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H∞ Control for 2-D Fuzzy Systems With Interval Time-Varying Delays and Missing Measurements Yuqiang Luo, Zidong Wang, Fellow, IEEE, Jinling Liang, Guoliang Wei, and Fuad E. Alsaadi
Abstract—In this paper, we consider the H∞ control problem for a class of 2-D Takagi–Sugeno fuzzy described by the second Fornasini–Machesini local state-space model with timedelays and missing measurements. The state delays are allowed to be time-varying within a known interval. The measurement output is subject to randomly intermittent packet dropouts governed by a random sequence satisfying the Bernoulli distribution. The purpose of the addressed problem is to design an outputfeedback controller such that the closed-loop system is globally asymptotically stable in the mean square and the prescribed H∞ performance index is satisfied. By employing a combination of the intensive stochastic analysis and the free weighting matrix method, several delay-range-dependent sufficient conditions are presented that guarantee the existence of the desired controllers for all possible time-delays and missing measurements. The explicit expressions of such controllers are derived by means of the solution to a class of convex optimization problems that can be solved via standard software packages. Finally, a numerical simulation example is given to demonstrate the applicability of the proposed control scheme. Index Terms—2-D systems, fuzzy systems, H∞ control, missing measurements, output feedback, time-varying delays.
I. I NTRODUCTION
N
ONLINEARITY is a ubiquitous phenomenon in the natural world that has been receiving ever-increasing research
Manuscript received May 2, 2015; revised August 24, 2015 and November 18, 2015; accepted January 4, 2016. This work was supported in part by the Royal Society of U.K., in part by the National Natural Science Foundation of China under Grant 61174136, Grant 61329301, and Grant 61374039, in part by the Hujiang Foundation of China under Grant C14002 and Grant D15009, in part by the Natural Science Foundation of Jiangsu province of China under Grant BK20130017, in part by the Programme for New Century Excellent Talents in University under Grant NCET-12-0117, and in part by the Alexander von Humboldt Foundation of Germany. This paper was recommended by Associate Editor H. Gao. (Corresponding author: Guoliang Wei.) Y. Luo and G. Wei are with the Shanghai Key Laboratory of Modern Optical System, Department of Control Science and Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China (e-mail: [email protected]). Z. Wang is with the Department of Computer Science, Brunel University London, Uxbridge UB8 3PH, U.K., and also with the Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia (e-mail: [email protected]). J. Liang is with the Department of Mathematics, Southeast University, Nanjing 210096, China. F. E. Alsaadi is with the Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia. 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.2016.2514846
attention from a variety of subject areas. Traditionally, rigorous mathematical analysis on nonlinear systems rely on precise description of the nonlinearities with some stringent assumptions, and this sometimes hinders the nonlinear systems theory from being applied to certain engineering practice, for example, those data-based applications where an exact mathematical model is hard to be established [4], [16], [26], [29]–[31], [38]. On the other hand, the fuzzy logic theory has proven to be effective in handling reasoning that is approximate rather than fixed and exact. Furthermore, after a decade of theoretical and practical development, the Takagi–Sugeno (T–S) model has been recognized as an efficient way to approximate certain nonlinear systems. By using a set of local linear models which are smoothly connected by nonlinear fuzzy member ship functions to present a nonlinear plant, the T–S fuzzy model has brought the analysis and synthesis of nonlinear systems into a unified framework. Moreover, owing to the peculiarity of the T–S model, a large portion of existing results for linear systems can be readily extended for some nonlinear systems. As pointed out in [12] and [32], the T–S model is able to approximate any smooth nonlinear function to any degree of accuracy in any convex compact region. As is well known, many practical systems can be ideally described by the 2-D systems that have received tremendous research attention because of their applications in thermal processes, seismic data sections, digitized photographic data, digital filtering, and magnetic maps, etc. [11]. For four decades, the theoretical investigations on 2-D systems have been attracting recurring research interests and fruitful research results have been available in the literature. As early as in 1970s, some basic behaviors and modeling issues were thoroughly examined for 2-D systems (see [11], [18]). Parallel to the rapid research development of the traditional 1-D systems, in the past few years, some important breakthroughs have been reported on the analysis and design issues for the 2-D control systems. For example, the stability analysis and stabilization problems for 2-D systems have been addressed in [7] and [13], and the filter/observer design problems have been investigated in [25]. In particular, some new kinds of 2-D models and new techniques have recently been put forward in the literature. In [23], a state estimation problem for 2-D complex networks with randomly occurring nonlinearities and randomly varying sensor delays has been considered, and a new synchronization problem has been dealt with for an array of 2-D coupled dynamical network in [24].
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Given the practical importance of 2-D systems in representing 2-D evolutions and the technical convenience of T–S fuzzy systems for handling affine nonlinearities, a seemingly natural research issue would be to investigate the dynamical behaviors as well as the estimator/controller design problems of 2-D T–S fuzzy systems. Unfortunately, a literature review has revealed that the corresponding results have been very few due primarily to the mathematical complexities for 2-D T–S fuzzy systems. On the other hand, time delays are very often a major concern in many practical applications such as communication line, electrical signal processing systems, network transmission systems, seismic wave, urban traffic management systems, etc. Time delays are well recognized as one of the main sources for poor performance or even instability of control systems. In the 1-D settings, the time-delay systems have recently received considerable research attention and a rich body of literature has appeared on this topic (see [3], [17]). Moreover, among the existing works, plenty of stability conditions are delay-dependent, which pose less conservativeness (see [19], [20]). With respect to 2-D timedelay systems, over the past few years, some initial results have been reported on the control and filtering problems. For example, the problem of delay-dependent H∞ control for 2-D discrete state delay systems has been investigated in [22], and a robust H∞ filter for 2-D discrete systems with time delays has been designed in [34]. It should be noted that most available results have been concerned with time-invariant delays. As a matter of fact, the phenomenon of missing measurements (packet losses or dropouts) is virtually inevitable in measurement processes (particularly within a networked environment), which is caused by some harsh working conditions and imperfect communications. Many factors have contributed to this kind of less-than-ideal situation. Such factors include, but are not limited to, the limited bandwidth of the communication channel, abnormity of swap device, random network congestion, accidental loss of some collected data in a very noisy environment. So far, the problem of missing measurements has been well studied for 1-D systems and a great deal of literature has been available (see [28], [33]). Nevertheless, the relevant research for 2-D systems is still in its early stage especially when both the time-varying delays and missing measurements are simultaneously present, not to mention the case when the 2-D systems are further complicated by the T–S fuzzy model. As such, the stability analysis and stabilization problem for 2-D time-delay fuzzy systems with missing measurements remains a challenging issue that motivates our current research, and the main task of this paper is to propose a general framework for handling such a challenge. In this paper, we endeavor to research into the H∞ control problem for a class of 2-D T–S fuzzy systems described by the second Fornasini–Machesini (FM) local state-space (LSS) model with time-delays and missing measurements. The main contributions of this paper can be boiled down as follows. 1) A 2-D T–S fuzzy model is considered, which is comprehensive to include time-varying delays, bounded noises,
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and probabilistic missing measurements, thereby reflecting engineering practice more closely. 2) For the purpose of stabilizing the addressed 2-D fuzzy systems, an energy-like functional is constructed and several delay-range-dependent stability criteria are obtained to accomplish the design goal. 3) The close-loop system has the expected disturbance attenuation level in terms of a prescribed H∞ performance index. Compared with the existing delay-independent or delay-dependent results for 1-D delayed systems, the delay-range-dependent method addressed in this paper is of less conservativeness. The adopted 2-D T–S model can be thought to be a universal approximator for nonlinear 2-D systems, and the corresponding controller can also be used to stabilize the complicated 2-D nonlinear plant. Potential applications of the investigated fuzzy control approach include the 2-D digital systems, image processing, and wireless communications, etc. Especially for the case of interval time-varying delays and randomly missing measurements, the proposed fuzzy controller will show its strength. The rest of this paper is organized as follows. Section II is devoted to the formulation of the H∞ control problem for the addressed 2-D fuzzy systems with interval time-varying delays, where the phenomenon of probabilistic missing measurements is characterized by a Bernoulli distribution model, and some notations and related definitions are also given. In Section III, with the aid of an energy-like functional and the delay-range-dependent method, both the analysis and the synthesis problems of the 2-D fuzzy control system are investigated. In Section IV, an example is given to validate the design approach of the proposed fuzzy control scheme, and some concluding remarks have been drawn in Section V. Notation: In this paper, Rn , Rn×m , and Z (Z+ , Z− ) denote, respectively, the n-dimensional Euclidean space, the set of all n × m real matrices, and the set of all integers (non-negative integers, negative integers). · refers to the Euclidean norm in Rn . In represents the identity matrix of dimension n × n. The notation X ≥ Y (respectively, X > Y), where X and Y are symmetric matrices, means that X − Y is positive semi-definite (respectively, positive definite). For a matrix M, M T , and M −1 represent its transpose and inverse, respectively. The shorthand diag{M1 , M2 , . . . , Mn } denotes a block diagonal matrix with diagonal blocks being the matrices M1 , M2 , . . . , Mn . In symmetric block matrices, the symbol “∗” is used as an ellipsis for terms induced by symmetry. Matrices, if they are not explicitly stated, are assumed to have compatible dimensions.
