Model-Dependent Energy-to-Peak Control for Switched Fuzzy Singular Systems With Dynamic Quantization

This paper is concerned with the energy-to-peak (E2P) feedback control for discrete-time switched Takagi-Sugeno fuzzy (STSF) singular systems with input quantized. The main challenge of this work comes from the difficulty of solving the semi-positive problem on the main diagonal in the E2P feedback control design. In order to further study the E2P control problem in the presence of dynamic quantization strategy and fuzzy switching mechanism, a new method is proposed to ensure the E2P performance index and stochastic stability of closed-loop singular (CLS) systems. Then, the design conditions of model-dependent proportional plus derivative (MDPPD) state feedback controllers and mode-dependent (MD) dynamic quantizers for STSF singular systems are obtained in the form of linear matrix inequalities. Finally, a simulation example is given to illustrate the effectiveness of the proposed control design method.


I. INTRODUCTION
Practical systems exhibiting nonlinear properties have attracted significant attention from researchers over the past few decades due to their wide applicability [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. Fuzzy systems are derived from fuzzy mathematics. As an effective method to deal with nonlinear systems, it has been developed rapidly and improved significantly. A Takagi-Sugeno (T-S) fuzzy model that uses a local linear model to approximate the treatment of nonlinear systems was proposed [11], which states that the T-S fuzzy controller can adapt to different system states in different linear subsystems, resulting in more accurate control [12], [13], [14], [15], [16]. In addition to nonlinearity, dynamical systems with switching characteristics, called STSF systems, are also receiving increasing attention [17], [18]. Reference [17] investigated the filtering problem of STSF systems and also gave design conditions for static quantizers. [18] discussed dissipative control of switched T-S fuzzy systems while considering the effect of actuator failure on system performance. To our knowledge, The associate editor coordinating the review of this manuscript and approving it for publication was Qi Zhou. the fuzzy switching mechanism mentioned in [17] and [18] was not applied to singular systems, which motivates the present work.
Singular systems are composed of differential and algebraic equations. Compared with traditional state space systems, singular systems have been widely studied by scholars because of their special structure, which can more accurately describe the dynamic and static properties of practical systems [19], [20], [21]. In [19], new admissibility analysis conditions for nonlinear discrete-time singular systems were given based on auxiliary matrices and the piecewise Lyapunov function. In [20], the free-weighting matrix method was used to study the real-time reachable set control of neutral singular Markov jump systems, and the effects of mixed delay and bounded perturbation were considered. The admissibility condition of fuzzy singular systems was given by means of generalized singular value decomposition based on a new model of actuator-saturated singular systems in [21]. As mentioned above, singular systems and STSF systems have received extensive attention. However, the study of the two systems discussed above has been unrelated to each other. How to combine their characteristics for an in-depth study is of great importance.
Networked control systems are receiving increasing attention from scholars with the widespread use of digital technology. As network bandwidth is limited, some strategies to save network transmission resources are gradually developed, such as event triggering mechanisms [22], [23], [24], communication protocol [25], [26], [27], quantization strategies [28], [29], [30], [31], etc. Among them, quantization strategies save network transmission resources by reducing the number of data bits [32]. However, signal quantization is lossy, resulting in a loss of information and introducing quantization errors, which can lead to adverse effects on control design. Moreover, some instabilities in the system itself and the interference of noisy signals would have a negative influence on the performance and stability of the system. Therefore, it is of great practical value and theoretical significance to study the stability of the system under quantized feedback control. In recent years, many design results on quantized nonlinear systems have been obtained based on the STSF model. Nevertheless, the problem of the E2P quantization control for nonlinear singular systems via the STSF model has been studied relatively scarce.
In practical applications, the controlled system that is only stable but does not meet expected performance requirements cannot perform excellently. Therefore, in addition to considering the stability of the system, the performance level of the system also needs to be analysed in relation to the actual application requirements. Thus, the H ∞ performance, the peak-to-peak performance, and the E2P performance have been widely used to study control and filtering problems for systems [33], [34], [35], [36], [37], [38], [39], [40], [41], [42]. In [41], the strictly passive H ∞ control for linear switched singular systems was studied. By using the proportional plus derivative feedback controller, the singular system was transformed into a normal system to ensure the regularity and causality of the singular system. The item E −1 σ will lead to unapplicable to STSF singular systems. In [42], the semi-positive definite problem of the main diagonal for singular linear systems was solved, and the item C T C on the main diagonal posed a new challenge to discrete-time STSF singular systems. It should be noted that the methods mentioned in [41] and [42] cannot be applied to discrete-time STSF singular systems, which is another motivation for this work. The main contributions of this paper can be expressed as follows: 1) This paper investigates the problem of E2P control for discrete-time STSF singular systems with input quantization; 2) The standard design conditions of MDPPD state feedback controllers and MD dynamic quantizers are given; 3) The semi-positive definite problem on the main diagonal of linear matrix inequality introduced by the Lyapunov matrix is solved in the E2P control design.
Notations: The symbols utilized in this paper are conventional. He{M} = M + M T . ( * ) denotes the ellipsis of the symmetric structure of the matrix. diag{ · · · } denotes a diagonal matrix. The square-integrable space is denoted as ℓ 2 [0, ∞). The identity matrix I and the zero matrix 0 are conformable with their respective rows and columns.

