ROBUST ADAPTIVE FUZZY FILTERS OUTPUT FEEDBACK CONTROL OF STRICT–FEEDBACK NONLINEAR SYSTEMS

DOI: 10.2478/v10006-010-0047-x

ROBUST ADAPTIVE FUZZY FILTERS OUTPUT FEEDBACK CONTROL OF STRICT–FEEDBACK NONLINEAR SYSTEMS

S HAOCHENG TONG, C HANGLIANG LIU, Y ONGMING LI

Department of Mathematics

Liaoning University of Technology, Jinzhou, 121001, People’s Republic of China e-mail: jztsc@sohu.com

In this paper, an adaptive fuzzy robust output feedback control approach is proposed for a class of single input single output (SISO) strict-feedback nonlinear systems without measurements of states. The nonlinear systems addressed in this paper are assumed to possess unstructured uncertainties, unmodeled dynamics and dynamic disturbances, where the unstructured uncertainties are not linearly parameterized, and no prior knowledge of their bounds is available. In recursive design, fuzzy logic systems are used to approximate unstructured uncertainties, and K-filters are designed to estimate unmeasured states.

By combining backstepping design and a small-gain theorem, a stable adaptive fuzzy output feedback control scheme is developed. It is proven that the proposed adaptive fuzzy control approach can guarantee the all the signals in the closed-loop system are uniformly ultimately bounded, and the output of the controlled system converges to a small neighborhood of the origin. The effectiveness of the proposed approach is illustrated by a simulation example and some comparisons.

Keywords: nonlinear systems, adaptive fuzzy control, backstepping, small-gain approach, K-filters.

1. Introduction

In the past decade, interest in adaptive control of non- linear systems has been increasing, and many significant developments have been achieved. As a breakthrough in nonlinear control, adaptive backstepping control was in- troduced to achieve global stability and asymptotic track- ing for a class of nonlinear systems in parametric strict- feedback form by Kanellakopopoulos et al. (1991). Later, the overparametrization problem was successfully elimi- nated by Kristic et al. (1992) through the tuning function method. In an effort to extend the backstepping control idea to larger classes of nonlinear systems, Kristic et al.

(1995) and Qian et al. (2002) studied the adaptive control problem of parametric strict-feedback systems, obtained local stability results, and proposed several adaptive ap- proaches to nonlinear systems with a triangular structure.

To accommodate uncertainties, a robust adaptive backstepping control has been developed for strict- feedback nonlinear systems with time-varying distur- bances and static or dynamic uncertainties by Jiang et al.

(1998; 1999) (to name a few). The advantages of back- stepping methodology include the facts that: (i) global stability can be achieved with ease, (ii) the transient per- formance can be guaranteed and explicitly analyzed, and

(iii) it has the flexibility to avoid unnecessary cancellation of useful nonlinearities compared with feedback lineariza- tion techniques. However, these schemes are only suit- able for nonlinear systems with nonlinear dynamics mod- els known exactly or with unknown parameters appearing linearly with respect to known nonlinear functions. If that kind of knowledge is not available a priori, these adaptive backstepping controllers cannot be applied.

Fuzzy logic systems have been widely used to model nonlinearities. A fuzzy logic system is a universal ap- proximator which, with the increased size of fuzzy rules, can approximate any nonlinearities with arbitrary preci- sion (Wang, 1994). Based on this capability, fuzzy logic systems are vastly adopted for nonlinear systems identifi- cation and control (Chen et al., 1996; Denai et al., 2002;

Boukezzoula et al., 2007; Qi et al., 2009). Most of them

use fuzzy logic systems as nonlinear models for the under-

lying nonlinearity. The stability issues for adaptive fuzzy

controllers are addressed by Lyapunov functions. How-

ever, these adaptive fuzzy controllers are only applied to a

relatively simple class of nonlinear systems. The key re-

quirement is that the unknown nonlinearities must satisfy

the matching conditions. If the unknown nonlinearities

do not satisfy the matching conditions, the adaptive fuzzy

controllers mentioned above cannot be implemented.

638

To handle the control problem of uncertain nonlin- ear systems without satisfying matching conditions, in recent years, many backstepping-based adaptive fuzzy controllers have been developed for nonlinear systems in strict-feedback form (Yang et al., 2005; Wang et al., 2007; Zou et al., 2008; Chen et al., 2005; 2007; Tong et al., 2010a; 2010b). Among them, are those for single- input and single-output (SISO) nonlinear systems (Yang et al., 2005; Wang et al., 2007; Zou et al., 2008; Tong et al., 2010a; 2010b), those for multiple-input and multiple- output (MIMO) nonlinear systems (Chen et al., 2005;

2007), and the ones for SISO nonlinear systems with dy- namics and dynamical disturbances (Tong et al., 2010a;

2010b).

In general, adaptive fuzzy backstepping control can provide a systematic methodology of solving tracking or regulation control problems, where fuzzy systems are used to approximate unknown nonlinear functions. Typ- ically, adaptive fuzzy controllers are constructed recur- sively in the framework of the traditional backstepping design technique. The main features of these adaptive ap- proaches include the facts that (i) they can deal with those nonlinear systems without satisfying the matching condi- tions, and (ii) they do not require unknown nonlinear func- tions being linearly parameterized (Kanellakopopoulos et al., 1991; Kristic et al., 1992; 1995; Qian et al., 2002;

Jiang et al., 1998; 1999). Therefore, approximator-based adaptive fuzzy backstepping control has attracted great in- terest in the intelligent control community.

Despite these efforts regarding adaptive fuzzy back- stepping control, the proposed adaptive fuzzy backstep- ping control methods are all based on the assumption that the states of the systems to be controlled can be mea- sured directly. As noted by Wang (1994), in practice, state variables are often unmeasurable for many nonlin- ear systems. In such cases, some output feedback con- trol schemes should be applied. It is worth mentioning that, in the case of linear systems, output-feedback con- trol problems can be solved by combining state-feedback controllers with the state observer. However, the separa- tion principle doses not hold for nonlinear systems (Kris- tic et al., 1995; Qian et al., 2002). Thus, the adaptive out- put feedback control design is more complex and difficult than the counterpart based on state feedback.

Motivated by the above observations, in this paper, a robust adaptive fuzzy output feedback control approach is proposed for a class of SISO strict-feedback nonlin- ear systems with modeled dynamics and dynamical distur- bances, without measurements of states. Fuzzy logic sys- tems are utilized to approximate unknown nonlinear func- tions, K-filters are used for estimating unmeasured states, and, combining the backstepping technique and the small- gain theorem, a new stable adaptive fuzzy output feedback robust control scheme is developed. The main advantages of the proposed control scheme are as follows: (i) by de-

signing K-filters as a state observer, the proposed control method does not require that all the states of the system be measured directly, which is a common assumption in the existing adaptive fuzzy backstepping controller (Yang et al., 2005; Wang et al., 2007; Tong et al., 2010a; 2010b);

(ii) by combining backstepping design with input-to-state practically stability (ISpS) and the small-gain theorem, the proposed control method has a strong robustness to the modeled dynamics and dynamical disturbances, and the stability of entire closed-loop systems can be guaranteed by the small-gain theorem.