II. P ROBLEM F ORMULATION Consider a 2-D discrete-time T–S fuzzy system with statedelays and stochastic perturbations described by the following FM LSS second model: Plant Rule i: IF θ1(k,l) is Fi1 , . . . , θj(k,l) is Fij , . . . , and θp(k,l) is Fip
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THEN ⎧ x(k + 1, l + 1) = A1i x(k, l + 1) + A2i x(k + 1, l) ⎪ ⎪ ⎪ ⎪ ⎪ + D1i x(k − τ1 (k, l), l + 1) ⎪ ⎪ ⎪ ⎪ ⎪ + D2i x(k + 1, l − τ2 (k, l)) ⎪ ⎨ + B1i u(k, l + 1) + B2i u(k + 1, l) ⎪ ⎪ ⎪ + E1i ω(k, l + 1) + E2i ω(k + 1, l) ⎪ ⎪ ⎪ ⎪ ⎪ y(k, l) = Ci x(k, l) + Hi ω(k, l), i∈S ⎪ ⎪ ⎪ ⎩ z(k, l) = Gi x(k, l) + Li ω(k, l)
(1)
where k, l ∈ Z+ , and x(k, l) ∈ Rnx is the state vector; y(k, l) ∈ Rny is the measured output; z(k, l) ∈ Rnz is the controlled output; u(k, l) ∈ Rnu is the control input vecthe disturbance input which belongs to tor; ω(k, l) ∈ Rnω is ∞ E{ω(k, l)2 } < ∞; τ1 (k, l) l2 (Z+ , Z+ ), namely, ∞ l=0 k=0 and τ2 (k, l) are interval time-varying delays along the horizontal direction and the vertical direction, respectively, which are subjected to τ 1 ≤ τ1 (k, l) ≤ τ 1 and τ 2 ≤ τ2 (k, l) ≤ τ 2 ; the (k,l)
input vector θj = [θj (k−1, l), θj (k, l−1)] ( j = 1, 2, . . . , p) represents the spatial premise variable at the location (k, l), which may be states or measurable variables, and θj (k, l) is (k,l) the corresponding component of the spatial input vector θj ; Fij is a spatial fuzzy set of rule i corresponding to the spa(k,l) tial input vector θj ; S = {1, 2, . . . , r} with r being the number of IF-THEN rules; Asi , Bsi , Ci , Dsi , Esi , Hi , Gi , and Li (s = 1, 2) are known real constant system matrices with compatible dimensions. The initial boundary condition is given by ⎧ ψ1 (k, l); if (k, l) ∈ [−τ 1 , 0] × [0, z1 ] ⎪ ⎪ ⎪ ⎨ ψ2 (k, l); if (k, l) ∈ [0, z2 ] × [−τ 2 , 0] (2) x(k, l) = ⎪ 0; if (k, l) ∈ [−τ 1 , 0] × (z1 , ∞) ⎪ ⎪ ⎩ 0; if (k, l) ∈ (z2 , ∞) × [−τ 2 , 0] with ψ1 (0, 0) = ψ2 (0, 0), where z1 and z2 are finite positive integers, ψ1 (k, l) and ψ2 (k, l) are given vectors which are independent of the stochastic input sequence {ω(k, l)}. (k,l) Let hi = [hi (k − 1, l), hi (k, l − 1)], where hi (k, l) is the normalized membership function defined by i (k, l) (3) hi (k, l) = r i=1 i (k, l) p (k,l) (k,l) where, i (k, l) = j=1 Fij (θj ) and Fij (θj ) ≥ 0 is the (k,l)
grade of membership of θj in Fij , which is also called the fuzzy basis function. It can be easily verified that 0 ≤ hi (k, l) ≤ 1,
r
hi (k, l) = 1,
∀ k, l ∈ Z+ .
respectively. As pointed out in [1] and [21], the membership function satisfying (3) can be viewed as one of the vertices of a polyhedron. It should be noted that the vector valued normalized membership function comprises the main features of the 2-D T–S fuzzy model, which is totally different from the 1-D case. Subsequently, by fuzzy blending, the T–S fuzzy system (1) can be transformed into ⎧ r r ⎪ ⎪ ⎪ hi (k, l + 1) hi (k + 1, l) ⎪x(k + 1, l + 1) = ⎪ ⎪ ⎪ i=1 i=1 ⎪ ⎪ ⎪ ⎪ ⎪ × A1i x(k, l + 1) + A2i x(k + 1, l) ⎪ ⎪ ⎪ ⎪ ⎪ + D1i x(k − τ1 (k, l), l + 1) ⎪ ⎪ ⎪ ⎪ ⎪ + D2i x(k + 1, l − τ2 (k, l)) ⎪ ⎨ + B1i u(k, l + 1) + B2i u(k + 1, l) (5)
⎪ ⎪ ⎪ + E1i ω(k, l + 1) + E2i ω(k + 1, l) ⎪ ⎪ ⎪ r ⎪ ⎪ ⎪ ⎪ ⎪ y(k, l) = hi (k, l) Ci x(k, l) + Hi ω(k, l) ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ r ⎪ ⎪ ⎪ ⎪ hi (k, l) Gi x(k, l) + Li ω(k, l) . z(k, l) = ⎪ ⎩ i=1
The actual signal received by the designed controller may contain missing measurements, which can be governed by
y(k, l) = η(k, l)y(k, l)
(6)
where η(k, l) ∈ R is a random white sequence taking values of 0 and 1 with Prob{η(k, l) = 1} = α,
Prob{η(k, l) = 0} = 1 − α (7)
and α ∈ [0, 1] is a known scalar. Throughout this paper, we assume that the stochastic variables η(k, l) and ω(i, j) (k, l, i, j ∈ Z+ ) are mutually independent. In this paper, we are interested in designing a 2-D fuzzy controller for system (1) of the following form: Controller Rule i: (k,l) (k,l) (k,l) is Fi1 , . . . , θj is Fij , . . . , and θp is Fip IF θ1 THEN u(k, l) = Ki y(k, l)
(8)
where Ki is the gain matrix of the designed controller to be determined. Similar to (5), the proposed 2-D fuzzy controller can also be fuzzily blended as
(4)
i=1
Remark 1: Compared with the usual model rule for 1-D (k,l) systems, the premise variable θj of the 2-D systems is actually in a spatial type. According to the FM model, the value of the state at location (k +1, l+1) is related to those at (k, l+1) and (k + 1, l). Therefore, it is reasonable to define the normalized membership function of the inferred fuzzy set to be a (k,l) two-element vector hi , where hi (k − 1, l) and hi (k, l − 1) are deemed to be the scalar normalized membership function regarding the premise variables θj (k − 1, l) and θj (k, l − 1),
3
u(k, l) =
r
hi (k, l)η(k, l)Ki y(k, l).
