunov function V¯ j can be designed for the x j -subsystem to satisfy the following conditions. (1) There exist κ∞ -functions o j1 and o j2 such that o j1 (kx j k) ≤ V¯ j (x j ) ≤ o j2 (kx j k),

∀x j ∈ Rn j .

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(2) There exist κ∞ -functions α0 , α0 and κ-functions Γ j , ν j and some constants b j ≥ 0 such that if V¯ 1 (x1 ) ≥ max{Γ1 (V¯ 2 (x2 )), ν1 (ku1 k) + b1 },

then

∇V¯ 1 (x1 ) f1 (x1 , x2 , u1 ) ≤ −α0 (V1 ).

If V¯ 2 (x2 ) ≥ max{Γ2 (V¯ 1 (x1 )), ν2 (ku2 k) + b2 }, then ∇V¯ 2 (x2 ) f2 (x1 , x2 , u2 ) ≤ −α0 (V2 ).

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Moreover, if there exists b˜ ≥ 0 such that Γ1 ◦ Γ2 (δ) < δ,

˜ ∀δ > b,

where the notation ”◦” represents the composition operator between two functions, then the given interconnected systems are ISpS. 2.3

Fuzzy logic systems

where

m k=1 Θk Πi=1 µFik (xi ) , P M Qm k=1 [ i=1 µFik (xi )]

Θk = max µGk (y), y∈R

Let

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y(x) =

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A fuzzy logic system can be constructed by the following If-Then fuzzy rules: Rl : If x1 is F1k and . . . and xm is F km , Then y is Gk , k = 1, 2, . . . , M where x¯m = [x1 , x2 , . . . , xm ]T ∈ Rm and y ∈ R are respectively the input and the output of the fuzzy system. Fuzzy sets Fik and Gk are associate with the fuzzy membership functions µFik (xi ) and µGk (y). M is the number of the rules. The output of the fuzzy system can be designed as [37]:

Θ = (Θ1 , Θ2 , . . . , Θ M )T .

Qm

then the fuzzy logic system can be rewritten as

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i=1 µF k (xi ) νl (x) = P M Qm i , k=1 [ i=1 µFik (xi )]

y(x) = ΘT ν(x),

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(9)

2.4

sup | f (x) − ΘT ν(x)| ≤ ε,

ε > 0.

(10)

x∈Ω

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where ν(x) = (ν1 (x), ν2 (x), . . . , ν M (x))T . Lemma 2 [37]. For any given real continuous function f (x) defined on a compact set Ω, there exists a fuzzy logic system (9) such that

Properties of Nussbaum function [38, 39]. A function N(ζ) can be defined as a Nussbaum-type function if it satisfies the following prop-

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Definition 5 erties

Z 1 s N(ζ)dζ = +∞, lim sup s→+∞ s 0 Z s 1 lim inf N(ζ)dζ = −∞. s→−∞ s 0

The commonly used Nussbaum functions are [36, 37]: ζ 2 cos(ζ), ζ 2 sin(ζ) and exp(ζ 2 ) cos( π2 ζ). Lemma 3 [40]. For a smooth Nussbaum-type function N(·), if there exist smooth functions V(t), ζ(t) defined on [0, t f ) with V(t) ≥ 0, ∀t ∈ [0, t f ), a suitable constant h0 , and a positive constant h1 such that Z t  ˙ −h1 (t−s) ds, V(t) ≤ h0 + g(x(s))N(ζ) + 1 ζe 0

(11)

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where g(x(s)) R t is a time-varying parameter which takes values in the unknown closed intervals I = [I− , I+ ] with 0 < I, then ˙ must be bounded on [0, t f ). V(t), ζ(t), 0 (g(x(s))N(ζ) + 1)ζds

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Fuzzy adaptive control design and stability analysis

3.1

Dynamic feedback design

Design an input-driven filter [41] as xˆ˙ j = xˆ j+1 − σ j xˆ1 , xˆ˙n = u − σn xˆ1 ,

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In this section, an input-driven filter is first constructed to estimate the unmeasured states. Then an adaptive fuzzy outputfeedback control scheme is presented through the backstepping technique. Finally, the input-to-state practical stability is analyzed.

1 ≤ j ≤ n − 1,

(12)

where xˆ j ( j = 1, 2, . . . , n) are the estimates of x j , and σ j is the design parameter such that the matrix

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  −σ1     .  A =  .. In−1      −σn · · · 0

is a strict Hurwitz matrix, i.e., given a matrix Q = QT > 0, there is a matrix P = PT > 0 satisfying the following equation AT P + PA = −Q.

(13)

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Define e = x− xˆ as the estimate error, according to (1) and (12), the estimate error system can be expressed as the following form

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¯ + ∆], e˙ = Ae + [F(x)

(14)

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¯ where F(x) = (F¯ 1 ( x¯1 ), . . . , F¯ n ( x¯n ))T with F¯ j ( x¯ j ) = f j ( x¯ j ) + σ j x1 , and ∆ = (∆1 , . . . , ∆n )T . Combining (1), (12) and (14), the composite system can be expressed as ¯ + ∆], e˙ = Ae + [F(x) ˙ 1 )( xˆ2 + e2 + f1 ( x¯1 ) + ∆1 ), y˙ = H(x x˙ˆ j = xˆ j+1 − σ j xˆ1 , xˆ˙n = u − σn xˆ1 .

1 ≤ j ≤ n − 1,

(15)

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The adaptive fuzzy output feedback backstepping algorithm consists of n steps, and each step is based on the following coordinate change: χ1 = y − yr , (16) χ j = xˆ j − α j−1 , j = 2, . . . , n,

where α j−1 is an intermediate control that will be determined at the jth step. Below, we define some notations: τ = max{kΘ j k2 , j = 1, 2, . . . , n},

q1 = max{|q j1 |, q2j1 , j = 1, 2, . . . , n}, q2 = max{|q j2 |, q2j2 , j = 1, 2, . . . , n}.

It is obvious that τ, q j ( j = 1, 2) are unknown positive constants, because kΘ j k, q j1 and q j2 are unknown. Define τˆ as the estimate of τ, τ˜ = τ − τˆ . qˆ 1 and qˆ 2 are the estimates of q1 and q2 , respectively. q˜ 1 = q1 − qˆ 1 and q˜ 2 = q2 − qˆ 2 .

