Sliding mode control of interval type-2 T-S fuzzy systems with redundant channels

This paper investigates the sliding mode control (SMC) of interval type-2 (IT2) T-S fuzzy systems. The measurement outputs are propagated via redundant channels for reducing the probability of packet loss and improving the reliability of data transmission. A key feature for the above problem is that the premise variables and the measurement signals may not be available by the controller, which brings difficulty to stabilize the nonlinear systems. Accordingly, a crucial issue is how to synthesize an implementable SMC law under the redundant channels. To this end, the characteristic of the redundant channels is firstly analyzed and the model of available measurement output signals is established. By employing these available measurements as the premise variables and utilizing the upper and lower bounds of the system membership functions (MFs), new MFs are constructed and the sliding mode controller is synthesized. By introducing some null terms carrying the information of MFs, sufficient conditions are derived in terms of nonlinear matrix inequalities to ensure the stochastically ultimate boundedness of the closed-loop system and the reachability of the specified sliding surface. Besides, a binary genetic algorithm (GA) is introduced to solve the nonlinear criteria via the objective function reflecting the control performance. Finally, a numerical example illustrates the effectiveness of the proposed methods.


Introduction
As one of the frequently confronted network-induced phenomena, packet loss has gained considerable attention in the past decades. Lots of efforts have been devoted to compensating its negative influences such as system performance degradation or instability [1][2][3]. For example, a redundant channel transmission (RCT) strategy was proposed in [4] to bring down the probability of packet dropout and enhance the system performance. Compared with the data transmitted via a single channel, the RCT allows two or more accesses available for data propagation. If one channel undergoes the failure data exchange, the redundant channels may function to deliver the information, thereby improving the reliability of the signal communication. Based on the RCT, some initial works have been published on the estimate/filtering [5][6][7] and control [8,9] problems under packet loss. In [6], the authors investigated the distributed filtering of the Markovian jump systems.
Two independent paths were considered in the communication network toward performance benefit. Taking the advantage of the sliding mode control (SMC) approach (e.g., fast response, insensitivity to matched disturbances and uncertainties as discussed in [10][11][12][13][14][15]), Song, et al. [8] investigated the control design of the fast sampling singularity perturbed systems. A sliding function was constructed by taking the special structure of RCT into account, and the SMC law was designed relying on the measurement outputs of N channels.
Takagi-Sugeno (T-S) fuzzy system is powerful in representing nonlinear complex systems [16][17][18][19][20]. Its main idea is to express nonlinear plants by "blending" several local linear subsystems via membership functions, which measure the contributions of each subsystem. Based on this fuzzy modeling approach, some preliminary efforts have been devoted to nonlinear systems with redundant channels [21,22]. For example, the authors in [21] focused on the control design of fuzzy systems subject to packet loss, in which the controller was designed via the parallel distributed compensation (PDC) strategy, endowing the controller with the same premise variables as the ones in the plant. However, it is worth noting that the premise variables of the fuzzy system under consideration may not be available for this controller due to the packet dropout. This means that the controller based on the frequently utilized PDC strategy may be physically infeasible.
On the other hand, one should note that the above T-S fuzzy systems, also termed as type-1 fuzzy systems, do well in handling nonlinearities but are not competent for directly dealing with uncertainties. This urges the study of interval type-2 (IT2) T-S fuzzy systems [23][24][25][26]. In the IT2 T-S fuzzy model, uncertain parameters are absorbed via the upper and lower membership functions (MFs); thus, the MFs for the system model become uncertain. This special structure brings challenges for control design of IT2 T-S fuzzy systems, since the uncertain MFs for the fuzzy system cannot be directly applied. In view of this, MFs distinct from the ones of the fuzzy system are usually required for an implementable controller. In [23], for the first time, the IT2 T-S fuzzy model under the model-based framework was proposed and the MFs of the control law were designed through the average normalized membership of upper and lower MFs. By introducing predefined nonlinear coefficients, the authors in [27] established new MFs to develop fuzzy controllers. In general, the imperfect premise concept [28] and membership-function-dependent analysis [29] play an important role to investigate the stability analysis and control synthesis problems in IT2 fuzzy-modelbased control systems. When signals are transmitted via an unreliable network, the control problem for IT2 T-S fuzzy systems under packet loss has also attracted some research interest [30,31]. However, the predictive controller in [31] ignored the effects of packet dropout on premise variables. Besides, the above results on IT2 T-S fuzzy systems dealt with packet dropout in a passive way, that is, only when packet dropout happened, some compensating measures would be taken. Apparently, a more positive alternative is using the RCT to reduce the probability of data loss as in [8] on singularly perturbed systems and [21,22] on T-S fuzzy systems. Thus, a potential topic is to design a reliable controller for IT2 T-S fuzzy systems via redundant communication channels.
Based on the above discussions, this paper aims to study the SMC problem of IT2 T-S fuzzy systems via redundant channels. Due to the characteristic of IT2 T-S fuzzy systems as pointed out in the above analysis, the existing results as in [8,21,22] cannot be directly applied. To this end, the model of available measurement output signals is established for the redundant channels. By introducing two predefined nonlinear functions, an implementable SMC law is synthesized under the constraint that only the bounds of system MFs are known. With the aid of some null terms, sufficient criteria are obtained for the stochastically ultimate boundedness of the resultant closed-loop IT2 T-S fuzzy system and the reachability of the prescribed sliding surface. By constructing a minimization problem reflecting the SMC performance, a binary genetic algorithm (GA) is utilized to handle the control design under nonlinear matrix inequalities. Finally, an example verifies the proposed SMC design method.
Notations: In this paper, λ min (Ω) (λ max (Ω)) denotes the smallest (largest) eigenvalue of Ω; R refers to the real number set. ( * ) means that the matrix can be deduced via symmetry for symmetric matrices. . is the Euclidean norm. E{a | b} stands for the conditional expectation of stochastic variable a on stochastic variable b, and E{a} represents the expectation of a. The convex hull of points x, y ∈ R n is co{x, y} = {ϑ 1 x + ϑ 2 y|ϑ 1 + ϑ 2 = 1}. In addition, if not specifically defined a matrix, it is supposed to have compatible dimension.