(9)
i=1
In what follows, for brevity, we define h´ i = hi (k, l + 1), h` i = hi (k + 1, l) and: r i1 ,i2 ,...,is =1
for s ∈ Z+ .
hi1 hi2 · · · his =
r i1 =1
hi1
r i2 =1
hi2 · · ·
r is =1
his
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⎧
r r ⎪ ⎪ ´ ⎪ x(k + 1, l + 1) = A h h´ j h´ m B1i η(k, l + 1) x(k, l + 1) + D x(k − τ (k, l), l + 1) + E ω(k, l + 1) + i 1i 1i 1 1i ⎪ ⎪ ⎪ ⎪ i=1 j,m=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ × Kj [Cm x(k, l + 1) + Hm ω(k, l + 1)] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
r ⎪ ⎨ ` hi A2i x(k + 1, l) + D2i x(k + 1, l − τ2 (k, l)) + ⎪ i=1 ⎪ ⎪ ⎪ r ⎪ ⎪ ⎪ ⎪ ` j h` m B2i η(k + 1, l) × Kj [Cm x(k + 1, l) + Hm ω(k + 1, l)] h ω(k + 1, l) + + E ⎪ 2i ⎪ ⎪ ⎪ ⎪ j,m=1 ⎪ ⎪ ⎪ r ⎪ ⎪ ⎪ ⎪ hi (k, l)[Gi x(k, l) + Li ω(k, l)] z(k, l) = ⎪ ⎩
(10)
i=1
Combining (9) with (5), the closed-loop 2-D fuzzy system with the static output feedback controller is governed by (10), shown at the top of this page, which can be rewritten as ⎧ r ⎪ ⎪ ⎪ x(k + 1, l + 1) = h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` 1 (k, l) ⎪ ⎪ ⎨ ´i´jm ´ `i`jm ` ´i,´j,m, ` ` ´ i,j,m=1 ` (11) r ⎪ ⎪ ⎪ ⎪ z(k, l) = hi (k, l)[Gi x(k, l) + Li ω(k, l)] ⎪ ⎩
III. M AIN R ESULTS Before proceeding, let us recall the following lemmas which will be used in the sequence. Lemma 1: Let R ∈ Rn×n be a symmetric positive definite n matrix. For any real vectors X´i´jm´ `i`jm` ∈ Rn and Xa´ b´ ´ ca` b` `c ∈ R ´ c´ , a` , b, ` c` ∈ S, we have with ´i, ´j, m, ´ `i, `j, m, ` a´ , b, r
i=1
h´ ´i h´ ´j h´ m´ h´ a´ h´ b´ h´ c´ h` `i h` `j h` m` h` a` h` b` h` c` × X´iT´jm´ `i`jm` RXa´ b´ ´ ca` b` `c
´ c,`i,`j,m,` ` c=1 ´i,´j,m,´ ´ a,b,´ ` a,b,`
where 1 (k, l)
r
≤
´i´jm ´ `i`jm `
´i,´j,m, ´ `i,`j,m=1 `
= A1´i x(k, l + 1) + A2`i x(k + 1, l) + D1´i x(k − τ1 (k, l), l + 1) + D2`i x(k + 1, l − τ2 (k, l)) + αB1´i K´j (Cm´ x(k, l + 1) + Hm´ ω(k, l + 1)) + (η(k, l + 1) − α)B1´i K´j (Cm´ x(k, l + 1) + E1´i ω(k, l + 1) + Hm´ ω(k, l + 1)) + αB2`i K`j (Cm` x(k + 1, l) + Hm` ω(k + 1, l)) + (η(k + 1, l) − α)B2`i K`j × (Cm` x(k + 1, l) + Hm` ω(k + 1, l)) + E2`i ω(k + 1, l). Definition 1: The 2-D fuzzy system (10) with ω(k, l) ≡ 0 is said to be globally asymptotically stable in the mean-square sense if 2 (12) lim E x(k, l) = 0 k+l→∞
holds for all the initial condition in the form of (2). In this case, the 2-D fuzzy controller (8) is said to be a globally asymptotically stable fuzzy controller for the 2-D system (1). Definition 2: Given scalar γ > 0, the controller (8) is said to be an H∞ fuzzy controller for the 2-D system (1) with output measurements (6) and disturbance attenuation level γ if the dynamics of system (10) is globally asymptotically stable in the mean-square sense and under the zero initial conditions, the following performance constraint: ∞ ∞ z(k, l + 1) 2 E z(k + 1, l) l=0 k=0 ∞ ∞ ω(k, l + 1) 2 2 ≤γ E (13) ω(k + 1, l) l=0 k=0
holds for all nonzero sequences {ω(·, ·)}.
h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` X´iT´jm´ `i`jm` RX´i´jm´ `i`jm`
where h´ t ≥ 0, h` t ≥ 0,
r
h´ t =
t=1
r
h` t = 1 with t ∈ S.
t=1
Proof: Similar to the proof of Lemma 2 in [15], based on the well-known inequality 2X T RY ≤ X T RX + Y T RY where X and Y are any vectors belonging to Rn , one can easily have r
2
h´ ´i h´ ´j h´ m´ h´ a´ h´ b´ h´ c´ h` `i h` `j h` m` h` a` h` b` h` c` × X´iT´jm´ `i`jm` RXa´ b´ ´ ca` b` `c
´ c,`i,`j,m,` ` c=1 ´i,´j,m,´ ´ a,b,´ ` a,b,` r
≤
h´ ´i h´ ´j h´ m´ h´ a´ h´ b´ h´ c´ h` `i h` `j h` m` h` a` h` b` h` c`
´ c,`i,`j,m,` ` c=1 ´i,´j,m,´ ´ a,b,´ ` a,b,`
× X´iT´jm´ `i`jm` RX´i´jm´ `i`jm` + X T´ =2
r
´ ca` b` `c ` c RXa´ b´ a´ b´ca` b`
h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` X´iT´jm´ `i`jm` RX´i´jm´ `i`jm`
´i,´j,m, ´ `i,`j,m=1 `
which completes the proof. In this section, both the stability analysis and the H∞ performance for the 2-D closed-loop fuzzy system will be discussed and then a detailed process for the design of the proposed H∞ controller will be further presented. Finally, a cone complementarity linearization (CCL) algorithm is employed to overcome the numerical difficulty caused by the several matrix equality constraints presented in the main results.
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A. Stability Analysis for the 2-D Fuzzy System With Missing Measurements
with
We begin with the stability analysis for the 2-D fuzzy system (10) when the controller gain Ki (i = 1, 2, . . . , r) are given. The following theorem presents a sufficient condition under which the closed-loop 2-D fuzzy system (10) is globally asymptotically stable in the mean-square sense. Theorem 1: The closed-loop 2-D fuzzy system (10) with the given controller structure (8) is globally asymptotically stable in the mean-square sense if there exist matrices Qh > 0, Qv > 0, Ph > 0, Pv > 0, Rh > 0, Rv > 0, Jkl , Lkl (k, l = 1, 2), and Kj ( j ∈ S) such that the matrix inequalities (14) and (15) hold Rh > Ph , Rv > Pv ∗ 0. constrains (14) ensures that Rh2 > 0 and Combining the inequalities (26) and (27) together and noting that f (k, l) = x(k + 1, l) − x(k, l), g(k, l) = x(k, l + 1) − x(k, l), it can be calculated along the closed-loop system (11) that I(k, l) ≤
Subsequently, it follows from Lemma 1 that: I(k, l) ≤
.
T 1 (k, l) Q 1 (k, l) ´ ca` b` `c a´ b´
2 (k, l) + 2 T (k, l) 3 T (k, l) + Rh1 Rv1
Rh1 2 (k, l) + 2 (k, l) T
´i´jm ´ `i`jm `
´i´jm ´ `i`jm `
´ ca` b` `c a´ b´
* −1 + −1 T ¯ h T ¯ ¯ Rh2 + τ1 φ J1 R2 J1 + L1 L¯ 1T φ $ % $ v %−1 T −1 + τ2 ϕ T J¯ 2 Rv2 J¯ 2T + L¯ 2 R2 L¯ 2 ϕ + 3 (k, l) Rv1 3 (k, l) T
´i´jm ´ `i`jm `
+ (k, l)
/
´ ca` b` `c a´ b´
(k, l)
1 (k, l) − x(k, l + 1)
´i´jm ´ `i`jm `
+ (η(k, l + 1) − α)B1´i K´j Cm´ x(k, l + 1) + (η(k + 1, l) − α)B2`i K`j Cm` x(k + 1, l) 3 (k, l) ´i´jm ´ `i`jm `
= 1 (k, l) − x(k + 1, l) ´i´jm ´ `i`jm `
and i (k, l) (i = 1, 2, 3) have the similar representations as ´ ca` b` `c a´ b´
1 (k, l) ´i´jm ´ `i`jm `
(i = 1, 2, 3) by just substituting ´i, ´j, m, ´ `i, `j, and m `
´ c´ , a` , b, ` and c` , respectively. with a´ , b,
1 (k, l) (k, l) Q ´i´jm ´ `i`jm `
/ (k, l)
+ A2`i + αB2`i K`j Cm` x(k + 1, l)
+ D1´i x(k − τ1 (k, l), l + 1) + D2`i x(k + 1, l − τ2 (k, l)) 1 (k, l) − x(k, l + 1) 2 (k, l) =
´i´jm ´ `i`jm `
´i´jm ´ `i`jm `
3 (k, l) = 1 (k, l) − x(k + 1, l) .
