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Robust Adaptive Prescribed Performance Dynamic Surface Control for Uncertain Nonlinear Pure-Feedback Systems Fujin Jia, Xiao Yan, Xuhuan Wang, Junwei Lu, Yongmin Li PII: DOI: Reference:

S0016-0032(19)30893-2 https://doi.org/10.1016/j.jfranklin.2019.12.006 FI 4323

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

8 April 2019 18 October 2019 2 December 2019

Please cite this article as: Fujin Jia, Xiao Yan, Xuhuan Wang, Junwei Lu, Yongmin Li, Robust Adaptive Prescribed Performance Dynamic Surface Control for Uncertain Nonlinear Pure-Feedback Systems, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.12.006

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Robust Adaptive Prescribed Performance Dynamic Surface Control for Uncertain Nonlinear Pure-Feedback Systems Fujin Jia, Xiao Yan, Xuhuan Wang, Junwei Lu, and Yongmin Li

Abstract—A robust adaptive prescribed performance dynamic surface control design procedure for a class of nonlinear purefeedback systems with both unknown vector parameters and unmodeled dynamics is presented. The unmodeled dynamics lie within some bounded functions, which are assumed to be partially known. A state transformation and an auxiliary system are proposed to avoid using the cumbersome formula to handle the non-affine structure. A radical-type Lyapunov function and L function are designed to ensure the prescribed performance of the pure-feedback system. Bounded value of finding the partial derivative of the non-affine function is avoided by designing special virtual controllers in using dynamic surface control. As illustrated by examples, the proposed adaptive prescribed performance dynamic surface control scheme is shown to guarantee the semi-global uniformly ultimately bounded, and the dynamic performance and steady-state performance of the tracking error depend on the prescribed performance functions. Index Terms—Pure-feedback systems, prescribed performance, dynamic surface, adaptive control, backstepping method

I

I. I NTRODUCTION

N recent decades, adaptive control theory of nonlinear systems has been widely concerned by scholars [1]–[7]. It is worth mentioning that [8] and [9] proposed new adaptive schemes that simplify multiple unknown parameters into one to solve the problem of making the closed-loop system more complicated due to multiple parameter estimators. In order to solve the adaptive control problem of uncertain nonlinear pure-feedback systems, the traditional method is to transform pure-feedback systems into strict-feedback systems, and the controller is designed with implicit function theorem [10]. However, this method ignores the essential properties of the system’s non-affine. Even if the model information is completely known, the resulting algorithm can only guarantee the semi-global stability of the closed-loop system. There are some methods for the pure-feedback systems with the non-affine function being in-differentiable which effectively overcome the difficulties that non-affine structures bring to the system controller design [11]–[13]. Despite these advances in Manuscript received ; revised. (Corresponding author: Fujin Jia) Fujin Jia is with the School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]). Xiao Yan is with the School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China ([email protected]). Xuhuan Wang is with the Department of Mathematics, Pingxiang University, Pingxiang 337500, China ([email protected]). Junwei Lu is with the School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210023, P.R. China ([email protected]). Yongmin Li is with the School of Science, Huzhou Teachers College, Huzhou 313000, Zhejiang, P.R. China ([email protected]).

nonlinear pure-feedback systems control, most of the literature so far deals with the ideal case where all the nonlinearities of the pure-feedback system are known [10]–[12], [32]; in other words, the uncertainty in the systems is assumed to be depend only on unknown parameters or disturbance functions [35]. It is rarely attempted to study the effects of both unmodeled dynamics in a singular perturbation framework and unknown parameters in the robust adaptive nonlinear purefeedback system control scheme. In practical engineering, it is difficult to establish accurate models for control objects. Therefore, It does make sense to study the control problems of pure-feedback systems with both unmodeled dynamics and unknown parameters. In practical applications, excessive overshoot may make the control scheme difficult to apply [13]–[16]. We need to consider the systems tracking problem more deeply; that is, the prescribed performance [17]. Traditional prescribed performance can limit greatly the dynamic process (convergence rate and overshoot) of tracking error and applied to the actual system [18]–[25], [38]–[41], [43]–[45]. However, the steadystate process is still based on the inherent properties of the system control method (semi-global/global control scheme); the choice of the prescribed performance functions can only be attached to the symmetric exponential functions, so the traditional prescribed performance algorithm has limitations and lacks flexibility. Therefore, it is an important and interesting problem to design relatively flexible prescribed performance functions which the dynamic performance and steadystate performance of the tracking error is depended on the prescribed performance functions. In this paper, we develop a robust adaptive prescribed performance dynamic surface control design for a class nonlinear uncertain pure-feedback systems. The main contributions of the paper are summarized as follows: 1) The uncertainty in the class of pure-feedback systems that we consider is due to both parameters vector uncertainty and unmodeled dynamics functions. These unmodeled dynamics functions could be to modeling errors, external disturbances, time variations, or a combination of these in the system. The assumption is that these unmodeled dynamics satisfy a triangular bounds condition. In particular, the unmodeled dynamics functions satisfy some growth conditions characterized by bounding functions composed of unknown parameters multiplied by known functions. 2) Dynamic surface control is used to avoid the cumbersomeness of formula recursion. However, when applying to

2

nonlinear pure feedback systems, it is usually assumed that the absolute value of derivative of the non-affine structure to the next subsystem state less than known parameters, that is, bounded [14], [15], [32], [34], [42]. However, such parameters are difficult to find, so the existed theory on the based of this assumption has great limitations. In this paper, it is avoided finding such parameters to satisfy the assumptions by designing special virtual controllers. 3) Due to the non-affine structure of the pure-feedback systems, the conventional coordinate transformation will bring difficulties to the design of the controller [26]–[31], [33], so a new coordinate transformation is applied and an auxiliary system is introduced [32]. 4) A radical-type Lyapunov function and L function are designed such that the dynamic performance and steadystate performance of the tracking error are depended on the prescribed performance functions. Based on the above discussion, a robust adaptive dynamic surface control method which preserved the prescribed performance is proposed for the nonlinear pure-feedback systems. Simultaneously, simulation results illustrate the effectiveness of the control scheme. The remainder of this paper is organized as follows. Section 2 presents the systems and problem statement. The prescribed performance control is addressed in Section 3. The controller design of nonlinear system is designed in Section 4. The stability analysis of nonlinear system is discussed in Section 5. Two examples are given to verify the effectiveness of the algorithm in Section 6, and the paper is concluded in Section 7.