II. PROBLEM FORMULATIONS A. NONLINEAR PLANT
Consider the following discrete-time STSF singular system: where ℘(k) = [ ℘ 1 (k), ℘ 2 (k), · · · , ℘ q (k) ] and ℘ o (k) are premise variables and measurable, W om are fuzzy sets, o = 1, 2, · · · q, m = 1, 2, . . . , l, l is the number of fuzzy rules. The state variable is x(k) ∈ R n x , the quantized control input isū p (k) ∈ R n u , the controlled output is z(k) ∈ R n z , and w(k) ∈ R n w is the noise signal belonging to ℓ 2 [0, ∞). The dynamic quantizers ϑ p (u p (k)) will be described in detail in the following part. The switching signals ι k are used to describe switching phenomena. The matrices E, A ι k ,m , B ι k ,m , F ι k ,m , and C ι k ,m are known system matrices with appropriate dimensions, and rank(E) = r < n x . Meanwhile, for p ∈ 1, 2, . . . , P and β 1 + β 2 + · · · + β P = 1, the variables ι k are state independent and satisfy where the pth subsystem is activated with probability β p , p ∈ 1, 2, . . . , P. The system switching mechanism β p and correspondingly more details in [17].
The fuzzy basis functions are given by Utilizing aforementioned notations, the system (1) is inferred as follows where Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.

B. MDPPD STATE FEEDBACK CONTROLLERS
As T-S fuzzy systems are composed of multiple subsystems, parallel distribution compensators are often used to control T-S fuzzy systems. It adopts the same membership function as the system. However, the control inputs are quantized in this paper, in which case the parallel distributed compensation solution will not be feasible. That is, prerequisite variables are quantified before transmission over the network, and prerequisite variables of the original system are not available to the controller. Then, considering the semi-positive definite problem of E2P control design caused by the derivative matrix of singular systems, the following form of MDPPD state feedback controllers is introduced where K d is a known matrix, and K ps are controller gains with appropriate dimensions to be designed.

C. MD DYNAMIC QUANTIZERS
Due to the multi-model characteristic of switched systems, u p (k) of each subsystem are obtained by using MD dynamic quantizers.
The quantizers ϑ p (u p (k)) of p subsystem are dynamic quantizers of the single-parameter family form as follows [32]: where ϑ p (u p (k)/ξ p (k)) are static quantizers, ξ p (k) > 0 are dynamic parameters, and the quantization errors satisfy where M p and p denote the quantization ranges and the quantization error bounds, respectively.

D. DISCRETE-TIME QUANTIZED CLOSED-LOOP STSF SYSTEM
To obtain the relation of quantization error, we reconstruct the system (2) as Then, we can rewrite (7) as follows from the definition of u p (k) in (4) where Combining (3), (4) with (8), it leads to the CLS system  (1) can be considered regular and causal for the CLS system (9). Definition 2: [17] Under the Definition 1, the CLS system (9) is stochastically stable with a predefined E2P performance γ > 0, if the following two requirements are satisfied: 1) (Stochastically stable) The CLS system (9) with w(k) ≡ 0 is called stochastically stable, such that 2) (E2P performance) Under the zero initial condition, for Lemma 1: [44] For any vectors ℵ ∈ R n ,h ∈ R n and any For matrices , ϒ 1 , ϒ 2 , and N with appropriate dimensions and scalar α. The inequality is fulfilled if the following condition holds:

