Global Stabilization of Stochastic Nonlinear Systems Via C1 and C

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 62, NO. 11, NOVEMBER 2017

Global Stabilization of Stochastic Nonlinear Systems Via C 1 and C ∞ Controllers Hui Wang and Quanxin Zhu Abstract—The problem of designing C1 or C∞ controllers for a class of stochastic nonlinear systems (SNSs) in lower-triangular form is studied in this note. By using the well-known backstepping method, the concept of homogeneity with monotone degrees (HWMD) and the sign function approach, we construct a C1 state feedback controller recursively. Meanwhile, by employing a polynomial Lyapunov function with sign functions, we prove that the solution to SNSs is globally asymptotically stable in probability. Furthermore, based on the concept of HWMD, it shows that C1 controllers for a class of three-dimensional SNSs can be modified to C∞ controllers under certain conditions. Finally, two simulation examples are given to demonstrate the effectiveness of the proposed controllers. Index Terms—C1 and C controllers, globally asymptotic stability, HWMD, polynomial Lyapunov function, sign function approach, stochastic nonlinear systems.

I. INTRODUCTION

In this technical note, let (Ω, F, P, {Ft }t≥0 ) be a complete probability space, Ω denote a sample space, F denote a σalgebra, P denote the probability measure and Ft be a filtration of sub-σ-fields of F. We consider a class of stochastic nonlinear systems (SNSs) in lower-triangular form i + fi (¯ xi ))dt dxi = (xpi+1

xi )dw, + giT (¯

i = 1, . . . , n − 1,

xn ))dt + gnT (¯ xn )dw, dxn = (up n + fn (¯

(1)

where pi ≥ 1, i = 1, . . . , n, are odd positive integers, x ¯n = (x1 , . . . , xn )T is a Ft -adapted, continuous Rn -valued measurable process, ω is a d-dimensional Ft -adapted in-

Manuscript received July 30, 2016; revised October 6, 2016; accepted December 12, 2016. Date of publication December 22, 2016; date of current version October 25, 2017. This work was jointly supported by the National Natural Science Foundation of China (61374080,11531006), the Natural Science Foundation of Jiangsu Province (BK20161552), the Alexander von Humboldt Foundation of Germany (Fellowship CHN/1163390), Qing Lan Project of Jiangsu Province and the Priority Academic Program Development of Jiangsu Higher Education Institutions and Scientific Innovation Research of College Graduate in Jiangsu Province (KYZZ15_0210). Recommended by Associate Editor N. Kazantzis. (Corresponding author: Prof. Quanxin Zhu.) H. Wang is with the School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing 210023, Jiangsu, China (e-mail: [email protected]). Q. Zhu is with the School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing 210023, Jiangsu, China, also with the Department of Mathematics, University of Bielefeld, Bielefeld D-33615, Germany (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2016.2644379

, Senior Member, IEEE dependent standard Wiener process, u ∈ R is the system’s input. Set x ¯i := (x1 , . . . , xi )T . Functions fi : Ri → R and gi : Ri → Rd , i = 1, . . . , n are locally Lipschitz with fi (0) = 0, gi (0) = 0. In the past decades, the global stabilization problem of system (1) has been paid a great deal of attention since many physical models and engineering devices such as underactuated systems with weak coupling [12], [18] and single-link robotic manipulator systems [13] can be described by such a system. In the existing literature, many techniques have been applied to solve this problem. For instance, the backstepping design method was introduced to analyze the global stabilization of one-order SNSs in [2] and [10]. The stabilization of high-order SNSs by using adding a power integrator was investigated in [8] and [17]. In [20], Xie et al. applied the homogeneous system theory and the sign function approach to achieve the globally asymptotic stabilization of SNSs under more general hypotheses. More works on this topic can be found in [3], [7], [18], [19] and the references therein. On the other hand, a better smoothness controller is always needed in practical applications (e.g., engineering works or mechanical systems). Compared to continuous controllers, C 1 or C ∞ controllers are more widely used since they can guarantee the uniqueness of solutions. In particular, C ∞ controllers can suppress fully the chattering phenomenon in trajectories of systems. In other words, a system with a better smoothness controller has higher control accuracy and lower cost. It is inspiring that a large number of results have been focused on constructing C 1 or C ∞ controllers. For instance, Polendo in [11] and Tian et al. in [15], [16] applied the homogeneous system theory to study the stability of uncertain nonlinear systems and obtained smooth controllers by using the concept of HWMD, rather than homogeneity with a unified degree. In [2] and [4], Deng and Krstić provided the construction of linear (smooth) feedback controllers for one-order SNSs in strict-feedback form via the backstepping design method. Based on the notion of homogeneity with a unified degree, Xie et al. in [20] constructed a C 1 output feedback controller for SNSs with time-varying delays under weaker assumptions. Naturally, some important problems are aroused: Can we use the notion of HWMD to construct a polynomial Lyapunov function and a C 1 state feedback controller for system (1)? How to achieve C ∞ controllers for SNSs? In this technical note, we will try to answer these questions. The rest of this technical note is organized as follows. Section II presents some preliminary results. In Section III, a polynomial Lyapunov function and a C 1 state feedback controller are constructed recursively for system (1). Section IV further discusses sufficient conditions for the existence of C ∞ state feedback controllers for a class of three-dimensional SNSs. Sec-

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tion V gives several simulation examples to illustrate our theoretical results. Finally, we conclude this technical note with some general remarks. II. PRELIMINARIES