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Step 1 Consider the following Lyapunov function candidate χ21 q˜ 2 q˜ 2 τ˜ 2 + + 1 + 2, 2 21 22 23

V1 = eT Pe +

(17)

where  j ( j = 1, 2, 3) are design constants. Differentiating V1 yields 1 1 1 V˙ 1 = eT (AT P + PA)e + 2eT P(F¯ + ∆) + χ1 χ˙ 1 − τ˜ τ˙ˆ − q˜ 1 q˙ˆ 1 − q˜ 2 q˙ˆ 2 . 1 2 3

(18)

F¯ j ( x¯ j ) = ΞTj0 ν0 (X¯ 0 ) + $ j0 (X¯ 0 ),

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According to Lemma 2, for any given ε j0 ≥ 0, a fuzzy logic system ΞTj0 ν0 (X¯ 0 ) can be employed to approximate the nonlinear function F¯ j ( x¯ j ) such that k$ j0 (X¯ 0 )k ≤ ε j0

where X¯ 0 = x = (x1 , x2 , . . . , xn ). Therefore ¯ F(x) = ΞT0 ν0 (X¯ 0 ) + $0 (X¯ 0 ),

k$0 (X¯ 0 )k ≤ ε0

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where

Ξ0 = (Ξ10 , . . . , Ξn0 ), $0 (X¯ 0 ) = ($10 , . . . , $n0 )T , q ε0 = ε210 + . . . + ε2n0 .

As ν0T ν0 ≤ 1, and from the definition of τ, we can see that kΞ0 k2 ≤ τ. According to Young’s inequality, we have 2eT PF¯ = 2eT P[ΞT0 ν0 (X¯ 0 ) + $0 (X¯ 0 )]

(19)

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≤ 2kek2 + kPk2 τ + kPk2 ε20 .

By utilizing Assumption 1 and Young’s inequality, we get

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2eT P∆ ≤ 2kek kPk k∆k ≤ 2

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Differentiating χ1 yields

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≤ 2kek2 + q1 kPk2

n X

j=1 n X j=1

kek kPk |∆ j |

n 2 2 X γ j2 (kzk) . γ j1 (| x¯ j |) + q2 kPk2

(20)

j=1

˙ 1 )( xˆ2 + e2 + f1 ( x¯1 ) + ∆1 ) − y˙ r χ˙ 1 = H(x

= β1 χ2 + β1 α1 + β1 e2 + β1 f1 ( x¯1 ) + β1 ∆1 − y˙ r ,

(21)

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˙ 1 ), |β1 | ≤ h. Now, we introduce a Nussbaum-type function N(ζ) to overcome the difficulty caused by the where β1 = H(x variability of β1 . Choose the first intermediate stabilizing function as follows α1 = −N(ζ)α¯ 1 ,

(22)

where α¯ 1 is the auxiliary virtual controller to be designed, and N(ζ) is a Nussbaum-type function which is defined as N(ζ) = ζ 2 cos(ζ),

ζ˙ = −ηα¯ 1 χ1 ,

(23)

where η > 0 is a design parameter. According to (22), we have χ1 β1 α1 = −χ1 β1 N(ζ)α¯ 1 − χ1 α¯ 1 + χ1 α¯ 1 = −χ1 (β1 N(ζ) + 1)α¯ 1 + χ1 α¯ 1 .

(24)

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In order to facilitate the design of the auxiliary virtual controller, we introduce the following lemma about the unknown function f1 (x1 ). Lemma 4 [31]. For the unknown smooth function f1 (x1 ) in system (1), there exist a smooth positive function ϑ(·) and a constant ϑ0 such that | f1 (x1 )| ≤ ϑ(|y|) + ϑ0 .

(25)

By using Young’s inequality, Assumptions 1, 5 and Lemma 4, we have 2

χ2 h χ1 β1 e2 ≤ 1 + kek2 , 4

2d12

+

χ21 h ϑ20 2d12

+ d12 ,

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χ1 β1 f1 ( x¯1 ) ≤ |χ1 |h(ϑ(|y|) + ϑ0 ) ≤

2 χ1 β1 ∆1 ≤ γ11 (| x¯1 |) +

(26)

2

2

χ21 h ϑ2 (|y|)

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Remark 2 The inequality (25) can be deduced by utilizing Assumption 4, see [31]. The relationship between the norm of f1 (x1 ) and the output y is revealed by the inequality (25). The difficulty caused by the unavailable state x1 is overcome, so the intermediate signal α j and the actual control u can be independent of the variable x1 in the following procedures.

2 q1 2 2 q2 2 χ h + γ12 (kzk) + χ21 h , 4 1 4

(27)

(28)

where d1 is an arbitrary positive constant. From (21)-(24) and (26)-(28), we get

Then V˙ 1 can be calculated as

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1 2 ¯ 1 ) + γ12 V˙ 1 ≤ −eT Qe + 5kek2 + χ1 (β1 χ2 + α¯ 1 + Λ (kzk) + (β1 N(ζ) + 1)ζ˙ η n X 2 q˜ 1 q˜ 2 τ˜ + (1 kPk2 − τ˙ˆ ) + (ω1 − q˙ˆ 1 ) + (k1 − q˙ˆ 2 ) + ς0 + qˆ 2 kPk2 γ j2 (kzk) , 1 2 3 j=1 where

(29)

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2 2 2   h ϑ20 h h ϑ2 (|y|) q2 2 q1 2 χ1 χ˙ 1 ≤ χ1 β1 χ2 + α¯ 1 + χ1 + h χ + h χ − y ˙ χ + χ + 1 1 r 1 1 4 4 4 2d12 2d12 1 2 2 +γ11 (| x¯1 |) + γ12 (kzk) + (β1 N(ζ) + 1)ζ˙ + kek2 + d12 . η

 2 ϑ2 ¯ 1 = Γ1 + 1 + ϑ (|y|) + 0 + Λ 4 2d12 2d12   2 n  P  2  qˆ kPk2 γ j1 (| x¯ j |) +γ11 (| x¯1 |)    1 j=1 , Γ1 =  χ1      0, ω1 = 2

k1 = 3

2 χ21 h

4

2 χ21 h

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+ 2 kPk2 + 3 kPk2

n X j=1

n X j=1

ς0 = kPk2 τˆ + kPk2 ε20 + d12 .

