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Universal Fuzzy Integral Sliding-Mode Controllers for Stochastic Nonlinear Systems Qing Gao, Member, IEEE, Lu Liu, Senior Member, IEEE, Gang Feng, Fellow, IEEE, and Yong Wang

Abstract—In this paper, the universal integral sliding-mode controller problem for the general stochastic nonlinear systems modeled by Itˆo type stochastic differential equations is investigated. One of the main contributions is that a novel dynamic integral sliding mode control (DISMC) scheme is developed for stochastic nonlinear systems based on their stochastic T-S fuzzy approximation models. The key advantage of the proposed DISMC scheme is that two very restrictive assumptions in most existing ISMC approaches to stochastic fuzzy systems have been removed. Based on the stochastic Lyapunov theory, it is shown that the closed-loop control system trajectories are kept on the integral sliding surface almost surely since the initial time, and moreover, the stochastic stability of the sliding motion can be guaranteed in terms of linear matrix inequalities. Another main contribution is that the results of universal fuzzy integral slidingmode controllers for two classes of stochastic nonlinear systems, along with constructive procedures to obtain the universal fuzzy integral sliding-mode controllers, are provided, respectively. Simulation results from an inverted pendulum example are presented to illustrate the advantages and effectiveness of the proposed approaches. Index Terms—Dynamic integral sliding mode control (DISMC), stochastic nonlinear systems, stochastic T-S fuzzy models, universal fuzzy integral sliding-mode controllers.

I. Introduction

T

AKAGI-SUGENO fuzzy models [1], or the so-called fuzzy dynamic models [2], have been widely utilized in control of complex nonlinear systems during the last few decades. In the T-S fuzzy model-based methodology, local dynamics of the original nonlinear system in different state space regions are described by linear dynamic models, and

Manuscript received May 20, 2013; revised November 12, 2013; accepted March 17, 2014. This work was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region of China under Project CityU/113212 and in part by the National Natural Science Foundation of China under Grant 61273122. This paper was recommended by Associate Editor H. Zhang. Q. Gao is with the School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600, Australia, and also with the Department of Automation, University of Science and Technology of China, Hefei 230026, China (e-mail: [email protected]). L. Liu and G. Feng are with the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong (e-mail: [email protected]; [email protected]). Y. Wang is with the Department of Automation, University of Science and Technology of China, Hefei 230026, China (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/TCYB.2014.2313028

the overall model of the system is then constructed by fuzzy blending of these local models through a set of fuzzy membership functions. This relatively simple structure provides great advantages in stability analysis and controller synthesis for T-S fuzzy systems in view of the powerful conventional control theory and techniques. T-S fuzzy models have been shown to be universal function approximators in the sense that they are able to approximate any smooth nonlinear functions to arbitrary degree of accuracy in any convex compact region [3]–[5]. All these results provide a solid theoretical foundation for modeling and control design of complex nonlinear systems based on T-S fuzzy models [6]–[21]. Readers can refer to several books and survey papers [22]–[24] and the references therein for the most recent advances on this topic. On one hand, the stochastic nonlinear systems described by Itˆo type stochastic differential equations have attracted increasing attention from the control community recently [25]. However, it is worth noting that most of the existing results on stochastic nonlinear control systems suffer from lack of systematic tools for control design due to difficulties in searching for appropriate Lyapunov functions [25]–[26]. Recently, in view of the great success achieved in T-S fuzzy model based approaches to controlling complex deterministic nonlinear systems, the ordinary deterministic T-S fuzzy models have been further extended to stochastic T-S fuzzy models where the local models are Itˆo type stochastic linear dynamic models instead of deterministic ones, to describe complex stochastic nonlinear systems [27]–[29]. By using quadratictype Lyapunov functions, it is shown that control design of stochastic T-S fuzzy systems can be accomplished by solving a set of linear matrix inequalities (LMIs) (see [27]–[29] for details). On the other hand, as a powerful robust control strategy, sliding mode control (SMC) [30] has various favorable features such as strong robustness and fast response. Generally speaking, the main idea of the normal SMC scheme is to utilize a discontinuous control law to drive the system trajectories onto a specified sliding manifold containing the origin, which is normally called the sliding surface, in finite time (this process is called reaching phase), and then to keep the system trajectories moving along the sliding surface toward the origin with desired performance (such motion is called sliding mode). As long as the sliding mode is achieved and maintained, the overall system dynamics are determined by the sliding surface which is often designed as linear hyper-planes of the system states. An improved SMC scheme, which is called integral

