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A New Design of Robust H∞ Sliding Mode Control for Uncertain Stochastic T–S Fuzzy Time-Delay Systems Qing Gao, Student Member, IEEE, Gang Feng, Fellow, IEEE, Zhiyu Xi, Member, IEEE, Yong Wang, and Jianbin Qiu, Member, IEEE
Abstract—In this paper, a novel dynamic sliding mode control scheme is proposed for a class of uncertain stochastic nonlinear time-delay systems represented by Takagi–Sugeno fuzzy models. The key advantage of the proposed scheme is that two very restrictive assumptions in most existing sliding mode control approaches for stochastic fuzzy systems have been removed. It is shown that the closed-loop control system trajectories can be driven onto the sliding surface in finite time almost certainly. It is also shown that the stochastic stability of the resulting sliding motion can be guaranteed in terms of linear matrix inequalities; moreover, the sliding-mode controller can be obtained simultaneously. Simulation results illustrating the advantages and effectiveness of the proposed approaches are also provided. Index Terms—Dynamic sliding-mode controllers, sliding mode control (SMC), stochastic nonlinear systems, Takagi–Sugeno (T–S) fuzzy models, time delay.
I. Introduction
R
ECENTLY, stochastic dynamic systems modeled by Itˆo stochastic differential equations have been gaining increasing attention from the control community since stochastic modeling plays an important role in many branches of
Manuscript received November 6, 2012; revised August 12, 2013; accepted October 31, 2013. Date of publication December 3, 2013; date of current version August 14, 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, in part by the National Natural Science Foundation of China under Grant 61374031, in part by the Program for New Century Excellent Talents in University under Grant NCET-12-0147, and in part by the Alexander von Humboldt Foundation of Germany. This paper was recommended by Associate Editor T. H. Lee. Q. Gao is with the Department of Automation, University of Science and Technology of China, Hefei 230026, China, and also with the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]). G. Feng is with the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong, and also with the Nanjing University of Science and Technology, Nanjing 210044, China (e-mail: [email protected]). Z. Xi is with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia (e-mail: [email protected]). Y. Wang is with the Department of Automation, University of Science and Technology of China, Hefei 230026, China (e-mail: [email protected]). J. Qiu is with the School of Astronautics, Harbin Institute of Technology, Harbin 150001, 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.2013.2289923
science and engineering [1]–[4]. However, it is worth noting that most of the existing results on stochastic nonlinear control systems, especially for those with high nonlinearity, suffer from a lack of systematic way of control design due to difficulties in finding appropriate Lyapunov functions [3], [4]. On one hand, Takagi–Sugeno (T–S) fuzzy models [5] have been widely used as a convenient tool for handling complex nonlinear systems [5]–[17]. T–S fuzzy models describe a nonlinear system as the blending of a set of local linear dynamic models. This relatively simple structure provides systematic stability analysis and controller design of T–S fuzzy control systems by exploiting conventional control theory. Recently, 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 [18]– [20]. Like the case of deterministic T–S fuzzy models, control design of stochastic fuzzy systems can be realized by solving a set of linear matrix inequalities, see [18]–[20] for details. On the other hand, as a powerful robust control strategy, sliding mode control (SMC) has been extensively studied and widely used for control of nonlinear systems due to its various attractive features such as strong robustness and fast response [21]– [33]. Generally speaking, SMC is attained by using a discontinuous control law to drive system state trajectories onto a predefined sliding manifold containing the origin, which is commonly termed the sliding surface or switching surface, in finite time (this process is called reaching phase), and then to force the state trajectories to move along the surface toward the origin with desired performance (such a motion is called sliding mode). The overall system dynamics are determined by the sliding surface that is normally designed as linear hyperplanes of the system states. More recently, some results on applying SMC to stabilize stochastic nonlinear systems have been reported (see, for example, [29]–[31]). In particular, Ho and Niu [31] addressed the SMC design for stochastic fuzzy systems. Stochastic systems considered in those works are given in the form of dx(t) = {f (x(t)) + Bu(t)}dt + g(x(t))dW(t)
c 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2267 See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
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where it is assumed that there exists a matrix S with appropriate dimension such that SB is nonsingular and Sg(x(t)) = 0 for all t ≥ 0.
(1)
In this way, stochastic perturbations do not need to be dealt with during the reaching phase, and SMC techniques for deterministic systems can thus be used. In addition, another restrictive assumption is required for the fuzzy SMC approach in [31], that is, all subsystems of the stochastic T–S fuzzy model have the same input matrix. It is noted that the mathematical models of many physical plants, such as the well known inverted pendulum on a cart [11], do not satisfy this assumption. In [28], a piecewise fuzzy SMC approach was developed for uncertain T–S fuzzy models by partitioning the premise state space into a set of subregions, and by designing individual sliding surfaces within each subregion. In the approach developed in [28], different local control input matrices are allowed for the T–S fuzzy models. However, the approach proposed in [28] is very difficult to implement in practice due to its high complexity. To the authors’ best knowledge, few effective SMC approaches to stochastic T–S fuzzy systems without the aforementioned two assumptions have been reported in the literature, which motivates us for this paper. In this paper, we propose a novel dynamic sliding mode control (DSMC) approach to a class of stochastic nonlinear time-delay systems represented by stochastic T–S fuzzy models. A key feature of the proposed DSMC approach is that the sliding surface function is defined to be linearly dependent on both of the system state vector x and the control input vector u. By utilizing a sliding-mode controller with a fuzzy dynamic feedback control form, both the system states x and control inputs u are driven onto the sliding surface in finite time almost certainly. It is shown that the stochastic asymptotic stability on the sliding surface can be guaranteed. It is also shown that the sliding surface and the dynamic sliding-mode controller can be obtained simultaneously in terms of a set of linear matrix inequalities (LMIs). Compared with the existing fuzzy SMC approaches to stochastic T–S fuzzy systems, for example, the approach in [31], the main advantages of the proposed DSMC approach include: 1) the two aforementioned restrictive assumptions that are required in the existing approaches have been removed, and 2) the proposed DSMC approach is able to deal with a much broader class of stochastic nonlinear systems than the existing approaches. The rest of this paper is organized as follows. Section II is devoted to model description and preliminaries. Details of the DSMC approach to robust stabilization and H∞ control for uncertain stochastic time delay T–S fuzzy models are given in Section III. Simulation results are provided in Section IV to demonstrate the advantages and effectiveness of the proposed approaches. Conclusions are given in Section V. Notations: The notations used in this paper are fairly standard. The notation in a matrix is used to indicate the terms that can be induced 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. The notation L2 [0, T ] will be used forvector-valued functions, i.e., ω : [0, T ] → k ∈ L2 [0, T ] T if 0 ωT (t)ω(t)dt < ∞ and its L2 − norm is defined as T T ||ω(t)||2 = 0 ω (t)ω(t)dt. 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 Preliminaries In this paper, we consider a general Itˆo type stochastic nonlinear system described by the following stochastic T–S fuzzy model with a constant time delay. Plant rule: Rl IF θ1 (t) is μl1 AND ... AND θg (t) is μlg ; THEN dx(t) = [(Al + Al )x(t) + (Bl + Bl )u(t) + Hl ω(t) +(El + El )x(t − τ)]dt + g(x(t), u(t))dW(t) y(t) = Cl x(t) + Dl u(t) + Cld x(t − τ) x(t) = ϕ(t), t ∈ [−τ, 0], l ∈ L := {1, 2, ...r}
(2)
where θ(t) = [θ1 (t), ..., θg (t)] is the premise variables vector, r is the number of fuzzy rules, Rl denotes the lth rule, μli are the fuzzy sets, τ is a constant time delay, x(t) ∈ n is the state vector, u(t) ∈ m is the control input vector, y(t) ∈ p is the output vector, ω(t) ∈ h is the exogenous disturbance belonging to L2 [0, ∞), ϕ(t) is a known bounded vector function, W(t) = [W1 (t), ..., Wq (t)]T is a q-dimensional Wiener process defined on the probability space (, F, P) with a natural filtration {Ft }t≥0 , and Al , Bl , Cl , Cld , Dl , El , Hl are system matrices with appropriate dimensions. The following two assumptions are made in this paper. 1) The parameter uncertainties Al , Bl , and El are norm bounded [31], that is [Al , Bl ] ≤ A , and El ≤ E
(3)
where A and E are known positive constants. 2) The function g(x, u) : n × m → q , is not exactly known but satisfies the following condition [31], [35]: trace[gT (x, u)g(x, u)] ≤ G[xT , uT ]T 2
(4)
where G is a known constant matrix. Via the standard fuzzy blending method, i.e., the centeraverage defuzzifier, product inference, and singleton fuzzifier, the stochastic T–S fuzzy system (2) can be expressed globally as ⎧ dx(t) = rl=1 μl (θ(t))[(Al + Al )x(t) ⎪ ⎪ ⎪ ⎪ +(Bl + Bl )u(t) + (El + El )x(t − τ) ⎨ +H (5) l ω(t)]dt + g(x(t), u(t))dW(t) r ⎪ ⎪ y(t) = μ (θ(t))[C x(t) + D u(t) + C x(t − τ)] ⎪ l l l ld l=1 ⎪ ⎩ x(t) = ϕ(t), t ∈ [−τ, 0]
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with the normalized fuzzy membership functions μl (θ(t)) defined by g l μ (θi (t)) (6) μl (θ(t)) = r i=1 g i l l=1 i=1 μi (θi (t)) satisfying μl (θ(t)) ≥ 0 and rl=1 μl (θ(t)) = 1. It is well known that the T–S fuzzy model (2) can be used to represent a large class of stochastic nonlinear systems [18], [19]. One can easily verify that the stochastic differential equation (5) with u(t) ≡ 0 and ω(t) ≡ 0 satisfies the so-called Lipschitz conditions and Linear growth conditions, and thus has a unique solution [1]. In [31], SMC design for a class of stochastic nonlinear systems represented by stochastic T–S fuzzy models as in (2) has been investigated. However, the approach proposed in [31] relies on two very restrictive assumptions: 1) all subsystems of the stochastic T–S fuzzy model (2) share the same input matrix, that is, Bl ≡ B, for all l ∈ L, and 2) there exists a matrix S with appropriate dimension that SB is nonsingular, and moreover Sg(x(t), u(t)) = 0, for all t ≥ 0.