qˆ 1 qˆ 2  2 + h χ1 − y˙ r , 4 4 χ1 , 0, χ1 = 0,

2 γ j1 (| x¯ j |) , 2 γ j2 (kzk) ,

(30)

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According to Lemma 2, for any given constant ε1 > 0, we can get a fuzzy logic system ΞT1 ν1 (X¯ 1 ) such that ¯ 1 = ΞT1 ν1 (X¯ 1 ) + $1 (X¯ 1 ), Λ

|$1 (X¯ 1 )| ≤ ε1 ,

(31)

where X¯ 1 = (qˆ 1 , qˆ 2 , y, yr , y˙r ). From the definition of τ and Young’s inequality, for any υ1 > 0, we have υ2 χ2 ε2 ¯ 1 ≤ τ χ21 ν1T ν1 + 1 + 1 + 1 . χ1 Λ 2 2 2 2υ21

(32)

Take the intermediate control function α¯ 1 as τˆ χ1 ν1T (X¯ 1 )ν1 (X¯ 1 ), 2υ21

where l1 is a positive design constant. Substituting (32), (33) into (30) results in

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α¯ 1 = −(l1 + 0.5)χ1 −

(33)

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τ˜ 1 2 V˙1 ≤ −eT Qe + 5kek2 − l1 χ21 + γ12 (kzk) + (β1 N(ζ) + 1)ζ˙ + β1 χ1 χ2 + (ρ1 − τ˙ˆ ) η 1 n X 2 q˜ 1 q ˜ 2 + (ω1 − q˙ˆ 1 ) + (k1 − q˙ˆ 2 ) + ς1 + qˆ 2 kPk2 γ j2 (kzk) 2 3 j=1

1 τ˜ 2 ≤ −(λmin (Q) − 5)kek2 − l1 χ21 + γ12 (kzk) + (β1 N(ζ) + 1)ζ˙ + β1 χ1 χ2 + (ρ1 − τ˙ˆ ) η 1 n 2 X q˜ 1 q˜ 2 γ j2 (kzk) , + (ω1 − qˆ˙ 1 ) + (k1 − qˆ˙ 2 ) + ς1 + qˆ 2 kPk2 2 3 j=1

where

ED

M

υ21 ε21 υ2 ε2 + = kPk2 τˆ + kPk2 ε20 + d12 + 1 + 1 , 2 2 2 2 1 = 1 kPk2 + 2 χ21 ν1T ν1 . 2υ1

ς1 = ς0 + ρ1

(34)

Step 2 We choose the following Lyapunov function

AC

CE

then

where

(35)

PT

1 V2 = V1 + χ22 . 2

α˙ 1 =

V˙ 2 = V˙ 1 + χ2 χ˙ 2 = V˙ 1 + χ2 ( xˆ3 − σ2 xˆ1 − α˙ 1 )

= V˙ 1 + χ2 (χ3 + α2 − σ2 xˆ1 − α˙ 1 ),

(36)

∂α1 ∂α1 ∂α1 ∂α1 β1 ( xˆ2 + e2 + f1 ( x¯1 ) + ∆1 ) + y˙ r + y¨ r + (−ηχ1 α¯ 1 ) ∂y ∂yr ∂˙yr ∂ζ ∂α1 ˙ ∂α1 ˙ ∂α1 ˙ τˆ + qˆ 1 + qˆ 2 . + ∂ˆτ ∂qˆ 1 ∂qˆ 2

Using the similar analysis as in (27) and (28), it follows 2 2 χ22 h ϑ2 (|y|)  ∂α1 2 χ22 h ϑ20  ∂α1 2 ∂α1 −χ2 β1 f1 ( x¯1 ) ≤ + + d22 , ∂y ∂y ∂y 2d22 2d22   2 ∂α1 2 ∂α1 q1 2 q2 2 −χ2 β1 ∆1 ≤ 2χ22 h + γ11 (| x¯1 |) + γ12 (kzk), ∂y ∂y 4 4

(37) (38)

ACCEPTED MANUSCRIPT

where d2 is a positive constant. From Young’s inequality, we get 2 χ22 h  ∂α1 2 2 g22 ∂α1 −χ2 β1 xˆ2 ≤ xˆ2 + , ∂y 2 2g22 ∂y  2 2 χ h ∂α1 2 ∂α1 e2 ≤ 2 + kek2 , −χ2 β1 ∂y 4 ∂y 2

β1 χ1 χ2 ≤

h χ21 χ22 2v22

+

v22 , 2

(39) (40) (41)

CR IP T

where g2 and v2 are arbitrary positive constants. According to the above calculation,

1 ¯ 2 + ϕ2 (X¯ 2 ) − ∂α1 τ˙ˆ ) V˙ 2 ≤ −(λmin (Q) − 6)kek2 − l1 χ21 + (β1 N(ζ) + 1)ζ˙ + χ2 (χ3 + α2 + Λ η ∂ˆτ n   X 2 τ˜ q˜ 1 q˜ 2 2 + (ρ1 − τ˙ˆ ) + (ω2 − q˙ˆ 1 ) + (k2 − q˙ˆ 2 ) + qˆ 2 kPk2 γ j2 (kzk) + γ12 (kzk) 1 2 3 j=1 g2 v2 qˆ 2 2 γ12 (kzk) + ς1 + d22 + 2 + 2 , 4 2 2

(42)

AN US

+ where

PT

ED

M

2  2 ϑ20 xˆ22  ∂α1 2 h χ21 2 ϑ (|y|) 9 ∂α1 ∂α1 ¯ + 2+ + 2 χ2 − σ2 xˆ1 + χ2 − y˙ r − y¨ r Λ2 = Γ 2 + h ∂yr ∂˙yr 2d22 2d2 4 2g2 ∂y 2v22 ∂α1 ∂α1 ˙ ∂α1 ˙ (ηχ1 α¯ 1 ) − qˆ 1 − qˆ 2 − ϕ2 (X¯ 2 ), + ∂ζ ∂qˆ 1 ∂qˆ 2  2 qˆ 1 γ11 (| x¯1 |)    , χ2 , 0,  4χ 2 Γ2 =     0, χ2 = 0, ∂α1 1 ∂α1 ∂α1 1 2 T + ϕ2 (X¯ 2 ) = −l0 τˆ − χ2 2 χ2 χ ν ν1 , ∂ˆτ ∂ˆτ ∂ˆτ 2υ21 1 1 2υ2 2 n X 2 γ2 (|x1 |) χ2 h γ2 (| x¯1 |) ω2 = ω1 + 2 11 = 2 kPk2 γ j1 (| x¯ j |) + 2 1 + 2 11 , 4 4 4 j=1