It is noted that, in recent years, several adaptive fuzzy backsteping control approaches have also been developed by Yang et al. (2005) and Tong et al. (2010a; 2010b) for some strict-feedback nonlinear systems based on small- gain theorem. However, the approach of Yang et al.

(2005) can only control a class of nonlinear systems with- out unmodeled dynamics or dynamical disturbances and requires that the states of the controlled systems must be measured. Although the approaches of Tong et al. (2010a;

2010b) have addressed the same class of nonlinear sys- tems as this paper, they also require that the states of the nonlinear systems must be measured. Therefore, they cannot be applied to nonlinear systems with unmeasured states.

2. Problem formulations and some preliminaries

2.1. Model description and basic assumptions. Con- sider a class of strict-feedback nonlinear systems with un- modeled dynamics and dynamical disturbances given by the following equations:

ζ = q(ζ, y), ˙

˙x 1 = x 2 + f 1,0 (y) + f 1 (y) + Δ 1 (ζ, y), .. .

˙x n−1 = x n + f n−1,0 (y) + f n−1 (y) + Δ _n−1 (ζ, y),

˙x n = b 0 σ(y)u + f n,0 (y) + f n (y) + Δ n (ζ, y), y = x 1 ,

(1)

where x = [x 1 , . . . , x n ] ^T ∈ R ⁿ is the state vector, u ∈ R is the control input, y ∈ R is the output; σ(y) is a known smooth nonlinear function (σ(y) = 0), and f i (y), 1 ≤ i ≤ n is an unknown smooth nonlinear func- tion; f i,0 (y), 1 ≤ i ≤ n is a known smooth nonlinear function; ζ represents unmodeled dynamics and Δ i (ζ, y), 1 ≤ i ≤ n represents disturbances related to unmodeled dynamics; b 0 = 0 is an unknown constant and the sign of b 0 is known. In this paper, it is assumed that only y = x 1

is available for control design.

In the sequel, the following assumptions are imposed

on the system (1):

639 Assumption 1. (Jiang et al., 1998; 1999) For each 1 ≤

i ≤ n, there exists an unknown positive constant p ^∗ _i such that

|Δ i | ≤ p ^∗ _i ψ i1 ( |y|) + p ^∗ _i ψ i2 ( |ζ|), (2) where ψ i1 and ψ i2 are two known nonnegative smooth functions. Without loss of generality, it is assumed that ψ i2 (0) = 0.

It is worth mentioning that Assumption 1 implies that the allowed class of uncertainties Δ _i (x, ς, t) satisfies the so-called triangular bounds condition in terms of x and ς.

The same or similar assumptions can be found in recent works (Jiang et al., 1998; 1999). Such a restriction is cru- cial in controller design.

Definition 1. (Kristic et al., 1995) A continuous function γ: [0, a) → R + is said to belong to class κ if it is strictly increasing and γ(0) = 0. It is said to belong to class κ ∞

if a = ∞ and γ(r) → ∞ as r → ∞.

Assumption 2. (Jiang et al., 1998; 1999) Unmodeled dy- namics are input-to-state practically stable (ISpS), i.e., the system ˙ ζ = q(ζ, y) has an ISpS Lyapunov function V 0 (ζ) such that

α 1 ( |ζ|) ≤ V ₀ (ζ) ≤ α 2 ( |ζ|),

∂V 0

∂ζ q(ζ, y) ≤ −α 0 ( |ζ|) + γ 0 ( |y|) + d 0 , (3) where α 0 , α 1 , α 2 and γ 0 are κ ∞ -functions defined by Kristic et al. (1995), d 0 is a nonnegative constant.

Control objectives: The control task is to design an adap- tive fuzzy controller using output y only of the form

χ = v(χ, y), ˙ u = μ(χ, y), (4) such that all the signals of the closed-loop systems (1) and (4) are uniformly ultimately bounded. Furthermore, the output can be forced to a small neighborhood of the origin.

Definition 2. (Coddington, 1989) Let f be a function defined for (x, y) in a set S. We say that f satisfies locally the Lipschitz condition on S if there exists a constant M >

0 such that

|f(x, y ₁ ) − f(x, y ₂ ) | ≤ M |y ₁ − y ₂ |

for all (x, y 1 ) , (x, y 2 ) in S. The constant M is called a Lipschitz constant.

Lemma 1. (Jiang et al., 1996) Given the interconnected systems

x ˙ 1 = f 1 (x 1 , x 2 , u 1 ), (5) x ˙ 2 = f 2 (x 1 , x 2 , u 2 ), (6) where, for i = 1, 2, x i ∈ R ⁿ

ⁱ

, u i ∈ R ^m

ⁱ

and f i : R ⁿ

¹

× R ⁿ

²

× R ^m

ⁱ

→ R ⁿ

ⁱ

locally satisfies the Lipschitz condition.

Assume that, for i = 1, 2, there exists an ISpS-Lyapunov function V i for the x i -subsystems such that the following holds:

1. there exist κ ∞ -functions ϑ i1 and ϑ i2 such that

ϑ i1 ( |x i |) ≤ V i (x i ) ≤ ϑ i2 ( |x i |), ∀x i ∈ R ⁿ

ⁱ

, (7) 2. there exist κ ∞ -function α ^ _i and κ-functions χ i , γ i and some constant c i ≥ 0 such that, if

V 1 (x 1 ) ≥ max{χ ₁ (V 2 (x 2 )), γ 1 ( |u ₁ |) + c ₁ }, then

∇V ₁ (x 1 )f 1 (x 1 , x 2 , u 1 ) ≤ −α ^ ₁ (V 1 ), (8) and, if

V 2 (x 2 ) ≥ max{χ 2 (V 1 (x 1 )), γ 2 ( |u 2 |) + c 2 }, then

∇V 2 (x 2 )f 2 (x 1 , x 2 , u 2 ) ≤ −α ^ ₂ (V 2 ). (9) A nonlinear small-gain condition is given by Jiang et al. (1996), under which an ISpS-Lyapunov function for the interconnected systems (5)–(6) may be expressed in terms of ISpS-Lyapunov functions for the two subsys- tems.

Theorem 1. (Jiang et al., 1996) Assume that, for i = 1, 2, the x i -subsystems have an ISpS-Lyapunov V i satisfying (7)–(9). If there exists c 0 ≥ 0 such that

χ 1 ◦ χ 2 (r) < r, ∀r > c 0 , (10) then the interconnected system (5)–(6) is ISpS. Further- more, if c 0 = c 1 = c 2 = 0, then the system is ISS.