(7)
Both of these two assumptions impose great limitations in real applications. In this paper, we will develop a new SMC scheme, which is not restricted by these two assumptions, to robustly stabilize the stochastic T–S fuzzy system (2) with a guaranteed H∞ performance. Specifically, the objective of this paper can be formulated as: construct an SMC law such that: 1) the closed-loop control system trajectories can be driven onto the sliding surface in finite time almost surely, and 2) given a positive constant γ > 0, the resulting sliding motion with ω(t) ≡ 0 is stochastically asymptotically stable ∞ ∞ and E t0 y(t)2 dt < γ 2 t0 ω(t)2 dt is satisfied for all nonzero ω(t) ∈ L2 [0, ∞) under zero initial condition x(φ) = 0, for all φ ∈ [tr , t0 ], where t0 = tr + τ and tr is the time when the sliding surface is reached.
A. Design of Sliding Surface and Sliding-Mode Controller For a stochastic nonlinear system in (2), a novel sliding surface is designed as s(t) = Sx x(t) + Su u(t) = S¯ x¯ (t) = 0 (8) m×n m×m ¯ , Su ∈ , S = [Sx , Su ], and x¯ (t) = where Sx ∈ [x1 (t), ..., xn (t), u1 (t), ..., um (t)]T . Su is assumed to be nonsingular. In this case, the following fuzzy dynamic sliding-mode controller can be employed: r u(t) ˙ =− μl (θ)Su−1 Sx (Al x(t) + El x(t − τ) + Bl u(t)) l=1
(9)
where σ(t) = Sx ( A [xT (t), uT (t)]T + E x(t − τ)) r + wl (θ(t))Hl ρ(t). l=1
The conditions on reachability of the sliding surface (8) are given in Theorem 1. Theorem 1: For the closed-loop control system (11), the sliding surface (8) can be reached in finite time almost surely, if there exists a positive constant δ such that ¯ GT G < δS¯ T S. (12) Moreover, β in (9) can be chosen as β ≥ 21 λmax (SxT Sx )δ. Proof: Consider the function S(t) = sT (t)s(t) for all t > 0. Then from Itˆo’s formula [1] with the fact that ¯ 1 − R2 Su−1 Sx ) = 0 and SR ¯ 2 Su−1 = I, the infinitesimal S(R generator r
¯ l x¯ (t) μl (θ(t))sT (t)S¯ (R1 − R2 Su−1 Sx )A LS(t)|(11) = 2 l=1
¯ l x¯ (t − τ) + R1 Hl ω(t) +(R1 − R2 Su−1 Sx )E ¯ l x¯ (t) + R1 E ¯ l x¯ (t − τ) +R1 A −2(α + σ(t))s(t) − 2βsT (t)s(t) ¯ 1 g(x, u)} +trace{gT (x, u)RT1 S¯ T SR ¯ 1 g(x(t), u(t))} ≤ trace{gT (x(t), u(t))RT1 S¯ T SR
III. Design of Dynamic Sliding Mode Control
−(α + σ(t))Su−1 sgn(s(t)) − βSu−1 s(t)
A and E are defined in (3), α > 0, ρ(t) is the known uniform upper bound of the disturbance ω(t), and β is a positive constant to be determined. ¯ l = [Al , Bl ], E ¯l = Denote R1 = [In , 0]T , R2 = [0, Im ]T , A ¯ ¯ [El , 0n×m ], Al = [Al , Bl ], El = [El , 0n×m ] and x¯ (t) = [xT (t), uT (t)]T , l ∈ L := {1, 2, ..., r}. Then by introducing an additional term 0 × u(t − τ) with u(t) = 0, t ∈ [−τ, 0] in both (2) and (9), the closed-loop control system consisting of (2) and⎧(9) can be rewritten in a compact form as
r −1 ⎪ ¯ ¯ (t) ⎪ d x¯ (t) = l=1 μl (θ(t)) (R1 − R2 Su Sx )Al x ⎪ ⎪ ⎪ −1 ⎪ ¯ l x¯ (t) ¯ l x¯ (t − τ) + R1 A +(R1 − R2 Su Sx )E ⎪ ⎪ ⎪ ⎨ ¯ l x¯ (t − τ) + R1 Hl ω(t) +R1 E (11) −1 ⎪ −R2 (α + σ(t))S ⎪ ⎪ u sgn(s(t)) ⎪ ⎪ ⎪ −βR2 Su−1 s(t) dt + R1 g(x(t), u(t))dW(t) ⎪ ⎪ ⎪ ⎩ T x¯ (t) = [φ (t), 0]T , t ∈ [−τ, 0].
(10)
−2βsT (t)s(t) − 2αs(t).
(13)
By using Lemma 1 in the Appendix based on (4) and (12), one has ¯ 1 g(x(t), u(t)) trace{gT (x(t), u(t))RT1 S¯ T SR ≤ λmax (SxT Sx )trace{gT (x(t), u(t))g(x(t), u(t))} ≤ λmax (SxT Sx )¯xT (t)GT G¯x(t) < λmax (SxT Sx )δ¯xT (t)S¯ T S¯ x¯ (t) ≤ 2βsT (t)s(t). Then, it follows from (13) and (14) that LS(t)|(11) < −αs(t) = −α S(t).
(14) (15)
From (15) and by using Itˆo’s formula again, one has Ls(t)|(11) = L S(t)|(11) 1 1 = √ LS(t)|(11) − √ 2 S(t) 2( S(t))3 ¯ 1 g(x(t), u(t))} ×trace{gT (x(t), u(t))RT1 S¯ T s(t)sT (t)SR 1 ≤ √ (16) LS(t)|(11) ≤ −α, for s(t) = 0. 2 S(t)
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Integrating from 0 to t on both sides of (16) results in 0 < Es(t) ≤ s(0) − αt. Thus, one has Es(t) = 0 for all t ≥ tr = s(0) . This implies s(t) = 0 almost surely for all α t ≥ tr . In other words, the sliding surface (8) can be reached in finite time almost surely. The proof is then completed. Remark 1: It is noted that the sliding variable s(t) is designed as a linear combination of both the state vector x and the control input vector u. In addition, the sliding-mode controller is in the form of a fuzzy dynamic state feedback control. All these features distinguish the proposed SMC scheme from existing ones [25]– [28]. This is the main reason why it is called the DSMC approach. Remark 2: Most of the existing SMC approaches for T–S fuzzy models, either stochastic or deterministic, suffer from a very restrictive assumption that all local systems of the T–S fuzzy models share the same input gain matrix while the DSMC approach does not. Remark 3: It is noted that assumption (7) required in [29]–[31] has been also removed. One can observe from the proof procedure of Theorem 1 that one can choose β = 0 in (9) if assumption (7) holds.
Proof: We first consider the stability of system (17) with ω(t) = 0. Consider the following Lyapunov–Krasovskii functional candidate: t T
V (t) = x¯ (t)X¯x(t) +
Theorem 2: Given a constant γ > 0, the sliding mode dynamics (17) is stochastically asymptotically stable with H∞ performance γ, if there exist two positive definite matrices P, Q ∈ (m+n)×(m+n) , two sets of matrices Wl1 , Wl2 ∈ m×(m+n) , a positive constant δg , and a set of positive constants εl , l ∈ L satisfying the LMIs (18) at the bottom of this page and P − δg I > 0
l∈L
(19)
¯ T RT + W T RT + Q + ε l R1 RT . ¯ l P + R2 W1l + P A where l = R1 A 1 l 1 1l 2 Moreover, the sliding surface matrix is given by S¯ = RT2 P −1 .
(20)
t−τ
for all t ≥ t0 = tr + τ, where X = P −1 and Y = P −1 QP −1 . Since (19) is equivalent to X < δ1g I, then based on Lemma 1 in the Appendix, (4), and the fact that RT1 R1 = I, one has trace{gT (x(t), u(t))RT1 XR1 g(x(t), u(t)) 1 ≤ trace{gT (x(t), u(t))g(x(t), u(t)) δg 1 ≤ x¯ T (t)GT G¯x(t). δg
(21)
It is noted that on the sliding surface, s(t) = S¯ x¯ (t) = RT2 X¯x(t) = 0, then one has
B. Stability Analysis of Sliding Motion It has been shown in Theorem 1 that the sliding surface in (8) can be reached in finite time almost surely. In this subsection, we proceed to analyze the stability of the sliding motion. ¯ l, First, we denote Hl = rl=1 μl (θ(t))Hl , C¯l = rl=1 μl (θ(t))C r ¯ ¯ ¯ μ (θ(t)) D and D¯ l = , where C = [C , C ] and D l l l ld l = l=1 l [Dl , 0n×m ]; then, the closed-loop control system (11) restricted on the sliding surface (8) becomes ⎧
r ⎪ ¯ l x¯ (t) d x ¯ (t) = μ (θ(t)) (R1 − R2 Su−1 Sx )A ⎪ l l=1 ⎪ ⎪ ⎪ ⎪ ¯ l x¯ (t) ¯ l x¯ (t − τ) + R1 A +(R1 − R2 Su−1 Sx)E ⎪ ⎪ ⎨ ¯ l x¯ (t − τ) + R1 Hl ω(t) dt +R1 E (17) ⎪ ⎪ +R g(x(t), u(t))dW(t), ⎪ 1 ⎪ ⎪ ⎪ y(t) = C¯l x¯ (t) + D¯ l x¯ (t − τ), ⎪ ⎪ ⎩ x¯ (t) = [φT (t), 0]T , t ∈ [−τ, 0].