IT2 T-S fuzzy systems
Consider an s-rule IT2 T-S fuzzy system as follows: . . .
Apparently, due to the uncertain coefficients, the MFs w i (ξ(k)) in (4) are unknown and cannot be used by the controller. This is just the feature and difficulty in controlling IT2 T-S fuzzy systems.

Redundant channels
As shown in Fig. 1, the communication paths are composed of N independent links, where the first one is the primary channel and the others are redundant channels. The RCT structure is effective for overcoming the effect of the packet dropout and improving the transmission reliability of the measurement signals (this will be further discussed in Remark 1 and illustrated by the simulation later). In this case, the actual measurement output z(k) received at the controller side is modeled as: where θ i (k) (i = 1, . . . , N ) are mutually independent Bernoulli processes. For θ i (k) = 0, it means that the packet is loss, while θ i (k) = 1 implies a successful data transmission. The probability distribution of θ i (k) is expressed as: whereθ i is a known constant taking value in [0, 1]. Denoting Remark 1 It can be seen from (5) that if a packet is successfully transmitted via the primary channel, i.e., i.e., the packet loss happens in the first channel, the measurement will be obtained from the other channel j satisfying θ ı (k) = 0 (ı = 1, 2, . . . , j − 1) and θ j (k) = 1, that is, z(k) = y j (k). Therefore, the RCT can effectively reduce the probability of packet loss from 1 −θ 1 to N q=1 (1 −θ q ), and improve the network reliability, which will be further illustrated in the simulation. Moreover, according to model (5) or (8), if the transmissions via all channels fail, i.e., θ ı (k) = 0 for ı = 1, 2, . . . , N , the measurement at instant k − 1 will be exploited to compensate for the effect of packet dropout. (1)
This work aims to synthesize a feasible SMC law for achieving the SUB of the resultant closed-loop IT2 T-S fuzzy systems under redundant channels.

SMC law design
Now, considering the structure of redundant channels (5) or (8), we design the following sliding function where G is the sliding gain to be designed later. Denotinḡ and using (2)-(3), the sliding function s(k) is further expressed as: In what follows, a desired SMC law will be synthesized. As discussed in Introduction section, new MFs are necessary for the controller design due to the following facts: (i) the MFs for the fuzzy model w i (ξ(k)) (i = 1, 2, . . . , s) are uncertain; (ii) the premise variable ξ(k) may be unavailable at the controller side when it is transmitted via unreliable links from the sensor to the controller. For the former, a feasible way is to use the bounds of the system MFs w i (ξ(k)) (i = 1, 2, . . . , s). For the latter, which is usually neglected in some existing works, one solution is to replace the premise variable ξ(k) by available information. Since only z(k), instead of y i (k), i = 1, . . . , N , is available at the controller side under the redundant channels, we construct the following MFs with z(k) as the premise variable for the controller: where ν j (z(k)) andν j (z(k)) are the predefined nonlinear functions satisfying ν j (z(k)) +ν j (z(k)) = 1 and ν j (z(k)),ν j (z(k)) ∈ [0, 1]. Based on the above MFs h j (z(k)) in (13), we synthesize the SMC law as follows: whereŝ(k) is the sliding variable s(k) in (12) with Ω 1 (k) and Ω 2 (k) replacedΩ 1 andΩ 2 , respectively. Substituting (8) and (14) into (1) yields the following closed-loop IT2 T-S fuzzy systems:

Stability and reachability analysis
In this part, a lemma is firstly provided to facilitate the analysis on the stability and reachability. By introducing some null terms, sufficient criteria are deduced subsequently for ensuring the SUB of the resultant closedloop IT2 T-S fuzzy system (15) and the reachability of the specified sliding surface s(k) = 0.