III. E2P CONTROL
The main goal of this section is to derive newly sufficient conditions for discrete-time STSF singular systems and achieve E2P control. These conditions ensure both stochastic stability and the expected E2P performance for the CLS system (9). The desired sufficient conditions are summarized below. Theorem 1: For given quantization ranges M p , quantization error bounds p , and scalar α, the CLS system (9) is stochastically stable and preserves the specified E2P performance index γ > 0, if there exist matrices P σ > 0, P σ + > 0, W pσ > 0, X , and scalars ς > 0, τ p > 0, such that P p=1 β p W pσ < P σ K T ps K ps < ςI  (18) and the on-line adjusting strategy for ξ p (k) is presented as: The proof is separated into two steps. We first to prove the stochastic stability of the CLS system (9). From (18) and (19), one has Then, it can be naturally established that Together (3) with (21) yields Then, it follows from (22) that . (23) According to Lemma 1 and (23), one has Considering conditions (15) and (24), we can deduce that Then, rewriting (25) as where For the CLS system (9), there exists a matrix X with appropriate dimension which guarantees the following formula: One can rewrite (27) as follows We choose the Lyapunov function as follows: For the CLS system (9), we give the following performance function: Thus, it follows from (14) and (30) that when w(k) = 0, from (16), (26), and (28), it holds that E V (x(k + 1)) − V (x(k)) < 0. Then, by the condition (10) of Definition 2, the CLS system (9) meets stochastic stability. Next, we aim to demonstrate that the CLS system (9) meets the E2P performance. It follows from (17) that It implies that We obtain the following inequalities from (16), (26), (28) and (31): for ı = 0, 1, . . . , k − 1, taking the sum of both sides, one can obtain which is From the zero initial condition we get V (x(0)) = 0, it can easy get that Observe (33) and (37), it will be established: (38) which means that (11) holds for any non-zero w(k) ∈ ℓ 2 [0, ∞) under zero initial conditions. Thus, the CLS system (9) meets the given E2P performance index. The proof is completed. Next, the main objective is to solve the coupling items XB pσ K ps and P σ + B pσ K ps for the controller gains in (16). In addition, we will propose standard design conditions that meet the design requirements for MDPPD state feedback controllers and MD dynamic quantizers. It can be found that the conditions (15) and (16) are not strictly linear and therefore must be linearised in order to be processed by the mathematical software. To this end, the linearization procedures are as follows.
Theorem 2: For given performance index γ > 0, scalar α, and quantization ranges M p , quantization error bounds p and the CLS system (9) is stochastically stable and preserves the specified E2P performance, if there exist matrices P m > 0, W pm > 0, X , V p , N p , and scalars ς > 0,τ p > 0, such that p and the controller gains K ps and quantized parameters τ p are given as Proof: Together (15) with (44) results in Applying the Schur complement and multiplying the two sides of (45) by diag I , N p and diag I , Considering the fact that −N p N T p ≤ −N p − N T p + I , this impiles (41) holds.

Remark 1: In
conditions of the controller were given. The proportional plus derivative feedback controller was used to avoid the semi-positive definite problem and ensure the regularity and causality of singular systems. However,Ē −1 σ (corresponds toĒ −1 pσ of this paper) were introduced in the transformation of singular systems to normal systems, which made it impossible to apply the design method of [41] to FTSF singular systems. More specifically, the STSF singular system proposed in this paper is assumed to consist of two switched subsystems, each of which has two fuzzy rules, In subsection A, we can know that β 1 + β 2 = 1 and σ 1 (℘(k)) + σ 2 (℘(k)) = . Therefore, the method proposed in [41] cannot be applied to STSF singular systems, and the proposed method is limited.
Remark 2: The problem of the E2P control for discrete-time singular systems was studied in [42]. The semi-positive definite problem for singular systems was solved by selecting a novel Lyapunov matrix E T (P + C T C)E (corresponds tō E pσ , P σ , C pσ of this paper), which will limit the applicability of the method proposed in STSF singular systems. Because the introduction of the Lyapunov matrix leads to the item −C T pσ C pσ appearing on the main diagonal, which makes it difficult to solve in STSF singular systems. That is, the item −C T pσ C pσ is not necessarily symmetric when different subsystems are coupled. And Schur complement cannot be applied to −C T pσ C pσ . The method proposed in [42] does not work in STSF singular systems.