x¯ T (ϕ)Y x¯ (ϕ)dϕ
LV (t)|(17) = 2
r
¯ l + A ¯ l )¯x(t) μl (θ(t))¯xT (t)XR1 [(A
l=1
¯ l )¯x(t − τ)] − 2 x¯ T (t)XR2 ¯ l + E +(E =0
×
r
¯ l x¯ (t) + E ¯ l x¯ (t − τ)] μl (θ(t))Su−1 Sx [A
l=1
+¯xT (t)Y x¯ (t) − x¯ T (t − τ)Y x¯ (t − τ) +trace gT (x(t), u(t))RT1 XR1 g(x(t), u(t)) r ¯ l + A ¯ l )¯x(t) ≤2 μl (θ(t))¯xT (t)XR1 [(A l=1
¯ l )¯x(t − τ)] + 1 x¯ T (t)GT G¯x(t) ¯ l + E +(E δg +¯xT (t)Y x¯ (t) − x¯ T (t − τ)Y x¯ (t − τ) r ¯ l + A ¯ l )¯x(t) =2 μl (θ(t))¯xT (t)XR1 [(A l=1
¯ l + E ¯ l )¯x(t − τ)] − 2 x¯ T (t)XR2 +(E =0
×
r
μl (θ(t))[K1l x¯ (t) + K2l x¯ (t − τ)]
l=1 T
+¯x (t)Y x¯ (t) − x¯ T (t − τ)Y x¯ (t − τ) 1 + x¯ T (t)GT G¯x(t). δg
(22)
⎡
⎤ l + γ12 R1 Hl HlT RT1 T T T T ⎢ PE ⎥ ⎢ ¯ l R1 + Wl2 R2 −Q ⎥ ⎢ ¯ ¯ C P D P −I ⎥ l l ⎢ ⎥ 0; then, one has LV (t)|(17) ≤
LV (t)|(17) + yT (t)y(t) − γ 2 ωT (t)ω(t) r
¯ l + R2 K1l )¯x(t) ≤2 μl (θ(t))¯xT (t)X (R1 A l=1
2 T +ε−1 ¯ (t − τ)¯x(t − τ) l E x 2 ε XR1 R1 X + ε−1 l A I = ζ T (t) l ζ(t) 2 0 ε−1 l E I
r
Along the trajectory of (17), one has
(23)
l=1
where
ϒl ζ(t) 2 ¯ l + R2 K2l )T X ε−1 (R1 E l E I − Y r +2 μl (θ(t))¯xT (t)XR1 Hl ω(t)
× ¯ l + R2 K1l ) + (R1 A ¯ l + R2 K1l )T X ϒl = X(R1 A 1 T 2 +εl XR1 RT1 X + ε−1 G G. l A I + Y + δg
Then LV (t)|(17) < 0 if (19) holds and ϒl < 0. 2 ¯ l + R2 K2l )T X ε−1 (R1 E l E I − Y
(24)
l=1
+yT (t)y(t) − γ 2 ωT (t)ω(t) (25)
Multiplying diag{P, P} with P = X−1 from both sides to (25) yields l t0 results in T 0 < E {V (t)} = E LV (t)dt t0 T T ≤ −E yT (t)y(t)dt + γ 2 ωT (t)ω(t)dt (37)
r l=1
⎤
2 ε−1 l E PP
When time delay, external disturbance, or random noises are not considered, the system in (2) reduces to a nominal deterministic T–S fuzzy model as follows:
t0
∞ ∞ which implies E t0 y(t)2 dt < γ 2 t0 ω(t)2 dt. The proof is thus completed. Actually, by choosing S¯ = RT2 P −1 , the condition in (12) can be rewritten in the form of LMIs as follows: R2 RT2 >0 (38) GP δI by which the positive parameter β in the sliding-mode controller (9) can be obtained. It then follows from Theorem 2 that the minimum H∞ performance index γmin can be obtained by the following convex optimization algorithm [36]. Algorithm 1: min − γ12 , subject to LMIs (18), P>0,Q>0,W1l ,W2l ,εl >0
(19), and (38). Remark 4: It is noted that the matrices Kil , i = 1, 2, in (22) are not feedback gain matrices. They are introduced in (22) to improve the feasibility of the LMI conditions in (18), by taking the advantage that RT2 X¯x(t) = 0 on the sliding surface. Remark 5: It is noted that the systems considered in this paper are more general than those in [31] in that the local input matrices of investigated stochastic T–S fuzzy systems are allowed to have unmatched uncertainties. Moreover, it has been proved in the authors’ previous work [20] that if the premise variable θ(t) contains both x and u, the stochastic T–S fuzzy models (2) can be used to approximate the so-called stochastic nonaffine nonlinear systems. With the dynamic sliding-mode controller employed, it is found that the proposed approach can also be applied to deal with stochastic nonaffine nonlinear systems. In both of these two cases, the approach in [31] cannot easily be applied. In other words, the proposed DSMC approaches can deal with a much broader class of nonlinear systems.
under which the sliding surface in (8) can be reached in finite time. The closed-loop control system restricted on the sliding surface is asymptotically stable if there exist a positive definite matrix P ∈ (m+n)×(m+n) and a set of matrices Wl ∈ m×(m+n) such that the following linear matrix inequalities are satisfied: ¯ Tl RT1 + R1 A ¯ l P + WlT RT2 + R2 Wl < 0 PA l ∈ L := {1, 2, ...r}.
(41)
Moreover, the sliding surface matrix is given by S¯ = RT2 P −1 . Remark 6: As indicated in Remark 2, most existing results fuzzy SMC approaches assume that all Bi in (39) to be identical. It can be observed that Corollary 1 is not restricted by such an assumption. An alternative SMC design approach to nominal T–S fuzzy models which avoided this assumption can be found in [27]. However, it is required in [27] that for r B the T–S fuzzy models as in (39), [Al , l=1r l ] is stabilizable for any l ∈ L, which is also very restrictive. Comparison between Corollary 1 and the results in [27] will be provided in the simulation section. Remark 7: In the proposed DSMC approaches, the dynamic sliding-mode controller (9) contains a switching term sgn(s(t)), which would cause chattering phenomenon when implemented. Chattering avoidance methods as in [33] can be adopted to address the problem.
IV. Simulation Studies In this section, simulation results will be given to demonstrate the advantages and effectiveness of the approaches proposed in this paper. In the simulations, the signum function s(t) sgn(s(t)) in DSMC control laws is replaced by s(t)+0.01 to reduce chattering in the control signals, following the result of [31]. Example 1: Control of the inverted pendulum is one of the benchmark examples to demonstrate nonlinear control scheme performances. To illustrate the effectiveness of the proposed approach, we consider the balancing problem of an inverted
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pendulum mounted on a cart. The dynamic of the pendulum is described by dx1 = x2 dt g sin(x1 ) − amlx22 sin(2x1 )/2 − a cos(x1 )v dx2 = + ω(t) dt 4l/3 − aml cos2 (x1 ) +[2x1 − x2 + u, x1 − x2 − u]dW where x1 denotes the angle of the pendulum from the vertical, x2 is the angular velocity, and W = [W1 , W2 ]T is a 2-D Wiener process. 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 , 2l is the length of the pendulum, and w is the disturbance. In this paper, we choose m = 2.0 kg, M = 8.0 kg, 2l = 1.0 m. The following uncertain stochastic T–S fuzzy model is widely used to approximate the original nonlinear model [11] dx(t) =
3
μl (x1 (t)){(Al + Al )x(t) + (Bl + Bl )u(t)
0 +Hl ω(t)}dt + dW1 (t) 2x1 − x2 + u 0 + dW2 (t) x1 − x2 − u l=1
y(t) =
3
μl (x1 (t)){Cl x(t) + Dl u(t)}
l=1
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 ! " 0 C1 = C2 = C3 = 0.5 1 , H1 = H2 = H3 = 1 D1 = 0.05, and D2 = D3 = 0.01. It is noted that this T–S fuzzy model is widely adopted in many investigations and has been confirmed in [11] to be a good approximate fuzzy model of the original nonlinear model. It is also noted that the mathematical model of the original pendulum plant can also be described by an affine nonlinear differential equation, i.e., x˙ (t) = f (x)+g(x)u, where f (x) = g sin x1 −amlx22 sin(2x1 )/2 −a cos x1 and g(x) = 4l/3−aml . In practice, it 4l/3−aml cos2 x1 cos2 x1 might be difficult to determine the exact upper bounds of approximationerrors. In this case study, we choose to calculate f (x)− 3l=1 μl (x1 )Al x γ1 (x) = | | and γ2 (x) = |g(x) − 3l=1 μl (x1 )Bl |} x at a number of vertex points (x1 , x2 ) ∈ [−π/2, π/2] × [−5, 5], within the operating range of the pendulum. It is found that γ1 (x) < 0.01 and γ2 (x) < 0.02. It is noted that only finite tests can be conducted. However, one can choose to test more points within the operating region thus to improve the precision of the obtained upper bounds.
Similar to [34], we assume that the delay terms are perturbed along values of the scalar λ ∈ [0, 1]; then, the system dynamic considered is described by dx(t) =
3
μl (x1 (t)){λ(Al + Al )x(t)
l=1
+(1 − λ)(Al + Al )x(t − τ) + (Bl + Bl )u(t) 0 dW1 (t) +Hl ω(t)}dt + 2x1 − x2 + u 0 + dW2 (t) x1 − x2 − u y(t) =
3
μl (x1 (t)){Cl x(t) + Dl u(t)}.
l=1
The objective here is to design a fuzzy dynamic slidingmode controller as in (9) such that the resulting closed-loop control system restricted to the sliding surface is stochastically asymptotically stable with H∞ performance γ. In this paper, λ is chosen to be 0.9. It is noted that the local control gain matrices of the obtained fuzzy models are not equal; thus, the SMC design results in [31] cannot be applied to this example. However, by applying Algorithm 1 with ¯ A and ¯ E chosen as 0.02 and 0.001, respectively, the minimum H∞ performance γmin = 3.7317 is obtained and the corresponding sliding surface matrix and positive definite matrices are, respectively, given by S¯ = [−0.0314 − 0.00440.0003] ⎡ ⎤ 0.043 −0.087 3.304 P = ⎣ −0.087 0.681 0.965 ⎦ 3.304 0.965 3848.540 ⎤ ⎡ 0.021 −0.069 −0.073 Q = ⎣ −0.069 0.826 0.927 ⎦ . −0.073 0.927 3270.423 It is found that β ≥ 2.28. In the simulation, Monte Carlo simulations have been conducted by using the discretization approach as in [20]. The simulation parameters used are as follows: the simulation interval t ∈ [0, T ] with T = 15, the normally distributed variance is δt = T/N with N = 3 ∗ 211 , 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 , u(0) = u0 = 0, and x(t) = [0, 1]T , 0 < t < τ with τ = 0.41 s. By using the fuzzy dynamic sliding-mode controller defined in (9) with α = 2.5 and β = 5, the state trajectories and the control inputs of the closed-loop control system with ω(t) = 0 along ten individual Wiener process paths are shown in Fig. 1(b) and (c), respectively. One can observe that the pendulum can be stochastically asymptotically stabilized. Next, we illustrate the disturbance attenuation performance. The response of the t T y (ψ)y(ψ)dψ ratio t0 wT (ψ)w(ψ)dψ of the closed-loop control system with the 0
disturbance given by ω(t) = e−0.02t sin(πt) and under zero conditions along ten individual Wiener process paths is shown in Fig. 1(d). One can see that the performance response
GAO et al.: NEW DESIGN OF ROBUST H∞ SLIDING MODE CONTROL
Fig. 1. Simulation results in Example 1. (a) Membership functions for Example 1. (b) State trajectories for Example 1. (c) Control input for Example 1. (d) Individual paths and the average of the ratio for Example 1. t
t T y (ψ)y(ψ)dψ
t0 0
wT (ψ)w(ψ)dψ
yT (ψ)y(ψ)dψ
is satisfactory and the average of the ratio t0 wT (ψ)w(ψ)dψ is 0 much less than the minimum disturbance attenuation level 2 γmin = 13.9256. Example 2: Consider a stochastic T–S fuzzy system with two fuzzy rules # 2 $ dx(t) = νl (θ(t))(Al + Al )x(t) + Bu(t) dt + g(x(t))dW(t) l=1
0 , the membership functions are x 1 + x2 shown in Fig. 3(a), and the system matrices are given as 0 1 0 1 0 A1 = A2 = B= . 2 −1 1 −2 1
where g(x(t)) =
It is noted that condition (1) is not satisfied, since for any given matrix S such that SB is nonsingular, Sg(x(t)) =
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Fig. 2. Simulation results in Example 2. (a) Membership functions for Example 2. (b) State trajectories for Example 2. (c) Control input for Example 2.