∂fn (u, x) ∂fi (xi+1 , Xi ) 6= 0, 6= 0, i = 1, · · · , n − 1. ∂xi+1 ∂u ∂fi (xi+1 ,Xi ) , ∂xi+1 ∂fi (xi+1 ,Xi ) | ∂xi+1 | ≤

Remark 1: In most research, functions h− i

(3) for

h+ i = 1, · · · , n needs to satisfy 0 < ≤ i , − + and hi and hi must be known parameters in using dynamic surface control theory [14], [15], [32], [34]. However, such + parameters h− i and hi are not readily available in the application. In contrast, this paper adopts different virtual control design algorithms to eliminate the search for such parameters. so this paper is not only more general, but also more practical. An auxiliary integral system is introduced to solve the nonaffine input of systems u˙ = ω,

(4)

where ω ∈ R is auxiliary control input. Lemma 1: For any parameter ψ 0 ∈ R and any functions χ(t) ∈ R, ψ(t) ∈ R, the following equation is established. χ(t)(ψ(t) − ψ 0 ) ≥ Proof:

1 1 2 χ (t) − (ψ(t) − χ(t) − ψ 0 )2 . 2 2

(5)

Due to

1 2 1 χ (t) + (ψ(t) − ψ 0 )2 − 2 2 1 1 = χ2 (t) + (ψ(t) − ψ 0 )2 − 2 2 + (ψ 0 − ψ(t))2 ]

1 (ψ(t) − χ(t) − ψ 0 )2 2 1 2 [χ (t) + 2χ(t)(ψ 0 − ψ(t)) 2

=χ(t)(ψ(t) − ψ 0 ).

II. S YSTEMS AND PROBLEM STATEMENT In this paper, we consider a class of uncertain nonlinear pure-feedback systems of the form x˙ i = fi (xi+1 , Xi ) + θ∗T ξi (Xi ) + ∆i (u, x, t), x˙ n = fn (u, x) + θ∗T ξn (x) + ∆n (u, x, t), (1) y = x1 , i = 1, . . . , n − 1, e = y − r,

where x = [x1 , . . . , xn ]T ∈ Rn and u ∈ R are the known system state and input, respectively, and Xi = [x1 , . . . , xi ]T ; y ∈ R is the system output; r ∈ R is a known tracking signal with 2th derivative bounded, and satisfy the compact set: S0 = {[r, r, ˙ r¨] : r2 + r˙ 2 + r¨2 ≤ H0 }, H0 is unknown positive constant; e ∈ R is the tracking error; fi (xi+1 , Xi ) and fn (u, x) are known smooth non-affine functions; ξi (Xi ) : Ri → Rp is known differentiable smooth functions with ξi (0) = 0; θ∗ ∈ Rp is an unknown vector of constant parameters, and ∆i is an unknown smooth function that contains unmodeled dynamics and external disturbances, and satisfies Assumption 1. Assumption 1 There exist unknown parameter values ψi∗ ≥ 0 and known smooth function pi (Xi ) such that, for all x ∈ Rn , u ∈ R and t ∈ R, |∆i (u, x, t)| ≤ ψi∗ pi (Xi ), 1 ≤ i ≤ n.

Assumption 2 For function fi (xi+1 , Xi ), fn (u, x), it is assumed at any time

(2)

(6) Therefore, it is obvious that we can obtain χ(t)(ψ(t) − ψ 0 ) ≥ 1 2 1 0 2 2 χ (t) − 2 (ψ(t) − χ(t) − ψ ) . Lemma 2: [35] For any bounded smooth function i fi (Xi , xi+1 ), fn (u, x), i = 1, · · · , n, if ∂x∂fi+1 6= 0 and ∂fn = 6 0, then x and u are bounded. i+1 ∂u Lemma 3: [36] The following inequality holds for any 0 > 0 and for any ` ∈ R ` 0 ≤ |`| − ` tanh( ) ≤ δ0 0 ,

(7)

where δ0 is a constant that satisfies δ0 = e−(δ0 +1) , i.e., δ0 = 0.2785. Problem Description: For the systems (1) and the auxiliary system (4), the controller is designed by the backstepping method and dynamic surface control, so that all signals of the closed-loop system are semi-global uniformly ultimately bounded, and the tracking error satisfies the prescribed performance K. III. P RESCRIBED PERFORMANCE CONTROL Definition 1: Prescribed performance control K: K = {(t ≥ 0, e(t)) ∈ R × R | a+b }, A(t) < e(t) < B(t), lim e(t) = t→∞ 2

(8)

3

where A(t) and B(t) are prescribed performance smooth functions that can be designed according to actual conditions, and the following conditions need to be satisfied: 1) Functions A(t), B(t) and their 2th derivatives are bounded functions; 2) Functions A(t) and B(t) have limits: limt→∞ A(t) = a, limt→∞ B(t) = b; 3) Stability condition: prescribed performance functions A(t) and B(t) are selected such that a + b = 0. (Non-essential conditions) Remark 2: If the control scheme is to make the system globally stable, we can choose a + b = 0 to make the tracking error stable. On the contrary, we can choose a + b = ς to compensate the system, so that the tracking error is more accurate, where ς is compensation constant. Therefore, the condition 3) is not necessary. Lemma 4: There are two radical-type functions below: ( √ √ Y1 (L) = δ − δ − L2 , (9) L2 Y2 (L) = √δ−L , 2 where δ ≤ 1 is a positive parameters; L is a function about e(t), A(t) and B(t). √ √ In the interval of − δ < L < δ, we have Y1 (L) ≤ Y2 (L).