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sliding mode control (ISMC), was proposed in [31]. The main appeal of ISMC is that the reaching phase required in the normal SMC scheme is eliminated by using a class of nonlinear sliding surfaces. Instead, the system trajectories always start from the sliding surface, which guarantees the robustness of the ISMC system throughout its entire trajectories starting from the initial time [31]–[33]. In recent years, a number of valuable results have been reported in using ISMC schemes for Itˆo type stochastic nonlinear systems (see [34]–[37]). ISMC approaches to stochastic nonlinear systems can be found in [34]–[36]. Ho and Niu [37] proposed an ISMC scheme for stochastic nonlinear systems via a way of T-S fuzzy modeling, where the ISMC design can be accomplished in terms of LMIs. Nevertheless, it is assumed in [34]–[37] that the sliding mode can be achieved and maintained without considering the stochastic perturbations. Moreover, the ISMC approach in [37] relies on an assumption that all subsystems of the stochastic T-S fuzzy models share the same input matrix. Both of these two assumptions impose great limitations in real applications since they are very hard to be satisfied for many practical stochastic systems. Thus, it is desirable to develop a new ISMC approach to stochastic fuzzy systems which is not restricted by either of these two assumptions. In addition, in spite of the results mentioned in [37], a critical question of the ISMC approaches to stochastic fuzzy systems is yet to be answered, that is, given a stochastic nonlinear system which can be stabilized by an appropriately defined state feedback controller, does there exist a fuzzy integral sliding-mode controller such that the sliding mode can be achieved and the resulting sliding motion is stable with desired performance? This is called the universal fuzzy integral sliding-mode controller problem in this paper. Furthermore, how to design the universal fuzzy integral sliding-mode controller if it exists? Unfortunately, to the best of our knowledge, very few results on the above issues for stochastic nonlinear systems have been reported in the literature, which motivates us for this paper. In this paper, the universal fuzzy integral sliding-mode controllers problem for stochastic nonlinear systems based on stochastic T-S fuzzy models is investigated. We first propose a new dynamic integral sliding mode control (DISMC) approach to stabilizing a stochastic nonlinear system via its stochastic T-S fuzzy approximation model, aiming to remove the restrictive assumptions required in most existing results. Then, based on the proposed DISMC approach, we obtain the results on universal fuzzy integral sliding-mode controllers for two classes of stochastic nonlinear systems, respectively. The rest of this paper is structured as follows. Section II is devoted to model description and problem formulation. In Section III, the new DISMC scheme is developed for stochastic nonlinear systems based on stochastic T-S fuzzy models. Then, based on the proposed DISMC approach, the results of universal fuzzy integral sliding-mode controllers for a class of stochastically uniformly exponentially stabilizable nonlinear systems and a class of stochastically uniformly asymptotically stabilizable nonlinear systems are given in Sections IV and V, respectively. Constructive procedures to obtain the universal fuzzy integral sliding-mode controllers are also provided in both sections.

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Simulation results are presented in Section VI to illustrate the advantages and effectiveness of the DISMC approach. Conclusions are given in Section VII. Notations: The notations used in this paper are fairly standard. The notation  in a matrix is used to indicate the terms that can be deduced by symmetry. The superscript T represents vector or matrix transpose. In and 0m×n are used to denote the n × n identity matrix and the m × n zero matrix, respectively. The subscripts n and m × n are omitted where the dimension is irrelevant or can be determined from the context. x stands for the Euclidean norm of vector x and A stands for the matrix induced norm of the matrix A. λmin (A) and λmax (A) denote the maximum and minimum eigenvalues of the matrix A, respectively. Let (, F, P) be a complete probability space with a natural filtration {Ft }t≥0 , and E{·} be the mathematical expectation operator with respect to the given probability measure P.

II. Model Description and Problem statement In this paper, we consider a general stochastic nonlinear system governed by the following Itˆo type stochastic differential equation: dx(t) = f (x(t), u(t))dt + g(x(t), u(t))dW(t)

(1)

where x(t) = [x1 (t), ..., xn (t)]T ∈ X ⊂ n , u(t) = [u1 (t), ..., um (t)]T ∈ U ⊂ m , X × U is a compact set on n × m containing the origin, and W(t) = [W1 (t), W2 (t), ..., Wq (t)]T is a q-dimensional Wiener process. Throughout this paper, it is always assumed that the mappings f (x, u) : n × m → n and g(x, u) : n × m → q both vanish at zero, that is, f (0, 0) = 0 and g(0, 0) = 0. It is also assumed that f and g satisfy the usual linear growth and local Lipschitz conditions for existence and uniqueness of solutions to (1). The function g(x, u) is assumed to be not exactly known but satisfy the following condition: trace[gT (x, u)g(x, u)] ≤ G[xT , uT ]T 2

(2)

where G is a known constant matrix. We also consider to approximate the stochastic nonlinear system (1) by the following stochastic T-S fuzzy model. Plant Rule: Rl IF θ1 (t) is μl1 AND ... AND θg (t) is μlg ; THEN dx(t) = [Al x(t) + Bl u(t)]dt + g(x(t), u(t))dW(t) l ∈ L := {1, 2, ...r}

(3)

where θ(t) = [θ1 (t), ..., θg (t)] is the premise variables vector, Rl denotes the lth rule, r is the total number of rules, μli are the fuzzy sets, and [Al , Bl ] are the system matrices of the lth local model. By the standard fuzzy blending method, the stochastic T-S fuzzy system (3) can be expressed globally as dx(t) = fˆ (x(t), u(t))dt + g(x(t), u(t))dW(t)

(4)

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terms of (3), which is not restricted by the assumptions in (11) or (12).

where fˆ (x(t), u(t)) =

r 

3

wl (θ(t))[Al x(t) + Bl u(t)]

(5)

l=1

with the normalized fuzzy basis functions wl (θ(t)) defined by g l μ (θi (t)) wl (θ(t)) = r i=1g i  μli (θi (t)) l=1 i=1

wl (θ(t)) ≥ 0, and

r 

wl (θ(t)) = 1.