SB(x1 + x2 ) = 0 for any nonzero states; thus, the approach in [31] cannot be applied to this example either. However, by applying Theorem 1 with ¯ A = 0.03 and ¯ E = 0, the corresponding sliding surface matrix and positive definite matrices are, respectively, given by S¯ = [0.8936, 1.0218, 0.1901] ⎡ ⎤ 0.0869 −0.0793 0.0179 P = ⎣ −0.0793 0.1385 −0.3716 ⎦ 0.0179 −0.3716 7.1732 ⎡ ⎤ 0.0173 −0.0180 0.0000 Q = ⎣ −0.0180 0.0666 −0.0000 ⎦ . 0.0000 −0.0000 0.9679 It is found that β can be chosen to be greater than 43.6355. By choosing Al = 0.03 sin(t) and following the similar simulation method in Example 1, a number of simulations have been conducted under the initial condition x(0) = x0 =
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IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 9, SEPTEMBER 2014
It can be seen from simulation results that this T–S fuzzy system is unstable. It is noted that the method in [27] cannot be used since [A1, (B1 + B2)/2] is not stabilizable. However, by using Corollary 1, one has the following feasible solutions: S¯ = [2.46794.13940.4832] ⎡ ⎤ 8.3380 −3.9657 −8.6131 P = ⎣ −3.9657 2.7719 −3.4907 ⎦ −8.6131 −3.4907 75.9592 which will lead to an asymptotically stable closed-loop control system. The simulation results with initial condition x(0) = [0.4, −1]T are shown in Fig. 3(b) where all states converge to the origin as time approaches infinity. V. Conclusion This paper proposes a novel DSMC approach to robust H∞ control for a class of uncertain stochastic T–S fuzzy time-delay systems. The proposed approach removes some restrictive assumptions that are required in most existing schemes. It is shown that the sliding surface and sliding-mode controller can be obtained in terms of linear matrix inequalities, and the stochastic asymptotic stability of the sliding motion can be guaranteed. Some interesting future topics include DSMC design based on piecewise Lyapunov functions or others. Appendix Lemma 1 [36]: 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).
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Lemma 2 [36]: For arbitrary matrices U and V with appropriate dimensions, the following matrix inequality holds: Fig. 3. Simulation results in Example 3. (a) Membership functions for Example 3. (b) State trajectories for Example 3. (c) Control input for Example 3.
UV + V T U T ≤ V T V + UU T .
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Acknowledgment [1, 0]T and u(0) = u0 = 0. The simulation results are shown in Fig. 2(b) and (c), which shows that the closed-loop control system is stochastically asymptotically stable. Finally, to show the advantage of the proposed approach over that in [27], we consider SMC design for the following nominal T–S fuzzy model. Example 3: Consider a T–S fuzzy model with two fuzzy rules x˙ (t) =
2
νl (θ(t)) [Al x(t) + Bl u(t)]
l=1
where the membership functions are shown in Fig. 3(a) and the system matrices are given as −1 1 2 0 A1 = A2 = 0 1 −0.2 −1 0 2 B1 = B2 = . 1 −1
The authors would like to thank the associate editor and reviewers for their constructive comments based on which the presentation of this paper has been greatly improved. References [1] X. Mao, Stochastic Differential Equations and Applications, 2nd ed. Chichester, U.K.: Horwood Publication, 2007. [2] W. Chen and L. C. Jiao, “Finite-time stability theorem of stochastic nonlinear systems,” Automatica, vol. 46, no. 12, pp. 2105–2108, Dec. 2010. [3] N. Berman and U. Shaked, “H∞ like control for nonlinear stochastic systems,” Syst. Control Lett., vol. 55, no. 3, pp. 123–135, Jun. 2004. [4] L. Wu and W. X. Zheng, “L2 − L∞ control of nonlinear fuzzy Itˆo stochastic delay systems via dynamic output feedback,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 39, no. 5, pp. 1308–1315, Oct. 2009. [5] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 15, no. 1, pp. 116–132, Jan.–Feb. 1985. [6] G. Feng, Analysis and Synthesis of Fuzzy Control Systems: A ModelBased Approach. Boca Raton, FL, USA: CRC Press, 2010.
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[7] C. L. Chen, G. Feng, D. Sun, and X.P. Guan, “H-infinity Output Feedback Control for Discrete Time Fuzzy Systems with Application to Chaos Control,” IEEE Trans. Fuzzy Syst., vol. 13, no. 4, pp. 531–543, Aug. 2005. [8] G. Feng, “H-infinity Controller Design of Fuzzy Dynamic Systems Based on Piecewise Lyapunov Functions,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 34, no. 1, pp. 283–292, Feb. 2004. [9] G. Feng and J. Ma, “Quadratic stabilization of uncertain discrete time fuzzy dynamic systems,” IEEE Trans. Circuits Syst. I, vol. 48, no. 11, pp. 1337–1344, Nov. 2001. [10] Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Reliable H∞ control for discretetime fuzzy systems with infinite-distributed delay,” IEEE Trans. Fuzzy Syst., vol. 20, no. 1, pp. 22–31, Feb. 2012. [11] 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. [12] H. K. Lam, “Stabilization of nonlinear systems using sampled-data output-feedback fuzzy controller based on polynomial-fuzzy-modelbased control approach,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 42, no. 1, pp. 258–267, Feb. 2012. [13] J. Qiu, G. Feng, and J. Yang, “A new design of delay-dependent robust H-infinity filtering for discrete-time T–S fuzzy systems with timevarying delay,” IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp. 1044–1058, Oct. 2009. [14] J. Qiu, G. Feng, and H. Gao, “Fuzzy-model-based piecewise H-infinity static output feedback controller design for networked nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 18, no. 5, pp. 919–934, Oct. 2010. [15] J. Qiu, G. Feng, and H. Gao, “Static output feedback H-infinity control of continuous-time T–S fuzzy affine systems via piecewise Lyapunov functions,” IEEE Trans. Fuzzy Syst., vol. 21, no. 2, pp. 245–261, Apr. 2013. [16] Q. Gao, X.-J. Zeng, G. Feng, Y. Wang, and J. Qiu, “T-S-fuzzymodel-based approximation and controller design for general nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 42, no. 4, pp. 1131–1142, Aug. 2012. [17] G. Feng, “A survey on analysis and design of model-based fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 14, no. 5, pp. 676–697, Oct. 2006. [18] S. Tong, Y. Li, Y. Li, and Y. Liu, “Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 6, pp. 1693–1704, Dec. 2011. [19] H. Dong, Z. Wang, J. Lam, and H. Gao, “Fuzzy-model-based robust fault detection with stochastic mixed time delays and successive packet dropouts,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 42, no. 2, pp. 365–376, Apr. 2012. [20] 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. [21] V. I. Utkin, “Variable structure systems with sliding modes,” IEEE Trans. Autom. Control, vol. 22, no. 2, pp. 212–222, Apr. 1977. [22] H. Sira-Ram´ırez, “On the sliding mode control of nonlinear systems,” Syst. Control Lett., vol. 19, no. 4, pp. 303–312, Oct. 1992. [23] K. Abidi, J. Xu, and X. Yu, “On the discrete-time integral sliding-mode control,” IEEE Trans. Autom. Control, vol. 52, no. 4, pp. 709–715, Apr. 2007. [24] Z. Galias and X. Yu, “Analysis of zero-order holder discretization of two-dimensional sliding-mode control systems,” IEEE Trans. Circuits Syst. II, Express Briefs, vol. 55, no. 12, pp. 1269–1273, Dec. 2008. [25] X. Yu, Z. Man, and B. Wu, “Design of fuzzy sliding-mode control systems,” Fuzzy Sets Syst., vol. 95, no. 3, pp. 295–306, May 1998. [26] H. H. Choi, “Robust stabilization of uncertain fuzzy-time-delay systems using sliding-mode-control approach,” IEEE Trans. Fuzzy Syst., vol. 18, no. 5, pp. 979–984, Oct. 2010. [27] H. H. Choi, “Robust stabilization of uncertain fuzzy systems using variable structure system approach,” IEEE Trans. Fuzzy Syst., vol. 16, no. 3, pp. 715–724, Jun. 2008. [28] Z. Xi, G. Feng, and T. Hesketh, “Piecewise sliding-mode control for T–S fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 19, no. 4, pp. 707–716, Aug. 2011. [29] Y. Niu, D. W. C. Ho, and X. Wang, “Robust H∞ control for nonlinear stochastic systems: A sliding-mode approach,” IEEE Trans. Autom. Control, vol. 53, no. 7, pp. 1695–1701, Aug. 2008. [30] 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.
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[31] 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. [32] 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. [33] 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. [34] C. Lin, Q. Wang, T. H. Lee, and Y. He, “Design of observer-based H∞ control for fuzzy time-delay systems,” IEEE Trans. Fuzzy Syst., vol. 16, no. 2, pp. 534–543, Apr. 2008. [35] W. Chen, Z. Guan, and X. Lu, “Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: An LMI approach,” Syst. Control Lett., vol. 54, no. 6, pp. 547–555, Jun. 2005. [36] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA, USA: SIAM, 1994.
