Fuzzy [Formula: see text] output-feedback control for the discrete-time system with channel fadings, sector nonlinearities, and randomly occurring interval delays and nonlinearities.

In this paper, the fuzzy [Formula: see text] output-feedback control problem is investigated for a class of discrete-time T-S fuzzy systems with channel fadings, sector nonlinearities, randomly occurring interval delays (ROIDs) and randomly occurring nonlinearities (RONs). A series of variables of the randomly occurring phenomena obeying the Bernoulli distribution is used to govern ROIDs and RONs. Meanwhile, the measurement outputs are subject to the sector nonlinearities (i.e. the sensor saturations) and we assume the system output is [Formula: see text], [Formula: see text]. The Lth-order Rice model is utilized to describe the phenomenon of channel fadings by setting different values of the channel coefficients. The aim of this work is to deal with the problem of designing a full-order dynamic fuzzy [Formula: see text] output-feedback controller such that the fuzzy closed-loop system is exponentially mean-square stable and the [Formula: see text] performance constraint is satisfied, by means of a combination of Lyapunov stability theory and stochastic analysis along with LMI methods. The proposed fuzzy controller parameters are derived by solving a convex optimization problem via the semidefinite programming technique. Finally, a numerical simulation is given to illustrate the feasibility and effectiveness of the proposed design technique.

Introduction

It is well known that the complexity and nonlinearity of the models are considered as ubiquitous in practical systems. The emergence of this fuzzy modeling approach is based on the Takagi-Sugeno (T-S) fuzzy system (see [1]), which provides a powerful tool for modeling complex nonlinear systems. The output-feedback control problem for the T-S fuzzy system has received considerable attention (see [2-4]). Nevertheless, the nonlinearity of the T-S fuzzy subsystem is inevitable in practical applications, along with the fact, that the T-S fuzzy model has been successfully used in complex nonlinear systems (see [5-8]). The authors of [8] assumed the nonlinear function in the T-S fuzzy cellular neural networks satisfied the 1-Lipschitz condition and researched the global exponential stability problem for T-S fuzzy cellular neural networks.

In the past decade, networked control systems (NCSs) have played an important role in many engineering applications such as remote militarization, remote medical service, and so on (see [9-11]). However, there are some unavoidable phenomena for the NCSs which may cause poor performance of the controlled systems, for instance, the signal is often transmitted through networks which might be subjected to the occurrence of the phenomenon of incomplete information. The considered incomplete information mainly includes the ROIDs (see [12]), RONs (see [13]), channel fadings (see [14, 15]), and sector nonlinearities (see [16, 17]). The nonlinear output-feedback controller design for polynomial system has been studied (see [18, 19]). A full-order dynamic output-feedback controller was designed by [20] for the time-varying delays case when all state variables are not available for the feedback. Further, the author of [21] has researched the output-feedback controller design problem for networked systems with random communication delays by using a linear matrix inequality (LMI) approach. The output-feedback control problem for a class of discrete-time systems with RONs has been investigated in [22], where random variables are adopted to characterize the RONs and satisfy the binary distribution. The designing output-feedback controller problems for the T-S fuzzy system with randomly occurring phenomena have been studied in [23, 24]. However, in the case when ROIDs and RONs appear simultaneously in a controlled system, the designing of the fuzzy output-feedback control problem has received little attention by researchers.

In practical applications, the phenomena of the channel fadings and sector nonlinearities based on unreliable communication networks could occur, which should not be ignored. Considering the situation of signal transmission in fading channels, the output-feedback control problem with channel fadings has been studied (see [25]). The channel fading has been modeled as a time-varying stochastic model which can describe the transmitted signal’s change in both the amplitude and the phase. The channel fadings with exogenous input disturbance in wireless mobile communications has not been researched extensively (see [12, 22]). On the other hand, the sector nonlinearities of the sensors are usually in order in practical industrial control systems, and this is the main factor that gives rise to the nonlinearity of control systems (see [22]). Since the sensor nonlinearity cannot be neglected and often leads to bad performance of the discrete-time control system, it has attracted the attention of researchers (see [26-28]).

In [22], the output-feedback control problem for a class of discrete-time systems with channel fadings and sector nonlinearities has been studied, and the existence of the desired controllers has been derived via using a combination of the stochastic analysis and Lyapunov function approach. The design of fuzzy controller problem for the fuzzy system with the probabilistic infinite-distributed delay and the channel fadings also has been investigated in [23], where the channel fading model can better reflect the reality of measurement transmission especially through a wireless sensor network. So far, to the best of the authors’ knowledge, the fuzzy output-feedback control problem for a class of discrete-time T-S fuzzy system with channel fadings, sector nonlinearities, ROIDs and RONs have not been investigated yet, and the main purpose of this paper is to bridge such a gap.