2.2. Fuzzy logic systems and system modeling. A fuzzy logic system (FLS) consists of four parts: a knowl- edge base, a fuzzifier, a fuzzy inference engine working on fuzzy rules, and a defuzzifier. The knowledge base for an FLS is composed of a collection of fuzzy If-then rules of the following form:

R ^l : If x 1 is F ₁ ^l and x 2 is F ₂ ^l and . . . and x n is F _n ^l , then y is G ^l , l = 1, 2, . . . , N ,

where x = [x 1 , . . . , x n ] ^T and y are the FLS input and output, respectively; F _i ^l and G ^l are fuzzy sets, associated with the fuzzy functions μ F

_i^l

(x i ) and μ G

^l

(y); N is the rule number.

Through the singleton function, center average de- fuzzification and product inference, the FLS can be ex- pressed as

y(x) =

 N l=1

y ¯ l

 n

i=1 μ _F

_i^l

(x i )

 N l=1

  n i=1

μ _F

_i^l

(x i )

, (11)

640

where ¯ y l = max

y∈R μ G

^l

(y).

Define the fuzzy basis functions as

ϕ l =

 n i=1

μ _F

_i^l

(x i )

 N l=1

(  ⁿ

i=1 μ _F

_i^l

(x i )) .

If we let θ = [¯ y 1 , ¯ y 2 , . . . , ¯ y N ] ^T = [¯ θ 1 , ¯ θ 2 , . . . , ¯ θ N ] ^T and ϕ ^T (x) = [ϕ 1 (x), . . . , ϕ N (x)] then the FLS (11) can be rewritten as

y(x) = θ ^T ϕ(x). (12)

It has been proved that the fuzzy logic system (12) can approximate any continuous function f (x) over a compact set Ω ⊂ R ^q to any arbitrary accuracy as

f (x) = θ ^∗T ϕ(x) + ε(x), ∀x ∈ Ω, (13) where θ ^∗ is an ideal constant parameter, and ε(x) is the fuzzy minimums approximation error, which is defined by Wang (1994) as

θ ^∗ = arg min

θ∈U {sup

y∈Ω

 f (x) − θ ^T ϕ(x)}.

By employing the FLS to approximate the unknown smooth function f i (y) in (1) and assuming that

f i (y) = θ ^∗T _i ϕ _i (y) + ε i (y), (14) denote the fuzzy minimums approximation error vector as ε(y) = [ ε 1 (y) . . . ε n (y) ] ^T .

Assumption 3. The fuzzy minimum approximation error vector ε(y) satisfies ε(y) ≤ β, where β is an unknown positive constant, and · represents the 2-norm of a vec- tor.

By substituting (14) into (1), the system (1) can be expressed as

ζ = q(ζ, y), ˙

˙x 1 = x 2 + f 1,0 (y) + θ ^∗T ₁ ϕ ₁ (y) + ε 1 (y) + Δ 1 (ζ, y), .. .

˙x n−1 = x n + f n−1,0 (y) + θ ^∗T _n−1 ϕ _n−1 (y) + ε n−1 (y) + Δ n−1 (ζ, y),

˙x n = b 0 σ(y)u + f n,0 (y) + θ n ^∗T ϕ n (y) + ε n (y) + Δ n (ζ, y),

y = x 1 .

(15)

Rewrite (15) as ζ = q(ζ, y), ˙

x = Ax + f ˙ 0 (y) + Φ ^T (y)θ + ε(y) +Δ + 

0 b 0 ,  _T σ(y)u y = C ₁ ^T x,

(16)

where

A =

⎡

⎢ ⎣ 0

.. . I n−1

0 · · · 0

⎤

⎥ ⎦ ,

f 0 (y) =

⎡

⎢ ⎣

f 1,0 (y) .. . f n,0 (y)

⎤

⎥ ⎦ ,

Φ ^T (y) =

⎡

⎢ ⎣ ϕ ^T ₁ (y)

. ..

ϕ ^T _n (y)

⎤

⎥ ⎦

n×l

,

l = l 1 + · · · + l n , C 1 = [1, 0, . . . , 0] ^T , θ ^T = 

θ ^∗ ₁ · · · θ ^∗ _n 

1×l , Δ = 

Δ ₁ · · · Δ n

 _T . The system (16) is further rewritten as

ζ = q(ζ, y), ˙ (17)

˙x = Ax + f 0 (y) + G ^T (y, u)ϑ + ε(y) + Δ, (18) y = C ₁ ^T x,

where

ϑ =

 b 0

θ



(l+1)×1 , G ^T (y, u) =

 0 _(n−1)×1 1



σ(y)u, Φ ^T (y)

 . Choose a vector k = [k 1 , · · · , k n ] ^T so that the ma- trix A 0 = A − kC ₁ ^T is a strict Hurwitz matrix, i.e., given a positive definite matrix Q = Q ^T > 0, there exists a positive definite matrix P = P ^T > 0 such that

P A 0 + A ^T ₀ P = −Q. (19)

3. Adaptive fuzzy controller design and stability analysis

Note that in the system (1) or (16), the states x 2 , x 3 , . . ., x n are an unmeasured, b 0 and θ are an unknown constant and unknown parameter vector, respectively. Thus, the states of the system (1), b 0 and θ should be estimated by using the filters given by Kristic et al. (1995) as well as Ye (2001). Define the virtual state estimate as

x = ξ + Ω ˆ ^T ϑ. (20) According to Kristic et al. (1995) and from (18), the K-filters may be defined as follows:

ξ = A ˙ 0 ξ + ky + f 0 (y),

Ω ˙ ^T = A 0 Ω ^T + G ^T (y, u). (21)

641 Note that the parameter vector ϑ is unknown, and, as such,

it cannot be used in control design. Therefore, an estimate ϑ of the parameter vector ϑ need to be obtained later. On ˆ the other hand, the virtual state estimate defined by (20) is not used in control design, and the actual state estimate should be ˆ x = ξ + Ω ˆ ^T ϑ. Denote by v ˆ 0 the first column of Ω ^T . The vector v 0 is governed by

˙v 0 = A 0 v 0 + C n σ(y)u, (22) where C n = [0, · · · , 0, 1] ^T . In view of (21) and (22), Ω is expressed as

Ω ^T = [v 0 , Ξ]. (23) From (21), one obtains

Ξ = A ˙ 0 Ξ + Φ ^T (y). (24) Define the observation error vector e as

e = [e 1 , e 2 , · · · , e n ] ^T = x − ˆ x

p ^∗ , (25) where p ^∗ = max 

p ^∗ _i , p ^∗2 _i , 1; 1 ≤ i ≤ n 

is an unknown constant. The time derivative of e can be expressed as

˙e = A 0 e + ε(y) + Δ

p ^∗ . (26)

From the second equation in (16), one obtains

y = x ˙ 2 + f 1,0 (y) + θ ₁ ^∗T ϕ 1 (y) + ε 1 (y) + Δ 1 . (27) Since x 2 is unavailable, it is replaced by available filter signals. From (18), one has

x = ξ + Ω ^T ϑ + x − ˆ x

= ξ + Ω ^T ϑ + p ^∗ e. (28) Therefore, using (28), x 2 is expressed as

x 2 = ξ 2 + Ω ^T ₍₂₎ ϑ + p ^∗ e 2

= b 0 v 0,2 + ξ 2 + [0, Ξ (2) ]ϑ + p ^∗ e 2 , (29) where Ω ^T ₍₂₎ and Ξ ₍₂₎ are the second rows of Ω ^T and Ξ, respectively.