Qing Gao (S’11) was born in Hubei Province, China. He 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. His current research interests include intelligent systems and control, variable structure control systems, fractional order control systems, and stochastic systems.
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 a Chair Professor and the Associate Provost. He is a ChangJiang Chair Professor with the Nanjing University of Science and Technology, Nanjing, awarded by the Ministry of Education, China. He was a Lecturer/Senior Lecturer with the School of Electrical Engineering, University of New South Wales, Kensington, NSW, 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. Prof. Feng 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, IEEE Transactions on Systems, Man and Cybernetics, Part C: Applications and Reviews, and the Journal of Control Theory and Applications. He was awarded an Alexander von Humboldt Fellowship in 1997 to 1998 and the IEEE Transactions on Fuzzy Systems Outstanding Paper Award in 2007.
Zhiyu Xi (M’11) received the B.Eng. degree in control science and engineering from the Harbin Institute of Technology, Harbin, China, in 2004, and the M.Eng. and Ph.D degrees in automatic control from the University of New South Wales, Sydney, NSW, Australia, in 2007 and 2011, respectively. She is currently an Associate Lecturer with the School of Electrical Engineering and Telecommunications, University of New South Wales. Her current research interests include sliding-mode control, stochastic systems, model predictive control, multiagent systems, and control applications.
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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 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.
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Jianbin Qiu (M’11) 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 2004 and 2009, respectively, and the Ph.D. degree in mechatronics engineering from the City University of Hong Kong, Kowloon, Hong Kong, in 2009. He has been with the School of Astronautics, Harbin Institute of Technology, Harbin, China, as an Associate Professor since 2009. From 2010 to 2011, he was a Senior Research Associate with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong. His current research interests include intelligent and hybrid control systems, signal processing, and robotics. Dr. Qiu is an Associate Editor of the Journal of The Franklin Institute, Journal of Intelligent and Fuzzy Systems, and so on. He was the recipient of the Outstanding Doctoral Thesis Award from the Anhui Province of China in 2011. He was awarded as the New Century Excellent Talents in University by the Ministry of Education of China in 2012, and was awarded an Alexander von Humboldt Fellowship of Germany in 2013.
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A New Design of Robust H∞ Sliding Mode Control for Uncertain Stochastic T–S Fuzzy Time-Delay Systems Qing Gao, Student Member, IEEE, Gang Feng, Fellow, IEEE, Zhiyu Xi, Member, IEEE, Yong Wang, and Jianbin Qiu, Member, IEEE
Abstract—In this paper, a novel dynamic sliding mode control scheme is proposed for a class of uncertain stochastic nonlinear time-delay systems represented by Takagi–Sugeno fuzzy models. The key advantage of the proposed scheme is that two very restrictive assumptions in most existing sliding mode control approaches for stochastic fuzzy systems have been removed. It is shown that the closed-loop control system trajectories can be driven onto the sliding surface in finite time almost certainly. It is also shown that the stochastic stability of the resulting sliding motion can be guaranteed in terms of linear matrix inequalities; moreover, the sliding-mode controller can be obtained simultaneously. Simulation results illustrating the advantages and effectiveness of the proposed approaches are also provided. Index Terms—Dynamic sliding-mode controllers, sliding mode control (SMC), stochastic nonlinear systems, Takagi–Sugeno (T–S) fuzzy models, time delay.
I. Introduction
R
ECENTLY, stochastic dynamic systems modeled by Itˆo stochastic differential equations have been gaining increasing attention from the control community since stochastic modeling plays an important role in many branches of
Manuscript received November 6, 2012; revised August 12, 2013; accepted October 31, 2013. Date of publication December 3, 2013; date of current version August 14, 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, in part by the National Natural Science Foundation of China under Grant 61374031, in part by the Program for New Century Excellent Talents in University under Grant NCET-12-0147, and in part by the Alexander von Humboldt Foundation of Germany. This paper was recommended by Associate Editor T. H. Lee. Q. Gao is with the Department of Automation, University of Science and Technology of China, Hefei 230026, China, and also with the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]). G. Feng is with the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong, and also with the Nanjing University of Science and Technology, Nanjing 210044, China (e-mail: [email protected]). Z. Xi is with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia (e-mail: [email protected]). Y. Wang is with the Department of Automation, University of Science and Technology of China, Hefei 230026, China (e-mail: [email protected]). J. Qiu is with the School of Astronautics, Harbin Institute of Technology, Harbin 150001, 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.2013.2289923
science and engineering [1]–[4]. However, it is worth noting that most of the existing results on stochastic nonlinear control systems, especially for those with high nonlinearity, suffer from a lack of systematic way of control design due to difficulties in finding appropriate Lyapunov functions [3], [4]. On one hand, Takagi–Sugeno (T–S) fuzzy models [5] have been widely used as a convenient tool for handling complex nonlinear systems [5]–[17]. T–S fuzzy models describe a nonlinear system as the blending of a set of local linear dynamic models. This relatively simple structure provides systematic stability analysis and controller design of T–S fuzzy control systems by exploiting conventional control theory. Recently, 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 [18]– [20]. Like the case of deterministic T–S fuzzy models, control design of stochastic fuzzy systems can be realized by solving a set of linear matrix inequalities, see [18]–[20] for details. On the other hand, as a powerful robust control strategy, sliding mode control (SMC) has been extensively studied and widely used for control of nonlinear systems due to its various attractive features such as strong robustness and fast response [21]– [33]. Generally speaking, SMC is attained by using a discontinuous control law to drive system state trajectories onto a predefined sliding manifold containing the origin, which is commonly termed the sliding surface or switching surface, in finite time (this process is called reaching phase), and then to force the state trajectories to move along the surface toward the origin with desired performance (such a motion is called sliding mode). The overall system dynamics are determined by the sliding surface that is normally designed as linear hyperplanes of the system states. More recently, some results on applying SMC to stabilize stochastic nonlinear systems have been reported (see, for example, [29]–[31]). In particular, Ho and Niu [31] addressed the SMC design for stochastic fuzzy systems. Stochastic systems considered in those works are given in the form of dx(t) = {f (x(t)) + Bu(t)}dt + g(x(t))dW(t)
c 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2267 See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
GAO et al.: NEW DESIGN OF ROBUST H∞ SLIDING MODE CONTROL
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where it is assumed that there exists a matrix S with appropriate dimension such that SB is nonsingular and Sg(x(t)) = 0 for all t ≥ 0.
(1)
In this way, stochastic perturbations do not need to be dealt with during the reaching phase, and SMC techniques for deterministic systems can thus be used. In addition, another restrictive assumption is required for the fuzzy SMC approach in [31], that is, all subsystems of the stochastic T–S fuzzy model have the same input matrix. It is noted that the mathematical models of many physical plants, such as the well known inverted pendulum on a cart [11], do not satisfy this assumption. In [28], a piecewise fuzzy SMC approach was developed for uncertain T–S fuzzy models by partitioning the premise state space into a set of subregions, and by designing individual sliding surfaces within each subregion. In the approach developed in [28], different local control input matrices are allowed for the T–S fuzzy models. However, the approach proposed in [28] is very difficult to implement in practice due to its high complexity. To the authors’ best knowledge, few effective SMC approaches to stochastic T–S fuzzy systems without the aforementioned two assumptions have been reported in the literature, which motivates us for this paper. In this paper, we propose a novel dynamic sliding mode control (DSMC) approach to a class of stochastic nonlinear time-delay systems represented by stochastic T–S fuzzy models. A key feature of the proposed DSMC approach is that the sliding surface function is defined to be linearly dependent on both of the system state vector x and the control input vector u. By utilizing a sliding-mode controller with a fuzzy dynamic feedback control form, both the system states x and control inputs u are driven onto the sliding surface in finite time almost certainly. It is shown that the stochastic asymptotic stability on the sliding surface can be guaranteed. It is also shown that the sliding surface and the dynamic sliding-mode controller can be obtained simultaneously in terms of a set of linear matrix inequalities (LMIs). Compared with the existing fuzzy SMC approaches to stochastic T–S fuzzy systems, for example, the approach in [31], the main advantages of the proposed DSMC approach include: 1) the two aforementioned restrictive assumptions that are required in the existing approaches have been removed, and 2) the proposed DSMC approach is able to deal with a much broader class of stochastic nonlinear systems than the existing approaches. The rest of this paper is organized as follows. Section II is devoted to model description and preliminaries. Details of the DSMC approach to robust stabilization and H∞ control for uncertain stochastic time delay T–S fuzzy models are given in Section III. Simulation results are provided in Section IV to demonstrate the advantages and effectiveness of the proposed approaches. Conclusions are given in Section V. Notations: The notations used in this paper are fairly standard. The notation in a matrix is used to indicate the terms that can be induced 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. The notation L2 [0, T ] will be used forvector-valued functions, i.e., ω : [0, T ] → k ∈ L2 [0, T ] T if 0 ωT (t)ω(t)dt < ∞ and its L2 − norm is defined as T T ||ω(t)||2 = 0 ω (t)ω(t)dt. 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 Preliminaries In this paper, we consider a general Itˆo type stochastic nonlinear system described by the following stochastic T–S fuzzy model with a constant time delay. Plant rule: Rl IF θ1 (t) is μl1 AND ... AND θg (t) is μlg ; THEN dx(t) = [(Al + Al )x(t) + (Bl + Bl )u(t) + Hl ω(t) +(El + El )x(t − τ)]dt + g(x(t), u(t))dW(t) y(t) = Cl x(t) + Dl u(t) + Cld x(t − τ) x(t) = ϕ(t), t ∈ [−τ, 0], l ∈ L := {1, 2, ...r}
(2)
where θ(t) = [θ1 (t), ..., θg (t)] is the premise variables vector, r is the number of fuzzy rules, Rl denotes the lth rule, μli are the fuzzy sets, τ is a constant time delay, x(t) ∈ n is the state vector, u(t) ∈ m is the control input vector, y(t) ∈ p is the output vector, ω(t) ∈ h is the exogenous disturbance belonging to L2 [0, ∞), ϕ(t) is a known bounded vector function, W(t) = [W1 (t), ..., Wq (t)]T is a q-dimensional Wiener process defined on the probability space (, F, P) with a natural filtration {Ft }t≥0 , and Al , Bl , Cl , Cld , Dl , El , Hl are system matrices with appropriate dimensions. The following two assumptions are made in this paper. 1) The parameter uncertainties Al , Bl , and El are norm bounded [31], that is [Al , Bl ] ≤ A , and El ≤ E
(3)
where A and E are known positive constants. 2) The function g(x, u) : n × m → q , is not exactly known but satisfies the following condition [31], [35]: trace[gT (x, u)g(x, u)] ≤ G[xT , uT ]T 2
(4)
where G is a known constant matrix. Via the standard fuzzy blending method, i.e., the centeraverage defuzzifier, product inference, and singleton fuzzifier, the stochastic T–S fuzzy system (2) can be expressed globally as ⎧ dx(t) = rl=1 μl (θ(t))[(Al + Al )x(t) ⎪ ⎪ ⎪ ⎪ +(Bl + Bl )u(t) + (El + El )x(t − τ) ⎨ +H (5) l ω(t)]dt + g(x(t), u(t))dW(t) r ⎪ ⎪ y(t) = μ (θ(t))[C x(t) + D u(t) + C x(t − τ)] ⎪ l l l ld l=1 ⎪ ⎩ x(t) = ϕ(t), t ∈ [−τ, 0]
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with the normalized fuzzy membership functions μl (θ(t)) defined by g l μ (θi (t)) (6) μl (θ(t)) = r i=1 g i l l=1 i=1 μi (θi (t)) satisfying μl (θ(t)) ≥ 0 and rl=1 μl (θ(t)) = 1. It is well known that the T–S fuzzy model (2) can be used to represent a large class of stochastic nonlinear systems [18], [19]. One can easily verify that the stochastic differential equation (5) with u(t) ≡ 0 and ω(t) ≡ 0 satisfies the so-called Lipschitz conditions and Linear growth conditions, and thus has a unique solution [1]. In [31], SMC design for a class of stochastic nonlinear systems represented by stochastic T–S fuzzy models as in (2) has been investigated. However, the approach proposed in [31] relies on two very restrictive assumptions: 1) all subsystems of the stochastic T–S fuzzy model (2) share the same input matrix, that is, Bl ≡ B, for all l ∈ L, and 2) there exists a matrix S with appropriate dimension that SB is nonsingular, and moreover Sg(x(t), u(t)) = 0, for all t ≥ 0.