The main contributions of this paper are summarized as follows. (1) Both the probabilistic interval time-varying delays and the randomly occurring nonlinearities are used for describing the discrete-time fuzzy model. (2) Moreover, a newly fuzzy control system as well as measurement model is put forward which can account for the randomly occurring incomplete information phenomena of the sensor saturation and the channel fadings. (3) A new fuzzy output-feedback controller has been designed.

Motivated by the above discussion, this paper intends to study the fuzzy output-feedback control problem for a class of discrete-time Takagi-Sugeno (T-S) fuzzy model system with channel fadings, sector nonlinearities, ROIDs, and RONs. The rest of this paper is organized as follows. In the next section, the problem descriptions of the discrete-time T-S fuzzy system with ROIDs, RONs, sector nonlinearities, and channel fadings are stated, and the necessary definitions and relevant lemmas are recalled. Section 3 presents the main results of this paper. Illustrative examples are provided in Section 4. Finally, conclusions are drawn in Section 5.

Model description and preliminaries

In this paper, we consider the following discrete-time fuzzy system with RONs and ROIDs is described by the following fuzzy IF-THEN rules:

Plant Rule i: IF is and is  , and is , THEN where , the system (1) is equivalent to a fuzzy combination of r subsystems. is the fuzzy set, is the premise variable vector. is the state vector with , is the control input vector, is the controlled output vector, is the exogenous disturbance input. is the initial state. , and are known real matrices with appropriate dimensions.

To characterize the phenomena of randomly occurring interval delays and randomly occurring nonlinearities, we employ the stochastic variables () and in (1), which are mutually independent Bernoulli-distributed white sequences with the following probability distribution: The variable () means the time-varying delays satisfying where and are real positive integers representing the lower bounds and the upper bounds on the communication delay, respectively.

Assumption 1

The nonlinear vector-valued function with is seen as continuous and satisfies the following sector-bounded condition: for all , where is a known positive scalar and G is a known real matrix.

Remark 1

In model (1), the stochastic variables is used for characterizing the phenomena of RONs. The T-S fuzzy model with RONs includes the fuzzy model with nonlinearity in [6, 8] as a special case where the values of are 1. Note that RONs is considered for the first time for fuzzy output-feedback control problem. On the other hand, the occurrence of the ROIDs is characterized by the random variables in a probabilistic way, which is more suitable for reflecting the network-induced phenomena. Meanwhile, it is worth of note that there are some results concerned with the continuous time-varying delays in [29, 30] and few results for randomly occurring interval delays, especially when the fuzzy output-feedback control problem becomes a research focus.

Let us now consider the case when the phenomena of the sector nonlinearities and the channel fadings may occur in signal transmission, where the system output is subject to sector-like bounds and the sensor signal sent to the actuator subject to channel fadings for the control purpose. The signal received by the actuator is modeled in the following: where represents the system output with , is the sector nonlinearity of the sensor, is the signal from the actuator, and is an external disturbance. () is the channel coefficient which is independent and conform Gaussian random variables distributed with mean and variance . In practice, the channel coefficients typically take values over the interval .

Assumption 2

The nonlinear function in (4) represents the sector nonlinearities satisfying the following sector condition: where and () are known real matrices with appropriate dimensions.

Remark 2

In this paper, the channel fadings and sector nonlinearities of the sensors can be described simultaneously by the model (4) in the measurement. In [31], this Rice fadings model can properly describe the phenomena of the channel fadings, time-delay, and date dropout, therefore the fadings model can be employed in this paper for the design of the fuzzy output-feedback controller of the discrete-time fuzzy system.

Remark 3

The sector nonlinearities of the sensors usually occur in practical network control systems and cause poor performance of the controlled system. The analysis and synthesis problems for a series of dynamics systems with sector nonlinearities has been investigated in [17, 32]. Particularly, the nonlinear function lies inside the sector in [17]. Furthermore, the Lipschitz condition can be concluded by the nonlinear description as a special case if or in Assumption 2.

Assumption 3

[22]

For technical convenience, the nonlinear function can be decomposed into a linear and a nonlinear parts as where the nonlinear part belongs to the set defined by with .

In this paper, we adopt a full-order fuzzy output-feedback controller for the discrete-time system by the fuzzy IF-THEN rules as follows:

Control rule i: IF is and is  , and is , THEN where is the state estimate of system (1), and are appropriately dimensioned parameters matrices to be determined.