CE

k2 = k1 + 3

2

n X 2 2 γ12 kzk χ2 h γ2 (kzk) = 3 kPk2 γ j2 (kzk) + 3 1 + 3 11 , 4 4 4 j=1

AC

where l0 and υ2 are arbitrary positive constants, and X¯ 2 = (y, yr , y¨ r , xˆ1 , xˆ2 , τˆ , qˆ 1 , qˆ 2 , ζ). From Lemma 2, for any given constant ε2 > 0, we can get a fuzzy logic system ΞT2 ν2 (X¯ 2 ) such that ¯ 2 = ΞT2 ν2 (X¯ 2 ) + $2 (X¯ 2 ), Λ

|$2 (X¯ 2 )| ≤ ε2 .

(43)

From the definition of τ and Young’s inequality, we have ¯2 ≤ χ2 Λ

υ22 χ22 ε22 τ 2 T ν ν + + + , χ 2 2 2 2 2υ22 2 2

(44)

where υ2 > 0 is a design parameter. Design an intermediate virtual controller α2 as α2 = −(l2 + 0.5)χ2 −

τˆ χ2 ν2T (X¯ 2 )ν2 (X¯ 2 ), 2υ22

(45)

ACCEPTED MANUSCRIPT

where l2 > 0 is a design constant. Substituting (43)-(45) into (42), then 2 X

1 τ˜ q˜ 1 l j χ2j + (β1 N(ζ) + 1)ζ˙ + χ2 χ3 + (ρ2 − τ˙ˆ ) + (ω2 − q˙ˆ 1 ) η  2 1 j=1 n X 2  q˜ 2 ∂α1 ˙  qˆ 2 2 2 + (k2 − q˙ˆ 2 ) + qˆ 2 kPk2 γ j2 (kzk) + γ12 (kzk) + ς2 + χ2 ϕ2 (X¯ 2 ) − τˆ , (kzk) + γ12 3 4 ∂ˆτ j=1

V˙ 2 ≤ −(λmin (Q) − 6)kek2 −

(46)

where 2

X 1 1 2 T 2 χ ν ν =  kPk + χ2 νT ν , 2 1 2 j j j 2υ22 2 2 2υ j j=1

ς2 = ς1 + d22 +

g22 v22 υ22 ε22 + + + . 2 2 2 2

Step m (3 ≤ m ≤ n − 1) Choose a Lyapunov function candidate

AN US

1 Vm = Vm−1 + χ2m . 2

CR IP T

ρ2 = ρ1 +

Differentiating Vm yields

(47)

V˙ m = V˙ m−1 + χm χ˙ m = V˙ m−1 + χm ( xˆm+1 − σm xˆ1 − α˙ m−1 )

M

= V˙ m−1 + χm (χm+1 + αm − σm xˆ1 − α˙ m−1 ),

where

m−1

m

X ∂αm−1 X ∂αm−1 ∂αm−1 y( j) β1 ( xˆ2 + e2 + f1 ( x¯1 ) + ∆1 ) + ( xˆ j+1 − σ j xˆ1 ) + = ( j−1) r ∂y ∂ x ˆ j j=1 j=1 ∂yr ∂αm−1 ∂αm−1 ˙ ∂αm−1 ˙ ∂αm−1 ˙ τˆ + (−ηχ1 α¯ 1 ) + qˆ 1 + qˆ 2 . ∂ζ ∂ˆτ ∂qˆ 1 ∂qˆ 2

PT

+

ED

α˙ m−1

(48)

Similarly, it is easy to show

AC

CE

2 2 χ2m h ϑ20  ∂αm−1 2 χ2m h ϑ2 (|y|)  ∂αm−1 2 ∂αm−1 2 −χm β1 f1 ( x¯1 ) ≤ + dm + , ∂y ∂y ∂y 2dm2 2dm2   2 ∂αm−1 2 ∂αm−1 q1 2 q2 2 −χm β1 ∆1 ≤ γ11 (| x¯1 |) + γ12 (kzk) + 2χ2m h , ∂y 4 4 ∂y

(49) (50)

where dm is a positive constant. By using Young’s inequality, we get 2 χ2 h  ∂αm−1 2 2 g2m ∂αm−1 xˆ2 ≤ m 2 xˆ2 + , ∂y ∂y 2 2gm 2 χ2m h  ∂αm−1 2 ∂αm−1 −χm β1 e2 ≤ + kek2 , ∂y 4 ∂y χ2 χ2m v2 χm−1 χm ≤ m−12 + m , 2 2vm

−χm β1

where gm , vm are arbitrary positive constants.

(51) (52) (53)

ACCEPTED MANUSCRIPT

According to (34), (46), (49)-(53), the time derivative of Vm is estimated as V˙ m ≤ −(λmin (Q) − (m + 4))kek2 −

m−1 X

1 ¯m l j χ2j + (β1 N(ζ) + 1)ζ˙ + χm (χm+1 + αm + Λ η j=1

∂αm−1 ˙ τ˜ q˜ 1 q˜ 2 2 +ϕm (X¯ m ) − τˆ ) + (ρm−1 − τ˙ˆ ) + (ωm − q˙ˆ 1 ) + (km − q˙ˆ 2 ) + γ12 (kzk) ∂ˆτ 1 2 3 n 2 X g2 v2 qˆ 2 2 γ j2 (kzk) + ςm−1 + (m − 1)γ12 (kzk) + dm2 + m + m +qˆ 2 kPk2 4 2 2 j=1  ∂α j−1  χ j ϕ j (X¯ j ) − τ˙ˆ , ∂ˆτ j=2

m−1 X

CR IP T

+ where

(54)

m−1  2 2 X ϑ2 χ2 xˆ2  ∂αm−1 ¯ m = Γm + h2 ϑ (|y|) + 0 + 9 + 2 ∂αm−1 χm + m−1 χm − Λ ( xˆ j+1 − σ j xˆ1 ) 2 2 2 2 ∂y ∂ xˆ j 2dm 2dm 4 2gm 2vm j=1