(6)

III. Dynamic Integral Sliding Mode Control A. Design of Integral Sliding Surface and Dynamic Sliding-Mode Controller For the stochastic nonlinear system (1) or its equivalent model (10), the following new type of integral sliding surface is designed: s(t) = Sx x(t) − Sx x(0) + Su u(t) − Su u(0)  t r − wl (θ(τ))Sx [Al x(τ) + Bl u(τ)]dτ

l=1

The universal function approximation capability of the T-S fuzzy models has been well studied in [3]–[5] and is described in Lemma 1. Lemma 1: [5] For any given continuously differentiable function f (x, u) in a compact region X × U with f (0, 0) = 0 and any positive constant f , there exists a T-S fuzzy model  fˆ (x, u) = rl=1 wl (θ(t))[Al x(t) + Bl u(t)] given in (5) such that for any (x, u) ∈ X × U f (x, u) = fˆ (x, u) + E(x, u) (7) r  = wl (θ(t))[(Al + Al )x(t) + (Bl + Bl )u(t)] (8) l=1

and [Al , Bl ] ≤ f .

l=1

(10)

In other words, ISMC design for stabilization of the stochastic nonlinear system (1) can be solved as an ISMC design problem for robust stabilization of its corresponding stochastic T-S fuzzy model (3) with the approximation error as an uncertainty term. Robust ISMC design for stochastic T-S fuzzy systems has been studied in [37]. However, it is assumed in [37] that: 1) all the subsystems of the stochastic fuzzy model (10) share the same input gain matrix, that is Bl ≡ B

(11)

for all l ∈ L , and 2) there exists a matrix S with appropriate dimensions such that SB is nonsingular, and moreover Sg(x(t), u(t)) = 0.

l=1

0

l=1

(12)

Both these two assumptions are very restrictive. In Section III, we will develop a novel fuzzy DISMC scheme for the stochastic nonlinear system (1) via a way of T-S fuzzy modeling in

wl (θ(τ))Su [Fl x(τ) + Gl u(τ)]dτ

(13)

where Sx ∈ m×n , Su ∈ m×m , and Fl ∈ m×n , Gl ∈ m×m A l Bl is Hurwitz for each are chosen such that the matrix F l Gl l ∈ L . Su is designed to be nonsingular. In this case, the following fuzzy dynamic sliding-mode controller is employed:  r du(t) = wl (θ(t))[Fl x(t) + Gl u(t)] l=1

(9)

Remark 1: It has been shown in [3]– [5] that the upper bound of the approximation error, that is, f in (9) can be made arbitrarily small by choosing large enough number of fuzzy rules in the T-S fuzzy models. Then, based on Lemma 1, one can easily conclude that the stochastic nonlinear system (1) can be represented in a compact region by a stochastic T-S fuzzy model with some norm bounded uncertainties as follows: r  dx(t) = wl (θ(t))[(Al + Al )x(t) + (Bl + Bl )u(t)]dt + g(x(t), u(t))dW(t).

−

0

 t r

−χ(t)Su−1 sgn(s(t))

 dt

(14)

where

⎧ ⎨ G[xT (t), uT (t)]T 2 β + α + ς(t), if s(t) = 0, χ(t) = (15) s(t) ⎩ α + ς(t) if s(t) = 0 ς(t) = f Sx [xT (t), uT (t)]T 

(16)

f is defined in (9), α is a given positive constant, and β = 1 λ {S T S }. 2 max x x ¯ l = [Al , Bl ], Denote R1 = [In , 0n×m ]T , R2 = [0m×n , Im ]T , A ¯ l = [Al , Bl ], and x¯ (t) = [xT (t), uT (t)]. ¯ l = [Fl , Gl ], A K Then, the closed-loop control system consisting of (10) and (15) can be rewritten in a compact form as r  ¯ l + R2 K ¯ l x¯ (t) ¯ l )¯x(t) + R1 A wl (θ(t)) (R1 A d x¯ (t) = l=1

 

G¯x(t)2 + α + ς(t) R2 Su−1 sgn(s(t)) dt − β s(t) (17) + R1 g(x(t), u(t))dW(t) and the integral sliding surface (13) can be rewritten as s(t) = S¯ x¯ (t) − x¯ (0) −

 t r 0

 ¯ l + R2 K ¯ l )¯x(ϕ)dϕ wl (θ(ϕ))(R1 A

l=1

where S¯ = [Sx , Su ]. Then, one has the following result.

(18)

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Theorem 1: The trajectories of the closed-loop control system (17) are kept on the sliding surface (13) or equivalently (18) almost surely since the initial time. Proof: By substituting the solution x¯ (t) of (17) into (18) ¯ 2 Su−1 = Im , one has with the fact that SR  t r ¯ 1 A ¯ l x¯ (ϕ) s(t) = wl (θ(ϕ))SR 0

l=1

 G¯x(ϕ)2 + α + ς(ϕ) sgn(s(ϕ))dϕ − β s(ϕ)  t ¯ 1 g(x(ϕ), u(ϕ))dW(ϕ) + SR

(19)

0

which implies that s(t) is also an Itˆo process satisfying ¯ 1 g(x(t), u(t))dW(t) ds(t) = a(t)dt + SR

(20)

where a(t) =

r 

¯ 1 A ¯ l x¯ (t) wl (θ(t))SR

l=1



G¯x(t)2 + α + ς(t) sgn(s(t)). − β s(t)

(21)

By choosing a Lyapunov function candidate as S(t) = sT (t)s(t), for all t > 0

(22)

and using Itˆo’s formula, one has that along the trajectories of (20) dS(t) = LS(t)|(20) dt + h(t)dW(t)

(23)

¯ 1 g(x(t), u(t)) and where h(t) = 2s (t)SR T

LS(t)|(20) = 2sT (t)a(t) + trace{gT (x(t), u(t))RT1 S¯ T SR1 g(x(t), u(t))} r  ¯ 1 A ¯ l − 2βG¯x(t)2 wl (θ(t))2sT (t)SR =

Using Lemma A3 in the Appendix with the fact that β = ¯ 1 yields and Sx = SR ¯ 1 g(x(t), u(t))} trace{gT (x(t), u(t))RT1 S¯ T SR T < 2βtrace{g (x(t), u(t))g(x(t), u(t))} (25) 