(7)
Both of these two assumptions impose great limitations in real applications. In this paper, we will develop a new SMC scheme, which is not restricted by these two assumptions, to robustly stabilize the stochastic T–S fuzzy system (2) with a guaranteed H∞ performance. Specifically, the objective of this paper can be formulated as: construct an SMC law such that: 1) the closed-loop control system trajectories can be driven onto the sliding surface in finite time almost surely, and 2) given a positive constant γ > 0, the resulting sliding motion with ω(t) ≡ 0 is stochastically asymptotically stable ∞ ∞ and E t0 y(t)2 dt < γ 2 t0 ω(t)2 dt is satisfied for all nonzero ω(t) ∈ L2 [0, ∞) under zero initial condition x(φ) = 0, for all φ ∈ [tr , t0 ], where t0 = tr + τ and tr is the time when the sliding surface is reached.
A. Design of Sliding Surface and Sliding-Mode Controller For a stochastic nonlinear system in (2), a novel sliding surface is designed as s(t) = Sx x(t) + Su u(t) = S¯ x¯ (t) = 0 (8) m×n m×m ¯ , Su ∈ , S = [Sx , Su ], and x¯ (t) = where Sx ∈ [x1 (t), ..., xn (t), u1 (t), ..., um (t)]T . Su is assumed to be nonsingular. In this case, the following fuzzy dynamic sliding-mode controller can be employed: r u(t) ˙ =− μl (θ)Su−1 Sx (Al x(t) + El x(t − τ) + Bl u(t)) l=1
(9)
where σ(t) = Sx ( A [xT (t), uT (t)]T + E x(t − τ)) r + wl (θ(t))Hl ρ(t). l=1
The conditions on reachability of the sliding surface (8) are given in Theorem 1. Theorem 1: For the closed-loop control system (11), the sliding surface (8) can be reached in finite time almost surely, if there exists a positive constant δ such that ¯ GT G < δS¯ T S. (12) Moreover, β in (9) can be chosen as β ≥ 21 λmax (SxT Sx )δ. Proof: Consider the function S(t) = sT (t)s(t) for all t > 0. Then from Itˆo’s formula [1] with the fact that ¯ 1 − R2 Su−1 Sx ) = 0 and SR ¯ 2 Su−1 = I, the infinitesimal S(R generator r
¯ l x¯ (t) μl (θ(t))sT (t)S¯ (R1 − R2 Su−1 Sx )A LS(t)|(11) = 2 l=1
¯ l x¯ (t − τ) + R1 Hl ω(t) +(R1 − R2 Su−1 Sx )E ¯ l x¯ (t) + R1 E ¯ l x¯ (t − τ) +R1 A −2(α + σ(t))s(t) − 2βsT (t)s(t) ¯ 1 g(x, u)} +trace{gT (x, u)RT1 S¯ T SR ¯ 1 g(x(t), u(t))} ≤ trace{gT (x(t), u(t))RT1 S¯ T SR
III. Design of Dynamic Sliding Mode Control
−(α + σ(t))Su−1 sgn(s(t)) − βSu−1 s(t)
A and E are defined in (3), α > 0, ρ(t) is the known uniform upper bound of the disturbance ω(t), and β is a positive constant to be determined. ¯ l = [Al , Bl ], E ¯l = Denote R1 = [In , 0]T , R2 = [0, Im ]T , A ¯ ¯ [El , 0n×m ], Al = [Al , Bl ], El = [El , 0n×m ] and x¯ (t) = [xT (t), uT (t)]T , l ∈ L := {1, 2, ..., r}. Then by introducing an additional term 0 × u(t − τ) with u(t) = 0, t ∈ [−τ, 0] in both (2) and (9), the closed-loop control system consisting of (2) and⎧(9) can be rewritten in a compact form as
r −1 ⎪ ¯ ¯ (t) ⎪ d x¯ (t) = l=1 μl (θ(t)) (R1 − R2 Su Sx )Al x ⎪ ⎪ ⎪ −1 ⎪ ¯ l x¯ (t) ¯ l x¯ (t − τ) + R1 A +(R1 − R2 Su Sx )E ⎪ ⎪ ⎪ ⎨ ¯ l x¯ (t − τ) + R1 Hl ω(t) +R1 E (11) −1 ⎪ −R2 (α + σ(t))S ⎪ ⎪ u sgn(s(t)) ⎪ ⎪ ⎪ −βR2 Su−1 s(t) dt + R1 g(x(t), u(t))dW(t) ⎪ ⎪ ⎪ ⎩ T x¯ (t) = [φ (t), 0]T , t ∈ [−τ, 0].
(10)
−2βsT (t)s(t) − 2αs(t).
(13)
By using Lemma 1 in the Appendix based on (4) and (12), one has ¯ 1 g(x(t), u(t)) trace{gT (x(t), u(t))RT1 S¯ T SR ≤ λmax (SxT Sx )trace{gT (x(t), u(t))g(x(t), u(t))} ≤ λmax (SxT Sx )¯xT (t)GT G¯x(t) < λmax (SxT Sx )δ¯xT (t)S¯ T S¯ x¯ (t) ≤ 2βsT (t)s(t). Then, it follows from (13) and (14) that LS(t)|(11) < −αs(t) = −α S(t).
(14) (15)
From (15) and by using Itˆo’s formula again, one has Ls(t)|(11) = L S(t)|(11) 1 1 = √ LS(t)|(11) − √ 2 S(t) 2( S(t))3 ¯ 1 g(x(t), u(t))} ×trace{gT (x(t), u(t))RT1 S¯ T s(t)sT (t)SR 1 ≤ √ (16) LS(t)|(11) ≤ −α, for s(t) = 0. 2 S(t)
GAO et al.: NEW DESIGN OF ROBUST H∞ SLIDING MODE CONTROL
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Integrating from 0 to t on both sides of (16) results in 0 < Es(t) ≤ s(0) − αt. Thus, one has Es(t) = 0 for all t ≥ tr = s(0) . This implies s(t) = 0 almost surely for all α t ≥ tr . In other words, the sliding surface (8) can be reached in finite time almost surely. The proof is then completed. Remark 1: It is noted that the sliding variable s(t) is designed as a linear combination of both the state vector x and the control input vector u. In addition, the sliding-mode controller is in the form of a fuzzy dynamic state feedback control. All these features distinguish the proposed SMC scheme from existing ones [25]– [28]. This is the main reason why it is called the DSMC approach. Remark 2: Most of the existing SMC approaches for T–S fuzzy models, either stochastic or deterministic, suffer from a very restrictive assumption that all local systems of the T–S fuzzy models share the same input gain matrix while the DSMC approach does not. Remark 3: It is noted that assumption (7) required in [29]–[31] has been also removed. One can observe from the proof procedure of Theorem 1 that one can choose β = 0 in (9) if assumption (7) holds.
Proof: We first consider the stability of system (17) with ω(t) = 0. Consider the following Lyapunov–Krasovskii functional candidate: t T
V (t) = x¯ (t)X¯x(t) +
Theorem 2: Given a constant γ > 0, the sliding mode dynamics (17) is stochastically asymptotically stable with H∞ performance γ, if there exist two positive definite matrices P, Q ∈ (m+n)×(m+n) , two sets of matrices Wl1 , Wl2 ∈ m×(m+n) , a positive constant δg , and a set of positive constants εl , l ∈ L satisfying the LMIs (18) at the bottom of this page and P − δg I > 0
l∈L
(19)
¯ T RT + W T RT + Q + ε l R1 RT . ¯ l P + R2 W1l + P A where l = R1 A 1 l 1 1l 2 Moreover, the sliding surface matrix is given by S¯ = RT2 P −1 .