Substituting (29) into (27) yields

y = b ˙ 0 v 0,2 +ξ 2 +f 1,0 (y)+¯ ω ^T ϑ+p ^∗ e 2 +ε 1 (y)+Δ 1 , (30) where the “regressor” ω and the “truncated regressor” ¯ ω are defined by Kristic et al. (1995) as follows

ω = [v 0,2 , Φ ^T ₍₁₎ (y) + Ξ (2) ] ^T , (31) ω = [0, Φ ¯ ^T ₍₁₎ (y) + Ξ (2) ] ^T . (32) From (22), we obtain

˙v 0,i = v 0,i+1 − k i v 0,1 , i = 2, . . . , n, −1, (33)

˙v 0,n = σ(y)u − k n v 0,1 . (34)

Define a change of coordinates as

z 1 = yλ ^ (y ² ), (35)

z i = v 0,i − π i−1 , i = 2, · · · , n, (36) where λ ^ (y ² ) is the derivative of a smooth class κ ∞ -function λ(y ² ) , and λ ^ (y ² ) = 0, which will be chosen later.

After the above preparations, adaptive fuzzy back- stepping control design is given by the following proce- dures.

Step 0: Consider the following Lyapunov function:

V 0 = e ^T P e. (37)

The time derivative of V 0 along (26) is V ˙ 0 = e ^T (A ^T ₀ P + P A 0 )e + 2

p ^∗ e ^T P (ε + Δ). (38) By Assumption 1 and Young’s inequality 2ab ≤ a ² + b ² and p ^∗ ≥ 1, we have

2

p ^∗ e ^T P Δ ≤ 2

p ^∗ e P  Δ

≤ 2 p ^∗

 n i=1

e P  |Δ i |

≤ 2 e P  (

 n k=1

ψ k1 ( |y|) +

 n k=1

ψ k2 ( |ζ|)), (39)

2 e P 

 n k=1

ψ k1 ( |y|)

≤ e ² + P  ² (

 n k=1

ψ k1 ( |y|)) ² . (40) Since ψ i1 is a smooth function, using the same proof of Jiang (1999), we get

(

 n k=1

ψ k1 ( |y|)) ² ≤ y ² φ 1 (y) + d ⁰ ψ , (41)

where φ 1 is a smooth nonnegative function, and d ⁰ _ψ = (  _n

i=1 ψ i1 (0)) ² is a constant.

Substituting (41) into (40) yields 2 e P   ⁿ

k=1 ψ k1 ( |y|)

≤ e ² + P  ² y ² φ 1 (y) + P  ² d ⁰ _ψ .

(42)

Using Young’s inequality, we have 2 e P   ⁿ

k=1

ψ k2 ( |ζ|)

≤ e ² + P  ² (  ⁿ

k=1

ψ k2 ( |ζ|)) ² ,

(43)

642 2

p ^∗ e ^T P ε ≤ 2 e P  ε ≤ e ² + P  ² β ² . (44) Substituting (42)–(44) into (38), we obtain

V ˙ 1 ≤ − [λ min (Q) − 3] e ² + P  ² (  ⁿ

k=1

ψ k2 ( |ζ|)) ² + P  ² y ² φ 1 (y) + P  ² β ² + P  ² d ⁰ _ψ .

(45)

Step 1: Consider the following Lyapunov function:

V 1 = V 0 + ¹ ₂ λ(y ² ) + ¹ ₂ ϑ ˜ ^T Γ ⁻¹ ϑ ˜

+ ¹ ₂ γ ₁ ⁻¹ β ˜ ² + ¹ ₂ γ ₂ ⁻¹ p ˜ ² + ¹ ₂ γ ₃ ⁻¹ |b ₀ | ˜κ ² , (46) where Γ = Γ ^T > 0, γ 1 > 0, γ 2 > 0 andγ 3 > 0 are design constants; ˜ ϑ = ϑ − ˆ ϑ, ˜ β = β − ˆ β, ˜ p = p − ˆ p and κ = κ − ˆ ˜ κ; ˆ ϑ, ˆ β, ˆ p and ˆ κ are the estimates of ϑ, β, p and κ, respectively. Here κ = b ⁻¹ ₀ is an unknown constant.

Define π 1 = ˆ κ¯ π 1 , where ¯ π 1 is a stabilizing function to be designed later.

The time derivative of V 1 along (30) is V ˙ 1 = ˙ V 0 + yλ ^ (y ² ) ˙ y − ˜ ϑ ^T Γ ⁻¹ ϑ ˙ˆ

− γ ₁ ⁻¹ β ˜ β − γ ˙ˆ ₂ ⁻¹ p ˙ˆ ˜ p − γ ₃ ⁻¹ |b 0 | ˜κ ˙ˆκ

= ˙ V 0 + yλ ^ (y ² )(b 0 v 0,2 + ξ 2

+ f 1,0 (y) + ¯ ω ^T ϑ + p ^∗ e 2

+ ε 1 (y) + Δ 1 ) − ˜ϑ ^T Γ ⁻¹ ϑ − γ ˙ˆ ₁ ⁻¹ β ˜ β ˙ˆ

− γ ₂ ⁻¹ p ˙ˆ ˜ p − γ ₃ ⁻¹ |b 0 | ˜κ ˙ˆκ.

(47)

Substituting (35) into (47) results in

V ˙ 1 = ˙ V 0 + z 1 (b 0 v 0,2 + ξ 2 + f 1,0 (y) + ¯ ω ^T ϑ + p ^∗ e 2 + ε 1 (y) + Δ 1 )

− ˜ϑ ^T Γ ⁻¹ ϑ − γ ˙ˆ ₁ ⁻¹ β ˜ β − γ ˙ˆ ₂ ⁻¹ p ˙ˆ ˜ p

− γ ₃ ⁻¹ |b ₀ | ˜κ ˙ˆκ

= ˙ V 0 + b 0 z 1 z 2 − z ₁ b 0 κ¯ ˜ π 1

+ z 1 (¯ π 1 + ξ 2 + f 1,0 (y) + ¯ ω ^T ϑ) + yλ ^ (y ² )(p ^∗ e 2 + ε 1 (y) + Δ 1 )

− ˜ϑ ^T Γ ⁻¹ ϑ − γ ˙ˆ ₁ ⁻¹ β ˜ β ˙ˆ

− γ ₂ ⁻¹ p ˙ˆ ˜ p − γ ₃ ⁻¹ |b 0 | ˜κ ˙ˆκ.