(10)

Proof: Let Y2 (L) subtract Y1 (L), Y (L = 0) = 0 √ √ δ − δ δ − L2 (11) √ Y2 (L) − Y1 (L) = , δ − L2 √ √ √ √ 2 in the√ interval √ − δ < L < δ, we can get δ − L ≤ δ ⇒ (δ − δ δ − L2 ) ≥ 0 ⇒ Y2 (L) − Y1 (L) ≥ 0 ⇒ Y1 (L) ≤ Y2 (L). Remark 3: Compared with the traditional prescribed performance design scheme [17]–[22], the design method of this paper has four advantages as follows. 1) The L function is proposed such that both the dynamic performance and the steady-state performance of the tracking error depend on prescribed performance functions. 2) The selection of prescribed performance functions can be symmetrical or asymmetric, so it is more flexible. 3) In the global control scheme, if prescribed performance functions are selected as ` = `1 e−`2 t + `∞ , where `1 , `2 and `∞ are normal numbers, and limt→∞ `(t) = `∞ > 0 [17]– [22]. Then, `∞ of this paper can be selected as `∞ ≥ 0. 4) According to (9), the parameter δ can be changed to adjust the dynamic performance and the steady-state performance of the tracking error. Lemma 5: For the first-order system x˙ = u, e = x − r, where r is tracking signal with√2th derivatives bounded. In the compact set Ω := {|L| < δ}, the Lyapunov function e−A e−B V = Y1 (L) is constructed, L = B−A + B−A . Then, we set 1 1 ˙ ˙ ˙ ˙ u = − 2 L(B−A)+ r+ ˙ 2 [A+ B+L(B− A)], which means that the first-order system have reached the prescribed performance K.

Proof: From the above Lyapunov V derivative of V is available

= Y1 (L), the

L ˙ √ [2e˙ − A˙ − B˙ − L(B˙ − A)] (B − A) δ − L2 L ˙ √ = [2u − 2r˙ − A˙ − B˙ − L(B˙ − A)]. (B − A) δ − L2 (12) 1 1 ˙ ˙ ˙ ˙ Due to u = − 2 L(B − A) + r˙ + 2 [A + B + L(B − A)] and according to Lemma 2 V˙ =

p √ L2 V˙ = − √ ≤ −( δ − δ − L2 ) = −V. δ − L2

(13)

From Equation (13), we can get the following two conclusions. 1) From (13) and Barbalat’s Lemma in [37], we know p √ lim ( δ − δ − L2 ) = 0.

t→∞

Due to L =

e−A B−A

+

e−B B−A ,

it can be deduce

2e(t) − A(t) − B(t) )=0 B(t) − A(t) 1 ⇒ lim e(t) = lim (A(t) + B(t)). t→∞ 2 t→∞

lim L = 0 ⇒ lim (

t→∞

t→∞

(14)

2) From (13), we can obtain V (0) V˙ (t) ≤ −V (t) ⇒ V (t) ≤ t e r r √ √ V (0) 2 V (0) ⇒ − δ − ( δ − t ) ≤ L ≤ δ − ( δ − t )2 e e √ 2e(t) − A(t) − B(t) √ ⇒−1≤− δ < < δ≤1 B(t) − A(t) ⇒A(t) < e(t) < B(t). (15) The equations (14) and (15) explain that the system x˙ = u, e = x−r has the prescribed performance K. According to the prescribed performance conditions 2) and 3), we can obtain the that the tracking error is asymptotically stable if a+b = 0. Remark 4: Recently, some new finite-time control techniques are propose in [46]–[48]. The differences between the methods in this paper and these results are shown below. 1) The prescribed performance control in this paper is an infinite time stabilization method. The finite time stabilization system means that the state of the system can always stabilize to the origin within finite time. 2) The prescribed performance control in this paper can make the dynamic performance and steady-state performance of the system depend on the prescribed performance functions. The finite time control can only adjust the dynamics performance of the systems by changing the controller parameters.

4

IV. C ONTROLLER DESIGN OF NONLINEAR SYSTEM

ω1 = p1 (x1 ) tanh[

Based on the above discussion, we define coordinate transformation as follows: z1 = x1 − r, zi+1 = fi − Di , , i = 1, · · · , n, zn+2 = ω − an+1 P ϕ1 = ξ1 , ϕi = ij=1 ∂f∂xi−1 ξj , j Pn (16) n ϕn+1 = j=1 ∂f ξ , i = 2, · · · , n, j ∂x Pj i ∂fi−1 Λ1 = ∆1 , Λi = j=1 ∂xj ∆j , Pn ∂fn Λ n+1 = j=1 ∂xj ∆j , i = 2, · · · , n,

where ai−1 and an+1 are stabilising functions obtained in the adaptive backstepping design, and θ denotes an estimate of θ∗ . From (2), Λi satisfies |Λi | ≤ ψi∗ pi (Xi ), where ψi∗ is an unknown parameter, and pi (Xi ) is a known function about Xi . Di are generated by the following low pass filters [31]: D˙ i = −ρi ei , ei = Di − ai , Di (0) = ai (0), i = 1, · · · , n, (17) where ρi is positive design parameters; ei is a boundary layer error. We start the adaptive backstepping from the dynamics of z1 . Step 1: From (1), (16) and (17), we can obtain z˙1 = z2 + a1 + e1 + θT ϕ1 − φT ϕ1 + Λ1 − r. ˙