 2 qˆ 1 γ11 (| x¯1 |)    , χm , 0,  4χ m Γm =     0, χm = 0,

ϕm (X¯ m ) = −l0 τˆ

m

X ∂αm−1 ∂αm−1 ˙ ∂αm−1 ˙ ∂αm−1 (ηχ1 α¯ 1 ) − y( j) − qˆ 1 − qˆ 2 , ( j−1) r ∂ζ ∂ q ˆ ∂qˆ 2 1 ∂y r j=1

AN US

−σm xˆ1 − ϕm (X¯ m ) +

m−1 m ∂αm−1 X 1 ∂α j−1 X ∂αm−1 1 2 T + − χm 2 χ j χ ν ν j, ∂ˆτ ∂ˆτ j=1 ∂ˆτ 2υ2j j j 2υm j=2 2

M

n X 2 χ2 h γ2 (| x¯1 |) γ2 (| x¯1 |) = 2 kPk2 γ j1 (| x¯ j |) + 2 1 + 2 (m − 1) 11 , ωm = ωm−1 + 2 11 4 4 4 j=1 2

ED

n X 2 γ2 kzk χ2 h γ2 (kzk) km = km−1 + 3 12 = 3 kPk2 γ j2 (kzk) + 3 1 + 3 (m − 1) 12 , 4 4 4 j=1

PT

where υm is an arbitrary positive constant, and X¯ m = ( xˆ1 , . . . , xˆm , y, τˆ , qˆ 1 , qˆ 2 , ζ, y˙ r , y¨ r , . . . , y(m) r ). According to Lemma 2, for any given constant εm > 0, there exists a fuzzy logic system ΞTm νm (X¯ m ) such that ¯ m = ΞTm νm (X¯ m ) + $m (X¯ m ), Λ

|$m (X¯ m )| ≤ εm .

(55)

AC

CE

By the definition of τ and Young’s inequality, we have ¯m ≤ χm Λ

υ2 χ2 ε2 τ 2 T χ m νm νm + m + m + m , 2 2 2 2 2υm

(56)

where υm > 0 is a design parameter. Design an intermediate virtual controller as αm = −(lm + 0.5)χm −

τˆ T ¯ (Xm )νm (X¯ m ), χ m νm 2υ2m

where lm is a positive constant. Similar to the above analysis, we have m X

τ˜ 1 l j χ2j + (β1 N(ζ) + 1)ζ˙ + χm χm+1 + (ρm − τ˙ˆ ) η  1 j=1 n 2 qˆ X q˜ 1 q˜ 2 2 2 2 + (ωm − q˙ˆ 1 ) + (km − q˙ˆ 2 ) + γ12 (kzk) + qˆ 2 kPk2 γ j2 (kzk) + (m − 1)γ12 (kzk) 2 3 4 j=1

V˙ m ≤ −(λmin (Q) − (m + 4))kek2 −

(57)

ACCEPTED MANUSCRIPT

+ςm +

 ∂α j−1  χ j ϕ j (X¯ j ) − τ˙ˆ , ∂ˆτ j=2

m X

(58)

where m

X 1 1 2 T 2 χ ν ν =  kPk + χ2 νT ν , m 1 2 j j j 2υ2m m m 2υ j j=1

ςm = ςm−1 + dm2 +

g2m v2m υ2m ε2m + + + . 2 2 2 2

CR IP T

ρm = ρm−1 +

Step n Choose the Lyapunov function candidate as 1 Vn = Vn−1 + χ2n . 2 Similar to the analysis in (47)-(53), we substitute n for m in (54) and obtain n−1 X

1 ¯ n + ϕn (X¯ n ) l j χ2j + (β1 N(ζ) + 1)ζ˙ + χn (u + Λ η j=1

AN US

V˙ n ≤ −(λmin (Q) − (n + 4))kek2 −

(59)

τ˜ q˜ 1 q˜ 2 ∂αn−1 ˙ 2 τˆ ) + (ρn−1 − τ˙ˆ ) + (ωn − q˙ˆ 1 ) + (kn − q˙ˆ 2 ) + γ12 (kzk) ∂ˆτ 1 2 3 n X 2 qˆ g2 v2 2 2 (kzk) + ςn−1 + dn2 + n + n +qˆ 2 kPk2 γ j2 (kzk) + (n − 1)γ12 4 2 2 j=1

−

 ∂α j−1  χ j ϕ j (X¯ j ) − τ˙ˆ , ∂ˆτ j=2

n−1 X

(60)

M

+ where

n X ∂αn−1

y( j) ( j−1) r ∂y r j=1

PT

−σn xˆ1 −

ED

n−1  2 2 X ϑ2 χ2 xˆ2  ∂αn−1 ¯ n = Γn + h2 ϑ (|y|) + 0 + 9 + 2 ∂αn−1 χn + n−1 χn − Λ ( xˆ j+1 − σ j xˆ1 ) ∂y ∂ xˆ j 2dn2 2dn2 4 2g2n 2v2n j=1

+

∂αn−1 ∂αn−1 ˙ ∂αn−1 ˙ (ηχ1 α¯ 1 ) − qˆ 1 − qˆ 2 − ϕn (X¯ n ), ∂ζ ∂qˆ 1 ∂qˆ 2

CE

 2 qˆ 1 γ11 (| x¯1 |)    , χn , 0,  4χ n Γn =     0, χn = 0,

∂αn−1 ϕn (X¯ n ) = −l0 τˆ − ∂ˆτ

n−1 1 ∂α j−1 X ∂αn−1 1 2 T + χn 2 χ j χ ν ν j, ∂ˆτ j=1 ∂ˆτ 2υ2j j j 2υn j=2

n X

2

AC

n X 2 γ2 (| x¯1 |) χ2 h γ2 (| x¯1 |) ωn = ωn−1 + 2 11 = 2 kPk2 γ j1 (| x¯ j |) + 2 1 + 2 (n − 1) 11 , 4 4 4 j=1 2

n X 2 γ2 kzk χ2 h γ2 (kzk) kn = kn−1 + 3 12 = 3 kPk2 γ j2 (kzk) + 3 1 + 3 (n − 1) 12 , 4 4 4 j=1

where υn is an arbitrary positive constant, and X¯ n = ( xˆ1 , . . . xˆn , y, τˆ , qˆ 1 , qˆ 2 , ζ, y˙ r , y¨ r , . . . , y(n) r ). Similar to the aforementioned steps, for any given constant εn > 0, the fuzzy logic system ΞTn νn (X¯ n ) is adopted to ¯ n such that approximate Λ ¯ n = ΞTn νn (X¯ n ) + $n (X¯ n ), Λ

|$n (X¯ n )| ≤ εn .