(48)

Using Assumption 1 and Young’s inequality, we have yλ ^ (y ² )(p ^∗ e 2 + Δ ₁ )

≤ p ^∗  yλ ^ (y ² ) |e ₂ | +  yλ ^ (y ² ) | Δ ₁ |

≤ p ^∗  yλ ^ (y ² ) |e 2 | + p ^∗ ₁  yλ ^ (y ² ) (ψ 11 ( |y|)

− ψ 11 ( |0|)) + p ^∗ ₁  yλ ^ (y ² )  ψ 12 ( |ζ|) + p ^∗ ₁  yλ ^ (y ² )  ψ 11 ( |0|)

≤ |e ₂ | ² + p ^∗2 ₁

4 (yλ ^ (y ² )) ² + p ^∗ ₁ y ² λ ^ (y ² ) ¯ ψ 11 ( |y|) + p ^∗2 ₁

2 (yλ ^ (y ² )) ² + ψ ² ₁₂ ( |ζ|) + ψ ² ₁₁ (0),

(49)

where ¯ ψ 11 ( |y|) =  ₁

0 ψ ^ ₁₁ (s |y|) ds.

Using the proof of Jiang (1999), given any d 11 > 0, there exists a smooth function ˆ ψ 11 with ˆ ψ 11 (0) = 0, such that

|y| ¯ ψ 11 ( |y|) ≤ y ˆ ψ 11 (y) + d 11 , ∀y ∈ R, Therefore, (49) can be rewritten as

yλ ^ (y ² )(p ^∗ e 2 + Δ ₁ )

≤ e ² + pφ 11 (y)(yλ ^ (y ² )) ² + ψ ₁₂ ² ( |ζ|) + d ² ₁₁ + ψ ₁₁ ² (0),

(50)

where p = (p ^∗ ) ² and φ 11 (y) = 1 + 1 

(2λ ^ (y ² )) + 1 

(2λ ^ (y ² )) ˆ ψ ² ₁₁ (y) is a smooth nonnegative function.

Note that, for ∀ς > 0, the following inequality holds:

|r| − r tanh(r/ς) ≤ 0.2785ς. (51) By (51), one has

|ε 1 (y)z 1 | − z 1 η 1 β tanh

 z 1 η 1

ς



≤ β 

|z ₁ | − z ₁ η 1 tanh

 z 1 η 1

ς



≤ 0.2785ςβ = ς ^ , (52)

where ς is an arbitrary small constant and η 1 = −1.

Substituting (45), (50) and (52) into (48) yields V ˙ 1 ≤ − [λ min (Q) − 4] e ² + z 1



¯ π 1 + ξ 2

+ f 1,0 (y) + ¯ ω ˆ ϑ + P  λ ^

2

yφ 1 (y) + ˆ pφ 11 (y)z 1 + η 1 β tanh ˆ

 z 1 η 1

ς



+ b 0 z 1 z 2 + ˜ ϑ ^T (¯ ωz 1 − Γ ⁻¹ ϑ) ˙ˆ + ˜ β



z 1 η 1 β tanh

 z 1 η 1

ς

 − γ ⁻¹ ₁ β ˙ˆ



+ ˜ p(φ 11 (y)z ₁ ² − γ ₂ ⁻¹ p) − ˜ ˙ˆ κ(z 1 b 0 ¯ π 1

+ γ ₃ ⁻¹ |b 0 | ˙ˆκ) + P  ²   ⁿ

k=1

ψ k2 ( |ζ|)  ₂ + ψ ² ₁₂ ( |ζ|) + P  ² β ² + P  ² d ⁰ _ψ + ψ ² ₁₁ (0) + ς ^ + d ² ₁₁ .

(53)

643 Choose the stabilizing control function ¯ π 1 , tuning func-

tions and parameters adaptation laws as π ¯ 1 = −yρ(y ² ) − ξ 2 − f 1,0 (y) − ¯ ω ˆ ϑ

− P  λ ^

2

yϕ 1 (y) − ˆ pφ 11 (y)z 1

− η 1 β tanh ˆ

 z 1 η 1

ς



, (54)

τ 1 = ¯ ωz 1 , (55)

σ 1 = z 1 η 1 β tanh

 z 1 η 1

ς



, (56)

λ ¯ 1 = φ 11 (y)z ₁ ² , (57)

˙ˆκ = −γ 3 (sgn(b 0 )¯ π 1 z 1 + μˆ κ), (58) where ρ(y ² ) is a smooth non-decreasing function with ρ(0) > 0, and μ > 0 is a design parameter. Substitut- ing (54)–(58) into (53) yields

V ˙ 1 ≤ − [λ _min (Q) − 4] e ² − z ₁ yρ(y ² ) + b 0 z 1 z 2 + ˜ ϑ ^T (τ 1 − Γ ⁻¹ ϑ) ˙ˆ + ˜ β(σ 1 − γ ₁ ⁻¹ ˙ˆβ) + ˜p(¯λ 1 − γ ₂ ⁻¹ p) ˙ˆ + μ ˜ κˆ κ + P  ² (

 n k=1

ψ k2 ( |ζ|)) ² + ψ ² ₁₂ ( |ζ|) + P  ² β ² + P  ² d ⁰ ψ

+ ς ^ + ψ ² ₁₁ (0) + d ² ₁₁ .

(59)

Step 2: The time derivative of z 2 along (36) is

˙z 2 = v 0,3 − k 2 v 0,1 − ∂π 1

∂y (ξ 2 + f 1,0 (y) + ω ^T ϑ + p ^∗ e 2 + Δ ₁ + ε 1 (y))

− ∂π 1

∂ξ (A 0 ξ + ky) − ∂π 1

∂Ξ (A 0 Ξ + Φ ^T (y))

− ∂π 1

∂v 0 ˙v 0 − ∂π 1

∂κ ˙κ − ∂π 1

∂ ˆ ϑ Γ(τ 1 − μ ˆϑ)

− ∂π 1

∂ ˆ β γ 1 (σ 1 − μ ˆβ) − ∂π 1

∂ ˆ p γ 2 (¯ λ 1 − μ ˆp)

− ∂π 1

∂ ˆ ϑ (˙ˆϑ − Γτ 1 + Γμ ˆ ϑ)

− ∂π 1

∂ ˆ β (˙ˆβ − γ 1 σ 1 + γ 1 μ ˆ β)

− ∂π 1

∂ ˆ p (˙ˆp − γ 2 λ ¯ 1 + γ 2 μ ˆ p).