LS(t)|(20) ≤ −2αs(t) = −2α

S(t).



 x˙ (t) = fˆ (x, u) = rl=1 wl (θ(t))[Al x(t) + Bl u(t)]  u(t) ˙ = fˆ c (x, u) = rl=1 wl (θ(t))[Fl x(t) + Gl u(t)]

(29)

r 

¯ l + R2 K ¯ l )¯x(t). wl (θ(t))(R1 A

(30)

l=1

1 λ {S T S } 2 max x x

Combination of (24) and (25) implies

(28)

which leads to a contradiction. Thus, one has Es(t) = 0 for all t ≥ 0. In other words, the system trajectories will stay on the integral sliding surface (18) almost surely since the initial time, which is a favorable feature of ISMC approaches. The proof is thus completed. Remark 2: It is noted that the integral sliding surface (13) is a nonlinear hyperplane dependent on both the system state vector x and the control input vector u. Moreover, the adopted sliding-mode controller is in the form of fuzzy dynamic state feedback control. Both of these key features distinguish the proposed integral sliding mode control scheme from the commonly used ones, such as those in [37]. This is also the main reason why it is termed as the DISMC approach in this paper. Remark 3: Most of the existing ISMC approaches for stochastic fuzzy systems, such as the one in [37], are restricted by the assumptions in (11) and (12), while the DISMC approach proposed in this paper is not. Remark 4: It is also noted that the sliding-mode controller 1 in (15) contains a term s(t) which is ill-defined when s(t) = 0. 1 1 To avoid this problem, one can replace the term s(t) by s(t)+ϑ where ϑ is a small positive constant. In this case Es(t) cannot be exactly zero but varies within a time varying band around zero instead, which will be illustrated in Section VI. One can observe from (13) that s(t) can be recognized as a weighted distance between the trajectories of the real closedloop control system (17) and the following nominal closedloop control system:

x¯˙ (t) = (24)

≤ 2βG¯x(t)2 .

Es(ts ) ≤ −α0 ts < 0

or equivalently

l=1

−2(α + ς(t))s(t) +trace{gT (x(t), u(t))RT1 S¯ T SR1 g(x(t), u(t))} ¯ 1 g(x(t), u(t))} ≤ trace{gT (x(t), u(t))RT1 S¯ T SR −2βG¯x(t)2 − 2αs(t).

Taking expectation on both sides of (27) yields

(26)

It is noted that Es(0) = s(0) = 0. Suppose that there exists a time ts > 0 such that Es(ts ) > 0. Then, it follows from (23) and (26) that:  ts −LS(t) |(20) √ 2α0 ts ≤ dt S(t) 0  ts  ts −1 h(t) √ √ = dS(t) + dW(t). (27) S(t) S(t) 0 0

While the sliding mode is achieved and maintained, both the system state vector x and the control input vector u of the real closed-loop control system intercept those of the nominal closed-loop control system (30), and the real closed-loop control system behaves as the nominal closed-loop control system (30) does. In other words, the system performance during the sliding mode can be guaranteed if the nominal closed-loop control system (30) is well designed. The stability analysis result of the nominal closed-loop control system (30) is shown in Lemma 2. Lemma 2: The nominal closed-loop control system (30) is asymptotically stable if there exist a positive definite matrix Q ∈ (n+m)×(n+m) and a set of matrices Wl ∈ m×(n+m) , l ∈ L such that the following LMIs are satisfied: ¯ l Q + R2 Wl + (R1 A ¯ l Q + R2 Wl )T < 0, l ∈ L . R1 A

(31)

¯ l = [Fl , Gl ] = Moreover, the controller gains are given by K −1 Wl Q .

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Proof: By using the Lyapunov function candidate V (t) = x¯ T (t)Q−1 x¯ (t), Lemma 1 can be easily obtained and the proof is thus omitted here. B. Stability Analysis of the Sliding Motion It has been shown in Section III-A that the sliding mode can be achieved and maintained almost surely since the initial time. In this subsection, the stability of the closed-loop dynamics restricted on the sliding surface, which corresponds to the ideal motion in the deterministic case counterpart, will be discussed. Substituting the solution x(t) of (10) into (13) yields  t r s(t) = wl (θ(s))Sx [Al x(s) + Bl u(s)]ds 0



l=1

+Su u(t) −  +

 t r 0

Sx g(x(s), u(s))dW(s)

l ⎢ RT P ⎢ 1T ⎣ R2 P f I(m+n)

⎤    ⎥ l   ⎥ 0. Then from the stability theory of stochastic differential equations [25], one can conclude that the stochastic nonlinear system (56) is semi-globally stochastically uniformly exponentially stable in rth mean on the compact set X × U. Thus, based on Definition 2, FISMC are universal fuzzy integral sliding-model controllers for the concerned class of stochastic nonlinear systems. The proof is thus completed. Consider a stochastic nonlinear system in the form of (1), if a stochastically uniformly exponentially stable ref¯ x(t))dW(t), where erence model d x¯ (t) = Fm (¯x(t))dt + G(¯ ¯ x(t)) = Fm (¯x(t)) = R1 f (x(t), u(t)) + R2 fcm (x(t), u(t)) and G(¯ R2 g(x(t), u(t)), is given, one can apply the construction schemes given in [3]–[5] to obtain the model reference fuzzy sliding-mode controller. That is, one can construct a fuzzy dynamic integral sliding-mode controller as in (55) such that for any given cm > 0 fˆ cm (x, u) = fcm (x, u) + cm (x, u)

(64)

cm (x, u) ≤ cm ¯x.