Throughout this technical note, R denotes the set of real numbers, R + denotes the set of positive real numbers, Z + denotes the set of positive integers, Rn denotes the real ndimensional space, C i denotes continuous functions who have continuous derivatives up to i th-order. For a vector or matrix A, AT denotes its transpose, tr{A} denotes its trace when A is a square matrix, and |A| is the Euclidean norm when A is a vector. For a ∈ R + , a denotes the integer part of a and a denotes the smallest integer larger than a. For fixed coordinates x = (x1 , . . . , xn ) ∈ Rn , its dilation Δ (x) is defined by Δ (x) = ( 1 x1 , . . . , n xn ) for any > 0, where i > 0 is the weights of the coordinates. Consider the following n-dimensional stochastic nonlinear system dx = f (x)dt + g(x)dw,

x0 = η,

(2)

where x ∈ Rn is the state, ω is a d-dimensional independent standard Wiener process, and functions f : Rn → Rn and g : Rn → Rn ×d are locally Lipschitz and satisfy f (0) = 0, g(0) = 0. Definition 1: ([2], [9]) The trivial solution of system (2) is globally asymptotically stable in probability (GAS-P) if for any t0 ≥ 0 and > 0, limx(t 0 )→0 P {supt≥t 0 |x(t)| > } = 0, and for any initial condition x(t0 ), P {limt→∞ x(t) = 0} = 1. Definition 2: ([15], [16]) (HWMD) For a dilation Δ and a series of real monotone numbers σ1 ≥ σ2 ≥ · · · ≥ σn , a continuous vector field ϕ : Rn → Rn is said to be homogeneous with monotone degrees σ1 , . . . , σn , if ∀x ∈ Rn \ {0}, ϕj (Δ x) = σ j + j ϕj (x), j = 1, . . . , n. For a C 2 Lyapunov function V , we use LV to define the differential operator of V as follows: LV (x) =

∂V (x) 1 ∂ 2 V (x) T tr g f (x) + (x) g(x) . ∂xT 2 ∂x2

(3)

Lemma 1: ([2], [5]) Suppose that there exists a positive definite, radially unbounded, C 2 Lyapunov function V : Rn → R such that the differential operator LV is negative definite. Then the trivial solution of (2) is GAS-P. Lemma 2: ([20]) Suppose that the sign function f (s) = sgn(s)|s|a = [s]a , s ∈ R + . Then the function f (s) is C 2 , for all a > 2. Furthermore, one has f (s) = a|s|a−1 and f (s) = a−2 a(a − 1)[s] . Lemma 3: ([20]) Suppose that p ∈ R + , x ∈ R, y ∈ R. If 0 < p ≤ 1, then |[x]p − [y]p | ≤ 21−p |x − y|p , (|x| + |y|)p ≤ |x|p + |y|p . Moreover, if p ≥ 1, we have |x + y|p ≤ 2p−1 |xp + y p |, |x − y|p ≤ 2p−1 |[x]p − [y]p |, |[x]p − [y]p | ≤ c|x − y|(|x − y|p−1 + |y|p−1 ), where c is a positive constant. Lemma 4: ([6]) Suppose that c and d are two positive real numbers, x, y ∈ R. Then, for any real-valued function γ(x, y) > 0, we have |x|c |y|d ≤ c/(c + d)γ(x, y)|x|c+d + d/(c + d)(γ(x, y))−c/d |y|c+d .

Fig. 1. Underactuated system with weak coupling under zero-gravity or micro-gravity circumstance.

III. C 1 CONTROLLERS FOR SYSTEM (1)

To proceed, two important assumptions are required in our design procedure. Motivated by [15], [16] and [20], we give them as below. Assumption 1: For i = 1, . . . , n, there exists a series of real numbers σ1 ≥ σ2 ≥ · · · ≥ σn and two known nonnegative C ∞ functions hf ,i (·) and hg ,i (·) satisfying ai ai xi )| ≤ |x1 | 1 + · · · + |xi | i hf ,i (¯ xi ), (4) |fi (¯ bi bi |gi (¯ xi )| ≤ |x1 | 2 1 + · · · + |xi | 2 i hg ,i (¯ xi ), (5) where two positive constants ai and bi satisfy ai ≥ i + σi and bi ≥ 2i + σi , the weights 1 = b > 0, i+1 = (i + σi )/pi . Assumption 2: Let m ax = max1≤i≤n {i } and δi = m ax /i , i = 1, . . . , n. Then the weights i satisfy the following conditions: (i) n + 1 ≥ m a x , if δi = 1 or δi ≥ 2 for all i = 1, . . . , n; (ii) n + 1 ≥ 2m a x , otherwise.

Remark 1: A distinct feature in Assumption 1 is that σi can take different value but not a unified value. That is, Assumption 1 is more general than those in the existing literature. Specifically, when pi = 1 and σi = 0, Assumption 1 is the same as in [2]. When σi = τ > 0, 1 = 1, ai = i + σi , bi = 2i + σi , hf ,i (·) = hg ,i (·) = c ∈ R + , Assumption 1 can be reduced to those used in [7], [19] and [20]. When σi = pi − 1 and 1 = 1, Assumption 1 comprises the conditions proposed in [1] and [14]. Remark 2: Assumption 2 is quite different from those in [7], [15] and [16]. The condition rn + τn ≥ max1≤i≤n {ri } is enough to guarantee the design procedure in [15] and [16]. However, it is invalid in our control scheme. So, we need a new condition imposed on system (1). Besides, we replace rn + τ in [7] by n +1 , which contributes to constructing C 1 controllers and even smooth controllers for high-order SNSs. Owing to the presence of noise terms and the high power pn , a more general system is considered in our technical note. Remark 3: System (1) with Assumptions 1 and 2 can describe many practical models. In what follows, we will take an underactuated system with weak coupling [12], [18] as an example. Throughout this technical note, we assume that this system is under zero-gravity or micro-gravity circumstance (e.g., in space or underwater). As demonstrated in Fig. 1, the mechanical system with two degrees of freedom is composed of two masses and two springs. The mass m1 is connected to the wall by an unstretched spring on a smooth horizontal surface, and the mass m2 is supported by a massless rod. The two masses are jointed

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with each other by an unstretched spring. Both of two springs are nonlinear springs which have cubic force-deformation relations. In this model, θ is the angle of the mass m2 away from the vertical, and x is the displacement of the mass m1 . Moreover, they satisfy θ = 0 at x = 0. Assume that there exists a control force u driving m1 and the angle θ is tiny. The equations of the system are k2 θ¨ = (x − lθ)3 , m2 l

x ¨=−

k1 3 k2 u x − (x − lθ)3 + . m1 m1 m1

According to [18], the white Gaussian noise can be introduced into this system. The spring coefficients k1 (t) with a nominal value k0 can take values in certain interval [p, p¯], where p, p¯ > 0. Let Δ(t) be the white Gaussian noise with zero mean and EΔ2 (t) = σ 2 . Set k1 (t) = Δ(t) + k0 . We choose a suitable value of σ to ensure that k1 (t) can not escape from [p, p¯] with a sufficiently large probability. This can be guaranteed by Chebyshev’s inequality. Then, under the following coordinates x1 = θ,

x2 = x˙ 1 ,

x3 = x − lθ,

LV1 ≤ [x1 ]

