Equivalent-Input-Disturbance Based Robust Control Design for Fuzzy Semi-Markovian Jump Systems via the Proportional-Integral Observer Approach

: This work focuses on the design of a uniﬁed control law, which enhances the accuracy of both the disturbance estimation and stabilization of nonlinear T-S fuzzy semi-Markovian jump systems. In detail, a proportional-integral observer based equivalent-input-disturbance (PIO-EID) approach is considered to model and develop the controller. The PIO approach includes a variable for relaxation in the system design along with an additional term for integration to improve the ﬂexibility of the design and endurance of the system. The proposed stability criteria are formulated in the form of matrix inequalities using Lyapunov theory and depend on the sojourn time for robust control design. Final analyses are performed using MATLAB software with simulations to endorse the theoretical ﬁndings of this paper.


Introduction
In the real world, nonlinearity poses many challenges during the analysis of the stability and stabilization of control systems. The study based on the Takagi-Sugeno (T-S) fuzzy model technique has attracted a lot of attention due to its remarkable ability to approximate complex nonlinear systems as the weighted sum of linearized subsystems at operating points using fuzzy membership functions and if-then rules. In recent decades, remarkable research attention has been given to the stability and stabilization for T-S fuzzy systems [1][2][3][4][5][6]. In [7,8], the authors looked into the H ∞ filtering problem for nonlinear switched systems with time-varying delay by incorporating T-S fuzzy model characteristics. The research findings in the design of fault-tolerant controller for fuzzy systems with actuator fault were presented in [3,9].
Meanwhile, random factors may also have an impact on physical systems that are actually in use, rapidly altering the structure of the system. Due to the fact that traditional systems are unable to accurately reflect the system models in this situation, MJSs are used to describe them. As a result of their composition as a collection of subsystems with Markov chains that select values from a finite set, MJSs are seen as an effective tool for modeling systems that are subject to rapid changes. To be specific, the sojourn time (ST) is the interval between two subsequent jumps. In reality, the ST of MJS is a random variable distributed according to a probability distribution, and the ε ι℘ (l) variable denotes the rate at which the system switches from mode i to mode j. The transition rate (TR) is usually taken to be • The EID based FSMJSs with a PIO are being taken into consideration for the first time. • A new set of LMI-based restrictions is derived using Lyapunov functional approaches to ensure the disturbance rejection and stochastic stability of FSMJSs. • The accompanying controller gain and observer gain are realized by addressing the LMI restrictions. The offered simulation results are able to unequivocally show the advantages and applicability of the produced theoretical conclusions.
The remainder of this article proceeds as follows: Section 2 illustrates the construction of an EID based FSMJS with a PIO. Through mathematical authentications, the major result is demonstrated in Section 3. The efficiency of the suggested control design is demonstrated by the theoretical findings in Section 4. Section 5 draws a brief conclusion regarding our study.

Variable Explanation
ℵ(t) and‫(א‬t) actual and reconstructed state vector, respectively u(t) control input £(t) and£(t) actual and reconstructed measured output, respectively predicted filter signal F(s) low-pass filter u f (t) enhanced control input L ςι and N ςι proportional and integral gains, respectively ℵ I (t) integral of the weighted output estimation K ηι fuzzy controller gain S −1 differentiator In Figure 1, the setup of an EID based control system with a PIO is depicted. It includes the plant, a state-feedback controller, a PIO, and an EID estimator. EID, as defined in [26], is a signal on the control input channel that affects the output in a manner similar to genuine disturbances. To start, we characterize the plant (2) using an EID υ e (t). This gives: where the equivalent-input-disturbance υ e (t) of υ d (t) means that the output of the EID υ e (t) is equal to the output produced by the real disturbance υ d (t). The PIO from [37] is utilized in this work to aid with system design. The PIO is represented in state space by: where the estimated reconstructed states of ℵ(t) and £(t) are indicated by‫(א‬t) ∈ R n , £(t) ∈ R r . u f (t) is the enhanced control input, and the proportional and integral gains, respectively, are L ςι ∈ R n×r and N ςι ∈ R r×r . A vector named ℵ I (t) represents the integral of the weighted output estimation.