´i´jm ´ `i`jm `
+ D1´i x(k − τ1 (k, l), l + 1) + D2`i x(k + 1, l − τ2 (k, l))
T
´i´jm ´ `i`jm `
where the relationship (7) is utilized in the last step when deriving the above inequality, in which 1 (k, l) = A1´i + αB1´i K´j Cm´ x(k, l + 1)
´i´jm ´ `i`jm `
+ A2`i + αB2`i K`j Cm` x(k + 1, l)
´i´jm ´ `i`jm `
´i´jm ´ `i`jm `
(28)
1 (k, l) = A1´i + αB1´i K´j Cm´ x(k, l + 1)
´i´jm ´ `i`jm `
3 (k, l) + 1 ×
where (k, l) = −xT (k, l+1)Qh x(k, l+1)−xT (k+1, l)Qv x(k+ 1, l)+2φ T (J¯ 1 I1 + L¯ 1 I2 )φ+2ϕ T (J¯ 2 I1 + L¯ 2 I2 )ϕ, 2 (k, l) =
´i´jm ´ `i`jm `
* −1 + −1 (k, l) + τ1 φ T J¯ 1 Rh2 Rh2 J¯ 1T + L¯ 1 L¯ 1T φ $ % $ v %−1 T −1 + τ2 ϕ T J¯ 2 Rv2 J¯ 2T + L¯ 2 R2 L¯ 2 ϕ T + α¯ B1´i K´j Cm´ x(k, l + 1) Q B1´i K´j Cm´ x(k, l + 1) T + α¯ B2`i K`j Cm` x(k + 1, l) Q B2`i K`j Cm` x(k + 1, l)
i j
´i´jm ´ `i`jm `
h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` E
.
r h´ ´h´ ´h´ m´ h´ a´ h´ ´ h´ c´ h` `h` `h` m`
× h` a` h` b` h` c` E
r
´i,´j,m, ´ `i,`j,m=1 `
×
i j b ´ c,`i,`j,m,` ` c=1 ´i,´j,m,´ ´ a,b,´ ` a,b,`
(27)
´i´jm ´ `i`jm `
´i´jm ´ `i`jm `
Defining ζ¯ (k, l) = [xT (k, l+1) xT (k+1, l) xT (k−τ 1 , l+1) + 1, l − τ 2 ) xT (k − τ1 (k, l), l + 1) xT (k + 1, l − τ2 (k, l)) xT (k − τ 1 , l + 1) xT (k + 1, l − τ 2 )]T , it can be observed from (29) that r 2 $ % h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` E x¯ (k, l) (k, l) I(k, l) ≤ λmax
xT (k
´i,´j,m, ´ `i,`j,m=1 `
(29) = + T R2 −1 I −1 2 + T R2 −1 I −1 3 + α
where
Q 4 + ¯ T4 2 3 T T T T α
¯ 5 Q 5 + 67 R1 67 + 8 Q 8 , x¯ (k, l) = [x (k, l + 1) ) is the maximal eigenvalue of
. It xT (k + 1, l)]T and λmax(
should be noted that the Schur complement lemma has been < 0 from the inequality condition < 0 utilized to get
in (15). For any integer M > 1 and N > 1, summing up both sides of the inequality (29) for k and l varying from 0 to, respectively,
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M and N, we obtain N M
N M $ % I(k, l) ≤ λmax
r
h´ ´i h´ ´j h´ m´ h` `i h` `j h` m`
l=0 k=0 ´i,´j,m, ´ `i,`j,m=1 `
l=0 k=0
2 × E x¯ (k, l) (k, l) (30)
which immediately infers the following inequality by further considering the non-negativeness of the energy-like functional V(k, l): N M
r
2 h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` E x¯ (k, l) (k, l)
l=0 k=0 ´i,´j,m, ´ `i,`j,m=1 `
N −1 $ % ≤ E V1h (0, l + 1) + V2h (0, l + 1) λmax
l=0 + V3h (0, l + 1) (k, l) +
M $ E V1v (k + 1, 0) + V2v (k + 1, 0) k=0
% + V3v (k + 1, 0) (k, l) .
(31)
Taking the mathematical expectation on both sides of (31) and further considering the finite initial boundary condition (2), we conclude that N M
r
2 h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` E x¯ (k, l) < ∞ (32)
l=0 k=0 ´i,´j,m, ´ `i,`j,m=1 `
which implies that
lim E{x(k, l)2 } k+l→∞
= 0. According to
Definition 1, it is concluded that the closed-loop 2-D fuzzy system (10) is globally asymptotically stable in the meansquare sense. The proof is complete. Remark 2: As is well known, delay-independent condition will inevitably bring conservativeness, especially when the delay is very large. Here, we choose a delaydependent Lyapunov functional associated with both the lower and the upper bound of the time-varying delay τi (k, l) (i = 1, 2), which will undoubtedly lead to less conservativeness. Moreover, the matrix inequality constraint (15) derived by introducing the free-weighting matrix method is actually a delay-range-dependent stability criterion, which has proven to be an effective approach in stability analysis of time-delay systems (see [35]). B. H∞ Performance Analysis for the 2-D Fuzzy System With Missing Measurements In Theorem 1, a general condition has been derived to guarantee the globally asymptotic stability of the closed-loop 2-D fuzzy system (10). Based upon the above stability analysis, we now deal with the capacity of disturbance attenuation and rejection as well as the tolerance level to the missing measurements with the designed 2-D controller (8). The sufficient condition obtained in the following theorem is to make sure that the fuzzy system (10) is asymptotically stable with a prescribed H∞ performance index for nonzero exogenous
disturbances and possible missing measurements under the zero-initial condition. Theorem 2: The closed-loop 2-D fuzzy system (10) with the designed controller structure (8) is globally asymptotically stable with disturbance attenuation level γ if there exist matrices Qh > 0, Qv > 0, Ph > 0, Pv > 0, Rh > 0, Rv > 0, Jkl , Lkl (k, l = 1, 2), and Kj ( j ∈ S) such that the matrix inequality (14) and the matrix inequalities (33) hold ¯ ∗ 0, Q v > 0, P h > 0, P v > 0, R h > 0, R v > 0, Qh > 0, Qv > 0, Ph > 0, Pv > 0, Rh > 0, Rv > 0, Jkl , Lkl (k, l = 1, 2), and Kj ( j ∈ S) such that the linear matrix inequality (14) and the inequalities/equalities (38)–(40) hold
¯ 1
in (37) as follows: Proof: Rewrite ¯ + T2 R2 −1 I −1 2 + T3 R2 −1 I −1 3 = + α(1 ¯ 6 ⊗ 4 )T diag Qh , Qv , Ph , Pv , τ1 Rh , τ2 Rv ¯ 6 ⊗ 5 )T × (16 ⊗ 4 ) + α(1 × diag Qh , Qv , Ph , Pv , τ1 Rh , τ2 Rv (16 ⊗ 5 ) + T67 diag Ph , Pv 67 + T67 I diag Rh , Rv 67 + (12 ⊗ 8 )T × diag Qh , Qv (12 ⊗ 8 ) + T910 910 < 0.