4 ρ −σ 1 − 1 1

dx1 = x2 dt, k2 3 x dt, m2 l 3

a1

LV1 ≤ [x1 ]

Define η1 , . . . , ηn and virtual controllers x∗1 , . . . , x∗n : λ

η1 = [x1 ] 1 ,

ηi = [xi ]

−

λ

[x∗i ] i

,

,

i = 2, . . . , n,

b 1 − 1 4ρ − σ1 − 1 |x1 | 1 h2g ,1 (x1 )). 21

4ρ

1 λ (xp2 1 − x∗p 2 ) − n|η1 | .

(7)

Step i-1. Assume that there exists a C , positive definite and radially unbounded Lyapunov function Vi−1 satisfying LVi−1 ≤ ([xi−1 ](4ρ−σ i −1 − i −1 )/ i −1 − [x∗i−1 ](4ρ−σ i −1 − i −1 )/ i −1 )(xpi i −1 − i −1 4ρ/λ ) − (n − i + 2) i−1 . x∗p i l=1 |ηl | Step i. Define a Lyapunov function Vi = Vi−1 + Ui with Ui =

4 ρ −σ i i 4ρ − σi − i ∗ 4 ρ−σ i |xi | i + |xi | i 4ρ − σi 4ρ − σi

− |xi |[x∗i ]

4 ρ −σ i − i i

.

(8)

Recalling that ρ ≥ max1≤i≤n {i + σi }, one gets functions Vi , i = 1, . . . , n, are C 2 , positive definite and radially unbounded functions. Then, the following inequality can be proved 4 ρ −σ i − i 4 ρ −σ i − i i i (xpi+1 − [x∗i ] i − x∗p LVi ≤ [xi ] i i+1 ) i

4ρ

|ηl | λ .

(9)

l=1

(i) λ = m a x , if m a x ≤ n + 1 < 2m a x ; (ii) λ is a positive real number satisfying n + 1 ≥ λ ≥ 2m a x , if n + 1 ≥ 2m a x .

λ i

4 ρ −σ 1 − 1 1

− (n − i + 1)

where v = u/m1 , f4 (·) = −k0 /m1 (x3 + lx1 )3 − (k2 (m1 + m2 ))/(m1 m2 )x33 and g4 (·) = −σ/m1 (x3 + lx1 )3 . By choosing σ1 = σ2 = σ3 = b, σ4 = 0, we see that Assumptions 1 and 2 hold with 1 = 3 = b and 2 = 4 = 5 = 2b. It means that the underactuated system with weak coupling under zerogravity or micro-gravity circumstance can be depicted by system (1) with Assumptions 1 and 2. Next, we will state one of our main results in this technical note. Theorem 1: Assume that Assumptions 1 and 2 hold. Then, there exists a C 1 state feedback controller u such that the trivial solution of system (1) is GAS-P. Proof: Let ρ be a positive integer satisfying ρ ≥ max1≤i≤n {i + σi } and λ be a positive constant. According to Assumption 2, the choice of λ can be given as follows:

p i −1 i λ

1 (x∗p 2

2

dx4 = (v + f4 (x1 , x2 , x3 ))dt + g4 (x1 , x2 , x3 )dw,

i −1 = −αi−1 (·)[ηi−1 ] x∗p i

4 ρ −σ 1 − 1 1

By defining α1 (x1 ) := n + (1 + |x1 |a 1 / 1 )hf ,1 (x1 ) + (4ρ − σ1 − 1 )/(21 )(1 + |x1 |b 1 / 1 )h2g ,1 (x1 ), it is easy to derive

dx3 = x4 dt,

x∗1 = 0,

1 (xp2 1 − x∗p 2 ) + [x1 ]

+ |x1 | 1 hf ,1 (x1 ) +

x4 = x˙ 3 ,

˙ x, x) on (θ, θ, ˙ ∈ (−π/2, π/2) × R3 , the above system can be transformed into the following system

dx2 =

σ1 )|x1 |(4ρ−σ 1 )/ 1 . Then, it follows from Assumption 1 that

Indeed, we compute directly LVi ≤ −(n − i + 2)

where α1 (·), . . . , αn −1 (·) are C ∞ positive functions to be determined later. Step 1. Let us choose a C 2 , positive definite and radially unbounded Lyapunov function V1 (x1 ) = 1 /(4ρ −

|ηl |

4ρ λ

+ Ξ1 + Ξ2 + Ξ3 + Ξ4 , (10)

l=1

where

Ξ1 =

[xi−1 ]

4 ρ −σ i −1 − i −1 i −1

−

[x∗i−1 ]

4 ρ −σ i −1 − i −1 i −1

i −1 × (xpi i −1 − x∗p ), i

Ξ2 =

i−1 ∂Ui l=1

∂xl

l (xpl+1 + fl (¯ xl )),

1 ∂ 2 Ui xi ) T Ξ4 = tr g¯iT (¯ 2 ∂x ¯i ∂ x ¯i

∂Ui p i (x + fi (¯ xi )), ∂xi i+1 g¯i (¯ xi ) . Ξ3 =

Note that ρ ≥ max1≤i≤n {i + σi }. It follows from Lemma 3 that there exists a continuous function α ˇ i−2 (·) > 0 satisfying

4 ρ −σ i −1 − i −1 4 ρ −σ i −1 − i −1

∗ i −1 i −1

[xi−1 ]

≤ c|ηi−1 | − [x ] i−1

(6)

i−1

|ηi−1 |

4 ρ −σ i −1 − i −1 λ

−1

+α ˇ i−2 (·)|ηi−2 |

4 ρ −σ i −1 − i −1 λ

−1

.