(65)

where

Then, based on Theorem 3, the constructed fuzzy slidingmode control law (55) guarantees that the trajectories of the closed-loop control system can be kept on the fuzzy integral sliding surface (54) almost surely since the initial time, and the resulting sliding motion is semi-globally stochastically uniformly exponentially stable in the compact region X × U.

V. Universal Fuzzy Integral Sliding-Mode Controllers for More General Stochastic Nonlinear Systems It has been shown in Section IV that the fuzzy integral sliding-mode controllers in the form of (15) are universal integral sliding-mode controllers for stochastic nonlinear systems as in (1) which are globally stochastically uniformly exponentially stabilizable in the rth mean. In this section, we will consider more general stochastic nonlinear systems as in (1) which are only globally stochastically uniformly asymptotically stabilizable. The following definitions are introduced first. Definition 3: A function α : + → + is said to belong to class K if it is continuous, strictly increasing and α(0) = 0. It is said to belong to class K∞ if it belongs to class K and α(s) → ∞ as s → ∞. A function β : + ×+ → + is said to belong to class KL if for each fixed t ≥ 0, the function β(·, t) is a K function and for each fixed s ≥ 0, β(s, t) is decreasing with respect to t and β(s, t) → 0 as t → ∞. Definition 4 [29]: A stochastic system in SNS is said to be globally stochastically uniformly asymptotically stabilizable,

if there exists a control law du(t) = fc (x(t), u(t))dt such that the closed-loop control system given by (57) is globally stochastically uniformly asymptotically stable, that is, there exists a KL function β such that given any initial states x¯ (0) the solution x¯ (t) of (57) exists for all t ≥ 0 and satisfies P { lim x(t) = 0} = 1. t→∞

(66)

Definition 5 [29]: A stochastic system in SNS is said to be semi-globally stochastically uniformly input-to-state stable (SISS) on a compact set X × U if for any given ε > 0, there exist a KL function β and a K function γ such that P {¯x(t) < β(¯x0 , t) + γ(sup ¯(¯x(τ)))} ≥ 1 − ε.

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τ>0

Definition 6: FISMC are said to be universal practical fuzzy integral sliding-mode controllers, if for any system in SNS which is globally stochastically uniformly asymptotically stabilizable, there exists a fuzzy dynamic integral sliding-mode controller in FISMC such that the resulting sliding motion as in (56) is semi-globally stochastically uniformly input-to-state stable, and the trajectories of the closed-loop control system can be kept on the integral sliding surface as in (54) almost surely since the initial time. Definition 7: FISMC are said to be universal asymptotic fuzzy integral sliding-mode controllers, if for any system in SNS which is globally stochastically uniformly asymptotically stabilizable there exists a fuzzy dynamic integral sliding-mode controller in FISMC such that the resulting sliding motion as in (56) is semi-globally stochastically uniformly asymptotically stable, and the trajectories of the closed-loop control system can be kept on the integral sliding surface as in (54) almost surely since the initial time. Theorem 4: FISMC are universal practical fuzzy integral sliding-model controllers for a class of stochastic nonlinear systems in SNS which are globally stochastically uniformly asymptotically stabilizable. Proof: Given a stochastic nonlinear system (1) which is globally stochastically uniformly asymptotically stabilizable, then there exists a control law du(t) = fc (x(t), u(t))dt, where fc (x(t), u(t)) ∈ C 1 , such that the closed-loop control system described by (57) is globally stochastically uniformly asymptotically stable. Based on Lemma 1, for any given positive constants f and fc , one can construct a nominal closed-loop fuzzy control system as in (29) in the compact region X × U such that fˆ (x, u) = f (x, u) + f (x, u) fˆc (x, u) = fc (x, u) + fc (x, u)

(68)

f (x, u) ≤ f [xT , uT ]T  fc (x, u) ≤ fc [xT , uT ]T .

(69)

where

Then, the integral sliding surface and the dynamic slidingmode controller can be designed as in (54) and (55), respectively. Following the proof procedure of Theorem 1, one can conclude that the trajectories of the closed-loop control system consisting of (1) and (55) can be kept on the integral sliding

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surface (54) almost surely since the initial time. Moreover, the sliding motion of the closed-loop control system with respect to the integral sliding surface (54) is given by (56). Since the closed-loop control system described by (57) is globally stochastically uniformly asymptotically stable, then based on the stochastic type Lyapunov converse theorem in [42], one has that there exist a Lyapunov function V (¯x(t)), K∞ functions α1 (·) and α2 (·), K function α3 (·) and a positive constant c such that α1 (¯x(t)) ≤ V (¯x(t)) ≤ α2 (¯x(t)) (70) ¯ LV (¯x(t))|(57) = DV (¯x(t))F (¯x(t)) 1 ¯ x(t))] ¯ T (¯x(t))Vxx (¯x(t))G(¯ + trace[G 2 ≤ −α3 (x(t)) (71) DV (¯x(t)) ≤ c. (72) It follows from (58) that the stochastic differential operator L associated with the stochastic nonlinear system (56) satisfies LV (¯x(t))|(56) = DV (¯x(t))(F¯ (¯x(t)) + ¯ (¯x(t))) 1 ¯ T (¯x(t))Vxx (¯x(t))G(¯ ¯ x(t))} + trace{G 2 ≤ −α3 (¯x(t)) + c(f Su−1 Sx  + fc )¯x(t).