(61)

ACCEPTED MANUSCRIPT

Similarly, from the definition of τ and Young’s inequality, for the above υn > 0, we get ¯n ≤ χn Λ

υ2 χ2 ε2 τ 2 T χn νn νn + n + n + n . 2 2 2 2 2υn

(62)

Choose the control law as u = −(ln + 0.5)χn −

τˆ χn νnT νn , 2υ2n

(63)

where ln is a positive constant. Then n X

where n

(64)

X 1 1 2 T 2 χ ν ν =  kPk + χ2 νT ν , n 1 n n 2 j j j 2υ2n 2υ j j=1

ςn = ςn−1 + dn2 +

AN US

ρn = ρn−1 +

CR IP T

1 τ˜ q˜ 1 q˜ 2 l j χ2j + (β1 N(ζ) + 1)ζ˙ + (ρn − τ˙ˆ ) + (ωn − q˙ˆ 1 ) + (kn − q˙ˆ 2 ) η 1 2 3 j=1 n n 2 qˆ X  X ∂α j−1  2 2 2 γ j2 (kzk) + (n − 1)γ12 χ j ϕ j (X¯ j ) − +γ12 (kzk) + qˆ 2 kPk2 (kzk) + ςn + τ˙ˆ , 4 ∂ˆτ j=1 j=2

V˙ n ≤ −(λmin (Q) − (n + 4))kek2 −

g2n v2n υ2n ε2n + + + . 2 2 2 2

Through the above steps, we can employ the following adaptive laws n  χ2 X 1 j j=1

2υ2j

νTj ν j − l0 τˆ ,

q˙ˆ 1 = ωn − lq1 qˆ 1 = 2 kPk2

(65)

2 2 γ2 (| x¯1 |) χ21 h + 2 (n − 1) 11 − lq1 qˆ 1 , γ j1 (| x¯ j |) + 2 4 4 j=1

(66)

χ2 h γ2 (kzk) γ j2 (kzk) + 3 1 + 3 (n − 1) 12 − lq2 qˆ 2 . 4 4 j=1

(67)

n X

n X

ED

CE

PT

q˙ˆ 2 = γn − lq2 qˆ 2 = 3 kPk2 From [42], we have

τˆ (0) ≥ 0,

M

τ˙ˆ = 1 kPk2 +

2

2

 ∂α j−1  χ j ϕ j (X¯ j ) − τ˙ˆ ≤ 0. ∂ˆ τ j=2

n X

(68)

AC

According to the analysis in [34], because each function γ j2 is smooth and γ j2 (0) = 0, a smooth κ∞ -function φ can be taken to satisfy the following inequality qˆ 2 kPk2

n X

2 qˆ 2 2 2 γ j2 (kzk) + γ12 (kzk) + (n − 1)γ12 (kzk) ≤ φ(kzk2 ). 4 j=1

(69)

Considering (64)-(69), we have V˙ n ≤ −(λmin (Q) − (n + 4))kek2 − +

n X

1 1 l j χ2j + (β1 N(ζ) + 1)ζ˙ + l0 τ˜ τˆ η 1 j=1

1 1 lq q˜ 1 qˆ 1 + lq2 q˜ 2 qˆ 2 + φ(kzk2 ) + ςn . 2 1 3

(70)

Theorem 1 For system (1) with unknown dead zone output, unmodeled dynamics, dynamic disturbances and unmeasurable states, if we choose the controller (63) with the intermediate virtual control signals (22), (23), (45) and (57), and

ACCEPTED MANUSCRIPT

the adaptive laws (65)-(67), then all signals that are contained in the closed-loop system (1) are semi-globally uniformly ultimately bounded. Furthermore, the tracking error χ1 = y − yr satisfies the following inequality lim χ21 ≤

t→∞

2(ς + φ(kzk2 )) 2` + , C η

(71)

where C and ς are defined in (76), and ` is defined in (80). Proof It is noted that

Substituting (72)-(74) into (70), we get n X

l j χ2j −

j=1

l 0 2 l q1 2 l q2 2 1 ˙ + τ + q + q + φ(kzk2 ) + ςn + (β1 N(ζ) + 1)ζ. 21 22 1 23 2 η Define

 − (n + 4) , 2l j , l0 , lq1 , lq2 > 0, j = 1, 2, . . . , n, λmax (P) l0 2 lq1 2 lq2 2 ς = ςn + τ + q + q 21 22 1 23 2 n v2 n υ2 n ε2 n n g2 X X X X X l0 2 lq1 2 lq2 2 j j j j + + + + = kPk2 τˆ + kPk2 ε20 + d2j + τ + q + q, 2 2 2 2 21 22 1 23 2 j=2 j=1 j=1 j=1 j=2 min (Q)

(74)

(75)

M

λ

ED

C = min

(73)

l0 2 lq1 2 lq2 2 τ˜ − q˜ − q˜ 21 22 1 23 2

AN US

V˙ n ≤ (λmin (Q) − (n + 4))kek2 −

(72)

CR IP T

τ˜ 2 τ2 + , 2 2 q˜ 21 q21 q˜ 1 qˆ 1 = q˜ 1 (q1 − q˜ 1 ) ≤ − + , 2 2 q˜ 22 q22 q˜ 2 qˆ 2 = q˜ 2 (q2 − q˜ 2 ) ≤ − + . 2 2 τ˜ τˆ ≤ τ˜ (τ − τ˜ ) ≤ −

then we have

PT

1 ˙ V˙ n ≤ −CVn + ς + φ(kzk2 ) + (β1 N(ζ) + 1)ζ. η

(76)

(77)

CE

Integrating (77) over [0, t], we can get the following inequality −Ct

AC

Vn (t) ≤ Vn (0)e

According to Lemma 2, Vn (t), ζ(t),

Rt 0

Z

0

ς + φ(kzk2 ) e−Ct + (1 − e−Ct ) + C η

Z

0

t

˙ Cs ds. (β1 N(ζ) + 1)ζe

(78)

˙ are bounded on [0, t f ). Moreover, the following inequality holds (β1 N(ζ) + 1)ζds t

˙ −C(t−s) ds ≤ (β1 N(ζ) + 1)ζe

Z

0

t

˙ (β1 N(ζ) + 1)ζds.