i −1 if λ ≥ i pi−1 , one has |xpi i −1 − x∗p |≤ i p i −1 i p i −1 /λ 2 |ηi | . If λ ≤ i pi−1 , we also have |xi − i −1 x∗p | ≤ c|ηi |(|ηi | i p i −1 /λ−1 + α ˘ i−1 (·)|ηi−1 | i p i −1 /λ−1 ), i where α ˘ i−1 (·) is a continuous positive function. Then applying

Besides,

1− i p i −1 /λ


Lemma 4 with i pi−1 = i−1 + σi−1 , we obtain 4ρ 4ρ 4ρ 1 Ξ1 ≤ |ηi−2 | λ + |ηi−1 | λ + a1,i (·)|ηi | λ , 4

(11)

where a1,i (·) is a C ∞ positive function. r /p For any given r ∈ R + , [x∗i ]r = −αi−1 i −1 (·)[ηi−1 ]r i /λ . With the aid of this equality and ρ ≥ max1≤i≤n {i + σi }, one has that for l = 1, . . . , i − 1, 4 ρ −σ i − i ∂ηi−1 ∂Ui 4ρ − σi − i |ηi−1 | λ −1 = ∂xl λ ∂xl i 4 ρ −σ i − i × |xi |¯ αi−1 (·) + [ηi−1 ] λ α ˜ i−1 (·) + [ηi−1 ] λ i ∂α ˜ i−1 (·) ∂α ¯ i−1 (·) 4ρ − σi − i λ × |xi | + [ηi−1 ] , ∂xl 4ρ − σi ∂xl (12)

(4ρ−σ − )/ p

˜ i−1 (·) := where α ¯ i−1 (·) := αi−1 i i i i −1 (·) and α (4ρ−σ i )/ i p i −1 ∞ αi−1 (·) are C positive functions. In what follows, we need to prove that for any l = 1, . . . , i − 1, there exists a C ∞ positive function bi−1,l (·) such that

i−1

∂ηi−1 l

|ηj |1− λ bi−1,l (·).

∂xl ≤

(13)

In fact, with the aid of λ ≥ max1≤i≤n {i }, we know that for i = 3 and l = 1, |∂η2 /∂x1 | ≤ |η1 |1− 1 /λ b2,1 (·), with a C ∞ positive function b2,1 (·). Assume that for any l = 1, . . . , i − 3, there exists a C ∞ positive function bi−2,l (·) satisfying |∂ηi−2 /∂xl | ≤ i−3 j =1 |ηj |1− l /λ bi−2,l (·). Then, we know that for any l = 1, . . . , i − 3,

∂ηi−1

∂α ˆ i−2 (·)

≤ ∂[ηi−2 ] α

∂xl ˆ i−2 (·) + |ηi−2 | ∂xl

∂xl

i−2

For Ξ3 , a straight computation leads to ∂Ui /∂xi = [xi ](4ρ−σ i − i )/ i − [x∗i ](4ρ−σ i − i )/ i . Then, it follows from Lemma 4 with (4) and (6) that

i−1

∂Ui

1 4ρ 4ρ

≤ f (¯ x ) |ηj | λ + a3,i (·)|ηi | λ , (15) i i

∂xi

4 j =1 where a3,i (·) is a C ∞ positive function. Hence, we have 4 ρ −σ i − i 4 ρ −σ i − i i i (xpi+1 − [x∗i ] i − x∗p Ξ3 ≤ [xi ] i i+1 ) 4 ρ −σ i − i 4 ρ −σ i − i i x∗p + [xi ] i − [x∗i ] i i+1 4ρ 4ρ 1 |ηj | λ + a3,i (·)|ηi | λ . 4 j =1

i−1

+

(16)

For Ξ4 , based on Assumption 2, we can obtain the following lemma. We omit the proof here since it can be computed directly. Lemma 5: For j = 1, . . . , i − 1 and l = 1, . . . , i − 1, Ui is a C 2 function and i ∂ 2 Ui = |xi |¯ αi−1 (·) + [ηi−1 ] λ α ˜ i−1 (·) ∂xj ∂xl 4 ρ −σ i − i ∂ 2 ηi−1 4ρ − σi − i |ηi−1 | λ −1 λ ∂xj ∂xl 4ρ − σi − i − 1 |xi |¯ + αi−1 (·) λ i 4ρ − σi λ − 1 [ηi−1 ] α ˜ i−1 (·) + λ

×

j =1

≤

5883

l

|ηj |1− λ bi−1,l (·),

4 ρ −σ i − i ∂ηi−1 ∂ηi−1 4ρ − σi − i [ηi−1 ] λ −2 λ ∂xj ∂xl i ∂α ˜ i−1 (·) ∂α ¯ i−1 (·) + |xi | + [ηi−1 ] λ ∂xj ∂xj

×

4 ρ −σ i − i 4ρ − σi − i ∂ηi−1 |ηi−1 | λ −1 λ ∂xl i ∂α ˜ i−1 (·) ∂α ¯ i−1 (·) + |xi | + [ηi−1 ] λ ∂xl ∂xl

×

j =1 λ/

where bi−1,l (·) and α ˆ i−2 (·) := αi−2 i −1 i −2 (·) are C ∞ positive functions. Besides, it is not difficult to show that for l = i − 1, |∂ηi−1 /∂xi−1 | ≤ (|ηi−1 |1− i −1 /λ + |ηi−2 |1− i −1 /λ ) bi−1,i−1 (·), where bi−1,i−1 (·) is a C ∞ positive function. Similarly, for l = i − 2, we have |∂ηi−1 /∂xi−2 | ≤ (|ηi−3 |1− i −2 /λ + |ηi−2 |1− i −2 /λ )bi−1,i−2 (·), where bi−1,i−2 (·) > 0 is a C ∞ function. In summary, we see that the inequality (13) holds. Furthermore, it follows from Lemma 3 again that for l = 1, . . . , n, there exists a positive constant cl ¯ f ,l (·) satisfying |xp l −1 | ≤ and a C ∞ positive function h l p l −1 l /λ p l −1 l /λ (|ηl | + αl−1 (·)|ηl−1 | )cl , |fl (¯ xl )| ≤ (|η1 |a l /λ + ¯ f ,l (·). Therefore, recalling that al ≥ pl l+1 and · · · + |ηl |a l /λ )h using the monotonicity of {σi }ni∈Z + , we use Lemma 4, (12) and (13) to derive p

4ρ 4ρ 1 |ηj | λ + a2,i (·)|ηi | λ , Ξ2 ≤ 4 j =1

4 ρ −σ i − i ∂ηi−1 4ρ − σi − i ∂ 2 Ui |ηi−1 | λ −1 = α ¯ i−1 (·) ∂xi ∂xj λ ∂xj

+ [ηi−1 ]

i−1

where a2,i (·) is a C ∞ positive function.