(73)

Since x¯ (t) ∈ X×U and X×U is a compact set on m ×n , there exists a positive constant σ such that ¯x(t) < σ for all x¯ (t) ∈ X × U. Then it follows from (73) that LV (¯x(t))|(56) ≤ −α3 (¯x(t)) + c(f Su−1 Sx  + fc )σ and there exists a pair of positive constants f and fc such that f Su−1 Sx  + fc ≤ α3 (σ) −1 . Thus if σ ≥ ¯x(t) ≥ α−1 3 (c(f Su Sx  + fc )σ), cσ LV (¯x(t))|(56) ≤ 0, which implies that V (¯x(t)) is a stochastic ISS-Lyapunov function for system (56) [43]. System (56) is thus semiglobally stochastically uniformly input-to-state stable on the compact set X×U. Then, based on Definition 4, FISMC are universal practical fuzzy integral sliding-model controllers. Theorem 5: FISMC are universal asymptotic fuzzy integral sliding-model controllers for a class of stochastic nonlinear systems in SNS which are globally stochastically uniformly asymptotically stabilizable, if for the K function α3 (·) given in (71), there exist a K function α4 (·) and a positive constant κ such that α3 (¯x(t)) − α4 (¯x(t)) inf ≥ κ. (74) ¯x(t)>0,¯x(t)∈X×U ¯x(t) Proof: From the proof of Theorem 4, by designing the integral sliding surface and the dynamic sliding-mode controller as (54) and (55), respectively, the sliding mode can be achieved and maintained almost surely since the initial time. Following the similar arguments as in Theorem 4, there exist a Lyapunov function V (¯x(t)), K∞ functions α1 (·) and α2 (·), K function α3 (·) and a positive constant c such that (70) to (73) hold. Then, one can choose f > 0 and fc > 0 small enough such that f Su−1 Sx +fc < κc . Combining (73) and (74) yields LV (¯x(t))|(56) ≤ −α4 (¯x(t)).

(75)

Then, one can conclude that the closed-loop control system (56) is semi-globally stochastically uniformly asymptotically

9

stable on the compact set X × U, and via Definition 5 FISMC are universal asymptotic fuzzy integral sliding-model controllers. Corollary 1: FISMC are universal asymptotic fuzzy integral sliding-model controllers for a class of stochastic nonlinear systems in SNS which are globally stochastically uniformly asymptotically stabilizable, if for the K function α3 (·) given in (71), there exists a positive constant γ > 0 such that α3 (¯x(t)) inf ≥ γ. (76) ¯x(t)>0,¯x(t)∈X×U ¯x(t) Remark 8: One can observe that for the class of stochastic nonlinear systems which are globally stochastically uniformly exponentially stablizable discussed in Section IV with r = 2, condition (74) or (76) always holds. In other words, universal fuzzy integral sliding-model controllers always imply universal practical fuzzy integral sliding-model controllers and universal asymptotic fuzzy integral sliding-model controllers. Consider a stochastic nonlinear system in the form of (1), if a stochastically uniformly asymptotically stable ref¯ x(t))dW(t), where erence model d x¯ (t) = Fm (¯x(t))dt + G(¯ ¯ x(t)) = Fm (¯x(t)) = R1 f (x(t), u(t)) + R2 fcm (x(t), u(t)) and G(¯ R2 g(x(t), u(t)), is given, one can apply the construction schemes given in [3]– [5] to obtain the model reference fuzzy sliding-mode controller. That is, one can construct a fuzzy dynamic integral sliding-mode controller as in (55) such that for any given cm > 0 ˆ (x, u) = fcm (x, u) + cm (x, u) fcm

(77)

cm (x, u) ≤ cm ¯x.

(78)

where

Then, based on Theorem 4, the constructed fuzzy slidingmode control law (55) guarantees that the trajectories of the closed-loop control system can be kept on the fuzzy integral sliding surface (54) almost surely since the initial time, and the resulting sliding motion is semi-globally stochastically uniformly input-to-state stable in the compact region X × U. VI. Simulation Studies Control of an inverted pendulum is one of benchmark examples to demonstrate the performance of nonlinear control schemes. In this section, we apply the proposed DISMC approach to solving the balancing problem of an inverted pendulum mounted on a cart. In the simulations, the signum function sgn(s(t)) in DISMC control laws is replaced by s(t) to avoid chattering in control signals. The dynamic s(t)+0.03 of the pendulum is described by dx1 (t) = x2 (t)dt dx2 (t) = f (x(t), u(t))dt +(−0.2x1 (t) + 0.3x2 (t) + u(t))dW1 (t) +(0.1x1 (t) − 0.1x2 (t) − u(t))dW2 (t) g sin(x )−amlx2 sin(2x )/2−a cos(x )u

1 1 1 2 where f (x, u) = , x1 denotes the 4l/3−aml cos2 (x1 ) angle of the pendulum from the vertical, x2 is the angular velocity, W(t) = [W1 (t), W2 (t)]T is a 2-D Wiener process,

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g = 9.8 m/s2 is the gravity constant, m is the mass of 1 pendulum, M is the mass of the cart, a = M+m , and 2l is the length of the pendulum. In this paper, we choose m = 2.0 kg, M = 8.0 kg, and 2l = 1.0 m. By using the T-S fuzzy modeling result in [38], one can easily obtain the following uncertain stochastic T-S fuzzy model for the original inverted pendulum plant: dx(t) =