L=

˙ 2z˙1 − A˙ − B˙ − L(B˙ − A) 2z1 − B − A ˙ ,L = , (20) B−A B−A

where Γ ∈ Rp×p is a positive definite matrix; γ1 > 0 is a scalar constant. Then, (18)-(20) explain that the time derivative of V1 is given as L √ [2a1 + 2θT ϕ1 − 2r˙ − A˙ − B˙ − L(B˙ (B − A) δ − L2 2L 2L ˙ + √ √ − A)] e1 + z2 (B − A) δ − L2 (B − A) δ − L2 2L 1 √ + Λ1 − σθ φT (θ − θ0 ) + χ1 ψ˙ 1 2 γ1 (B − A) δ − L T −1 ˙ + φ Γ (θ − τ1 ), (21) where σθ and θ0 are positive design constants, τ1 = √ (Γ (B−A)2L ϕ − Γσθ (θ − θ0 )). δ−L2 1 The virtual control law a1 and ψ˙ 1 (t) are chosen as V˙1 =

L 1 √ a1 = − [c1 L(B − A) + ] − θ T ϕ1 2 (B − A) δ − L2 (22) 1 ˙ ˙ ˙ ˙ + r˙ + [A + B + L(B − A)] − ψ1 ω1 ), 2 2L √ ψ˙ 1 = γ1 [ ω1 (x1 ) − σ1 (ψ1 − ψ10 )], (23) (B − A) δ − L2

(24)

where c1 , 1 , and σ1 are positive design constants. From (22)-(24), then V˙ 1 can be rewritten as L L √ √ ] (B − A) δ − L2 (B − A) δ − L2 2L 2L √ √ + e1 + z2 2 (B − A) δ − L (B − A) δ − L2 2L 2L √ √ − χ 1 ω1 − ψ1M ω1 + 2 (B − A) δ − L (B − A) δ − L2 2L √ Λ1 − σθ φT (θ − θ0 ) + φT Γ−1 (θ˙ − τ1 ) (B − A) δ − L2 2L √ + χ1 [ ω1 − σ1 (ψ1 − ψ10 )]. (B − A) δ − L2 (25) From Lemma 1, Lemma 3 and (2), we can obtain as

V˙ 1 = − [c1 L(B − A) +

L L √ √ ] 2 (B − A) δ − L (B − A) δ − L2 1 1 2L √ − σθ φ2 − σ1 χ21 + e1 + λ 1 2 2 (B − A) δ − L2 2L √ z2 + φT Γ−1 (θ˙ − τ1 ), + (B − A) δ − L2 (26)

V˙ 1 ≤ − [c1 L(B − A) +

(18)

where φ = θ − θ∗ , and now let χ1 = ψ1 − ψ1M , ψ1M = max{ψ1∗ , ψ10 }; the constant ψ10 ≥ 0 appears in the adaptive law for ψ1 (t). Design Lyapunov function V1 as p √ 1 1 2 V1 = δ − δ − L2 + φT Γ−1 φ + χ , (19) 2 2γ1 1

2L √ p1 (x1 )], ε1 (B − A) δ − L2

where 1 1 σθ φ2 − σθ (θ∗ − θ0 )2 , 2 2 1 1 σ1 χ1 (ψ1 − ψ10 ) ≥ σ1 χ21 − σ1 (ψ1M − ψ10 )2 , 2 2 2L 2L √ √ p1 tanh[ p1 ] (27) − 2 (B − A) δ − L ε1 (B − A) δ − L2 2L √ +| p 1 | ≤ δ 0 ε1 , (B − A) δ − L2 1 1 λ1 = σθ (θ∗ − θ0 )2 + σ1 (ψ1M − ψ10 )2 + ψ1M δ0 ε1 . 2 2 σθ φT (θ − θ0 ) ≥

Step 2: From (1), (16) and (17), the dynamics of z2 is given as, z˙2 =

∂f1 ∂f1 f1 + (z3 + e2 + a2 ) ∂x1 ∂x2 + ϕ2 (θT − φT ) + Λ2 + ρ1 e1 .

(28)

Consider the Lyapunov function 1 1 2 V2 = V1 + z22 + χ , 2 2γ2 2

(29)

where γ2 > 0 is a scalar constant, χ2 = ψ2 − ψ2M , ψ2M = max(ψ2∗ , ψ20 ), ψ20 ≥ 0 appears in the adaptive law for ψ2 .

5

From (26) and (29), the time derivative of V2 is given as L L √ √ V˙ 2 ≤ − [c1 L(B − A) + ] 2 (B − A) δ − L (B − A) δ − L2 1 1 1 2L √ − σθ φ2 − σ1 χ21 + e1 + χ2 ψ˙2 2 2 2 γ (B − A) δ − L 2 2L T −1 ˙ √ z2 + φ Γ (θ − τ1 ) + λ1 + + (B − A) δ − L2 ∂f1 ∂f1 z2 [ f1 + (z3 + e2 + a2 ) + ϕ2 θT − ϕ2 φT ∂x1 ∂x2 + Λ2 + ρ1 e1 ]. (30) Stabilising function a2 is designed as follows 1 a2 = ∂f1 [−c2 z2 −

2L ∂f1 √ − f1 2 ∂x (B − A) δ − L 1 ∂x2 1 ∂f1 − ϕ2 θT − ρ1 e1 − ψ2 ω2 ] − z2 , 2 ∂x2 ψ˙ 2 = γ2 [z2 ω2 − σ2 (ψ2 − ψ 0 )], 2

z2 p2 ω2 = p2 tanh[ ], ε2 where c2 , 2 , and σ2 are positive design parameters. From (31)-(33), then V˙ 2 can be rewritten as

(31)

(32)