5
Q8 + T9 9 + T10 10 . (37) + T67 R1 67 + T8
Q h Qh = I,
P h Ph = I,
R h Rh = I
(38)
Q Q = I, ∗
1
H∞ Control for 2-D Fuzzy Systems With Interval Time-Varying Delays and Missing Measurements Yuqiang Luo, Zidong Wang, Fellow, IEEE, Jinling Liang, Guoliang Wei, and Fuad E. Alsaadi
Abstract—In this paper, we consider the H∞ control problem for a class of 2-D Takagi–Sugeno fuzzy described by the second Fornasini–Machesini local state-space model with timedelays and missing measurements. The state delays are allowed to be time-varying within a known interval. The measurement output is subject to randomly intermittent packet dropouts governed by a random sequence satisfying the Bernoulli distribution. The purpose of the addressed problem is to design an outputfeedback controller such that the closed-loop system is globally asymptotically stable in the mean square and the prescribed H∞ performance index is satisfied. By employing a combination of the intensive stochastic analysis and the free weighting matrix method, several delay-range-dependent sufficient conditions are presented that guarantee the existence of the desired controllers for all possible time-delays and missing measurements. The explicit expressions of such controllers are derived by means of the solution to a class of convex optimization problems that can be solved via standard software packages. Finally, a numerical simulation example is given to demonstrate the applicability of the proposed control scheme. Index Terms—2-D systems, fuzzy systems, H∞ control, missing measurements, output feedback, time-varying delays.
I. I NTRODUCTION
N
ONLINEARITY is a ubiquitous phenomenon in the natural world that has been receiving ever-increasing research
Manuscript received May 2, 2015; revised August 24, 2015 and November 18, 2015; accepted January 4, 2016. This work was supported in part by the Royal Society of U.K., in part by the National Natural Science Foundation of China under Grant 61174136, Grant 61329301, and Grant 61374039, in part by the Hujiang Foundation of China under Grant C14002 and Grant D15009, in part by the Natural Science Foundation of Jiangsu province of China under Grant BK20130017, in part by the Programme for New Century Excellent Talents in University under Grant NCET-12-0117, and in part by the Alexander von Humboldt Foundation of Germany. This paper was recommended by Associate Editor H. Gao. (Corresponding author: Guoliang Wei.) Y. Luo and G. Wei are with the Shanghai Key Laboratory of Modern Optical System, Department of Control Science and Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China (e-mail: [email protected]). Z. Wang is with the Department of Computer Science, Brunel University London, Uxbridge UB8 3PH, U.K., and also with the Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia (e-mail: [email protected]). J. Liang is with the Department of Mathematics, Southeast University, Nanjing 210096, China. F. E. Alsaadi is with the Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia. 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.2016.2514846
attention from a variety of subject areas. Traditionally, rigorous mathematical analysis on nonlinear systems rely on precise description of the nonlinearities with some stringent assumptions, and this sometimes hinders the nonlinear systems theory from being applied to certain engineering practice, for example, those data-based applications where an exact mathematical model is hard to be established [4], [16], [26], [29]–[31], [38]. On the other hand, the fuzzy logic theory has proven to be effective in handling reasoning that is approximate rather than fixed and exact. Furthermore, after a decade of theoretical and practical development, the Takagi–Sugeno (T–S) model has been recognized as an efficient way to approximate certain nonlinear systems. By using a set of local linear models which are smoothly connected by nonlinear fuzzy member ship functions to present a nonlinear plant, the T–S fuzzy model has brought the analysis and synthesis of nonlinear systems into a unified framework. Moreover, owing to the peculiarity of the T–S model, a large portion of existing results for linear systems can be readily extended for some nonlinear systems. As pointed out in [12] and [32], the T–S model is able to approximate any smooth nonlinear function to any degree of accuracy in any convex compact region. As is well known, many practical systems can be ideally described by the 2-D systems that have received tremendous research attention because of their applications in thermal processes, seismic data sections, digitized photographic data, digital filtering, and magnetic maps, etc. [11]. For four decades, the theoretical investigations on 2-D systems have been attracting recurring research interests and fruitful research results have been available in the literature. As early as in 1970s, some basic behaviors and modeling issues were thoroughly examined for 2-D systems (see [11], [18]). Parallel to the rapid research development of the traditional 1-D systems, in the past few years, some important breakthroughs have been reported on the analysis and design issues for the 2-D control systems. For example, the stability analysis and stabilization problems for 2-D systems have been addressed in [7] and [13], and the filter/observer design problems have been investigated in [25]. In particular, some new kinds of 2-D models and new techniques have recently been put forward in the literature. In [23], a state estimation problem for 2-D complex networks with randomly occurring nonlinearities and randomly varying sensor delays has been considered, and a new synchronization problem has been dealt with for an array of 2-D coupled dynamical network in [24].
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Given the practical importance of 2-D systems in representing 2-D evolutions and the technical convenience of T–S fuzzy systems for handling affine nonlinearities, a seemingly natural research issue would be to investigate the dynamical behaviors as well as the estimator/controller design problems of 2-D T–S fuzzy systems. Unfortunately, a literature review has revealed that the corresponding results have been very few due primarily to the mathematical complexities for 2-D T–S fuzzy systems. On the other hand, time delays are very often a major concern in many practical applications such as communication line, electrical signal processing systems, network transmission systems, seismic wave, urban traffic management systems, etc. Time delays are well recognized as one of the main sources for poor performance or even instability of control systems. In the 1-D settings, the time-delay systems have recently received considerable research attention and a rich body of literature has appeared on this topic (see [3], [17]). Moreover, among the existing works, plenty of stability conditions are delay-dependent, which pose less conservativeness (see [19], [20]). With respect to 2-D timedelay systems, over the past few years, some initial results have been reported on the control and filtering problems. For example, the problem of delay-dependent H∞ control for 2-D discrete state delay systems has been investigated in [22], and a robust H∞ filter for 2-D discrete systems with time delays has been designed in [34]. It should be noted that most available results have been concerned with time-invariant delays. As a matter of fact, the phenomenon of missing measurements (packet losses or dropouts) is virtually inevitable in measurement processes (particularly within a networked environment), which is caused by some harsh working conditions and imperfect communications. Many factors have contributed to this kind of less-than-ideal situation. Such factors include, but are not limited to, the limited bandwidth of the communication channel, abnormity of swap device, random network congestion, accidental loss of some collected data in a very noisy environment. So far, the problem of missing measurements has been well studied for 1-D systems and a great deal of literature has been available (see [28], [33]). Nevertheless, the relevant research for 2-D systems is still in its early stage especially when both the time-varying delays and missing measurements are simultaneously present, not to mention the case when the 2-D systems are further complicated by the T–S fuzzy model. As such, the stability analysis and stabilization problem for 2-D time-delay fuzzy systems with missing measurements remains a challenging issue that motivates our current research, and the main task of this paper is to propose a general framework for handling such a challenge. In this paper, we endeavor to research into the H∞ control problem for a class of 2-D T–S fuzzy systems described by the second Fornasini–Machesini (FM) local state-space (LSS) model with time-delays and missing measurements. The main contributions of this paper can be boiled down as follows. 1) A 2-D T–S fuzzy model is considered, which is comprehensive to include time-varying delays, bounded noises,
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and probabilistic missing measurements, thereby reflecting engineering practice more closely. 2) For the purpose of stabilizing the addressed 2-D fuzzy systems, an energy-like functional is constructed and several delay-range-dependent stability criteria are obtained to accomplish the design goal. 3) The close-loop system has the expected disturbance attenuation level in terms of a prescribed H∞ performance index. Compared with the existing delay-independent or delay-dependent results for 1-D delayed systems, the delay-range-dependent method addressed in this paper is of less conservativeness. The adopted 2-D T–S model can be thought to be a universal approximator for nonlinear 2-D systems, and the corresponding controller can also be used to stabilize the complicated 2-D nonlinear plant. Potential applications of the investigated fuzzy control approach include the 2-D digital systems, image processing, and wireless communications, etc. Especially for the case of interval time-varying delays and randomly missing measurements, the proposed fuzzy controller will show its strength. The rest of this paper is organized as follows. Section II is devoted to the formulation of the H∞ control problem for the addressed 2-D fuzzy systems with interval time-varying delays, where the phenomenon of probabilistic missing measurements is characterized by a Bernoulli distribution model, and some notations and related definitions are also given. In Section III, with the aid of an energy-like functional and the delay-range-dependent method, both the analysis and the synthesis problems of the 2-D fuzzy control system are investigated. In Section IV, an example is given to validate the design approach of the proposed fuzzy control scheme, and some concluding remarks have been drawn in Section V. Notation: In this paper, Rn , Rn×m , and Z (Z+ , Z− ) denote, respectively, the n-dimensional Euclidean space, the set of all n × m real matrices, and the set of all integers (non-negative integers, negative integers). · refers to the Euclidean norm in Rn . In represents the identity matrix of dimension n × n. The notation X ≥ Y (respectively, X > Y), where X and Y are symmetric matrices, means that X − Y is positive semi-definite (respectively, positive definite). For a matrix M, M T , and M −1 represent its transpose and inverse, respectively. The shorthand diag{M1 , M2 , . . . , Mn } denotes a block diagonal matrix with diagonal blocks being the matrices M1 , M2 , . . . , Mn . In symmetric block matrices, the symbol “∗” is used as an ellipsis for terms induced by symmetry. Matrices, if they are not explicitly stated, are assumed to have compatible dimensions.