IV. SIMULATION
In this section, a numerical example is provided to illustrate the effectiveness and feasibility of MDPPD state feedback controllers designed according to Theorem 2. Here, we consider the STSF singular systems consisting of two subsystems. The specific parameters are expressed as follows: Mode 1: Mode 2: In order to prove the validity of the obtained results, we adopt the following normalized fuzzy membership function: σ 1 (℘(k)) = 1 + cos(x 1 (k)) 2 , σ 2 (℘(k)) = 1 − σ 1 (℘(k)). and the disturbance is supposed as Choose the zero initial condition as x(0) = [ 0 0 ] T , quantization error bounds as 1 = 0.02, 2 = 0.01, quantization ranges as M 1 = M 2 = 40, switching probabilities as Then, the main purpose of this paper is to design MDPPD state feedback controllers (3) and MD dynamic quantizers (4) that satisfy conditions (39), (40), (41), (42), (43), under given fuzzy rules and switching signals. To do this, setting the given E2P performance index γ = 1, the following solutions can be obtained and quantized parameters τ 1 = 0.0350 and τ 2 = 0.0431. We readily obtain the feasible controller gains as Simulation results of the discrete-time STSF singular system are shown in Figs. 1-6. A randomly generated switching signal satisfying the above regulations is given in Fig. 1. The state response of the closed-loop system is displayed in Fig. 2, which converges to zero and presents that the stochastic stability has been guaranteed. Figs. 3-5 present the corresponding results of controlled output z(k), quantized control inputs u 1 (k), u 2 (k), and dynamic parameters ξ 1 (k), ξ 2 (k). Moreover, introducē The evolution ofγ (k) is displayed in Fig. 6. It is obvious that the actual E2P performance index is always below the given E2P performance index γ = 1, which verifies the VOLUME 11, 2023  effectiveness of the developed quantized feedback control scheme in this paper. Remark 3: In this paper, the dynamic parameters ξ p (k) are also sent over the communication channel and the same adjusting rule as [43] to update the dynamic parameters ξ p (k), it is shown as: 1 ≤ a where a = ȷ p ∥u p (k)∥, e = min{e ∈ N + (2a × 10 e ) > 1} and the function floor(g) = [g] where [g] ≤ g < [g] + 1. In this paper, we select ȷ p = 1/ τ p . Remark 4: Define D as the number of undetermined variables and L as the lines of the matrix inequalities to be solved. Therefore, the conditions for designing MDPPD state feedback controllers and MD dynamic quantizers are in polynomial time with complexity proportional to C = D 3 L, where D = 1 2 Prn x (n x +1)+ 1 2 rn x (n x +1)+Pn u n x +Pn 2 u +n 2 x +P +1 and L = r 3 P(4n x + 2n u + n w + n z ) + Prn x + P(n u + n u ) + P.
Comparative explanations: In this paper, the proposed method provides an effective way for deigning MDPPD   state feedback controllers and MD dynamic quantizers of discrete-time STSF singular systems. Compared with many existing results on the E2P control (filtering) 96814 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  design [39], [40], [41], [42], the main advantages can be summarised as: 1) The E2P performance analysis conditions of systems were studied in [39] and [40]. For the methods in [39] and [40], the Lyapunov matrix inevitably appears on the main diagonal. In other words, if the above methods are applied to singular systems, the semi-positive problem of the Lyapunov matrix will lead to unacceptable nonlinearities. This paper addresses this problem by MDPPD state feedback controllers. Then, the standard control design conditions of MDPPD state feedback controllers and MD dynamic quantizers are given for discrete-time STSF singular systems.
2) The design method in this paper is more common than existing results of linear singular systems [41], [42] for the E2P control problem. Comparing the discussion in Remark 1 and Remark 2, this paper presents a new form of E2P control design for discrete-time STSF singular systems, which solves the problem of semi-positive definite on the main diagonal. In addition, the proposed method is also applicable to STSF singular systems.

V. CONCLUSION
The problem of E2P feedback control for discrete-time STSF singular systems has been investigated in this paper. The main objective is to design MDPPD state feedback controllers and MD dynamic quantizers so that the semi-positive definite problem can be solved in singular systems. A new method has been given to guarantee the stochastic stability of CLS systems and satisfy the E2P performance index. Then, the standard design conditions of MDPPD state feedback controllers and MD dynamic quantizers for STSF singular systems have been established. Finally, a simulation example has been given to illustrate the effectiveness of the proposed control design method.