Consider the Lyapunov function 1 1 2 χ , Vi = Vi−1 + zi2 + 2 2γi i

(38)

where γi > 0 is a scalar constant, χi = ψi − ψiM , ψiM = max(ψi∗ , ψi0 ), ψi0 ≥ 0 appears in the adaptive law for ψi . From (35) and (38), the time derivative of Vi is given as V˙ i ≤ −[c1 L(B − A) + −

i−1 X

L L √ √ ] 2 (B − A) δ − L (B − A) δ − L2

1 ∂fj−1 2 2L √ e1 [cj zj + ( ) zj ]zj + 2 ∂x (B − A) δ − L2 j j=2

i−1 i−1 X 1 ∂fi−2 1X ∂fj−1 − σθ φ2 − σj χ2j + zj ej + zi−1 zi 2 2 j=1 ∂xj ∂xi−1 j=2 i−1 X ∂fi−1 ∂fi−1 + λi−1 + φT Γ−1 (θ˙ − τi−1 ) + zi ( fj + ai ∂x ∂xi j j=1

∂fi−1 ∂fi−1 zi+1 + ei + ϕi θT − ϕi φT + Λi + ρi−1 ei−1 ) ∂xi ∂xi 1 + χi ψ˙ i , γi (39) Stabilising function ai is designed as follows +

(33)

L L √ √ ] 2 (B − A) δ − L (B − A) δ − L2 1 ∂f1 2 1 1 − [c2 z2 + ( ) z2 ]z2 − σθ φ2 − σ1 χ21 + ψ2M |z2 p2 | 2 ∂x2 2 2 2L ∂f1 ∂f1 √ + e1 + z2 e2 + z2 z3 + λ1 − ∂x2 ∂x2 (B − A) δ − L2 z2 p2 z2 ψ2M p2 tanh[ ] − σ2 χ2 (ψ2 − ψ20 )] + φT Γ−1 (θ˙ − τ2 ), ε2 (34) where τ2 = τ1 + Γϕ2 z2 . From Lemma 1, Lemma 3 and (2), we can obtain V˙ 2 ≤ −[c1 L(B − A) +

i−1 X ∂fi−2 ∂fi−1 1 zi−1 − fj ai = ∂fi−1 [−ci zi − ∂xi−1 ∂xj j=1 ∂xi

(40)

1 ∂fi−1 zi , − ϕi θ − ρi−1 ei−1 − ψi ωi ] − 2 ∂xi T

ψ˙ i = γi [zi ωi − σi (ψi − ψi0 )], ωi = pi tanh[

zi pi ], εi

(41) (42)

where ci , i , and σi are positive design constants. L L From Lemma 1, Lemma 3 and (40)-(42), we can obtain √ √ ] (B − A) δ − L2 (B − A) δ − L2 L L √ √ ] V˙ i ≤ −[c1 L(B − A) + 1 ∂f1 2 1 2L 2 (B − A) δ − L2 √ − [c2 z2 + ( ) z2 ]z2 − σθ φ2 + e1 (B − A) δ − L 2 2 ∂x2 2 (B − A) δ − L i X 1 ∂fj−1 2 2L 2 √ − [cj zj + ( ) zj ]zj + e1 ∂f ∂f 1X 1 1 2 T −1 ˙ 2 ∂xj − σj χj + z2 e2 + z2 z3 + λ2 + φ Γ (θ − τ2 ), (B − A) δ − L2 j=2 2 j=1 ∂x2 ∂x2 i i X (35) 1X ∂fj−1 ∂fi−1 1 2 2 σ φ − σ χ + zj ej + zi zi+1 − θ j j where 2 2 j=1 ∂x ∂xi j j=2 1 1 σ2 χ2 (ψ2 −ψ20 ) ≥ σ2 χ22 − σ2 (ψ2M − ψ20 )2 , + λi + φT Γ−1 (θ˙ − τi ), 2 2 (43) z2 p2 |z2 p2 | − z2 p2 tanh[ ] ≤ δ0 ε2 , (36) where τi = τi−1 + Γϕi zi , and ε2 1 1 1 M λ2 = λ1 + ψ2 δ0 ε2 + σ2 (ψ2M − ψ20 )2 . σi χi (ψi − ψi0 ) ≥ σi χ2i − σi (ψiM − ψi0 )2 , 2 2 2 zi p Step i (3 ≤ i ≤ n): The dynamics of zi can be written as |zi pi | − zi pi tanh[ i ] ≤ δ0 εi , (44) εi i−1 X ∂fi−1 ∂fi−1 ∂fi−1 ∂fi−1 1 z˙i = fj + ai + zi+1 + ei λi =λi−1 + ψiM δ0 εi + σi (ψiM − ψi0 )2 . ∂x ∂x ∂x ∂x 2 j i i i j=1 V˙ 2 ≤ −[c1 L(B − A) +

+ ϕi θT − ϕi φT + Λi + ρi−1 ei−1 .

(37)

Step n+1: In the fine step, the actual control u and adaptive θ˙ will be determined.

6

From (1), (4), (16) and zn+1 = fn − an , we have z˙n+1 =

n X ∂fn j=1

∂xj

fj +

∂fn ω + ϕn+1 θT − ϕn+1 φT ∂u

From (48), (50) and (52), we obtain dynamics of V as

(45)

+ Λn+1 + ρn en .