(60)

Consider the Lyapunov function V 2 = V 1 + 1

2 z ₂ ² . (61)

The time derivative of V 2 along the solutions of (60) is V ˙ 2 ≤ ˙V 1 + z 2 [z 3 + π 2 − k 2 v 0,1 − ∂π 1

∂y (ξ 2

+ f 1,0 (y) + ω ^T ϑ) − ∂π 1

∂ξ (A 0 ξ + ky)

− ∂π 1

∂v 0 ˙v 0 − ∂π 1

∂Ξ (A 0 Ξ + Φ ^T (y))

− ∂π 1

∂ ˆ ϑ Γ(τ 1 − μ ˆϑ) − ∂π 1

∂ ˆ β γ 1 (σ 1 − μ ˆβ)

− ∂π 1

∂ ˆ p γ 2 (¯ λ 1 − μ ˆp)

− ∂π 1

∂ ˆ ϑ ( ϑ − Γτ ˙ˆ 1 + Γμ ˆ ϑ)

− ∂π 1

∂ ˆ β ( β − γ ˙ˆ 1 σ 1 + γ 1 μ ˆ β)

− ∂π 1

∂ ˆ p ( ˙ˆ p − γ 2 ¯ λ 1 + γ 2 μ ˆ p) − ∂π 1

∂κ ˙κ]

+ 

 ∂π 1

∂y p ^∗ e 2 z 2

  + 

 ∂π 1

∂y Δ ₁ z 2

 

+ 

 ∂π 1

∂y ε 1 (y)z 2 

.

(62)

By Assumption 1 and Young’s inequality, using the similar derivations in Step 1, one obtains the following inequalities:

  ∂π 1

∂y p ^∗ e 2 z 2 

 ≤ e ^T e + p

 ∂π 1

∂y

 ₂

z ₂ ² , (63)

  ∂π 1

∂y Δ ₁ z 2 



≤ 

 ∂π 1

∂y (p ^∗ ₁ ψ 11 ( |y|) + p ^∗ ₁ ψ 12 ( |ζ|)) z 2 



≤ p ^∗ ₁ 

 ∂π 1

∂y z 2 

ψ ¹¹ ( |y|) + 1

4 p( ∂π 1

∂y ) ² z ² ₂ + ψ ² ₁₂ ( |ζ|)

= p ^∗ ₁ 

 ∂π 1

∂y z 2 

(ψ ¹¹ ( |y|) − ψ 11 (0)) + p ^∗ ₁ 

 ∂π 1

∂y z 2 

ψ ¹¹ (0) + 1

4 p( ∂π 1

∂y ) ² z ² ₂ + ψ ² ₁₂ ( |ζ|)

≤ p ^∗ ₁ 

 ∂π 1

∂y z 2

 |y| ¯ψ ¹¹ ( |y|) + 1 2 p( ∂π 1

∂y ) ² z ₂ ² + ψ ² ₁₂ ( |ζ|) + ψ ₁₁ ² (0),

(64)

p ^∗ ₁ 

 ∂π 1

∂y z 2 

|y| ¯ψ ¹¹ ( |y|)

≤ p ^∗ ₁ 

 ∂π 1

∂y z 2 

(y ˆψ ¹¹ (y) + d 11 )

≤ p  ∂π 1

∂y

 ₂

z ² ₂ ψ ˆ ² ₁₁ (y) + 1 4 y ² + 1

2 p

 ∂π 1

∂y

 ₂ z ₂ ² + 1

2 d ² ₁₁ ,

(65)

where d 11 > 0 is a known constant, ˆ ψ 11 is a known

644

smooth function with ˆ ψ 11 (0) = 0.

Substituting (65) into (64) yields

  ∂π 1

∂y Δ ₁ z 2 

 ≤ p( ∂π 1

∂y ) ² z ² ₂ ( ˆ ψ ₁₁ ² (y) + 1) + 1

4 y ² + ψ ₁₂ ² ( |ζ|) + ψ ₁₁ ² (0) + 1

2 d ² ₁₁ . (66) Note that

  ∂π 1

∂y ε 1 (y)z 2 

 − z ² η 2 β tanh

 z 2 η 2

ς



≤ 0.2785ςβ = ς ^ , (67) where η 2 = −∂π 1 /∂y.

Substituting (63), (66) and (67) into (62), we obtain V ˙ 2 ≤ − [λ _min (Q) − 5] e ² − z ₁ yρ(y ² )

+ 1

4 y ² + z 2 [z 3 + π 2 + ˆ b 0 z 1 − k ₂ v 0,1

− ∂π 1

∂y (ξ 2 + f 1,0 (y) + ω ^T ϑ) + H ˆ 2 ]

− z 2 ∂π 1

∂ ˆ ϑ ( ϑ − Γτ ˙ˆ 1 + Γμ ˆ ϑ)

− z 2 ∂π 1

∂ ˆ β ( β − γ ˙ˆ 1 σ 1 + γ 1 μ ˆ β)

− z 2 ∂π 1

∂ ˆ p ( ˙ˆ p − γ 2 ¯ λ 1 + γ 2 μ ˆ p) + ˜ ϑ ^T (τ 1 − ∂π 1

∂y ωz 2 + z 2 − Γ ⁻¹ ϑ) ˙ˆ + ˜ β



σ 1 + η 2 z 2 tanh

 z 2 η 2

ς

 − γ ₁ ⁻¹ β ˙ˆ



+ ˜ p(¯ λ 1 + ( ∂π 1

∂y ) ² z ₂ ² ( ˆ ψ ² ₁₁ (y) + 2) − γ ₂ ⁻¹ p) ˙ˆ + μ ˜ κˆ κ + P  ² (

 n k=1

ψ k2 ( |ζ|)) ²

+ 2ψ ² ₁₂ ( |ζ|) + P  ² β ² + P  ² d ⁰ _ψ + 2ς ^ + 2ψ ² ₁₁ (0) + 3

2 d ² ₁₁ ,

(68)

where H 2 = − ∂π 1

∂ξ (A 0 ξ + ky) − ∂π 1

∂Ξ (A 0 Ξ + Φ ^T (y))

− ∂π 1

∂v 0 ˙v 0 − ∂π 1

∂κ ˙κ − ∂π 1

∂ ˆ ϑ Γ(τ 1 − μ ˆϑ)

− ∂π 1

∂ ˆ β γ 1 (σ 1 − μ ˆβ) − ∂π 1

∂ ˆ p γ 2 (¯ λ 1 − μ ˆp)

− ˆβη 2 tanh

 z 2 η 2

ς



+ ˆ p( ∂π 1

∂y ) ² z ₂ ² ( ˆ ψ ₁₁ ² (y) + 2),

 = [ z 1 0 . . . 0 ] ^T .