Set where denotes the grade of membership of in . Obviously, , and . To ease the presentation, we use instead of .

Above all, the T-S fuzzy system (1) model can be constructed as follows:

Furthermore, the fuzzy control system can be described by

Combining (4), (6), (10), and (11), the fuzzy control system can be represented by where with Obviously Particularly, we can see from Assumption 1 that the nonlinear function satisfies the following formula: where .

To describe our main result more precisely, we first introduce the following definition and lemmas.

Definition 1

Exponentially mean-square stability [33]

The T-S fuzzy control system with channel fadings in (12) and every initial conditions ϕ, the zero solution is said to be exponentially mean-square stable if, in the case of , then exist constants and such that

With Definition 1, the aim of this paper is to design a robust output-feedback controller in the form of (11) such that the fuzzy discrete-time system (12) is exponentially mean-square stable and the performance is satisfied or, more specifically, the following two requirements are satisfied simultaneously:

(R1) The fuzzy discrete-time system (12) is exponentially mean-square stable.

(R2) Under the zero-initial condition, the controlled output satisfies for all nonzero , where is a prescribed scalar.

Lemma 1

Schur complement [34]

We have the linear matrix inequality where and are equivalent to

Lemma 2

[35]

For a symmetric positive definite S, and any real matrices with appropriate dimensions, we can get

Lemma 3

[36]

Given any matrices , a matrix , and a positive scalar ϵ, then we have

Main result

In this part, the following theorem provides a sufficient condition for the discrete-time T-S fuzzy system (12) to be exponentially mean-square stable and the controlled output to satisfy the disturbance reject requirement in (15).

Theorem 1

Let a scalar and the controller parameters matrix , and () be given. The fuzzy closed-loop system (12) is exponentially men-square stable and the controlled output satisfies (15), if there exist matrices , and (; ), and a positive scalar and satisfying where ()

Proof

We choose the following Lyapunov function: where The difference of along the trajectory of the system (12) is We have Considering Lemma 2 and taking the elementary inequality into consideration, we obtain Also, it can be seen that and For notational convenience, we have In the first place, we will prove the exponential stability of the system (12) with , considering (5), (13), Lemma 2, and Lemma 3, we can get where By utilizing Lemma 1, we know that (16) and (17) implies is true. Moreover, we can draw the conclusion that the nominal control system (10) with is exponentially mean-square stable as can be seen in the same way as in [33].

Now let us dispose of the performance for the system (12). For this purpose, we establish a cost function There is no doubt that we can show under the zero-initial condition, which is our purpose.

Along the trajectory of the fuzzy discrete-time system (12) and taking (25) into consideration, we have where

By using the Schur complement lemma, the conclusion can be drawn from (16) and (17) that . Letting , it follows from the above inequality that which completes the proof of Theorem 1. □

Through the above-mentioned analysis results for the control problem, we will deal with the problem of designing the desired fuzzy output-feedback controller in the following theorem.

Theorem 2

Let the disturbance attenuation level be given. A desired controller of the form (11) exists if there exist matrices , , (), (), matrices , , (), and a positive scalar and satisfying where () the controller parameters in the form of (8) are given in the following: where the matrix derives from the factorization , and then the fuzzy discrete-time closed-loop system (12) is exponentially mean-square stable and the controlled output satisfies (15).

For the purpose of design desired controller parameters , and from Theorem 1, we partition P and as where the partitioning of P and are appropriately dimensioned to be determined by , and in (12).

Define and then we have and . Now we define the controller parameters from (29) as follows: By applying the congruence transformation to (16) and (17), we can have On the other hand, it follows from that Furthermore, again applying the congruence transformation to (30), we have where . By combination (31) and (32), if (27) and (28) are satisfied, the inequality (33) holds. Therefore the sufficient condition (16) and (17) of Theorem 1 is effective.

Next, let us calculate the desired controller parameters. We can obtain from By , and if (27) and (28) are feasible, we can infer . So is nonsingular. Hence one can always find square and nonsingular and satisfying (34) [37]. In this case, we can obtain , and via solving (30). Now it can be concluded from Theorem 2 that the fuzzy closed-loop system (12) is exponentially mean-square stable and the controlled output satisfies (15) with the controller parameters given by (29). □

Remark 4

The model considered in this paper is more general than some existing ones [22, 23, 25]. For example, when the model does not take into consideration the randomly occurring interval delay and randomly occurring nonlinearities, it reduces to the model in [25]. The results derived in this paper also contain the two theorems in [25] as special cases. Moreover, randomly occurring nonlinearities have not been considered in [22], and sector nonlinearities have not been studied in [23].