4 ρ −σ i − i ∂ηi−1 4ρ − σi − i |ηi−1 | λ −1 λ ∂xj 4 ρ −σ i − i ¯ i−1 (·) 4ρ − σi − i ∂2 α + [ηi−1 ] λ + |xi | ∂xj ∂xl 4ρ − σi i ∂2 α ˜ i−1 (·) × [ηi−1 ] λ , ∂xj ∂xl

×

(14)

4 ρ −σ i − i λ

∂α ¯ i−1 (·) , ∂xj

4 ρ −σ i −2 i ∂ 2 Ui 4ρ − σi − i = |xi | i . 2 ∂ xi i

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2

2

To proceed our design, we need to estimate | ∂∂x jη∂i −1 x l |, j = 1, . . . , i − 1 and l = 1, . . . , i − 1. Indeed, for i = 3 and j = l = 1, a straight computation leads to

2

∂ η2

c2,1,1 (·), λ = 1 ,

∂x2 ≤ |η1 |1− 2 λ1 c2,1,1 (·), otherwise,

of | ∂∂x jη∂i −1 x l |, i.e.,

where c2,1,1 (·) > 0 is a C ∞ function. Assume that for j = l = 1, . . . , i − 3, there exists a C ∞ function ci−2,j,j (·) > 0 satisfying

∂2 η c λ = j , i−2,j,j (·),

i−2

2j

≤ 2 1− λ

∂xj

ci−2,j,j (·), otherwise. |ηj |

where ci−1,j,l > 0 is a C ∞ function. Applying Lemma 3 with (5) and (6), we have that for l = 1, . . . , i − 1,

Then, combining (6) and (13), we know that for j = l = 1, . . . , i − 3

∂2 η ∂2 α ˆ i−2 (·) ∂α ˆ i−2 (·) ∂|ηi−2 |

i−1

|ηi−2 | + 2

≤

∂x2j

∂x2j ∂xj ∂xj

¯ g ,l (·) is a C ∞ positive function. Hence, by (17), (18), where h Lemmas 4 and 5, we see that there exists a C ∞ positive function a4,i (·) satisfying

1

+α ˆ i−2 (·) ≤

⎧ ⎪ ci−1,j,l (·), j = l and λ = j ,

2

⎪

∂ ηi−1 ⎨ i−1

j +l

∂xj ∂xl ≤ ⎪ |ηk |1− λ ci−1,j,l (·), otherwise, ⎪ ⎩ k =1

bl bl ¯ g ,l (·), xl )| ≤ |η1 | 2 λ + · · · + |ηl | 2 λ h |gl (¯

∞

where ci−1,j,j > 0 is a C function. Similarly, by the definition 2 of ηi−1 and ηi−2 , it is easy to obtain the estimations of | ∂∂ xη2i −1 |,

LVi ≤ − (n − i + 1)

| ∂∂ xη2i −1 | and we omit the details here. 2

For i = 4, j = 1 and l = 2, noting that ∂∂x 1[η∂ 2x]2 = 0, it follows from (13) that

2 2

∂ η3 1 +2

≤

|ηk |1− λ c3,1,2 (·),

∂x1 ∂x2

k =1

where c3,1,2 (·) > 0 is a C ∞ function. Assume that there exists a C ∞ function ci−2,j,l (·) > 0 satisfying for j = 1, . . . , i − 3, l = 1, . . . , i − 3 and j = l,

i−2

2 j +l

∂ ηi−2

|ηk |1− λ ci−2,j,l (·).

∂xj ∂xl ≤ k =1

Therefore, combining (6) and (13), we get

2

∂ ηi−1 ∂ 2 α ˆ i−2 (·) ∂α ˆ i−2 (·)

∂[ηi−2 ]

≤ |η | + i−2

∂xj ∂xl

∂xj ∂xl ∂xl ∂xj

2

∂ [ηi−2 ]

∂α ˆ i−2 (·)

∂[ηi−2 ]

+ +α ˆ i−2 (·)

∂xj ∂xl

∂xj ∂xl

≤

|ηk |

j +l λ

|ηl |

4ρ λ

+ ai (·)|ηi |

4ρ λ

l=1

4 ρ −σ i − i i

where ai (·) := a1,i (·) + a2,i (·) + a3,i (·) + a4,i (·) is a C ∞ positive function. Thus, we choose the (i + 1)th virtual controller i x∗p i+1 = −αi (·)[ηi ]

pi i+ 1 λ

,

(21)

where αi (·) := 2(4ρ−σ i − i )/λ−1 (n − i + 1 + ai (·)) is a smooth positive function. Noting that ρ ≥ max1≤i≤n {i + σi }, it follows from Lemma 3 that ([xi ](4ρ−σ i − i )/ i − [x∗i ](4ρ−σ i − i )/ i ) i 1−(4ρ−σ i − i )/λ |ηi |4ρ/λ αi (·). Combining this with x∗p i+1 ≤ −2 (20), we see that the inequality (9) holds. Step n. Choosing the Lyapunov function Vn = Vn −1 + Un , we have 4 ρ −σ n −1 − n −1 4 ρ −σ n −1 − n −1 n −1 n −1 − [x∗n −1 ] LVn ≤ [xn −1 ] n −1 )−2 × (xpnn −1 − x∗p n

ci−1,j,l (·).

n −1

|ηj |

4ρ λ

j =1

k =1 2

i−1

4 ρ −σ i − i i i (xpi+1 + [xi ] − [x∗i ] i − x∗p i+1 ) 4 ρ −σ i − i 4 ρ −σ i − i i x∗p + [xi ] i − [x∗i ] i (20) i+1 ,

i −1

1−

(19)