V˙ ≤ −[c1 L(B − A) + −

The auxiliary control input ω in the dynamics zn+1 , in the term zn+2 . we obtain the auxiliary control input by setting zn+2 = 0, which gives ω =an+1 n X ∂fn−1 ∂fn zn − fj = ∂fn [−cn+1 zn+1 − ∂xn ∂xj ∂u j=1

1

(47)

(48) 2L √ ϕ1 − σθ (θ − θ0 )), zj ϕj + =Γ( 2 (B − A) δ − L j=2

∂fn−1 zn zn+1 − φT ϕn+1 zn+1 ∂xn M − χn+1 zn+1 ωn+1 + ψn+1 δ0 εn+1 , (50)

2 zn+1 z˙n+1 ≤ − cn+1 zn+1 −

(51)

1 σn+1 χ2n+1 2 1 M 0 − σn+1 (ψn+1 − ψn+1 )2 , (54) 2 1 M 0 M − ψn+1 )2 . = λn + ψn+1 δ0 εn+1 + σn+1 (ψn+1 2

λn+1

Definition set [30] Si ={(z1 , z2 , · · · , zi+1 , e1 , e2 , · · · , ei ) : i+1

i

1X 2 1X 2 z + e ≤ h} ∈ R2i+1 , 2 j=1 j 2 j=1 j

The main results are summarized as the following theorem. Theorem 1: Consider the closed-loop systems (1) and (4), and Assumption 1 and Assumption 2 are held. In the compact set Ω, the controller (46) and adaptive law (47)(48) are designed by adaptive backstepping to ensure the semi-global boundedness of all the signals and guarantee the prescribed performance; that is, tracking error e(t) satisfies A(t) < e(t) < B(t) and asymptotically close to 12 (a + b) by adjusting appropriate the parameters. Proof: Now we define a Lyapunov function as n

(52)

M where γn+1 > 0 is a scalar constant, χn+1 = ψn+1 − ψn+1 , M 0 0 ∗ ψn+1 = max(ψn+1 , ψn+1 ), ψn+1 ≥ 0 appeared in the adaptive law for ψn+1 .

(55)

Then, for any H0 > 0 and h > 0, the sets Sd and Si are the compact sets in R3 and R2i+1 respectively, then Sd × Si is also the compact set in R2i+4 , so the continuous functions Hi (·) = −a˙ i have a maximum value Ni in the set Sd × Si . Therefore, we can obtain 1 1 ei e˙ i ≤ −ρi e2i + e2i + Ni , 2 2

(56)

The time dynamics of V (t) can be rewritten as n+1 n+1 X L2 1 1X V˙ ≤ −c1 √ − cj zj2 − σθ φ2 − σj χ2j 2 2 j=1 δ − L2 j=2

V. S TABILITY ANALYSIS

1 2 1 1X 2 V = Vn + zn+1 + χ2n+1 + e , 2 2γn+1 2 i=1 i

(53)

0 σn+1 χn+1 (ψn+1 − ψn+1 )≥

where zn+1 pn+1 ] ≤δ0 εn+1 . εn+1

ei e˙ i ,

i=1

(49)

where cn+1 , εn+1 , σn+1 , and γn+1 are set of positive design 0 parameters, ψn+1 ≥ 0 appears in the adaptive law for ψn+1 . From (46), (47) and (49), we know

|zn+1 pn+1 | − zn+1 pn+1 tanh[

n X

where

n+1 X

zn+1 pn+1 ], εn+1

n+1 n X 1 1X ∂fj−1 2 − cn+1 zn+1 − σθ φ2 − σj χ2j + zj ej 2 2 j=1 ∂xj j=2

(46)

θ˙ =τn+1 = τn + Γzn+1 ϕn+1

ωn+1 = pn+1 tanh[

n X 2L 1 ∂fj−1 2 √ e1 ) zj ]zj + [cj zj + ( 2 ∂xj (B − A) δ − L2 j=2

+ λn+1 +

− ϕn+1 θT − ρn en − ψn+1 ωn+1 ],

0 ψ˙ n+1 = γn+1 [zn+1 ωn+1 − σn+1 (ψn+1 − ψn+1 )],

L L √ √ ] 2 (B − A) δ − L (B − A) δ − L2

n X 3 (ρj − 1)e2j + λ, − (ρj − )e2j − 2 j=1 j=1 n X

(57)

Pn where λ = λn+1 + 21 j=1 Nj . From Lemma 4, we can deduce that

n+1 p X √ 1 V˙ ≤ − c1 ( δ − δ − L2 ) − cj zj2 − σθ φ2 2 j=2 n+1 n X 1X − σj χ2j − (ρj − 1)e2j + λ, 2 j=1 j=1

(58)

this yields V˙ ≤ −cV + λ, where the constants c > 0 are defined as σθ c = min[2c1 , 2c2 , 2c3 , · · · , 2cn+1 , , σi γi , λmin (Γ−1 ) (59) 3 2(ρ1 − ), 2(ρ2 − 1), · · · , 2(ρn − 1)], 2

7

where λmin (Γ−1 ) is minimum eigenvalue of the positive definite matrix Γ. Then V (t) satisfies V (t) ≤

λ λ + (V (0) − )e−ct . c c

(60)

p√ √ δ − δ − L2 , z2 , z3 , · · · , zn+1 , θ(t), ψ(t) Therefore, and also x are semi-global uniformly ultimately bounded. √ Furthermore, given µ∗ ∈ (0, 2 δ) and µ∗√> 2λ , there exists c√ T such that, for all t ≥ T , we have ( δ − δ − L2 ) ≤ √ µ , |zi (t)| ≤ µ∗ . The compact set Θ = {zi ∈ Rn : p∗√ √ √ √ δ − δ − L2 ≤ µ∗ , |zi (t)| ≤ µ∗ } can be made as small as desired by an appropriate choice of the design constants. √ √ Based on this discussion, we can know ( δ − δ − L2 ) ≤ µ∗ , according to (15) can deduce that p √ δ − δ − L2 ≤ µ∗ ⇒ δ − L2 ≥ ( δ − µ∗ )2 q q √ √ ⇒ − δ − ( δ − µ∗ )2 ≤ L ≤ δ − ( δ − µ∗ )2 q √ 2z1 − A(t) − B(t) ⇒ − δ − ( δ − µ∗ )2 ≤ B(t) − A(t) q √ ≤ δ − ( δ − µ∗ )2 q √ B(t) + A(t) − (B(t) − A(t)) δ − ( δ − µ∗ )2 ⇒ ≤ e(t) 2 q √ B(t) + A(t) + (B(t) − A(t)) δ − ( δ − µ∗ )2 ≤ 2 (61) According to (61), we can obtain √

q

√ √ δ − ( δ − µ∗ )2 < δ ≤ 1 q √ ⇒(B(t) − A(t)) δ − ( δ − µ∗ )2 < B(t) − A(t) q √ B(t) + A(t) + (B(t) − A(t)) δ − ( δ − µ∗ )2 ⇒ < B(t), 2 (62) similarly q √ B(t) + A(t) − (B(t) − A(t)) δ − ( δ − µ∗ )2