Choose the tuning functions and parameters adapta- tion laws as follows:

τ 2 = τ 1 − z ₂  ∂π 1

∂y ω − 



, (69)

τ i = τ i−1 − z i ∂π i−1

∂y ω, i = 3, . . . , n, (70) σ i = σ i−1 + z i η i tanh

 z i η i

ς



, i = 2, . . . , n, (71)

¯ λ i = ¯ λ i−1 + ∂π i−1

∂y

 ₂

z _i ² ( ˆ ψ ₁₁ ² (y) + 2), i = 2, . . . , n, (72)

ϑ = Γ(τ ˙ˆ n − μ ˆϑ), (73)

β = γ ˙ˆ 1 (σ n − μ ˆβ), (74)

p = γ ˙ˆ 2 (¯ λ n − μ ˆp), (75)

where

η i = − ∂π i−1

∂y , i = 2, . . . , n.

Define

− ∂π 1

∂ ˆ ϑ ( ϑ − Γτ ˙ˆ 1 + Γμ ˆ ϑ) =

 n j=2

Δ _1,j z j , (76)

− ∂π 1

∂ ˆ β ( β − γ ˙ˆ 1 σ 1 + γ 1 μ ˆ β) =

 n j=2

Λ _1,j z j , (77)

− ∂π 1

∂ ˆ p ( ˙ˆ p − γ 2 λ ¯ 1 + γ 2 μ ˆ p) =

 n j=2

A _1,j z j , (78)

where

Δ _1,j = ∂π 1

∂ ˆ ϑ Γ ∂π j−1

∂y ω, Λ _1,j = ∂π 1

∂ ˆ β γ 1 η j tanh

 η j z j

ς

 , A _1,j = ∂π 1

∂ ˆ p γ 2

 ∂π j−1

∂y

 ₂

z ² _j ( ˆ ψ ² ₁₁ (y) + 2),

i = 2, . . . , n.

Choose the stabilizing control function π 2 as

π 2 = −ˆb ₀ z 1 − c ₂ z 2 + ∂π 1

∂y (ξ 2

+ f 1,0 (y) + ω ^T ϑ) + k ˆ 2 v 0,1

− (Δ _1,2 + Λ _1,2 + A _1,2 ) − H ₂ ,

(79)

where c 2 > 0 is a design constant.

645 Substituting (69) and (71)–(79) into (68) yields

V ˙ 2 ≤ − [λ min (Q) − 5] e ² − z 1 yρ(y ² ) + 1

4 y ² + z 2 z 3 + ˜ ϑ ^T (τ 2 − Γ ⁻¹ ϑ) + μ ˜ ˙ˆ κˆ κ + ˜ β(σ 2 − γ ₁ ⁻¹ β) + ˜ ˙ˆ p(¯ λ 2 − γ ₂ ⁻¹ p) ˙ˆ + P  ² (

 n k=1

ψ k2 ( |ζ|)) ² + 2ψ ² ₁₂ ( |ζ|)

+ P  ² β ² +

 n j=3

(Δ _1,j + Λ _1,j + A _1,j )z 2 z j

+ P  ² d ⁰ _ψ + 2ς ^ + 2ψ ₁₁ ² (0) + 3 2 d ² ₁₁ .

(80)

Step i (i = 3, . . . , n − 1): A similar procedure in Step 2 is employed recursively for consemtive steps. The time derivative of z i along (36) is

˙z i = v 0,i+1 − k _i v 0,1 − ∂π i−1

∂y (ξ 2 + f 1,0 (y) + ω ^T ϑ + p ^∗ e 2 + Δ ₁ + ε 1 (y))

− ∂π i−1

∂ξ (A 0 ξ + ky)

− ∂π i−1

∂Ξ (A 0 Ξ + Φ ^T (y)) − ∂π i−1

∂v 0 ˙v 0

− ∂π i−1

∂κ ˙κ − ∂π i−1

∂ ˆ ϑ Γ(τ i−1 − μ ˆϑ)

− ∂π i−1

∂ ˆ β γ 1 (σ i−1 − μ ˆβ)

− ∂π i−1

∂ ˆ p γ 2 (¯ λ i−1 − μ ˆp)

− ∂π i−1

∂ ˆ ϑ ( ϑ − Γτ ˙ˆ i−1 + Γμ ˆ ϑ)

− ∂π i−1

∂ ˆ β ( β − γ ˙ˆ 1 σ i−1 + γ 1 μ ˆ β)

− ∂π i−1

∂ ˆ p ( ˙ˆ p − γ 2 ¯ λ i−1 + γ 2 μ ˆ p).

(81)

Consider the following Lyapunov function:

V i = V i−1 + 1

2 z _i ² . (82) The time derivative of V i along the solutions of (81) is

V ˙ i ≤ ˙V _i−1 + z i [z i+1 + π i − k _i v 0,1

− ∂π i−1

∂y (ξ 2 + f 1,0 (y) + ω ^T ϑ)

− ∂π i−1

∂ξ (A 0 ξ + ky)

− ∂π i−1

∂Ξ (A 0 Ξ + Φ ^T (y)) − ∂π i−1

∂v 0 ˙v 0

− ∂π i−1

∂κ ˙κ − ∂π i−1

∂ ˆ ϑ Γ(τ i−1 − μ ˆϑ)

− ∂π i−1

∂ ˆ β γ 1 (σ i−1 − μ ˆβ)

− ∂π i−1

∂ ˆ p γ 2 (¯ λ i−1 − μ ˆp)

− ∂π i−1

∂ ˆ ϑ ( ϑ − Γτ ˙ˆ i−1 + Γμ ˆ ϑ)

− ∂π i−1

∂ ˆ β ( β − γ ˙ˆ 1 σ i−1 + γ 1 μ ˆ β)

− ∂π i−1

∂ ˆ p ( ˙ˆ p − γ 2 λ ¯ i−1 + γ 2 μ ˆ p)]

+ 

 ∂π i−1

∂y p ^∗ e 2 z i

  + 

 ∂π i−1

∂y Δ ₁ z i

 

+ 

 ∂π i−1

∂y ε 1 (y)z i

 .