3 

μl (x1 (t))[(Al + Al )x(t) + (Bl + Bl )u(t)]dt

l=1

+[0, −0.2x1 + 0.3x2 + u]T dW1 (t) +[0, 0.1x1 − 0.1x2 − u]T dW2 (t) where the membership functions are shown in Fig. 1(a), and     0 1 0 1 A1 = A2 = 17.2941 0 5.8512 0   0 1 A3 = 0.3593 0     0 0 B1 = B2 = −0.1765 −0.0779   0 B3 = . −0.0052 The objective here is to design a fuzzy dynamic integral sliding-mode controller as in (15) such that the sliding mode can be achieved, and the resulting closed-loop control system on the sliding surface is stochastically asymptotically stable. It is noted that the local control input matrices of the obtained fuzzy models are not equal; thus, the ISMC scheme for stochastic T-S fuzzy models in [37] cannot be applied in this example. However, by using the design procedure provided in Section III-B with f being chosen as 0.03, the corresponding controller gains and sliding surface matrix can be, respectively, given by ¯1 K ¯2 K ¯3 K S¯

= [24989.7118, 23750.8419, 131.8404] = [13085.2783, 12441.4975, −69.0723] = [818.8426, 788.2468, −4.3954] = [−5.5600, −5.9957, 0.0291].

It can also be found that β = 33.4313. In the simulation, Monte Carlo simulations have been conducted by using the discretization approach as in [29]. The simulation parameters are chosen as follows: the simulation interval t ∈ [0, T ] with T = 15, the normally distributed variance is δt = 5 ∗ 2−11 , the step size is t = 2δt, and the initial condition for the fuzzy dynamic sliding-mode controller is u0 = 0. A number of simulations have been conducted under the same initial condition x(0) = x0 = [80◦ , 0]T and u(0) = u0 = 0. By using the fuzzy dynamic sliding-mode controller defined in (15) with α = 5 and β = 33.4313, the state trajectories of the closed-loop control system along ten individual Wiener process paths are shown in Fig. 1(b), while the control input is shown in Fig. 1(c) and the sliding variable is shown in Fig. 1(d). It can be observed that the pendulum can be stochastically asymptotically stabilized.

Fig. 1. Simulation results. (a) Membership functions. (b) State trajectories. (c) Control input. (d) Sliding variable s(t).

VII. Conclusion A novel DISMC scheme has been developed for stochastic nonlinear systems based on stochastic T-S fuzzy models in this paper. The proposed DISMC scheme removes two very restrictive assumptions required in most existing ISMC approaches to stochastic T-S fuzzy systems. It has been shown that the sliding mode can be achieved and maintained almost surely since the initial time, and sufficient conditions to guarantee

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the stochastic stability of the sliding motion is given in terms of LMIs. It has been also shown that the proposed sliding-mode controllers are universal fuzzy integral slidingmode controllers for stochastic nonlinear systems which are globally stochastically uniformly exponentially stabilizable or globally uniformly stochastically asymptotically stabilizable under some sufficient conditions. Constructive procedures to obtain universal fuzzy integral sliding-mode controllers are also provided. Simulation results from an example of balancing an inverted pendulum are provided to demonstrate the advantages and effectiveness of the approaches proposed in this paper.

Acknowledgment The authors are grateful to the Associate Editor and reviewers for their constructive comments based on which the presentation of this paper has been greatly improved.

Appendix Lemma A1 [40]: For arbitrary matrices U and V with appropriate dimensions, the following matrix inequality holds for any positive constant ε: UV + V T U T ≤ ε−1 V T V + εUU T . Lemma A2 [40]: Let Q, E, H, and F (t) be real matrices of appropriate dimensions with F (t) satisfying F T (t)F (t) ≤ I. Then one has Q + EF (t)H + H T F (t)ET < 0 if and only if there exists some scalar ε > 0 such that Q + εEET + ε−1 H T H < 0. Lemma A3 [45]: For a pair of constant matrices G ∈ p×p and M ∈ p×q , if G ≥ 0, then trace(M T GM) ≤ λmax (G)trace(M T M) where λmax (G) is the maximum eigenvalue of G. Lemma A4 [40]: S-procedure. Let T0 , ..., Tp ∈ Rn×n be symmetric matrices. Then, the following condition on T0 , ..., Tp , ξ T T0 ξ > 0 for all ξ = 0 such that ξ T Ti ξ ≥ 0,

i = 1, ..., p,

holds if there exists τ1 ≥ 0, ..., τp ≥ 0 such that T0 −

p 

τi Ti > 0.