> A(t). (63) From (62) and (63), tracking error e(t) is satisfied: 2

A(t) < e(t) < B(t),

(64)

From (64), we can know that the tracking error e(t) can be between A(t) and B(t) by designing reasonable functions A(t) and B(t), which can satisfy the prescribed performance K in Definition 1, indicating that the nonlinear pure-feedback system (1) has prescribed performance. Furthermore, we can make µ∗ as close to zero as desired by adjusting the parameters, then the tracking error e(t) close to 21 (a + b). The structure of the proposed prescribed performance trajectory tracking control system is illustrated in Fig.1.

3UHVFULEHG 3HUIRUPDQFH . U

7UDMHFWRU\ *HQHUDWRU

H

$GDSWLYH %DFNVWHSSLQJ 0HWKRG

$X[LOLDU\ 6\VWHP

3XUH / $X[LOLDU\ ˶ (UURU &RQWURO/DZ )HHGEDFN 7UDQVIRUPDQFH V\VWHPV

\

Fig. 1. Schematic of the proposed prescribed performance control system

VI. S IMULATION RESULTS Example 1: We consider a simulation example similar to [32] as follows −0.5x1 + x2 + 0.5 sin x2 + ex2 , x˙ 1 = x1 e 1−e−x2 2 (65) x˙ = x1 + 1+e−x2 + u + 0.1 sin u, 2 y = x1 , e = y − r,

where tracking signal r = sin(2t) − cos(t); the value of can be taken as 0 or 1. To illustrate the performance of the proposed control scheme, and compare with [32] to illustrate the superiority of prescribed performance, we discuss the situation from = 1 and = 0 respectively: 1) The nonlinear non-affine system (65) is the same as [32] when = 0. According to (65), f1 = x1 e−0.5x1 + x2 + 0.5 sin x2 , f2 = −x2 2 + u + 0.1 sin u, therefore x1 + 1−e 1+e−x2 ∂f1 ∂f2 =1 + 0.5 cos x2 ≤ 1.5, = 1 + 0.1 cos x2 ≤ 1.1. ∂x2 ∂u (66) + Let h+ = 1.5, h = 1.1, then we know that the assumptions 1 2 i+1 ,Xi ) | ≤ h+ of [32] are satisfied: 0 < h− ≤ | ∂fi (x i . i ∂xi+1 + + This shows that it is easy to find h1 and h2 to satisfy the assumptions of [32] when = 0. Now we consider the change of coordinates z1 = x1 − r, z2 = f1 − D1 , z3 = f2 − D2 and z4 = ω − a3 . According to the theoretical idea of this paper, the auxiliary control law and virtual control laws of system (66) can be calculated as follows 1 L √ a1 = − [c1 (B − A)L + ] + r˙ 2 (B − A) δ − L2 1 ˙ + [A˙ + B˙ − L(B˙ − A)], 2 1 2L ∂f1 √ a2 = ∂f1 [−c2 z2 − f1 − ρ1 e 1 ] − 2 ∂x (B − A) δ − L 1 ∂x 2

1 ∂f1 z2 , 2 ∂x2 1 ∂f2 ∂f2 ∂f1 ω =a3 = ∂f2 [−c3 z3 − z2 − f1 − f2 − ρ2 e2 ]. ∂x ∂x ∂x 2 1 2 ∂x2 (67) Simulation results corresponding to the following initial conditions and design parameters are shown in Figs.2-3. −

x1 (0) = −0.6, x2 (0) = 1, u(0) = 0, A(t) = −0.05,

B(t) = 0.05 + e−t , c1 = 3, c2 = 2, c3 = 4, δ = 0.9.

(68)

8

1.2 e(t) B(t) A(t) e(t) of [32]

0.8 0.6

e(t)

e(t) B(t) A(t)

1

Prescribed performance

1

0.4 0.2 0 -0.2

0.8 0.6 0.4 0.2 0 -0.2

0

2

4

6

8

10

12

14

0

2

4

Fig. 2. Tracking error e(t) comparison with [32] for = 0

8

10

12

14

Fig. 4. Prescribed performance e(t) for = 1

15

15

x2

x2

System state x 2 and control law u

system state and control law

6

Time(sec)

Time(sec)

u

10

5

0

-5

u

10 5 0 -5 -10

-10 0

5

10

-15

15

0

2

4

6

8

10

12

14

Time(sec)

Time(sec)

Fig. 3. System state x2 and control input u for = 0

Fig. 5. System state x2 and control input u for = 1

It can be seen from the above theoretical and simulation experiments that the controller (67) which has prescribed performance can make the system (65) semi-global bounded, and the system tracking error is satisfied to the prescribed performance, i.e., e(t) satisfies A(t) < e(t) < B(t). In the same system (65), compared with the method in [32], accuracy of the tracking error is higher in the method of this paper. 2) When = 1, we know f1 = x1 e−0.5x1 + x2 + −x2 0.5 sin x2 + ex2 , f2 = x21 + 1−e + u + 0.1 sin u. Due 1+e−x2 ∂f1 x2 to ∂x = 1 + 0.5 cos x + e , so it is not easy to find h+ 2 1 2 + and h2 to satisfy the assumptions of [15], [16], [32], [34]: ∂fi (xi+1 ,Xi ) 0 < h− | ≤ h+ i ≤| i . This illustrates that the control ∂xi+1 algorithm in [15], [16], [32], [34] is not suitable for system (65). However, this paper does not have this assumption, so system (65) can be processed using the prescribed performance control algorithm herein. Simultaneously, it also shows that the algorithm of this paper is more general. Simulation results corresponding to the following initial conditions and design parameters are shown in Figs.4-5.