(83)

By Young’s inequality and Assumption 1, one ob- tains the following inequalities:

  ∂π i−1

∂y p ^∗ e 2 z i

  ≤ e ^T e + p( ∂π i−1

∂y ) ² z _i ² , (84)

  ∂π i−1

∂y Δ ₁ z i

 

≤ 

 ∂π i−1

∂y (p ^∗ ₁ ψ 11 ( |y|) + p ^∗ ₁ ψ 12 ( |ζ|)) z _i 



≤ p ^∗ ₁ 

 ∂π i−1

∂y z i

 ψ ¹¹ ( |y|) + 1

4 p( ∂π i−1

∂y ) ² z ² _i + ψ ₁₂ ² ( |ζ|)

≤ p ^∗ ₁ 

 ∂π i−1

∂y z i

 |y| ¯ψ ¹¹ ( |y|) + 1

2 p( ∂π i−1

∂y ) ² z ² _i + ψ ₁₂ ² ( |ζ|) + ψ ₁₁ ² (0) (85) p ^∗ ₁ 

 ∂π i−1

∂y z i

 |y| ¯ψ ¹¹ ( |y|)

≤ p ^∗ ₁ 

 ∂π i−1

∂y z i

 (y ˆψ ¹¹ (y) + d 11 )

≤ p( ∂π i−1

∂y ) ² z i ² ψ ˆ ² ₁₁ (y) + 1

4 y ² + 1

2 p( ∂π i−1

∂y ) ² z ² _i + 1

2 d ² ₁₁ (86)

  ∂π i−1

∂y Δ ₁ z i

 

≤ p( ∂π i−1

∂y ) ² z i ² ( ˆ ψ ² ₁₁ (y) + 1) + 1

4 y ² + ψ ₁₂ ² ( |ζ|) + ψ ² ₁₁ (0) + 1

2 d ² ₁₁ (87)

646 

 ∂π i−1

∂y ε 1 (y)z i

  − z ⁱ η i β tanh( z i η i

ς )

≤ 0.2785ςβ = ς ^ , (88)

where η i = −∂π i−1 /∂y.

Substituting (84), (87) and (88) into (83) gives V ˙ i ≤ − [λ min (Q) − (i + 3)] e ² − z 1 yρ(y ² )

+ i − 1

4 y ² + z i [z i+1 + π i − k i v 0,1

− ∂π i−1

∂y (ξ 2 + f 1,0 (y) + ω ^T ϑ) + H ˆ i ]

− z i ∂π i−1

∂ ˆ ϑ ( ϑ − Γτ ˙ˆ i−1 + Γμ ˆ ϑ)

− z _i ∂π i−1

∂ ˆ β ( β − γ ˙ˆ 1 σ i−1 + γ 1 μ ˆ β)

− z i ∂π i−1

∂ ˆ p ( ˙ˆ p − γ 2 λ ¯ i−1 + γ 2 μ ˆ p) + ˜ ϑ ^T (τ i−1 − ∂π i−1

∂y ωz i − Γ ⁻¹ ϑ) ˙ˆ + ˜ β(σ i−1 + η i z i tanh

 z i η i

ς

 − γ ₁ ⁻¹ β) ˙ˆ

+ ˜ p(¯ λ i−1 +

 ∂π i−1

∂y

 ₂

z ² _i ( ˆ ψ ² ₁₁ (y) + 2)

− γ ₂ ⁻¹ p) + μ ˜ ˙ˆ κˆ κ −

 i−1 j=1

c j z _j ²

+ P  ² (

 n k=1

ψ k2 ( |ζ|)) ² + iψ ₁₂ ² ( |ζ|) + P  ² β ² + P  ² d ⁰ _ψ + iς ^ + iψ ² ₁₁ (0) + i + 1

2 d ² ₁₁ ,

(89)

H i = − ∂π i−1

∂ξ (A 0 ξ + ky) − ∂π i−1

∂Ξ (A 0 Ξ + Φ ^T (y))

− ∂π i−1

∂v 0 ˙v 0 − ∂π i−1

∂κ ˙κ − ∂π i−1

∂ ˆ ϑ Γ(τ i−1 − μ ˆϑ)

− ∂π i−1

∂ ˆ β γ 1 (σ i−1 − μ ˆβ) − ∂π i−1

∂ ˆ p γ 2 (¯ λ i−1 − μ ˆp)

− ˆβη i tanh

 z i η i

ς

 + ˆ p

 ∂π i−1

∂y

 ₂

z i ² ( ˆ ψ ₁₁ ² (y) + 2).

Define

− ∂π i−1

∂ ˆ ϑ ( ϑ − Γτ ˙ˆ i−1 + Γμ ˆ ϑ) =

 n j=i

Δ _i,j z j , (90)

− ∂π i−1

∂ ˆ β ( β − γ ˙ˆ 1 σ i−1 + γ 1 μ ˆ β) =

 n j=i

Λ _i,j z j , (91)

− ∂π i−1

∂ ˆ p ( ˙ˆ p − γ 2 ¯ λ i−1 + γμ ˆ p) =

 n j=i

A _i,j z j , (92)

where Δ _i,j =

 n j=i

∂π i−1

∂ ˆ ϑ Γ ∂π j−1

∂y ω, Λ _i,j =

 n j=i

∂π i−1

∂ ˆ β γη j tanh

 η j z j

ς

 ,

A _i,j =

 n j=i

∂π i−1

∂ ˆ ϑ γ 2

 ∂π j−1

∂y

 ₂

z ² _j ( ˆ ψ ² ₁₁ (y) + 2).

Choose the stabilizing control function π i as π i = −z i−1 − c i z i + k i v 0,1

+ ∂π i−1

∂y (ξ 2 + f 1,0 (y) + ω ^T ϑ) ˆ

−

 i−1 k=2

(Δ _k,i + Λ _k,i + A _k,i )z k − H i ,

(93)

where c i > 0 is a design constant. Substituting (90)–(93) into (89) and repeating procedures in Step 2, we have

V ˙ i ≤ − [λ _min (Q) − (i + 3)] e ² − z ₁ yρ(y ² )

+ i − 1 4 y ² −

 i j=1

c j z ² _j + ˜ ϑ ^T (τ i − Γ ⁻¹ ϑ) ˙ˆ

+ ˜ β(σ i − γ ₁ ⁻¹ β) + ˜ ˙ˆ p(¯ λ i − γ ⁻¹ ₂ p) + μ ˜ ˙ˆ κˆ κ +

 n j=i+1

 i k=2

(Δ _k−1,j + Λ _k−1,j )z k z j

+ P  ² (

 n k=1

ψ k2 ( |ζ|)) ² + z i z i+1

+ iψ ₁₂ ² ( |ζ|) + P  ² β ² + P  ² d ⁰ _ψ + iς ^ + iψ ₁₁ ² (0) + i + 1

2 d ² ₁₁

(94)

Step n: In the final design step, the actual control input u appears. Consider the overall Lyapunov function as

V n = V n−1 + 1

2 z ² _n . (95) Using (33) and (34), the time derivative of V n is

V ˙ n ≤ − [λ min (Q) − (n + 3)] e ² − z 1 yρ(y ² ) + n − 1

4 y ² + z n [σ(y)u − k n v 0,1

− ∂π n−1

∂y (ξ 2 + f 1,0 (y) + ω ^T ϑ) + H ˆ n ]

− ∂π n−1

∂ ˆ ϑ ( ϑ − Γτ ˙ˆ n−1 + Γμ ˆ ϑ)

− ∂π n−1

∂ ˆ β ( β − γ ˙ˆ 1 σ n−1 + γ 1 μ ˆ β)