II. P ROBLEM F ORMULATION Consider a 2-D discrete-time T–S fuzzy system with statedelays and stochastic perturbations described by the following FM LSS second model: Plant Rule i: IF θ1(k,l) is Fi1 , . . . , θj(k,l) is Fij , . . . , and θp(k,l) is Fip
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THEN ⎧ x(k + 1, l + 1) = A1i x(k, l + 1) + A2i x(k + 1, l) ⎪ ⎪ ⎪ ⎪ ⎪ + D1i x(k − τ1 (k, l), l + 1) ⎪ ⎪ ⎪ ⎪ ⎪ + D2i x(k + 1, l − τ2 (k, l)) ⎪ ⎨ + B1i u(k, l + 1) + B2i u(k + 1, l) ⎪ ⎪ ⎪ + E1i ω(k, l + 1) + E2i ω(k + 1, l) ⎪ ⎪ ⎪ ⎪ ⎪ y(k, l) = Ci x(k, l) + Hi ω(k, l), i∈S ⎪ ⎪ ⎪ ⎩ z(k, l) = Gi x(k, l) + Li ω(k, l)
(1)
where k, l ∈ Z+ , and x(k, l) ∈ Rnx is the state vector; y(k, l) ∈ Rny is the measured output; z(k, l) ∈ Rnz is the controlled output; u(k, l) ∈ Rnu is the control input vecthe disturbance input which belongs to tor; ω(k, l) ∈ Rnω is ∞ E{ω(k, l)2 } < ∞; τ1 (k, l) l2 (Z+ , Z+ ), namely, ∞ l=0 k=0 and τ2 (k, l) are interval time-varying delays along the horizontal direction and the vertical direction, respectively, which are subjected to τ 1 ≤ τ1 (k, l) ≤ τ 1 and τ 2 ≤ τ2 (k, l) ≤ τ 2 ; the (k,l)
input vector θj = [θj (k−1, l), θj (k, l−1)] ( j = 1, 2, . . . , p) represents the spatial premise variable at the location (k, l), which may be states or measurable variables, and θj (k, l) is (k,l) the corresponding component of the spatial input vector θj ; Fij is a spatial fuzzy set of rule i corresponding to the spa(k,l) tial input vector θj ; S = {1, 2, . . . , r} with r being the number of IF-THEN rules; Asi , Bsi , Ci , Dsi , Esi , Hi , Gi , and Li (s = 1, 2) are known real constant system matrices with compatible dimensions. The initial boundary condition is given by ⎧ ψ1 (k, l); if (k, l) ∈ [−τ 1 , 0] × [0, z1 ] ⎪ ⎪ ⎪ ⎨ ψ2 (k, l); if (k, l) ∈ [0, z2 ] × [−τ 2 , 0] (2) x(k, l) = ⎪ 0; if (k, l) ∈ [−τ 1 , 0] × (z1 , ∞) ⎪ ⎪ ⎩ 0; if (k, l) ∈ (z2 , ∞) × [−τ 2 , 0] with ψ1 (0, 0) = ψ2 (0, 0), where z1 and z2 are finite positive integers, ψ1 (k, l) and ψ2 (k, l) are given vectors which are independent of the stochastic input sequence {ω(k, l)}. (k,l) Let hi = [hi (k − 1, l), hi (k, l − 1)], where hi (k, l) is the normalized membership function defined by i (k, l) (3) hi (k, l) = r i=1 i (k, l) p (k,l) (k,l) where, i (k, l) = j=1 Fij (θj ) and Fij (θj ) ≥ 0 is the (k,l)
grade of membership of θj in Fij , which is also called the fuzzy basis function. It can be easily verified that 0 ≤ hi (k, l) ≤ 1,
r
hi (k, l) = 1,
∀ k, l ∈ Z+ .
respectively. As pointed out in [1] and [21], the membership function satisfying (3) can be viewed as one of the vertices of a polyhedron. It should be noted that the vector valued normalized membership function comprises the main features of the 2-D T–S fuzzy model, which is totally different from the 1-D case. Subsequently, by fuzzy blending, the T–S fuzzy system (1) can be transformed into ⎧ r r ⎪ ⎪ ⎪ hi (k, l + 1) hi (k + 1, l) ⎪x(k + 1, l + 1) = ⎪ ⎪ ⎪ i=1 i=1 ⎪ ⎪ ⎪ ⎪ ⎪ × A1i x(k, l + 1) + A2i x(k + 1, l) ⎪ ⎪ ⎪ ⎪ ⎪ + D1i x(k − τ1 (k, l), l + 1) ⎪ ⎪ ⎪ ⎪ ⎪ + D2i x(k + 1, l − τ2 (k, l)) ⎪ ⎨ + B1i u(k, l + 1) + B2i u(k + 1, l) (5)
⎪ ⎪ ⎪ + E1i ω(k, l + 1) + E2i ω(k + 1, l) ⎪ ⎪ ⎪ r ⎪ ⎪ ⎪ ⎪ ⎪ y(k, l) = hi (k, l) Ci x(k, l) + Hi ω(k, l) ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ r ⎪ ⎪ ⎪ ⎪ hi (k, l) Gi x(k, l) + Li ω(k, l) . z(k, l) = ⎪ ⎩ i=1
The actual signal received by the designed controller may contain missing measurements, which can be governed by
y(k, l) = η(k, l)y(k, l)
(6)
where η(k, l) ∈ R is a random white sequence taking values of 0 and 1 with Prob{η(k, l) = 1} = α,
Prob{η(k, l) = 0} = 1 − α (7)
and α ∈ [0, 1] is a known scalar. Throughout this paper, we assume that the stochastic variables η(k, l) and ω(i, j) (k, l, i, j ∈ Z+ ) are mutually independent. In this paper, we are interested in designing a 2-D fuzzy controller for system (1) of the following form: Controller Rule i: (k,l) (k,l) (k,l) is Fi1 , . . . , θj is Fij , . . . , and θp is Fip IF θ1 THEN u(k, l) = Ki y(k, l)
(8)
where Ki is the gain matrix of the designed controller to be determined. Similar to (5), the proposed 2-D fuzzy controller can also be fuzzily blended as
(4)
i=1
Remark 1: Compared with the usual model rule for 1-D (k,l) systems, the premise variable θj of the 2-D systems is actually in a spatial type. According to the FM model, the value of the state at location (k +1, l+1) is related to those at (k, l+1) and (k + 1, l). Therefore, it is reasonable to define the normalized membership function of the inferred fuzzy set to be a (k,l) two-element vector hi , where hi (k − 1, l) and hi (k, l − 1) are deemed to be the scalar normalized membership function regarding the premise variables θj (k − 1, l) and θj (k, l − 1),
3
u(k, l) =
r
hi (k, l)η(k, l)Ki y(k, l).