Finally, substituting (11), (14), (16) and (19) into (10) yields

ci−1,j,j (·), λ = j , 2j i−2 1− λ ci−1,j,j (·), otherwise, k =1 |ηk |

i−2

(18)

i−1

∂ |ηi−2 | ∂x2j

i −2

4ρ 4ρ 1 |ηj | λ + a4,i (·)|ηi | λ . 4 j =1

Ξ4 ≤

2

2

(17)

2

η i −1 ∂ η i −1 Similarly, the estimations of | ∂ ∂x i −2 ∂ x l | and | ∂ x j ∂ x i −2 |, for j = i − 2 and l = 1, . . . , i − 3 (or l = i − 2 and j = 1, . . . , i − 3), can be easily obtained and so we omit them for brevity. For j = i − 1 and l = i − 1 (or l = i − 1 and j = i − 1), we have 2 η i −1 ∂ 2 η i −1 | ∂ ∂x i −1 ∂ x l | = | ∂ x j ∂ x i −1 | = 0. So far, we complete the estimation

+

n −1 ∂Un j =1

∂xj

p

j (xj +1 + fj (¯ xj )) +

∂Un p n (u + fn (¯ xn )) ∂xn

∂ 2 Un 1 T xn ) T g¯n (¯ xn ) . + tr g¯n (¯ 2 ∂x ¯n ∂ x ¯n

(22)


Similar to the estimations of Ξ1 , Ξ2 , and Ξ4 , we derive that there exist C ∞ positive functions ai,n (·) (i = 1, 2, 3) satisfying

4 ρ −σ n −1 − n −1 4 ρ −σ n −1 − n −1

∗ n −1 n −1

[xn −1 ] − [xn −1 ]

n −1 × |xpnn −1 − x∗p | n 4ρ 4ρ 4ρ 1 |ηn −2 | λ + |ηn −1 | λ + a1,n (·)|ηn | λ , ≤ 4

n −1 n −1

1

∂Un p j 4ρ

≤ (x + f (¯ x )) |ηj | λ j n

j +1

∂x 4 j

j =1 j =1 4ρ

+ a2,n (·)|ηn | λ , n −1 4ρ ∂ 2 Un 1 1 T tr g¯n (¯ xn ) T g¯n (¯ xn ) ≤ |ηj | λ 2 ∂x ¯n ∂ x ¯n 4 j =1 4ρ

+ a3,n (·)|ηn | λ .

(23)

(24)

n λ+ 1

(27)

with αn (·) := 2(4ρ−σ n − n )/λ−1 (5/4 + a1,n (·) + · · · + a4,n (·)) is a smooth positive function. Obviously, the state feedback controller (27) is at least C 1 due to Assumption 2and the choice of λ. Then, it follows from Lemma 4 and (27) that 4 ρ −σ n − n 4ρ ∂Un p n λ u ≤ −21− |ηn | λ βn (·). ∂xn

In this section, we consider the following three-dimensional SNSs

(26)

n +1

λ

}, then system (1) with the C state feedback controller (27) is GAS-P. Remark 7: It should be noted that the value range of σ in Corollary 1 is more larger than that in [20]. According to [20], 2 p 3 −1 the homogeneous degree σ should satisfy σ ≥ max{ p2p1 p12p+p , 1 +1 2p 2 p 3 −1 2p 3 −1 p 1 p 2 −2p 2 p 3 +p 1 +1 , p 1 p 2 +p 1 +1−2p 3 −2p 1 p 3 }, which is included in the interval given in Corollary 1 when 0 < b ≤ 1.

x2 )dt + g2 (¯ x2 )dw, dx2 = xp3 2 dt + f2 (¯

u = − αnp n (·)[ηn ] λ 1 λ λ = − αnp n (·) [xn ] n + α ˆ n −1 (·)[xn −1 ] n −1 + · · · +α ˆ n −1 (·) · · · α ˆ 1 (·)[x1 ] 1

(2p 2 p 3 −1)b (2p 3 −1)b p 1 p 2 −2p 2 p 3 +p 1 +1 , p 1 p 2 +p 1 +1−2p 3 −2p 1 p 3 1

dx1 = xp2 1 dt + f1 (¯ x1 )dt + g1 (¯ x1 )dw,

where a4,n (·) is a C ∞ positive function. Now, we choose the following state feedback controller 1

Next, we consider the three-dimensional case of system (1) (i.e., n = 3). This corollary is directly derived from Theorem 1, and so the proof is omitted here. Corollary 1: If 1+ p1 + p1 p2 > max{2p2 p3 , 2p3 (1 + p1 )}, assume that fi and gi satisfy (4) and (5) with 1 = b > 0 and ho1 p 2 p 3 −1)b mogeneous degrees σi satisfying σi = σ ≥ max{ (2p p 1 p 2 +p 1 +1 ,

IV. C ∞ CONTROLLERS FOR A CLASS OF THREE-DIMENSIONAL SNSS

(25)

Besides, similar to the proof of (15), we can prove

n

∂Un

3 4ρ 4ρ

≤ f (¯ x ) |ηj | λ + a4,n (·)|ηn | λ ,

∂xn n n 4 j =1

5885

(28)

Substituting (23)–(28) into (22) yields LVn ≤ −1/2(|η1 |4ρ/λ + · · · + |ηn |4ρ/λ ), which implies that LVn is negative definite. Therefore, by Lemma 1, we know that the trivial solution of system (1) is GAS-P. Remark 4: System (1) with (27) satisfies the locally Lipschitz condition. Actually, functions fi (·) and gi (·), i = 1, . . . , n, are assumed to be locally Lipschitz. Moreover, the control law (27) is at least C 1 , which implies that (27) is locally Lipschitz. Remark 5: Compared with [20], we design a C 1 state feedback controller for system (1) by introducing a polynomial Lyanunov function and the notion of HWMD. This method makes it possible to upgrade (27) to C ∞ . Remark 6: It is true that the authors in [15] and [16] have studied the problem of constructing C 1 controller for nonlinear systems. However, since the noise terms occur in system (1), we have to skillfully deal with the second-order differential term Ξ4 in (10) under Assumptions 1 and 2. It should be noticed that Assumption 2plays an important role in the estimation of Ξ4 .