Theorem 1 (Stability) Given scalars τ i satisfying
the resultant closed-loop IT2 T-S fuzzy system (15) is SUB with the bound D as: where δ is a positive scalar.
Remark 2 As discussed in Introduction, the MFs of system (1) and the ones for controller (14) are unmatched due to the special structure of IT2 T-S fuzzy systems and the unreliable channels. These different MFs simultaneously exist in the closed-loop systems (15). By using the null terms in (23) and the inequalities h i (z(k)) − τ i w i (ξ(k)) > 0 involving in these MFs, the favorable symmetry property is satisfied as shown in Eq. (33) and the relaxed stability conditions can be derived.
In what follows, the reachability of the specified sliding surface s(k) = 0 will be further analyzed.

satisfying (24)-(25) and the following inequalities
the states of the closed-loop IT2 T-S fuzzy system (15) will be forced into the following domain around the specified sliding surface s(k) = 0 in the mean square sense: Proof Applying Schur complement to inequalities (45)-(46) yields Replacing Γ i by Ξ i in (23) and following the similar lines from (32) to (35), we have by (47)-(49) wherê Set V 2 (k) = V 1 (k) + s T (k)Zs(k) as the Lyapunov function. By means of (44), one has for the second term of V 2 (k) By means of expressions (18)- (20) in Lemma 1 and inequalities (51)-(52), we can derive from (54) that Owing to Z > 0, expression (55) can be further developed as Furthermore, one can obtain from (38)-(40) and (56) that When s(k) > σ with σ defined below (50), we have E{ΔV 2 (k)|x(k), z(k − 1), s(k)} < 0 by (53). Taking expectations on both sides of E{ΔV 2 (k)|x(k), z(k − 1), s(k)} < 0 leads to E{ΔV 2 (k)} < 0. This implies that outside the domain O defined in (50), the Lyapunov function V 2 (k) is monotonically decreasing in the mean-square sense. Thus, it can be concluded that the states can be driven into the sliding domain O in the mean square.
The elements g u 1 v 1 and ℘ u 2 v 2 in ω are encoded as binary strings with lengths l g u 1 v 1 and l ℘ u 2 v 2 , respectively, over the ranges Note that ℘ u 2 v 2 > 0 for u 2 = v 2 due to P −1 > 0. Then, the precisions q g u 1 v 1 and ℘ u 2 v 2 can be yielded via the linearly mapped coding method as Step 2. Population initialization: Produce an initial population withN chromosomes ω ι (ι = 1, 2, . . . ,N ) stochastically. Denote p m and p c as the single-bit mutation probability and the single-point crossover probability, respectively.
Step 5. Stop criterion: Repeat Steps 3 and 4 in each generation until reaching the maximum generation G max . The real values of the sliding gain G and the Lyapunov variable P are decoded via the best chromosome ω b .
Step 6. SMC law design: The SMC law (14) can be constructed via the sliding gain G and the control gain F i obtained in Step 5.
Remark 3 It can be seen from (24)-(28), (45)-(49) and (50) that the value of σ may be indirectly affected by the choices of P and G via the decision variables Z , λ i and γ i . By solving the above optimization problem (58), a smaller sliding domain O defined in (50) may be obtained and a better SMC performance can be expected.
As a comparison, we further consider the case that only one communication channel is utilized. In this case, the actual measurement will be z(k) = θ 1 (k)y 1 (k) + (1 − θ 1 (k))z(k − 1). The corresponding results are shown in Figs. 5 and 6, from which it is seen that the control performances may become worse using only one channel.

Conclusion
This work has addressed the SMC of IT2 T-S fuzzy systems under redundant channels. The main challenge has been the premise variables and the measurement outputs unavailable for the controller. An implementable SMC scheme has been designed by constructing the premise variables and the MFs of the controller. The corresponding conditions on the stability and reachability analysis have been derived. Moreover, an optimizing algorithm via GA has been provided for solving the proposed control design problem subject to nonconvex constraints. Data availability All data generated or analyzed during this study are included within the article.

Declarations
Conflict of interest All the authors in this manuscript declare that they have no conflict of interest.