(9)
i=1
In what follows, for brevity, we define h´ i = hi (k, l + 1), h` i = hi (k + 1, l) and: r i1 ,i2 ,...,is =1
for s ∈ Z+ .
hi1 hi2 · · · his =
r i1 =1
hi1
r i2 =1
hi2 · · ·
r is =1
his
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⎧
r r ⎪ ⎪ ´ ⎪ x(k + 1, l + 1) = A h h´ j h´ m B1i η(k, l + 1) x(k, l + 1) + D x(k − τ (k, l), l + 1) + E ω(k, l + 1) + i 1i 1i 1 1i ⎪ ⎪ ⎪ ⎪ i=1 j,m=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ × Kj [Cm x(k, l + 1) + Hm ω(k, l + 1)] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
r ⎪ ⎨ ` hi A2i x(k + 1, l) + D2i x(k + 1, l − τ2 (k, l)) + ⎪ i=1 ⎪ ⎪ ⎪ r ⎪ ⎪ ⎪ ⎪ ` j h` m B2i η(k + 1, l) × Kj [Cm x(k + 1, l) + Hm ω(k + 1, l)] h ω(k + 1, l) + + E ⎪ 2i ⎪ ⎪ ⎪ ⎪ j,m=1 ⎪ ⎪ ⎪ r ⎪ ⎪ ⎪ ⎪ hi (k, l)[Gi x(k, l) + Li ω(k, l)] z(k, l) = ⎪ ⎩
(10)
i=1
Combining (9) with (5), the closed-loop 2-D fuzzy system with the static output feedback controller is governed by (10), shown at the top of this page, which can be rewritten as ⎧ r ⎪ ⎪ ⎪ x(k + 1, l + 1) = h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` 1 (k, l) ⎪ ⎪ ⎨ ´i´jm ´ `i`jm ` ´i,´j,m, ` ` ´ i,j,m=1 ` (11) r ⎪ ⎪ ⎪ ⎪ z(k, l) = hi (k, l)[Gi x(k, l) + Li ω(k, l)] ⎪ ⎩
III. M AIN R ESULTS Before proceeding, let us recall the following lemmas which will be used in the sequence. Lemma 1: Let R ∈ Rn×n be a symmetric positive definite n matrix. For any real vectors X´i´jm´ `i`jm` ∈ Rn and Xa´ b´ ´ ca` b` `c ∈ R ´ c´ , a` , b, ` c` ∈ S, we have with ´i, ´j, m, ´ `i, `j, m, ` a´ , b, r
i=1
h´ ´i h´ ´j h´ m´ h´ a´ h´ b´ h´ c´ h` `i h` `j h` m` h` a` h` b` h` c` × X´iT´jm´ `i`jm` RXa´ b´ ´ ca` b` `c
´ c,`i,`j,m,` ` c=1 ´i,´j,m,´ ´ a,b,´ ` a,b,`
where 1 (k, l)
r
≤
´i´jm ´ `i`jm `
´i,´j,m, ´ `i,`j,m=1 `
= A1´i x(k, l + 1) + A2`i x(k + 1, l) + D1´i x(k − τ1 (k, l), l + 1) + D2`i x(k + 1, l − τ2 (k, l)) + αB1´i K´j (Cm´ x(k, l + 1) + Hm´ ω(k, l + 1)) + (η(k, l + 1) − α)B1´i K´j (Cm´ x(k, l + 1) + E1´i ω(k, l + 1) + Hm´ ω(k, l + 1)) + αB2`i K`j (Cm` x(k + 1, l) + Hm` ω(k + 1, l)) + (η(k + 1, l) − α)B2`i K`j × (Cm` x(k + 1, l) + Hm` ω(k + 1, l)) + E2`i ω(k + 1, l). Definition 1: The 2-D fuzzy system (10) with ω(k, l) ≡ 0 is said to be globally asymptotically stable in the mean-square sense if 2 (12) lim E x(k, l) = 0 k+l→∞
holds for all the initial condition in the form of (2). In this case, the 2-D fuzzy controller (8) is said to be a globally asymptotically stable fuzzy controller for the 2-D system (1). Definition 2: Given scalar γ > 0, the controller (8) is said to be an H∞ fuzzy controller for the 2-D system (1) with output measurements (6) and disturbance attenuation level γ if the dynamics of system (10) is globally asymptotically stable in the mean-square sense and under the zero initial conditions, the following performance constraint: ∞ ∞ z(k, l + 1) 2 E z(k + 1, l) l=0 k=0 ∞ ∞ ω(k, l + 1) 2 2 ≤γ E (13) ω(k + 1, l) l=0 k=0
holds for all nonzero sequences {ω(·, ·)}.
h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` X´iT´jm´ `i`jm` RX´i´jm´ `i`jm`
where h´ t ≥ 0, h` t ≥ 0,
r
h´ t =
t=1
r
h` t = 1 with t ∈ S.
t=1
Proof: Similar to the proof of Lemma 2 in [15], based on the well-known inequality 2X T RY ≤ X T RX + Y T RY where X and Y are any vectors belonging to Rn , one can easily have r
2
h´ ´i h´ ´j h´ m´ h´ a´ h´ b´ h´ c´ h` `i h` `j h` m` h` a` h` b` h` c` × X´iT´jm´ `i`jm` RXa´ b´ ´ ca` b` `c
´ c,`i,`j,m,` ` c=1 ´i,´j,m,´ ´ a,b,´ ` a,b,` r
≤
h´ ´i h´ ´j h´ m´ h´ a´ h´ b´ h´ c´ h` `i h` `j h` m` h` a` h` b` h` c`
´ c,`i,`j,m,` ` c=1 ´i,´j,m,´ ´ a,b,´ ` a,b,`
× X´iT´jm´ `i`jm` RX´i´jm´ `i`jm` + X T´ =2
r
´ ca` b` `c ` c RXa´ b´ a´ b´ca` b`
h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` X´iT´jm´ `i`jm` RX´i´jm´ `i`jm`
´i,´j,m, ´ `i,`j,m=1 `
which completes the proof. In this section, both the stability analysis and the H∞ performance for the 2-D closed-loop fuzzy system will be discussed and then a detailed process for the design of the proposed H∞ controller will be further presented. Finally, a cone complementarity linearization (CCL) algorithm is employed to overcome the numerical difficulty caused by the several matrix equality constraints presented in the main results.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. LUO et al.: H∞ CONTROL FOR 2-D FUZZY SYSTEMS WITH INTERVAL TIME-VARYING DELAYS AND MISSING MEASUREMENTS
A. Stability Analysis for the 2-D Fuzzy System With Missing Measurements
with
We begin with the stability analysis for the 2-D fuzzy system (10) when the controller gain Ki (i = 1, 2, . . . , r) are given. The following theorem presents a sufficient condition under which the closed-loop 2-D fuzzy system (10) is globally asymptotically stable in the mean-square sense. Theorem 1: The closed-loop 2-D fuzzy system (10) with the given controller structure (8) is globally asymptotically stable in the mean-square sense if there exist matrices Qh > 0, Qv > 0, Ph > 0, Pv > 0, Rh > 0, Rv > 0, Jkl , Lkl (k, l = 1, 2), and Kj ( j ∈ S) such that the matrix inequalities (14) and (15) hold Rh > Ph , Rv > Pv ∗ 0. constrains (14) ensures that Rh2 > 0 and Combining the inequalities (26) and (27) together and noting that f (k, l) = x(k + 1, l) − x(k, l), g(k, l) = x(k, l + 1) − x(k, l), it can be calculated along the closed-loop system (11) that I(k, l) ≤
Subsequently, it follows from Lemma 1 that: I(k, l) ≤
.
T 1 (k, l) Q 1 (k, l) ´ ca` b` `c a´ b´
2 (k, l) + 2 T (k, l) 3 T (k, l) + Rh1 Rv1
Rh1 2 (k, l) + 2 (k, l) T
´i´jm ´ `i`jm `
´i´jm ´ `i`jm `
´ ca` b` `c a´ b´
* −1 + −1 T ¯ h T ¯ ¯ Rh2 + τ1 φ J1 R2 J1 + L1 L¯ 1T φ $ % $ v %−1 T −1 + τ2 ϕ T J¯ 2 Rv2 J¯ 2T + L¯ 2 R2 L¯ 2 ϕ + 3 (k, l) Rv1 3 (k, l) T
´i´jm ´ `i`jm `
+ (k, l)
/
´ ca` b` `c a´ b´
(k, l)
1 (k, l) − x(k, l + 1)
´i´jm ´ `i`jm `
+ (η(k, l + 1) − α)B1´i K´j Cm´ x(k, l + 1) + (η(k + 1, l) − α)B2`i K`j Cm` x(k + 1, l) 3 (k, l) ´i´jm ´ `i`jm `
= 1 (k, l) − x(k + 1, l) ´i´jm ´ `i`jm `
and i (k, l) (i = 1, 2, 3) have the similar representations as ´ ca` b` `c a´ b´
1 (k, l) ´i´jm ´ `i`jm `
(i = 1, 2, 3) by just substituting ´i, ´j, m, ´ `i, `j, and m `
´ c´ , a` , b, ` and c` , respectively. with a´ , b,
1 (k, l) (k, l) Q ´i´jm ´ `i`jm `
/ (k, l)
+ A2`i + αB2`i K`j Cm` x(k + 1, l)
+ D1´i x(k − τ1 (k, l), l + 1) + D2`i x(k + 1, l − τ2 (k, l)) 1 (k, l) − x(k, l + 1) 2 (k, l) =
´i´jm ´ `i`jm `
´i´jm ´ `i`jm `
3 (k, l) = 1 (k, l) − x(k + 1, l) .