dx3 = up 3 dt + f3 (¯ x3 )dt + g3 (¯ x3 )dw,

(29)

where the powers p1 , p2 , p3 are given odd integers, functions fi (·) and gi (·) are locally Lipschitz. Motivated by [15] and [16], we know that the state feedback controller u is C ∞ if powers λ/i (i = 1, . . . , n) and n +1 /λ in (27) are positive integers. In the sequel, we will design a smooth state feedback controller u for system (29) by employing the concept of HWMD. Theorem 2: (i) If p1 ≥ 2p2 ≥ 2p3 − 1, fi and gi satisfy (4) and (5) with σ1 = b(p1 − 1), σ2 = b(2p2 − 1) and σ3 = 2b(p3 − 1). Then there exists a C ∞ state feedback controller u ensuring that system (29) is GAS-P. (ii) If 2p2 > p1 ≥ 3p3 and p2 > 2p3 , fi and gi satisfy (4) and (5) with σ1 = b(p1 q − 1), σ2 = b(2p2 − q), σ3 = 2b(p3 q − 1), where q = (2p2 + 2)/(2p3 + 1). Then, there exists a C ∞ state feedback controller u to ensure that system (29) is GAS-P. Proof: (i) With the aid of p1 ≥ 2p2 ≥ 2p3 − 1, we have σ1 ≥ σ2 ≥ σ3 . It shows that Assumption 1 holds. A direct computation leads to 2 = b, 3 = 2b, 4 = 2b, which means that Assumption 2(i) is satisfied. Then, by Theorem 1, system (29) with (27) is GAS-P. Moreover, we choose λ = 2b according to the discussion in Section III. Then, we have λ/1 = 2, λ/2 = 2, λ/3 = 1 and 4 /λ = 1. It shows that λ/i (i = 1, 2, 3) and 4 /λ in (27) are positive integers, i.e., the C 1 state feedback controller (27) can be upgraded to C ∞ . (ii) We first prove the following inequality 2p2 + 2 2p2 + 1 . (30) q= > 2p3 + 1 p1 + 1 Indeed, when p1 ≥ 3p3 and p2 ≥ 3p3 + 2, it follows from the inequality x > x − 1 that q−

p1 (2p2 − 2p3 + 1) − 4p3 − 4p2 p3 2p2 − p1 > p1 + 1 (2p3 + 1)(p1 + 1) ≥

p3 (2p2 − 6p3 − 1) > 0, (2p3 + 1)(p1 + 1)

5886


Fig. 2. (a) The state response in Example 1. (b) The control input in Example 1. (c) The state response in Example 2. (d) The control input in Example 2. (e) The state response in Example 2. (f) The control input in Example 2.

that is, q > (2p2 + 1)/(p1 + 1). When p1 ≥ 3p3 and 2p3 < p2 ≤ 3p3 , we have (2p2 + 2)/(2p3 + 1) > 2, (2p2 + 1)/(p1 + 1) < 2. It means that (30) holds. Then, by (30), we know that σ1 > σ2 ≥ σ3 . So functions fi and gi , i = 1, 2, 3, satisfy Assumption 1. It follows from the definition of i that 2 = bq, 3 = 2b and 4 = 2bq. Owing to q ≥ 2, we know that Assumption 2is satisfied. Then, using Theorem 1, system (29) with the C 1 state feedback controller (27) is GAS-P.

Moreover, choosing λ = 2bq, a direct computation leads to λ/1 = 2q, 4 /2 = 2, λ/3 = q and 4 /λ = 1, which implies that the powers λ/i (i = 1, 2, 3) and 4 /λ are positive integers. In other words, (27) is a C ∞ controller. The proof is completed. Remark 8: Since the noise terms and the high power p3 appear simultaneously in (29), a new relationship of pi (i = 1, 2, 3) should be estabilished during the construction of C ∞


controllers. Besides, the choice of q is also different from that in [15] and [16] due to the presence of p3 . V. SIMULATION EXAMPLES

In this section, we present two examples to illustrate our theoretical results. Example 1: Let us consider the practical example in Remark 3. Similar to the discussion in [18], we assume that l = 10, m1 = 1, m2 = 0.5, k0 = 1.5, k2 = 5, p = 0.75, p¯ = 2.25, σ = 0.125 and the initial data x1 (0) = 0.1, x2 (0) = 0.1, x3 (0) = 0, x4 = 0.1. Then, by Theorem 1 and choosing b = 2, λ = 4, one can achieve the following smooth state feedback controller v = −α4 (·)([x4 ] + α ˆ 3 (·)[x3 ]2 + α ˆ 3 (·)ˆ α2 (·)[x2 ] + α ˆ 3 (·)ˆ α2 (·)ˆ α1 (·)[x1 ]2 ), where functions 4/( +σ ) α ˆ i (·) denote αi i i (·), i = 1, 2, 3, α1 (·) = 20, α2 (·) = 8 + 1.439|x1 | + 2.116|x2 |, α3 (·) = 4 + |x1 | + 3.254|x2 | + 1.501x23 , and α4 (·) = 4 + x41 + x42 + 3.221x43 + 2.834x44 . The effectiveness of the control scheme is shown in Figs 2 (a) and (b). Example 2: Consider the following stochastic nonlinear system dx1 = x92 dt + x81 ln(1 + x21 )dw, dx2 = x73 dt + x51 sin3 x2 dw, dx3 = u3 dt + x41 x3 ln(1 + x22 )dw,

(31)

with the initial data x1 (0) = −1, x2 (0) = 1 and x3 (0) = −1. It is easy to see that the inequality 1 + p1 + p1 p2 > max{2p2 p3 , 2p3 (1 + p1 )} holds. Note that sin x = O(|x|) and ln(1 + x2 ) = O(|x|2 ), where O(x) denotes the function of x satisfying supx |O(x)|/|x| < ∞. It is easy to verify that the other conditions in Corollary 1 are satisfied with 1 = 2/3 λ/( +σ ) and σ = 2. Set α ˆ i (·) = αi i (·), i = 1, 2. Then, using Corollary 1 with λ = 4 , we see that system (31) with the C 1 state feedback controller 1