i=1

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[29] Q. Gao, G. Feng, Y. Wang, and J. Qiu, “Universal fuzzy models and universal fuzzy controllers for stochastic non-affine nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 21, no. 2, pp. 328–341, Apr. 2013. [30] V. I. Utkin, “Variable structure systems with sliding modes,” IEEE Trans. Automatic Control, vol. 22, no. 2, pp. 212–222, Apr. 1977. [31] V. I. Utkin and J. Shi, “Integral sliding mode in systems operating under uncertainty conditions,” presented at the 35th Confernce on Decision Control, Kobe, Japan, Dec. 1996. [32] J. Chang, “Dynamic output integral sliding-mode control with disturbance attenuation,” IEEE Trans. Automat. Control, vol. 54, no. 11, pp. 2653–2658, Nov. 2009. [33] H. H. Choi, “LMI-based sliding surface design for integral sliding mode control of mismatched uncertain systems,” IEEE Trans. Automat. Control, vol. 52, no. 4, pp. 736–742, Apr. 2007. [34] Y. Niu, D. W. C. Ho, and X. Wang, “Robust H ∞ control for nonlinear stochastic systems: A sliding-mode approach,” IEEE Trans. Automat. Control, vol. 53, no. 7, pp. 1695–1701, Aug. 2008. [35] Y. Niu, D. W. C. Ho, and J. Lam, “Robust integral sliding mode control for uncertain stochastic systems with time-varying delay,” Automatica, vol. 41, no. 5, pp. 873–880, May 2005. [36] Y. Niu, D. W. C. Ho, and X. Wang, “Sliding mode control for Itˆo stochastic systems with Markovian switching,” Automatica, vol. 43, no. 10, pp. 1784–1790, Oct. 2007. [37] D. W. C. Ho and Y. Niu, “Robust fuzzy design for nonlinear uncertain stochastic systems via sliding-mode control,” IEEE Trans. Fuzzy Syst., vol. 15, no. 3, pp. 350–358, Jun. 2007. [38] M. Chen, G. Feng, H. Ma, and G. Chen, “Delay-dependent H ∞ filter design for discrete time fuzzy systems with time-varying delays,” IEEE Trans. Fuzzy Syst., vol. 17, no. 3, pp. 604–616, Jun. 2009. [39] E. Punta, G. Bartolini, A. Pisano, and E. Usai, “A survey of applications of second-order sliding mode control to mechanical systems,” Int. J. Control, vol. 76, nos. 9–10, pp. 875–892, 2003. [40] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA, USA: SIAM, 1994. [41] J. Tsinias and J. Spiliotis, “A converse Lyapunov theorem for robust exponential stochastic stability,” in Proc. Workshop Nonlinear Control, Lecture Notes in Control and Information Sciences, vol. 246. Berlin, Germany: Springer, 1999, pp. 355–374. [42] F. Abedi, M. A. Hassan, and N. M. D. Arifin, “Control Lyapunov function for feedback stabilization of affine in the control stochastic timevarying systems,” Int. J. Math. Anal., vol. 5, no. 4, pp. 175–188, 2011. [43] J. Tsinias, “Stochast input-to-state stability and applications to global feedback stabilization,” Int. J. Control, vol. 71, no. 5, pp. 907–930, 1998. [44] S. Liu, J. Zhang, and Z. Jiang, “A notion of stochastic input-to-state stability and its application to stability of cascaded stochastic nonlinear systems,” Acta Math. Appl. Sin., vol. 24, no. 1, pp. 141–156, 2008. [45] L. Huang and X. Mao, “SMC design for robust H ∞ control of uncertain stochastic delay systems,” Automatica, vol. 46, no. 2, pp. 405–412, Feb. 2010.

Qing Gao (M’13) received the B.Eng. and Ph.D. degrees in mechanical and electrical engineering from the University of Science and Technology of China, Hefei, China, in 2008 and 2013, respectively, and the Ph.D. degree in mechatronics engineering from the City University of Hong Kong, Kowloon, Hong Kong, in 2013. Since 2014, he has been a Research Associate at the School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, Australia. His current research interests include quantum control, intelligent systems and control, and variable structure control. Dr. Gao was the recipient of the Chinese Academy of Sciences Presidential Scholarship (Special Prize) in 2013. He was also the recipient of the Outstanding Research Thesis Award from the City University of Hong Kong in 2013.

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Lu Liu (SM’13) received the Ph.D. degree from the Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, Shatin, Hong Kong, in 2008. From 2009 to 2012, she was an Assistant Professor with the Department of Information Physics and Computing, University of Tokyo, Tokyo, Japan, and then a Lecturer at the Department of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, Nottingham, U.K. She is currently an Assistant Professor at the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong. Her current research interests include primarily networked dynamical systems, control theory, and applications and biomedical devices. Dr. Liu received the Best Paper Award (Guan Zhaozhi Award) in the 27th Chinese Control Conference in 2008.

Gang Feng (F’09) received the B.Eng. and M.Eng. degrees in automatic control from the Nanjing Aeronautical Institute, Nanjing, China, in 1982 and 1984, respectively, and the Ph.D. degree in electrical engineering from the University of Melbourne, Melbourne, Australia, in 1992. He has been with the City University of Hong Kong, Kowloon, Hong Kong, since 2000, where he is currently a Chair Professor of mechatronic engineering with the Department of Mechanical and Biomedical Engineering. He is with King Abdulaziz University, Saudi Arabia. He is also a ChangJiang Chair Professor at the Nanjing University of Science and Technology, awarded by the Ministry of Education, China. He was a Lecturer and Senior Lecturer at the School of Electrical Engineering, University of New South Wales, Australia, from 1992 to 1999. His current research interests include hybrid systems and control, modeling and control of energy systems, and intelligent systems and control. Dr. Feng was awarded an Alexander von Humboldt Fellowship in 1997 to 1998, and the IEEE Transactions on Fuzzy Systems Outstanding Paper Award in 2007. He is an Associate Editor of the IEEE Transactions on Fuzzy Systems and Mechatronics, and was an Associate Editor of the IEEE Transactions on Automatic Control, the IEEE Transactions on Systems, Man and Cybernetics, Part C, and the Journal of Control Theory and Applications.

Yong Wang received the B.Eng. degree in automatic control from the University of Science and Technology of China, Hefei, China, in 1982, and the M.Eng. and Ph.D. degrees in navigation, guidance and control from the Nanjing Aeronautical Institute, Nanjing, China. He has been with the Department of Automation, University of Science and Technology of China, Hefei, China, since 2001, where he is currently a Professor. He leads several research groups focusing on vehicle control and vibration control supported by the National Science Foundation of China and the 863 Project. His current research interests include active vibration control and vehicle guidance and control. Dr. Wang is a member of the Motion Control Committee of the Chinese Association of Automation.