´i´jm ´ `i`jm `
+ D1´i x(k − τ1 (k, l), l + 1) + D2`i x(k + 1, l − τ2 (k, l))
T
´i´jm ´ `i`jm `
where the relationship (7) is utilized in the last step when deriving the above inequality, in which 1 (k, l) = A1´i + αB1´i K´j Cm´ x(k, l + 1)
´i´jm ´ `i`jm `
+ A2`i + αB2`i K`j Cm` x(k + 1, l)
´i´jm ´ `i`jm `
´i´jm ´ `i`jm `
(28)
1 (k, l) = A1´i + αB1´i K´j Cm´ x(k, l + 1)
´i´jm ´ `i`jm `
3 (k, l) + 1 ×
where (k, l) = −xT (k, l+1)Qh x(k, l+1)−xT (k+1, l)Qv x(k+ 1, l)+2φ T (J¯ 1 I1 + L¯ 1 I2 )φ+2ϕ T (J¯ 2 I1 + L¯ 2 I2 )ϕ, 2 (k, l) =
´i´jm ´ `i`jm `
* −1 + −1 (k, l) + τ1 φ T J¯ 1 Rh2 Rh2 J¯ 1T + L¯ 1 L¯ 1T φ $ % $ v %−1 T −1 + τ2 ϕ T J¯ 2 Rv2 J¯ 2T + L¯ 2 R2 L¯ 2 ϕ T + α¯ B1´i K´j Cm´ x(k, l + 1) Q B1´i K´j Cm´ x(k, l + 1) T + α¯ B2`i K`j Cm` x(k + 1, l) Q B2`i K`j Cm` x(k + 1, l)
i j
´i´jm ´ `i`jm `
h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` E
.
r h´ ´h´ ´h´ m´ h´ a´ h´ ´ h´ c´ h` `h` `h` m`
× h` a` h` b` h` c` E
r
´i,´j,m, ´ `i,`j,m=1 `
×
i j b ´ c,`i,`j,m,` ` c=1 ´i,´j,m,´ ´ a,b,´ ` a,b,`
(27)
´i´jm ´ `i`jm `
´i´jm ´ `i`jm `
Defining ζ¯ (k, l) = [xT (k, l+1) xT (k+1, l) xT (k−τ 1 , l+1) + 1, l − τ 2 ) xT (k − τ1 (k, l), l + 1) xT (k + 1, l − τ2 (k, l)) xT (k − τ 1 , l + 1) xT (k + 1, l − τ 2 )]T , it can be observed from (29) that r 2 $ % h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` E x¯ (k, l) (k, l) I(k, l) ≤ λmax
xT (k
´i,´j,m, ´ `i,`j,m=1 `
(29) = + T R2 −1 I −1 2 + T R2 −1 I −1 3 + α
where
Q 4 + ¯ T4 2 3 T T T T α
¯ 5 Q 5 + 67 R1 67 + 8 Q 8 , x¯ (k, l) = [x (k, l + 1) ) is the maximal eigenvalue of
. It xT (k + 1, l)]T and λmax(
should be noted that the Schur complement lemma has been < 0 from the inequality condition < 0 utilized to get
in (15). For any integer M > 1 and N > 1, summing up both sides of the inequality (29) for k and l varying from 0 to, respectively,
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M and N, we obtain N M
N M $ % I(k, l) ≤ λmax
r
h´ ´i h´ ´j h´ m´ h` `i h` `j h` m`
l=0 k=0 ´i,´j,m, ´ `i,`j,m=1 `
l=0 k=0
2 × E x¯ (k, l) (k, l) (30)
which immediately infers the following inequality by further considering the non-negativeness of the energy-like functional V(k, l): N M
r
2 h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` E x¯ (k, l) (k, l)
l=0 k=0 ´i,´j,m, ´ `i,`j,m=1 `
N −1 $ % ≤ E V1h (0, l + 1) + V2h (0, l + 1) λmax
l=0 + V3h (0, l + 1) (k, l) +
M $ E V1v (k + 1, 0) + V2v (k + 1, 0) k=0
% + V3v (k + 1, 0) (k, l) .
(31)
Taking the mathematical expectation on both sides of (31) and further considering the finite initial boundary condition (2), we conclude that N M
r
2 h´ ´i h´ ´j h´ m´ h` `i h` `j h` m` E x¯ (k, l) < ∞ (32)
l=0 k=0 ´i,´j,m, ´ `i,`j,m=1 `
which implies that
lim E{x(k, l)2 } k+l→∞
= 0. According to
Definition 1, it is concluded that the closed-loop 2-D fuzzy system (10) is globally asymptotically stable in the meansquare sense. The proof is complete. Remark 2: As is well known, delay-independent condition will inevitably bring conservativeness, especially when the delay is very large. Here, we choose a delaydependent Lyapunov functional associated with both the lower and the upper bound of the time-varying delay τi (k, l) (i = 1, 2), which will undoubtedly lead to less conservativeness. Moreover, the matrix inequality constraint (15) derived by introducing the free-weighting matrix method is actually a delay-range-dependent stability criterion, which has proven to be an effective approach in stability analysis of time-delay systems (see [35]). B. H∞ Performance Analysis for the 2-D Fuzzy System With Missing Measurements In Theorem 1, a general condition has been derived to guarantee the globally asymptotic stability of the closed-loop 2-D fuzzy system (10). Based upon the above stability analysis, we now deal with the capacity of disturbance attenuation and rejection as well as the tolerance level to the missing measurements with the designed 2-D controller (8). The sufficient condition obtained in the following theorem is to make sure that the fuzzy system (10) is asymptotically stable with a prescribed H∞ performance index for nonzero exogenous
disturbances and possible missing measurements under the zero-initial condition. Theorem 2: The closed-loop 2-D fuzzy system (10) with the designed controller structure (8) is globally asymptotically stable with disturbance attenuation level γ if there exist matrices Qh > 0, Qv > 0, Ph > 0, Pv > 0, Rh > 0, Rv > 0, Jkl , Lkl (k, l = 1, 2), and Kj ( j ∈ S) such that the matrix inequality (14) and the matrix inequalities (33) hold ¯ ∗ 0, Q v > 0, P h > 0, P v > 0, R h > 0, R v > 0, Qh > 0, Qv > 0, Ph > 0, Pv > 0, Rh > 0, Rv > 0, Jkl , Lkl (k, l = 1, 2), and Kj ( j ∈ S) such that the linear matrix inequality (14) and the inequalities/equalities (38)–(40) hold
¯ 1
in (37) as follows: Proof: Rewrite ¯ + T2 R2 −1 I −1 2 + T3 R2 −1 I −1 3 = + α(1 ¯ 6 ⊗ 4 )T diag Qh , Qv , Ph , Pv , τ1 Rh , τ2 Rv ¯ 6 ⊗ 5 )T × (16 ⊗ 4 ) + α(1 × diag Qh , Qv , Ph , Pv , τ1 Rh , τ2 Rv (16 ⊗ 5 ) + T67 diag Ph , Pv 67 + T67 I diag Rh , Rv 67 + (12 ⊗ 8 )T × diag Qh , Qv (12 ⊗ 8 ) + T910 910 < 0.
5
Q8 + T9 9 + T10 10 . (37) + T67 R1 67 + T8
Q h Qh = I,
P h Ph = I,
R h Rh = I
(38)
Q Q = I, ∗