731

731

731

u = −α33 (·)([x3 ] 3 0 3 + α ˆ 2 (·)[x2 ] 2 3 1 + α ˆ 2 (·)ˆ α1 (·)[x1 ] 1 8 9 ) is GAS-P. Choosing suitable parameters α1 = 8 + 1.353x41 , α2 = 2.183 + 3.012x21 + 3.221x22 and α3 = 2.611 + 1.057x41 + 2.690x42 + 4.724x23 , the simulation is shown in Fig 2(c) and (d). Moreover, system (31) also satisfies the conditions in Theorem 2 (ii) with 1 = 2, σ1 = 34, σ2 = 24 and σ3 = 20. According to Theorem 2 (ii), we know that the above controller u can be modified to C ∞ . Denote smooth posi8/( +σ ) tive functions α ˆ i (·) = αi i i (·), i = 1, 2. Then, we obtain that system (31) with the C ∞ state feedback con1/3 ˆ 2 (·)[x2 ]2 + α ˆ 2 (·)ˆ α1 (·)[x1 ]4 ) is troller u = −α3 (·)([x3 ]2 + α GAS-P. Choosing suitable parameters α1 = 30 + 2.436x81 , α2 = 8.337 + 1.156x41 + 4.379x22 , α3 = 2.012 + 1.887x41 + 2.621x42 + 3.112x23 , the simulation is shown in Fig 2(e) and (f). VI. CONCLUDING REMARKS

In this technical note, we have investigated how to construct C 1 or C ∞ controllers for a class of SNSs with the aid of HWMD and the sign function approach. We have also presented several sufficient conditions for the existence of C 1 controllers and C ∞ controllers, respectively. A polynomial Lyapunov function with sign functions is constructed explicitly in our design proce-

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dure. Moreover, several simulation examples are given to show the application of our theoretical results. There are still two remaining problems to be solved: 1) How to give the differential geometric conditions for system (1); 2) How to construct C 1 or C ∞ controllers for high-order stochastic feedforward systems. ACKNOWLEDGMENT The authors would like to thank the associate editor and reviewers for their insightful suggestions. REFERENCES [1] W. Chen, J. Wu, and L. Jiao, “State-feedback stabilization for a class of stochastic time-delay nonlinear systems,” Int. J. Robust Nonlin. Control, vol. 22, no. 17, pp. 1921–1937, 2012. [2] H. Deng and M. Krstić, “Stochastic nonlinear stabilization-I: A backstepping design,” Syst. Control Lett., vol. 32, no. 3, pp. 143–150, 1997. [3] H. Deng and M. Krstić, “Output-feedback stochastic nonlinear stabilization,” IEEE Trans. Autom. Control, vol. 44, no. 2, pp. 328–333, 1999. [4] H. Deng, M. Krstić, and R. Williams, “Stabilization of stochastic nonlinear systems driven by noise of unknown covariance,” IEEE Trans. Autom. Control, vol. 46, no. 8, pp. 1237–1253, 2001. [5] R. Z. Khasminski, Stochastic Stability of Differential Equations. Kluwer Academic Publishers: Norwell, MA, 1980. [6] W. Lin and C. Qian, “Adding one power integrator: A tool for global stabilization of high-order lower-triangular systems,” Syst. Control Lett., vol. 39, no. 5, pp. 339–351, 2000. [7] W. Li, X. Xie, and S. Zhang, “Output-feedback stabilization of stochastic high-order nonlinear systems under weaker conditions,” SIAM J. Control Optim., vol. 49, no. 3, pp. 1262–1282, 2011. [8] L. Liu and X. Xie, “Output-feedback stabilization for stochastic highorder nonlinear systems with time-varying delay,” Automatica, vol. 47, no. 12, pp. 2772–2779, 2011. [9] X. Mao, Stochastic Differential Equations and Their Applications. Chichester, U.K.: Horwood Publishing, 2007. [10] Z. Pan and T. Bas¸ar, “Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion,” SIAM J. Control Optim., vol. 37, no. 3, pp. 957–995, 1999. [11] J. Polendo, Global Synthesis of Highly Nonlinear Dynamic Systems with Limited and Uncertain Information. Ph.D. dissertation, University of Texas at San Antonio, 2006. [12] C. Qian and W. Lin, “A continuous feedback approach to global strong stabilization of nonlinear systems with uncontrollable,” IEEE Trans. Autom. Control, vol. 46, no. 7, pp. 1061–1079, 2001. [13] Z. Sun, T. Li, and S. Yang, “A unified time-varying feedback approach and its applications in adaptive stabilization of high-order uncertain nonlinear systems,” Automatica, vol. 70, pp. 249–257, 2016. [14] J. Tian and X. Xie, “Adaptive state-feedback stabilization for high-order stochastic non-linear systems with uncertain control coefficients,” Int. J. Control, vol. 80, no. 9, pp. 1503–1516, 2007. [15] W. Tian, Generalized Homogeneous Methodologies and New Solutions to Control Problems of Nonlinear Systems. Ph.D. dissertation, , University of Texas at San Antonio, 2013. [16] W. Tian, C. Zhang, C. Qian, and S. Li, “Global stabilization of inherently non-linear systems using continuously differential controllers,” Nonlinear Dynamics, vol. 77, no. 3, pp. 739–752, 2014. [17] X. Xie and J. Tian, “State-feedback stabilization for high-order stochastic nonlinear systems with stochastic inverse dynamics,” Int. J. Robust Nonlin. Control, vol. 17, no. 14, pp. 1343–1362, 2007. [18] X. Xie and N. Duan, “Output tracking of high-order stochastic nonlinear systems with application to benchmark mechanical system,” IEEE Trans. Autom. Control, vol. 55, no. 5, pp. 1197–1202, 2010. [19] X. Xie and L. Liu, “A homogeneous domination approach to a class of stochastic time-varying delay nonlinear systems,” IEEE Trans. Autom. Control, vol. 58, no. 2, pp. 494–499, 2013. [20] X. Xie, N. Duan, and C. Zhao, “A combined homogeneous domination and sign function approach to output-feedback stabilization of stochastic high-order nonlinear systems,” IEEE Trans. Autom. Control, vol. 59, no. 5, pp. 1303–1309, 2014.