Remark 5

The design of controller directly affects the stability and performance of the discrete-time closed-loop system. Compared with [7, 20, 21], it should be pointed out that the fuzzy controller designing arithmetic in Theorem 2 has more generality than the usual controller, that is to say, the controller designing arithmetic in [20, 21] cannot be available for the design of fuzzy output-feedback controller for a class of discrete-time T-S fuzzy systems with channel fadings, sector nonlinearities, ROIDs, and RONs. To design the controller and complete the proof of Theorem 2, P and are in the form of (∗), which can be found in [7, 25]. The conditions as regards P in [7] are more conservative than ours because one not only needs , but also and . Therefore, Theorem 2 has less conservatism.

Remark 6

In Theorem 2, the sufficient conditions involved in the randomly occurring nonlinearities, the probabilistic interval delays, sector nonlinearities, and channel fadings were first established for the desired fuzzy output-feedback controller. The fuzzy output-feedback controller is designed such that the discrete-time system (12) is exponentially mean-square stable and, under the zero-initial condition, the proposed performance index can be satisfied. Particularly, with the designed of fuzzy controllers, the robustness of our developed controller operation algorithms of the discrete-time fuzzy system includes the traditional controller algorithms. In other words, the traditional controller algorithms means that we have the membership function (, ) in the discrete-time fuzzy system. Obviously, the developed controller algorithms work better than the traditional algorithms in dealing with the occurrence probability of randomly occurring nonlinearities, interval delays, sector nonlinearities, and channel fadings, which appropriately avoid the deterioration of the performance.

Numerical example

In this section, we present illustrative examples to show the effectiveness of the proposed controller design approach.

Consider the following discrete-time T-S fuzzy model from (10): Consider the model parameters as follows: with the initial value (). The nonlinear vector-valued function is as follows: Besides, it can easily be noticed that Assumption 1 satisfies and the sensor nonlinearity is given as where

In the meantime, the output measurement is described as follows: Here, the order of channel fading is , the mathematical expectations of the channel coefficients are and , and the variances of the channel coefficients are , and . Assume that for the time-varying delays, and are, respectively, uniformly distributed in the intervals and , and the stochastic variables . Other stochastic variables are .

To further illustrate the effectiveness of the designed fuzzy controller, the exogenous disturbance inputs are assumed to be The membership functions are shown in Figure 1. The formulated T-S fuzzy model is an approximation of the original nonlinear model has been verified in [38]. In Section 2, we saw that the premise viable space can be divided into two regions from the partition method, as shown in Figure 1.

Figure 1

Membership function.

Applying Theorem 2 and the LMI toolbox, we can obtain the desired controller parameter matrices in the form of (11) such that the fuzzy system (12) is exponentially mean-square stable with the norm bound as follows:

The simulation results are shown in Figures 2-4 where the states of the system and the fuzzy controller are shown in Figure 2. We can conclude that although the discrete-time fuzzy system and the full-order output-feedback controller are subject to RONs, ROIDs, and channel fadings as well as sector nonlinearities, respectively, the fuzzy controller can estimate the state well. Moreover, we can conclude that the designed fuzzy filter ensures the exponentially mean-square stable of the filtering error and obtains disturbances rejection level γ. Figure 3 shows the results of the uncontrolled fuzzy system, which are clearly unstable. Figure 4 shows the consequence of the closed-loop fuzzy system, which is indeed exponentially mean-square stable. All the simulation results have confirmed that the designed fuzzy output-feedback control performs very well.

Figure 2

The states of the system and the fuzzy controller.

Figure 4

State evolution of controlled fuzzy system.

Figure 3

State evolution of uncontrolled fuzzy system.

Conclusions

In this paper, a fuzzy output-feedback controller has been designed for a class of fuzzy discrete-time systems with sector nonlinearities, channel fadings, randomly occurring interval delays as well as randomly occurring nonlinearities. A sufficient condition for the robust exponential stability of the fuzzy discrete-time system has been obtained by a Lyapunov stability analysis approach and stochastic analysis theory. Moreover, by using the LMI technique, a clear expression of the desired fuzzy output-feedback controller can be obtained and the proposed -norm bound constraint has been guaranteed. At last, the developed fuzzy controller design approach has been checked by a numerical simulation example. Further research topics might include the development of our results to more complex and more varied cases with sector nonlinearities and channel fadings by using a stochastic analysis approach, such as multi-agent systems based on the T-S fuzzy model, descriptor systems, and affine fuzzy systems.