(20)
t−τ
for all t ≥ t0 = tr + τ, where X = P −1 and Y = P −1 QP −1 . Since (19) is equivalent to X < δ1g I, then based on Lemma 1 in the Appendix, (4), and the fact that RT1 R1 = I, one has trace{gT (x(t), u(t))RT1 XR1 g(x(t), u(t)) 1 ≤ trace{gT (x(t), u(t))g(x(t), u(t)) δg 1 ≤ x¯ T (t)GT G¯x(t). δg
(21)
It is noted that on the sliding surface, s(t) = S¯ x¯ (t) = RT2 X¯x(t) = 0, then one has
B. Stability Analysis of Sliding Motion It has been shown in Theorem 1 that the sliding surface in (8) can be reached in finite time almost surely. In this subsection, we proceed to analyze the stability of the sliding motion. ¯ l, First, we denote Hl = rl=1 μl (θ(t))Hl , C¯l = rl=1 μl (θ(t))C r ¯ ¯ ¯ μ (θ(t)) D and D¯ l = , where C = [C , C ] and D l l l ld l = l=1 l [Dl , 0n×m ]; then, the closed-loop control system (11) restricted on the sliding surface (8) becomes ⎧
r ⎪ ¯ l x¯ (t) d x ¯ (t) = μ (θ(t)) (R1 − R2 Su−1 Sx )A ⎪ l l=1 ⎪ ⎪ ⎪ ⎪ ¯ l x¯ (t) ¯ l x¯ (t − τ) + R1 A +(R1 − R2 Su−1 Sx)E ⎪ ⎪ ⎨ ¯ l x¯ (t − τ) + R1 Hl ω(t) dt +R1 E (17) ⎪ ⎪ +R g(x(t), u(t))dW(t), ⎪ 1 ⎪ ⎪ ⎪ y(t) = C¯l x¯ (t) + D¯ l x¯ (t − τ), ⎪ ⎪ ⎩ x¯ (t) = [φT (t), 0]T , t ∈ [−τ, 0].
x¯ T (ϕ)Y x¯ (ϕ)dϕ
LV (t)|(17) = 2
r
¯ l + A ¯ l )¯x(t) μl (θ(t))¯xT (t)XR1 [(A
l=1
¯ l )¯x(t − τ)] − 2 x¯ T (t)XR2 ¯ l + E +(E =0
×
r
¯ l x¯ (t) + E ¯ l x¯ (t − τ)] μl (θ(t))Su−1 Sx [A
l=1
+¯xT (t)Y x¯ (t) − x¯ T (t − τ)Y x¯ (t − τ) +trace gT (x(t), u(t))RT1 XR1 g(x(t), u(t)) r ¯ l + A ¯ l )¯x(t) ≤2 μl (θ(t))¯xT (t)XR1 [(A l=1
¯ l )¯x(t − τ)] + 1 x¯ T (t)GT G¯x(t) ¯ l + E +(E δg +¯xT (t)Y x¯ (t) − x¯ T (t − τ)Y x¯ (t − τ) r ¯ l + A ¯ l )¯x(t) =2 μl (θ(t))¯xT (t)XR1 [(A l=1
¯ l + E ¯ l )¯x(t − τ)] − 2 x¯ T (t)XR2 +(E =0
×
r
μl (θ(t))[K1l x¯ (t) + K2l x¯ (t − τ)]
l=1 T
+¯x (t)Y x¯ (t) − x¯ T (t − τ)Y x¯ (t − τ) 1 + x¯ T (t)GT G¯x(t). δg
(22)
⎡
⎤ l + γ12 R1 Hl HlT RT1 T T T T ⎢ PE ⎥ ⎢ ¯ l R1 + Wl2 R2 −Q ⎥ ⎢ ¯ ¯ C P D P −I ⎥ l l ⎢ ⎥ 0; then, one has LV (t)|(17) ≤
LV (t)|(17) + yT (t)y(t) − γ 2 ωT (t)ω(t) r
¯ l + R2 K1l )¯x(t) ≤2 μl (θ(t))¯xT (t)X (R1 A l=1
2 T +ε−1 ¯ (t − τ)¯x(t − τ) l E x 2 ε XR1 R1 X + ε−1 l A I = ζ T (t) l ζ(t) 2 0 ε−1 l E I
r
Along the trajectory of (17), one has
(23)
l=1
where
ϒl ζ(t) 2 ¯ l + R2 K2l )T X ε−1 (R1 E l E I − Y r +2 μl (θ(t))¯xT (t)XR1 Hl ω(t)
× ¯ l + R2 K1l ) + (R1 A ¯ l + R2 K1l )T X ϒl = X(R1 A 1 T 2 +εl XR1 RT1 X + ε−1 G G. l A I + Y + δg
Then LV (t)|(17) < 0 if (19) holds and ϒl < 0. 2 ¯ l + R2 K2l )T X ε−1 (R1 E l E I − Y
(24)
l=1
+yT (t)y(t) − γ 2 ωT (t)ω(t) (25)
Multiplying diag{P, P} with P = X−1 from both sides to (25) yields l t0 results in T 0 < E {V (t)} = E LV (t)dt t0 T T ≤ −E yT (t)y(t)dt + γ 2 ωT (t)ω(t)dt (37)
r l=1
⎤
2 ε−1 l E PP
When time delay, external disturbance, or random noises are not considered, the system in (2) reduces to a nominal deterministic T–S fuzzy model as follows:
t0
∞ ∞ which implies E t0 y(t)2 dt < γ 2 t0 ω(t)2 dt. The proof is thus completed. Actually, by choosing S¯ = RT2 P −1 , the condition in (12) can be rewritten in the form of LMIs as follows: R2 RT2 >0 (38) GP δI by which the positive parameter β in the sliding-mode controller (9) can be obtained. It then follows from Theorem 2 that the minimum H∞ performance index γmin can be obtained by the following convex optimization algorithm [36]. Algorithm 1: min − γ12 , subject to LMIs (18), P>0,Q>0,W1l ,W2l ,εl >0
(19), and (38). Remark 4: It is noted that the matrices Kil , i = 1, 2, in (22) are not feedback gain matrices. They are introduced in (22) to improve the feasibility of the LMI conditions in (18), by taking the advantage that RT2 X¯x(t) = 0 on the sliding surface. Remark 5: It is noted that the systems considered in this paper are more general than those in [31] in that the local input matrices of investigated stochastic T–S fuzzy systems are allowed to have unmatched uncertainties. Moreover, it has been proved in the authors’ previous work [20] that if the premise variable θ(t) contains both x and u, the stochastic T–S fuzzy models (2) can be used to approximate the so-called stochastic nonaffine nonlinear systems. With the dynamic sliding-mode controller employed, it is found that the proposed approach can also be applied to deal with stochastic nonaffine nonlinear systems. In both of these two cases, the approach in [31] cannot easily be applied. In other words, the proposed DSMC approaches can deal with a much broader class of nonlinear systems.
under which the sliding surface in (8) can be reached in finite time. The closed-loop control system restricted on the sliding surface is asymptotically stable if there exist a positive definite matrix P ∈ (m+n)×(m+n) and a set of matrices Wl ∈ m×(m+n) such that the following linear matrix inequalities are satisfied: ¯ Tl RT1 + R1 A ¯ l P + WlT RT2 + R2 Wl < 0 PA l ∈ L := {1, 2, ...r}.
(41)
Moreover, the sliding surface matrix is given by S¯ = RT2 P −1 . Remark 6: As indicated in Remark 2, most existing results fuzzy SMC approaches assume that all Bi in (39) to be identical. It can be observed that Corollary 1 is not restricted by such an assumption. An alternative SMC design approach to nominal T–S fuzzy models which avoided this assumption can be found in [27]. However, it is required in [27] that for r B the T–S fuzzy models as in (39), [Al , l=1r l ] is stabilizable for any l ∈ L, which is also very restrictive. Comparison between Corollary 1 and the results in [27] will be provided in the simulation section. Remark 7: In the proposed DSMC approaches, the dynamic sliding-mode controller (9) contains a switching term sgn(s(t)), which would cause chattering phenomenon when implemented. Chattering avoidance methods as in [33] can be adopted to address the problem.
IV. Simulation Studies In this section, simulation results will be given to demonstrate the advantages and effectiveness of the approaches proposed in this paper. In the simulations, the signum function s(t) sgn(s(t)) in DSMC control laws is replaced by s(t)+0.01 to reduce chattering in the control signals, following the result of [31]. Example 1: Control of the inverted pendulum is one of the benchmark examples to demonstrate nonlinear control scheme performances. To illustrate the effectiveness of the proposed approach, we consider the balancing problem of an inverted
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pendulum mounted on a cart. The dynamic of the pendulum is described by dx1 = x2 dt g sin(x1 ) − amlx22 sin(2x1 )/2 − a cos(x1 )v dx2 = + ω(t) dt 4l/3 − aml cos2 (x1 ) +[2x1 − x2 + u, x1 − x2 − u]dW where x1 denotes the angle of the pendulum from the vertical, x2 is the angular velocity, and W = [W1 , W2 ]T is a 2-D Wiener process. 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 , 2l is the length of the pendulum, and w is the disturbance. In this paper, we choose m = 2.0 kg, M = 8.0 kg, 2l = 1.0 m. The following uncertain stochastic T–S fuzzy model is widely used to approximate the original nonlinear model [11] dx(t) =
3
μl (x1 (t)){(Al + Al )x(t) + (Bl + Bl )u(t)
0 +Hl ω(t)}dt + dW1 (t) 2x1 − x2 + u 0 + dW2 (t) x1 − x2 − u l=1
y(t) =
3
μl (x1 (t)){Cl x(t) + Dl u(t)}
l=1
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 ! " 0 C1 = C2 = C3 = 0.5 1 , H1 = H2 = H3 = 1 D1 = 0.05, and D2 = D3 = 0.01. It is noted that this T–S fuzzy model is widely adopted in many investigations and has been confirmed in [11] to be a good approximate fuzzy model of the original nonlinear model. It is also noted that the mathematical model of the original pendulum plant can also be described by an affine nonlinear differential equation, i.e., x˙ (t) = f (x)+g(x)u, where f (x) = g sin x1 −amlx22 sin(2x1 )/2 −a cos x1 and g(x) = 4l/3−aml . In practice, it 4l/3−aml cos2 x1 cos2 x1 might be difficult to determine the exact upper bounds of approximationerrors. In this case study, we choose to calculate f (x)− 3l=1 μl (x1 )Al x γ1 (x) = | | and γ2 (x) = |g(x) − 3l=1 μl (x1 )Bl |} x at a number of vertex points (x1 , x2 ) ∈ [−π/2, π/2] × [−5, 5], within the operating range of the pendulum. It is found that γ1 (x) < 0.01 and γ2 (x) < 0.02. It is noted that only finite tests can be conducted. However, one can choose to test more points within the operating region thus to improve the precision of the obtained upper bounds.