Remark 1.
In this study, it is assumed that the state variables in (4), which represents the measured output £(t), are unavailable. In order to estimate the measured output, the system in (4) is manually replicated with the same behavior to form the estimated output£(t) in (5). In particular, while replicating the actual system, a negligible error component To be specific, when t → ∞, ∆£(t) → 0; thus, it is obvious that £(t) =£(t). Hence, when£(t) converges to 0, £(t) will also converge to 0.
Further, the error dynamics between state (2) and the observer (5) are denoted by: and substituting it to (4) yields: If we add a control input ∆υ(t), such that: then substituting (8) into (7) and allowing the estimated EID υ e (t) to be: allows us to denote the plant as: It should be noticed that the FSMJSs (4) may contain disturbances, which may lead to the poor performance or instability of the system. Moreover, the EID approach does not need prior information about the disturbances, and it may efficiently estimate and reject both matched and unmatched disturbances. In this FSMJSs, an EID technique is employed in the control channel, which yields satisfactory disturbance rejection performance. The optimal EID estimated disturbance can be expressed by: where E + ι is the pseudo inverse of E ι . It is commonly known that external disruptions typically have frequencies in the low-frequency range. Therefore, developing a method to determine the disturbance in a particular low-frequency band makes sense. The frequency range for the disturbance calculation is chosen using a low-pass filter, F(s) to achieve this.
Here, r is the highest angular frequency required for the EID estimation. It should be noted that the estimated total disturbance may contain some measurement noise. To eliminate this noise fromυ e (t), the low-pass-filter F(s) is integrated together with the state observer. Then, the state-space form of F(s) is considered as: where the state of F(s) is represented by ℵ ν (t). The predicted noise signal is shown asυ e (t). D ν , E ν , and J ν are constant parameters.

Remark 2.
It should be noted that the abovementioned conditions of the EID are used to construct an EID estimator, which estimates and rejects the unknown external disturbances. In particular, it shows the difference between the fuzzy state-observer error dynamics and its system parameters. This difference is also treated as the disturbance, since it is caused by modeling the error dynamics and unknown inputs. Further, it is added to the estimated disturbance and filtered by a first order low-pass filter. Finally, this disturbance is completely suppressed by the EID estimator. Therefore, the construction of this assumption is reasonable.

Remark 3.
It should be noticed that the PIO has an additional integrating term, x I (t), which raises the order of the state observer and enables quick and precise estimation of the system state. Additionally, the system flexibility is improved by an additional gain matrix N ςι introduced in the PIO. The PIO combined with the EID offers stable and accurate disturbance rejection performance as well as a trustworthy disturbance estimate.
Next, we consider the stabilization problem of FSMJSs (2). In order to stabilize system (2), a fuzzy state feedback controller u f (t) of η-th rule is designed as follows.
Control rule η : where K ηι ∈ R m×n indicates the fuzzy controller gain. Eventually, the noise estimated disturbanceυ e (t) together with the state feedback control law u f (t) yields a new controller, which is given by Furthermore, the system internal stability condition does not depends on exogenous signals; hence, the exogenous signals υ e (t) are assumed to be zero.

Remark 4.
The primary objective of this work is to stabilize the system (1); therefore, a state feedback control u f (t) is introduced in (14), which is a function of the estimated observer statê ℵ(t) . The developed control law u(t) in (15) is a combination of u f (t) and an EID estimatorυ e (t), which forces‫(א‬t) to zero for stabilizing the system. Therefore, the convergence of ℵ to zero implies the convergence of the measured output £ to zero from (1), and the developed controller significantly converges the estimate‫א‬ to zero.
Moreover, the following equation is obtained when incorporating (15) in (13): Furthermore, we define the error between the state (2) and observer (5) with the configuration ∆ℵ(t) = ℵ(t) −‫(א‬t). Then, the corresponding error system can be written as: For the sake of simplicity, we assume that the closed-loop system must satisfy the following stability requirement: υ d (t) = 0.
The enhanced closed loop system resulting from the aforementioned equations with T can be defined as: T is the output of the augmented closed loop system (19).

Remark 5.
It is noted that the augmented system ϕ in (19) combines the vectors‫(א‬t) and ∆ℵ(t) but does have the term ℵ(t). In addition, from (14) and (15), ϕ converges to zero. It is obvious that the terms‫(א‬t) and ∆ℵ(t) of the error dynamics in (6) converge to zero, achieving the convergence of ℵ(t) to zero.

Main Results
The issue of the stabilization and disturbance rejection for FSMJSs based on the EID control via the PIO method is covered in this section. It is possible to create a new set of necessary conditions based on the LMI that ensure the closed-loop system (19) is stochastically stable by building a correct Lyapunov-Krasovskii functional (LKF). Theorem 1. For fixed positive scalars ϑ 1 , ϑ 2 , and ϑ 3 , the closed-loop system (19) is stochastically stable, if there exist symmetric positive definite matrices P ι , Q ι , R ι , and S, such that the following inequalities are satisfied: where the elements of Π are: Φ ςη ι1,1 = Sym{ 1 Proof. Obtaining the stability criterion for the enhanced system (19) suffices to verify the stochastic stability for the considered FSMJSs (1). Let us build an LKF with the following structure for the enhanced system (19) for this purpose: S} are the positive definite matrices. By using the infinitesimal operator L{.} together with the solution of the augmented system (19) along with the mathematical expectation, we can have: (24)-(27), we can obtain the LMI (21). Furthermore, it is obvious and simple to draw the conclusion that E {LV (ϕ(t), t, ι)} < 0, if the condition in (21) is true. In addition, the closed-loop system (19) is stochastically stable according to the Lyapunov stability theory and the aforementioned assessments.