(79)

Rt ˙ −C(t−s) ds is bounded on [0, t f ). From (77)-(79), for C > 0, all signals of the closed-loop system Therefore, 0 (β1 N(ζ) + 1)ζe (1) can be guaranteed to be semi-globally uniformly ultimately bounded. Define Z t −C(t−s) ˙ ` = max (β1 N(ζ) + 1)ζe ds . (80) t∈[0,t f ]

0

ACCEPTED MANUSCRIPT

Combining (78) with (80), we have χ21 ς + φ(kzk2 ) ` ≤ Vn (t) ≤ Vn (0)e−Ct + (1 − e−Ct ) + , 2 C η

(81)

which implies χ21 ≤ 2Vn (0)e−Ct +

2ς + 2φ(kzk2 ) 2` (1 − e−Ct ) + . C η

(82)

3.2

AN US

CR IP T

Let t → ∞, the inequality (71) is derived. Remark 3 The larger l j ( j = 1, 2, . . . , n), η,  j ( j = 1, 2, 3) and smaller υ j , d j , g j , v j can make the tracking error smaller. However, from (71), we can see that the tracking error is associate with the unknown constants τ, q1 , q2 and kPk. So we can only guarantee that the tracking error is in a small neighborhood of the origin, this is because that neither the system functions nor their bounds are known. Remark 4 In the conventional fuzzy backstepping design, for the n-order nonlinear control system, there will be nN adaptive parameters to be estimated online. Eventually, the complexity of the controller and the computation burden increase drastically as the order of the system goes up. Motivated by [13,31,34,35,43], we use the estimation of the vector norm of the unknown parameters in this paper, which can reduce the computation burden significantly. Consequently, there are only three parameters to be estimated online regardless of the number of fuzzy rules and the dimension of the system. Analysis of ISpS stability

¯ if From Lemma 1, we know that, for any 0 < b0 < b,

¯ n + ς + φ(kzk2 ) + 1 (β1 N(ζ) + 1)ζ˙ ≤ 0, b0 Vn − bV η

ED

M

then

(83)

V˙ n ≤ −b0 Vn .

(84)

φ(kzk2 ) ς¯ + , b¯ − b0 b¯ − b0

(85)

Furthermore, the inequality (83) is equivalent to

PT

Vn ≥

CE

˙ From (4), we have where ς¯ = ς + 1η (β1 N(ζ) + 1)ζ. ¯ kzk ≤ α−1 (V(z)).

(86)

AC

According to (83)-(86), we know that if Vn ≥ max

 2φ(α−1 (V(z)) 2 2 ¯ ¯ ) ) 2ς¯  φ(α−1 (V(z)) ς¯ , ≥ + , b¯ − b0 b¯ − b0 b¯ − b0 b¯ − b0

(87)

then (84) holds. On the other hand, for any 0 < b1 < 1 and δ > 0, choose κ∞ -function ψ0 that satisfies ¯ ψ(|y|) ≤ ψ0 (Vn ), ψ0 (δ) <

1 − b1 α ◦ α−1 2



s

 b¯ − b  0 αφ−1 δ . 4

(88) (89)

Substituting (89) into (4), we get ∂V¯ ¯ q(z, x) ≤ −α(kzk) + ψ(|y|) + %0 ≤ −α(kzk) + ψ0 (Vn ) + %0 . ∂z

(90)

ACCEPTED MANUSCRIPT

If we obtain α(kzk) ≥

ψ0 (Vn ) %0 + , 1 − b1 1 − b1

(91)

or the following inequality  ψ (V ) %0  0 n , V¯ ≥ α ◦ α−1 + 1 − b1 1 − b1

(92)

V¯ ≤ −b1 α(kzk) + b1 α(kzk) − α(kzk) + ψ0 (Vn ) + %0 ≤ −b1 α(kzk).

(93)

CR IP T

then, we get

According to (91)-(93), we note that if

(94)

 2ψ (δ)  2φ(α−1 (δ2 )) 0 , Γ2 (δ) = α ◦ α−1 . 1 − b1 b¯ − b0

(95)

Γ1 (δ) =

AN US

then (93) holds. Now, we choose

 2ψ (V )   2%   0 n 0 V¯ ≥ max α ◦ α−1 , α ◦ α−1 , 1 − b1 1 − b1

From (89), (94) and (95), we get

Γ1 ◦ Γ2 (δ) =

δ < δ. 2

(96)

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According to Lemma 1, we know that the closed-loop system is ISpS.

4 Simulation example

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In this section, an example is given to verify the effectiveness of the proposed scheme. Example 1 Consider the following second-order nonlinear system z˙ = −z + x12 ,

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x˙1 = x2 + x12 sin(x1 ) + 0.5z2 cos(x1 ), x˙2 = u + x12 cos(x2 ) + x12 + z2 sin(x1 x2 ), (97)

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y = H(x1 ).

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The dead zone H(x1 ) is defined as follows   1.2(x1 − 0.1)1.3 , x1 ≥ 0.1,      y = H(x1 ) =  0, − 0.06 < x1 < 0.1,      0.1(x + 0.06)1.2 , x ≤ −0.06. 1 1

The output dead zone breakpoints are br = 0.1, bl = −0.06. The unmodeled dynamics are given by q(z, x) = −z + x12 . ∆1 = 0.5z2 cos(x1 ) and ∆2 = z2 sin(x1 x2 ) are nonlinear dynamic disturbances. f1 ( x¯1 ) = x12 sin(x1 ), f2 ( x¯2 ) = x12 cos(x2 ) + x12 . Take ¯ ¯ γ11 (| x¯1 |) = γ21 (| x¯2 |) = 0, γ12 (kzk) = γ22 (kzk) = z2 , V(z) = z2 , α(kzk) = 0.5z2 , α(kzk) = 1.5z2 , α(kzk) = z2 , ψ(|y|) = 105 y3 , %0 = 0.0001. The reference signal is defined as yr = 0.1 sin(0.1t) + 0.1 sin(1.2t). Define the following six fuzzy membership functions: µF 1j = exp