Similar to [34], we assume that the delay terms are perturbed along values of the scalar λ ∈ [0, 1]; then, the system dynamic considered is described by dx(t) =
3
μl (x1 (t)){λ(Al + Al )x(t)
l=1
+(1 − λ)(Al + Al )x(t − τ) + (Bl + Bl )u(t) 0 dW1 (t) +Hl ω(t)}dt + 2x1 − x2 + u 0 + dW2 (t) x1 − x2 − u y(t) =
3
μl (x1 (t)){Cl x(t) + Dl u(t)}.
l=1
The objective here is to design a fuzzy dynamic slidingmode controller as in (9) such that the resulting closed-loop control system restricted to the sliding surface is stochastically asymptotically stable with H∞ performance γ. In this paper, λ is chosen to be 0.9. It is noted that the local control gain matrices of the obtained fuzzy models are not equal; thus, the SMC design results in [31] cannot be applied to this example. However, by applying Algorithm 1 with ¯ A and ¯ E chosen as 0.02 and 0.001, respectively, the minimum H∞ performance γmin = 3.7317 is obtained and the corresponding sliding surface matrix and positive definite matrices are, respectively, given by S¯ = [−0.0314 − 0.00440.0003] ⎡ ⎤ 0.043 −0.087 3.304 P = ⎣ −0.087 0.681 0.965 ⎦ 3.304 0.965 3848.540 ⎤ ⎡ 0.021 −0.069 −0.073 Q = ⎣ −0.069 0.826 0.927 ⎦ . −0.073 0.927 3270.423 It is found that β ≥ 2.28. In the simulation, Monte Carlo simulations have been conducted by using the discretization approach as in [20]. The simulation parameters used are as follows: the simulation interval t ∈ [0, T ] with T = 15, the normally distributed variance is δt = T/N with N = 3 ∗ 211 , 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 , u(0) = u0 = 0, and x(t) = [0, 1]T , 0 < t < τ with τ = 0.41 s. By using the fuzzy dynamic sliding-mode controller defined in (9) with α = 2.5 and β = 5, the state trajectories and the control inputs of the closed-loop control system with ω(t) = 0 along ten individual Wiener process paths are shown in Fig. 1(b) and (c), respectively. One can observe that the pendulum can be stochastically asymptotically stabilized. Next, we illustrate the disturbance attenuation performance. The response of the t T y (ψ)y(ψ)dψ ratio t0 wT (ψ)w(ψ)dψ of the closed-loop control system with the 0
disturbance given by ω(t) = e−0.02t sin(πt) and under zero conditions along ten individual Wiener process paths is shown in Fig. 1(d). One can see that the performance response
GAO et al.: NEW DESIGN OF ROBUST H∞ SLIDING MODE CONTROL
Fig. 1. Simulation results in Example 1. (a) Membership functions for Example 1. (b) State trajectories for Example 1. (c) Control input for Example 1. (d) Individual paths and the average of the ratio for Example 1. t
t T y (ψ)y(ψ)dψ
t0 0
wT (ψ)w(ψ)dψ
yT (ψ)y(ψ)dψ
is satisfactory and the average of the ratio t0 wT (ψ)w(ψ)dψ is 0 much less than the minimum disturbance attenuation level 2 γmin = 13.9256. Example 2: Consider a stochastic T–S fuzzy system with two fuzzy rules # 2 $ dx(t) = νl (θ(t))(Al + Al )x(t) + Bu(t) dt + g(x(t))dW(t) l=1
0 , the membership functions are x 1 + x2 shown in Fig. 3(a), and the system matrices are given as 0 1 0 1 0 A1 = A2 = B= . 2 −1 1 −2 1
where g(x(t)) =
It is noted that condition (1) is not satisfied, since for any given matrix S such that SB is nonsingular, Sg(x(t)) =
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Fig. 2. Simulation results in Example 2. (a) Membership functions for Example 2. (b) State trajectories for Example 2. (c) Control input for Example 2.
SB(x1 + x2 ) = 0 for any nonzero states; thus, the approach in [31] cannot be applied to this example either. However, by applying Theorem 1 with ¯ A = 0.03 and ¯ E = 0, the corresponding sliding surface matrix and positive definite matrices are, respectively, given by S¯ = [0.8936, 1.0218, 0.1901] ⎡ ⎤ 0.0869 −0.0793 0.0179 P = ⎣ −0.0793 0.1385 −0.3716 ⎦ 0.0179 −0.3716 7.1732 ⎡ ⎤ 0.0173 −0.0180 0.0000 Q = ⎣ −0.0180 0.0666 −0.0000 ⎦ . 0.0000 −0.0000 0.9679 It is found that β can be chosen to be greater than 43.6355. By choosing Al = 0.03 sin(t) and following the similar simulation method in Example 1, a number of simulations have been conducted under the initial condition x(0) = x0 =
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IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 9, SEPTEMBER 2014
It can be seen from simulation results that this T–S fuzzy system is unstable. It is noted that the method in [27] cannot be used since [A1, (B1 + B2)/2] is not stabilizable. However, by using Corollary 1, one has the following feasible solutions: S¯ = [2.46794.13940.4832] ⎡ ⎤ 8.3380 −3.9657 −8.6131 P = ⎣ −3.9657 2.7719 −3.4907 ⎦ −8.6131 −3.4907 75.9592 which will lead to an asymptotically stable closed-loop control system. The simulation results with initial condition x(0) = [0.4, −1]T are shown in Fig. 3(b) where all states converge to the origin as time approaches infinity. V. Conclusion This paper proposes a novel DSMC approach to robust H∞ control for a class of uncertain stochastic T–S fuzzy time-delay systems. The proposed approach removes some restrictive assumptions that are required in most existing schemes. It is shown that the sliding surface and sliding-mode controller can be obtained in terms of linear matrix inequalities, and the stochastic asymptotic stability of the sliding motion can be guaranteed. Some interesting future topics include DSMC design based on piecewise Lyapunov functions or others. Appendix Lemma 1 [36]: 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).
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Lemma 2 [36]: For arbitrary matrices U and V with appropriate dimensions, the following matrix inequality holds: Fig. 3. Simulation results in Example 3. (a) Membership functions for Example 3. (b) State trajectories for Example 3. (c) Control input for Example 3.
UV + V T U T ≤ V T V + UU T .
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Acknowledgment [1, 0]T and u(0) = u0 = 0. The simulation results are shown in Fig. 2(b) and (c), which shows that the closed-loop control system is stochastically asymptotically stable. Finally, to show the advantage of the proposed approach over that in [27], we consider SMC design for the following nominal T–S fuzzy model. Example 3: Consider a T–S fuzzy model with two fuzzy rules x˙ (t) =
2
νl (θ(t)) [Al x(t) + Bl u(t)]
l=1
where the membership functions are shown in Fig. 3(a) and the system matrices are given as −1 1 2 0 A1 = A2 = 0 1 −0.2 −1 0 2 B1 = B2 = . 1 −1
The authors would like to thank the associate editor and reviewers for their constructive comments based on which the presentation of this paper has been greatly improved. References [1] X. Mao, Stochastic Differential Equations and Applications, 2nd ed. Chichester, U.K.: Horwood Publication, 2007. [2] W. Chen and L. C. Jiao, “Finite-time stability theorem of stochastic nonlinear systems,” Automatica, vol. 46, no. 12, pp. 2105–2108, Dec. 2010. [3] N. Berman and U. Shaked, “H∞ like control for nonlinear stochastic systems,” Syst. Control Lett., vol. 55, no. 3, pp. 123–135, Jun. 2004. [4] L. Wu and W. X. Zheng, “L2 − L∞ control of nonlinear fuzzy Itˆo stochastic delay systems via dynamic output feedback,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 39, no. 5, pp. 1308–1315, Oct. 2009. [5] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 15, no. 1, pp. 116–132, Jan.–Feb. 1985. [6] G. Feng, Analysis and Synthesis of Fuzzy Control Systems: A ModelBased Approach. Boca Raton, FL, USA: CRC Press, 2010.
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[31] 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. [32] 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. [33] 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. [34] C. Lin, Q. Wang, T. H. Lee, and Y. He, “Design of observer-based H∞ control for fuzzy time-delay systems,” IEEE Trans. Fuzzy Syst., vol. 16, no. 2, pp. 534–543, Apr. 2008. [35] W. Chen, Z. Guan, and X. Lu, “Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: An LMI approach,” Syst. Control Lett., vol. 54, no. 6, pp. 547–555, Jun. 2005. [36] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. Philadelphia, PA, USA: SIAM, 1994.
Qing Gao (S’11) was born in Hubei Province, China. He 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. His current research interests include intelligent systems and control, variable structure control systems, fractional order control systems, and stochastic systems.
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 a Chair Professor and the Associate Provost. He is a ChangJiang Chair Professor with the Nanjing University of Science and Technology, Nanjing, awarded by the Ministry of Education, China. He was a Lecturer/Senior Lecturer with the School of Electrical Engineering, University of New South Wales, Kensington, NSW, 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. Prof. Feng 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, IEEE Transactions on Systems, Man and Cybernetics, Part C: Applications and Reviews, and the Journal of Control Theory and Applications. He was awarded an Alexander von Humboldt Fellowship in 1997 to 1998 and the IEEE Transactions on Fuzzy Systems Outstanding Paper Award in 2007.
Zhiyu Xi (M’11) received the B.Eng. degree in control science and engineering from the Harbin Institute of Technology, Harbin, China, in 2004, and the M.Eng. and Ph.D degrees in automatic control from the University of New South Wales, Sydney, NSW, Australia, in 2007 and 2011, respectively. She is currently an Associate Lecturer with the School of Electrical Engineering and Telecommunications, University of New South Wales. Her current research interests include sliding-mode control, stochastic systems, model predictive control, multiagent systems, and control applications.
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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 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.
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Jianbin Qiu (M’11) 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 2004 and 2009, respectively, and the Ph.D. degree in mechatronics engineering from the City University of Hong Kong, Kowloon, Hong Kong, in 2009. He has been with the School of Astronautics, Harbin Institute of Technology, Harbin, China, as an Associate Professor since 2009. From 2010 to 2011, he was a Senior Research Associate with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong. His current research interests include intelligent and hybrid control systems, signal processing, and robotics. Dr. Qiu is an Associate Editor of the Journal of The Franklin Institute, Journal of Intelligent and Fuzzy Systems, and so on. He was the recipient of the Outstanding Doctoral Thesis Award from the Anhui Province of China in 2011. He was awarded as the New Century Excellent Talents in University by the Ministry of Education of China in 2012, and was awarded an Alexander von Humboldt Fellowship of Germany in 2013.