to the prescribed performance, i.e., e(t) satisfies A(t) < e(t) < B(t). In theory, because of [32] has the assumption: ∂fi (xi+1 ,Xi ) | ≤ h+ 0 < h− i . the algorithm of [15], [16], i ≤ | ∂xi+1 [32], [34] is not suitable for the system (65) when = 1, so the algorithm of this paper is more superior. Example 2: We consider Chua’s oscillator in [6], [49] as follows C1 − iD , C1 V˙ C1 = VC2 −V R C1 (70) C2 V˙ C2 = − VC2 −V + iL , R ˙ LiL = −VC2 − RL iL + u,

x1 (0) = 0, x2 (0) = 0, u(0) = 2, A(t) = −0.1,

B(t) = 0.1 + e−t , c1 = 3, c2 = 2, c3 = 4, δ = 0.9.

(69)

It can be seen theoretical in the paper which has prescribed performance can make the system (65) semi-global bounded when = 1, and the system tracking error is satisfied

where VC1 and VC2 are voltages through the capacitors C1 and C2 , respectively. iL is the current across the inductor L and voltage source u is the control input. iD is the current through Chus’s diode D. Now let x1 = VC1 , x2 = VC2 , and x3 = iL be the system states. Then, system (70) can be written as 1 1 x˙ 1 = RC1 x2 − RC1 x1 + ∆(x1 ), 1 1 1 x˙ 2 = C2 x3 − RC2 x2 + RC x1 , (71) 2 RL 1 1 x˙ 3 = L u − L x2 − L x3 ,

where ∆(x1 ) = − C11 iD is an unknown nonlinear function. Now let ∆(x1 ) = 0.4 sin(t). The parameters are chosen as: R = 0.2Ω, RL = 0.01Ω, C1 = 0.9F , C2 = 0.1F , L = 0.25H, c1 = 10, c2 = 10, c3 = 10, r = 0, c1 = 10, A(t) = −0.1, B(t) = 0.1 + e−t . The initial conditions are x1 (0) = 1,

9

x2 (0) = 1, x3 (0) = 0.5, ψ1 (0) = 0, D1 (0) = 1, D2 (0) = 1. Then, the adaptive law ψ˙ 1 and controller u are designed, respectively, as

RC1 C2 x3 − D2 .

1 RC1 x2 − D1 ,

and z3 = where z1 = x1 − r, z2 = The state trajectory of the system are shown in Figs. 6-8.

6

Control law u

2L √ ω1 (x1 ) − σ1 (ψ1 − ψ10 )], (B − A) δ − L2 (72) C3 L RC1 1 RL ˙ u= [−c3 z3 + ( x2 + x3 ) + D2 − z2 ]. RC1 C3 L L

ψ˙ 1 =γ1 [

8

4 2 0 -2 -4

0

1

B(t) e(t) A(t)

Prescribed performance

2

3

4

5

6

Time(sec)

1.5

Fig. 8. Control law u

1

0.5

0

-0.5

0

1

2

3

4

5

6

7

8

Time(sec)

System states and adaptive law 1

Fig. 6. Prescribed performance e(t)

2 1 0 -1 -2 -3

x2

0

1

2

3

4

5

x3

6

VIII. ACKNOWLEDGMENT

1

7

proposed and a first order auxiliary system for control input is introduced. Secondly, a radical-type Lyapunov function is designed to ensure the prescribed performance of the purefeedback system. Then, a new robust adaptive control method which preserving the prescribed performance is proposed for the nonlinear pure-feedback systems. Simultaneously, simulation results illustrate the effectiveness of the control scheme. The future research work can be carried out in the following two aspects. 1) In recent years, the study of the all state constraints of nonlinear systems have attracted scholar’s attention. However, the existing results are only available for system with static state constraints [40]. Therefore, it is possible that the prescribed performance method in this paper can solve the fullstate time-varying constraint problem of the system. 2) In most of the results of the prescribed performance control, it is generally assumed that the tracking signal is known. Therefore, how to solve the prescribed performance of the system output regulation problem is also an important issue [31].

8

Time(sec)

This work was supported by the National Natural Science Foundation of China (grant 11661065).

Fig. 7. System states and adaptive laws ψ1

It can be seen from the above theoretical and simulation experiments that the controller (72) can make the non-affine system (70) semi-global uniformly ultimately bounded. the accuracy of the tracking error e(t) is asymptotically close to stability and guarantee the prescribed performance of the system. VII. C ONCLUSIONS In this paper, an issue with regards to the adaptive prescribed performance control of a class of nonlinear pure-feedback system with both unknown vector parameters and unmodeled dynamics is studied. Firstly, to avoid using the cumbersome formula to handle the non-affine structure of the pure-feedback system, a type of non-traditional state transformation was

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Dear Editor-in-Chief, It is my pleasure to submit our manuscript entitled ‘ Robust Adaptive Prescribed Performance Dynamic Surface Control for Uncertain Nonlinear Pure-Feedback Systems’ for possible publication in Journal of The Franklin Institute. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Sincerely with best wishes, Authors: Fujin Jia, Xiao Yan, Xuhuan Wang and Junwei Lu, Yongmin Li. Institutions: Fujin Jia is with the School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China. Xiao Yan is with the School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China. Xuhuan Wang is with the Department of Mathematics, Pingxiang University, Pingxiang 337500, China. Junwei Lu is with the School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210023, P.R. China. Yongmin Li is with the School of Science, Huzhou Teachers College, Huzhou 313000, Zhejiang, P.R. China. E-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]. October 19, 2019.

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