 −(x + 2)2  16

, µF 2j = exp

 −(x + 1.5)2  16

,

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µF 3j = exp µF 5j = exp

 −(x + 0.5)2  16  −(x − 1.5)2  16

 −(x − 0.5)2  , 16  −(x − 2)2  = exp . 16

, µF 4j = exp , µF 6j

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Now, we employ the intermediate virtual functions (22), (23), (33) and (45), the actual control (63) and the parameter adaptive laws (65)-(67). To illustrate the effects of the design parameters υ j and l j ( j = 1, 2), we choose the design parameters under the following two cases. Case 1: l1 = 38, l2 = 40, υ1 = 5, υ2 = 8, l0 = 50, 1 = 0.02, 2 = 0.01, 3 = 0.01, h = 5, lq1 = 600, lq2 = 100, η = 30, σ1 = 144, σ2 = 24 and Q = [10, 0; 0, 10]. Case 2: l1 = 55, l2 = 57, υ1 = 3, υ2 = 5, and the other design parameters are as in Case 1. For the above two cases, the initial conditions are all selected as [x1 (0), x2 (0), z(0)] = [0.2, 0.01, −0.02], [ xˆ1 (0), xˆ2 (0)] = [−0.2, −0.1] and [ˆτ(0), qˆ 1 (0), qˆ 2 (0), ζ(0)] = [1, 0.5, 0.1, 1.2]. The simulation results are shown in Figs. 1-14.

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Fig. 1 Trajectories of y (solid line) and yr (dashed line) for Case 1.

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Fig. 2 Control u for Case 1.

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1 0.9 0.8 0.7

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Fig. 3 Adaptive parameter τˆ for Case 1.

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Fig. 4 Adaptive parameter qˆ 1 for Case 1.

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Fig. 5 Adaptive parameter qˆ 2 for Case 1.

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−1

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Fig. 6 Trajectories of x1 (dashed line) and x2 (solid line) for Case 1.

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Fig. 7 Trajectory of z for Case 1.

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Fig. 8 Trajectories of y (solid line) and yr (dashed line) for Case 2.

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Fig. 9 Control u for Case 2.

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Fig. 10 Adaptive parameter τˆ for Case 2.

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Fig. 11 Adaptive parameter qˆ 1 for Case 2.

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Fig. 12 Adaptive parameter qˆ 2 for Case 2.

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Fig. 13 Trajectories of x1 (dashed line) and x2 (solid line) for Case 2.

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Fig. 14 Trajectory of z for Case 2.

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PN In order to see the influence of the parameters υ j and l j ( j = 1, 2), we make a comparison in Table 1, where k=1 [y(k) − P N yr (k)]2 and k=1 [u(k)2 ] are defined as the indexes of the output and control law, respectively, and N is the number of sampling data. Table 1 Performance comparisons between Case 1 and Case 2. Indexes PN 2 k=1 [y(k) − yr (k)] PN 2 k=1 [u(k) ]

Case 1

Case 2

0.0014

0.0009

7.6192e+02

1.3246e+03

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From Table 1, it can be seen that the larger l j and the smaller υ j make the tracking performance better but make the control energy becomes larger. Therefore, we have to make a tradeoff between the tracking performance and the control energy in practical applications.

5 Conclusion

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This paper has been focused on the adaptive fuzzy tracking control for a class of uncertain SISO nonlinear systems with dynamic disturbances, unmodeled dynamics, unmeasurable states and unknown dead zone output. Combining the Nussbaum function with an auxiliary virtual controller, the problem of unknown virtual control coefficient caused by the unknown dead zone output has been resolved. The fuzzy logic system has been used to approximate the unknown control signals rather than the unknown nonlinear functions. Because only three online adaptive parameters need to be determined, the computation burden has been reduced greatly by our proposed algorithm. By using the backstepping technique and the small-gain approach, this presented fuzzy adaptive controller can not only make the tracking error as small as possible but also make the closed-loop system bounded. The simulation example has been given to illustrate the effectiveness of our design scheme. It is worth mentioning that we have only done preliminary research on the proposed method on SISO nonlinear systems. In the further work, the presented control algorithm will be extended to MIMO systems. Furthermore, stochastic control can be considered to investigate more applications of this presented method in engineering.

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Acknowledgements

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This work was supported by the National Natural Science Foundation of China (No. 61573227), State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources (Grant No. LAPS16011), the Research Fund for the Taishan Scholar Project of Shandong Province of China, and SDUST Research Fund (No.2015TDJH105). References

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Biography

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Hang Su received the Bachelor degree of Engineering from Shandong University of Science and Technology (Taishan College of Science and Technology), Taian, China, in 2011, the Master degree of Engineering from Shandong University of Science and Technology, Qingdao, China, in 2014. She is currently working toward the Ph.D. degree at Shandong University of Science and Technology, Qingdao, China. Her research interests include fuzzy control and adaptive control.

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Tianliang Zhang received his Bachelor of Engineering from Shandong University of Science and Technology, Qingdao, China, in 2014. He is now toward Master degree of Engineering at China University of Petroleum (East China). His research interests include adaptive control, mean-field stochastic systems and nonlinear stochastic control.

Weihai Zhang received the M.S. degree from Hangzhou University, China, and the Ph.D. degree from Zhejiang University, Hangzhou, China, in 1994 and 1998, respectively. He was a Postdoctoral Researcher from May 2001 to July 2003 and a visiting Professor from September 2010 to February 2011 and from February 2015 to January 2016 at National Tsing Hua University, Hsinchu, Taiwan. He is currently a Professor with the Shandong University of Science and Technology, Qingdao, China. His research interests include linear and nonlinear stochastic control, robust H infinity control, and stochastic stability and stabilization. Dr. Zhang is now an Associate Editor of the Asian Journal of Control. His representative paper “W. Zhang, B.S. Chen,

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SIAM J. Control and Optimization, 44(6), pp.1973–1991, 2006” has been selected as a featured fast moving front paper by Thomson Reuters Science Watch and has been selected as the most-cited paper in the research area of Mathematics by Essential Science Indicator from Thomson Reuters; see http://sciencewatch.com/dr/fmf/2010/10sepfmf/10sepfmfZhan/.