Theorem 2.
For fixed positive scalars κ 1 , ϑ 1 , ϑ 2 , and ϑ 3 , the closed-loop system (19) is stochastically stable, if there exist symmetric positive definite matrices, X ι , Y ι , Z ι , and M, and appropriate dimensional matrices, V ηι , U ςι , and H ςι , such that the following inequalities are satisfied: where the elements ofΠ are: Moreover, if the derived inequality (28) is feasible, then the stabilizing controller gain is given by K ηι = V ηι X −1 ι , and the observer gains are given by L ςι = U ςιȲ −1 ι and N ςι = H ςιȲ −1 ι .
Proof. It appears that inequality (21) has been transformed into an LMI-based constraint because it does not take the linear form. Let us define a few terms to that end: 1 Further, pre-and post-multiplying (21) by diag{ν 1 X ι , ν 2 Y ι , ν 3 Z ι , M, I, I, I}, and setting up V ηι = K ηι X ι , U ςι = L ςιȲι , H ςι = N ςιȲι , we can easily retrieve (28). Since it is not a linear equation, it is difficult to solve the equation C ςι Y ι =Ȳ ι C ςι using the MATLAB LMI toolbox. In order to circumvent the problem, the optimization strategy algorithm is taken into account for the assumptions C ςι Y ι =Ȳ ι C ςι . It can thus be understood equivalently as a given positive scalar that is sufficiently small. Then, by using the Schur complement, the aforementioned inequality can be equivalently turned into (29). The closed-loop system (19) is stochastically stable, if the relations (28) and (29) are met. As a result, the closed-loop system (19) meets the prerequisites for stochastic stability, which has been proved in Theorem 2. Since the TR matrix ε ιι (l) is time varying, it might be necessary to test an infinite number of LMIs to fulfill the criteria (28) in Theorem 2, which would be computationally inefficient. The next step is to create numerically feasible finite LMIs from the inequalities in (28). This problem is resolved by the subsequent theorem.
Proof. Theorem 2 makes it clear that assuming condition (28) holds, and the closed-loop system (19) accompanied by the time-varying TR matrix ε ι℘ (l) is robustly stochastically stable. The time-varying term ε ι℘ (l) is allowed to have a lower bound ε ι℘ and an upper bound ε ι℘ in order to ensure numerical tractability. There is no disagreement that this is a successful approach.
As can be seen from (31), it is clear that ε ι℘ (l) takes any value in ε ι℘ , ε ι℘ . In parallel, for a specific l, ε ι℘ (l) is a convex combination, as demonstrated below: where 0 ≤ β ≤ 1, because ε ι℘ (l) in (32) is linearly dependent on β, and (28) only has to be satisfied for β = 0 and β = 1; that is, (28) holds if the inequalities in (30) hold. The proof of this theorem is also completed by splitting the sojourn time s into w pieces.

Numerical Simulation
To demonstrate the theoretical viability of our strategy, we assume a system in this section that has two operating system modes and two fuzzy rules. The parameters are chosen in accordance with the following, and its model is identical to system (19): ϑ 1 = 1, ϑ 2 = 0.00001, and ϑ 3 = 1. A disturbance υ e (t) = e −0.0001t sin(2t) is injected to the system. We select r to be π. The parameters of the low-pass filter F(s) are chosen as D ν = −101, E ν = 100, and J ν = 1, which satisfies (12). By utilizing the MATLAB LMI toolbox, a set of feasible solutions for Theorem 2 is discovered via LMIs (28) and (29), and the associated admissible gain matrices are started as K 11 = [1.  Figure 2a, it is clear that the actual disturbance is completely estimated by the estimated disturbance, and its corresponding estimation error is displayed in Figure 2b. Figure 3a,b shows, respectively, the system's genuine state trajectory and associated error trajectory. Figure 4 shows the open-loop state trajectories. The state response curves of system along with its observer are presented in Figure 5a,b. Figure 6a displays the control responses. Furthermore, Figure 6b displays the control responses in the absence of a disturbance estimationυ e (t). The associated measured and observation output responses are displayed in Figure 7. In Figure 8, the membership functions are displayed. The jumping mode's trajectory is also shown in Figure 9. Additionally, Figure 10 shows the outcomes of output response curves in the control absenteeism.

Conclusions
This work has examined FSMJSs in terms of the stability and disturbance rejection, utilizing the EID technique. A PIO was utilized in conjunction with an EID estimator to improve the accuracy of the state system estimation. LMIs were formulated to design the stability condition from the gain of the feedback controller and PIO. The simulation results showed that the proposed PIO-based EID technique achieved appropriate stability in addition to enhanced disturbance estimation and rejection capabilities. The intriguing subject of our future study will be how we may incorporate the PIO to further extend the suggested method for event-triggered fuzzy sliding-mode control with semi-Markovian switchings using EID for networked control systems. Data Availability Statement: Data sharing is not applicable to this article, as no datasets were generated or analyzed during the current study.

Conflicts of Interest:
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Abbreviations
The following abbreviations are used in this manuscript: FSMJSs Fuzzy semi-Markovian jump systems EID Equivalent-input disturbance PIO Proportional-integral observer ST Sojourn time RT Transition rate