Adaptive Optimal Terminal Sliding Mode Control for T-S Fuzzy-Based Nonlinear Systems

Soltanian, Farzad; Valadbeigi, Amir Parviz; Tavoosi, Jafar; Aazami, Rahmat; Shasadeghi, Mokhtar; Shirkhani, Mohammadamin; Azizi, Amirreza

doi:https://doi.org/10.1155/2024/7126978

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Research Article | Open Access

Volume 2024 | Article ID 7126978 | https://doi.org/10.1155/2024/7126978

Adaptive Optimal Terminal Sliding Mode Control for T-S Fuzzy-Based Nonlinear Systems

Farzad Soltanian,^1,2Amir Parviz Valadbeigi,^1,3Jafar Tavoosi ,⁴Rahmat Aazami,⁴Mokhtar Shasadeghi,²Mohammadamin Shirkhani,⁴and Amirreza Azizi⁵

Academic Editor: Xiaoping Liu

Received13 Dec 2023

Revised10 Mar 2024

Accepted08 Apr 2024

Published25 Apr 2024

Abstract

This study utilizes the Takagi–Sugeno fuzzy model to represent a subset of nonlinear systems and presents an innovative adaptive approach for optimal dynamic terminal sliding mode control (TSMC). The systems under consideration encompass bounded uncertainties in parameters and actuators, as well as susceptibility to external disturbances. Performance evaluation entails the design of an adaptive terminal sliding surface through a two-step process. Initially, a state feedback gain and controller are developed using Linear Matrix Inequality (LMI) techniques, grounded on -performance and partial eigenstructure assignment. Dynamic sliding gain is subsequently attained via convex optimization, leveraging the derived state feedback gain and the designed terminal sliding mode (TSM) controller. This approach diverges from conventional methods by incorporating control effort and estimating actuator uncertainty bounds, while also addressing sliding surface and TSM controller design intricacies. The TSM controller is redefined into a strict feedback form, rendering it suitable for addressing output-tracking challenges in nonlinear systems. Comparative simulations validate the effectiveness of the proposed TSM controller, emphasizing its practical applicability.

1. Introduction

Sector nonlinearity and approximation are two approaches which widly used to describe the Takagi–Sugeno (T-S) fuzzy model of a nonlinear system. the sector nonlinearity approach present an exact model of the nonlinear system. Additionally, it transforms the nonlinear system into a collection of local linear models through fuzzy blending [1–3]. This T-S fuzzy model proves valuable for characterizing uncertain nonlinear systems. Furthermore, addressing added uncertainties, like those associated with actuators, recent work has proposed integrating T-S model representation with sliding mode control (SMC) designs [4–7].

Sliding mode control is a well-established approach in robust control theory, particularly for handling known but bounded-matched uncertainty [8–10]. To extend its applicability to scenarios with unknown upper bounds [11, 12], adaptive methods have been explored for estimating uncertainty boundaries [13–15]. An example is presented in [16], which introduces an adaptive SMC design for a nonlinear suspension system with unknown bound uncertainty. Additionally, the Terminal Sliding Mode Control (TSMC) technique is frequently used to rapidly converge system dynamics to the sliding surface. Proper design of TSMC parameters ensures infinite stability of the closed-loop system [17–19]. In [20], an adaptive TSMC is developed for a variable load DC-DC buck converter, where nonlinear sliding surface coefficients are adjusted to account for load resistance changes. An adaptive law is integrated to manage load fluctuations and achieve dynamic sliding along the surface. TSMC offers advantages such as swift response, disturbance resistance, and ease of implementation. Consequently, the focus of our literature analysis revolves around TSMC design for linear systems or TSFM, the fuzzy blending of local linear subsystems.

Various controller design methods exist for linear systems, such as Linear Quadratic Regulator (LQR), Pole Placement, Eigenvalue Assignment, and eigenstructure assignment techniques [21–24]. A comparative study [25] demonstrated that eigenstructure assignment yields fast, non-overshooting responses, outperforming other methods by enabling eigenvalue and eigenvector assignment. Combining different state feedback and robust approaches can enhance robust controller performance and achieve desired transient behavior in the presence of uncertainty or disturbance [26–29]. In [26], a blend of pole placement objectives an H_∞ performance is proposed. Also, combines eigenstructures assignment and H₂-characterization is presented in [27] for state feedback gain calculation, which is used to achieve sliding surface and control inputs. The definition of sliding surface significantly influences control system performance. In [28], a constrained bilinear quadratic regulator which is known as an optimal quadratic control problem was designed by a performance index. In [29], a combination of the sliding mode control (SMC) and nonlinear control law is proposed and the proposed controller parameter were optimized utilizing the particle swarm optimization (PSO) method [29].

Conversely, addressing output tracking control in nonlinear systems has long been a challenge [30–32]. Output signal convergence to reference signal was achieved in [33–35] through output feedback control. However, robustness against external disturbances was not fully considered. State and input constraints were incorporated in [36] for static output feedback control of T-S fuzzy systems. New design criteria were established using Finsler’s lemma and fuzzy Lyapunov functions to ensure stabilizing controller existence. Further relaxed LMI conditions were obtained using slack variables. Nonetheless, this study lacked robustness against uncertainties, external disruptions, or optimized performance. Addressing these concerns, Köhler et al. [37] introduced a method for designing static output feedback controllers for constrained T-S fuzzy systems. The fuzzy control framework integrated Lyapunov stability theory to consider state and input constraints. The fuzzy control design was reformulated as an optimization problem with LMI constraints. Despite producing positive results, this study did not fully address controller robustness against uncertainty and external disturbances.

In the realm of control theory, recent advancements have addressed critical challenges in various domains. Paper [38] introduces a hybrid Q-learning method for adaptive fuzzy control in discrete-time nonlinear Markov jump systems, offering innovative solutions to address complexities in solving fuzzy game-coupled algebraic Riccati equations (FGCAREs) [38]. However, despite its merits, this approach may encounter limitations in computational complexity and generalizability beyond Takagi–Sugeno fuzzy models [38]. Meanwhile, paper [39] proposes nonfragile synchronization for discrete-time T-S fuzzy Markov jump systems, leveraging a novel matrix transformation method and nonfragile controller design [39]. While promising, challenges such as computational complexity and conservatism in criteria underscore the need for further research to enhance practical applicability and robustness [39]. Furthermore, investigations in sliding mode control (SMC) for discrete-time T-S fuzzy networked singularly perturbed systems, as explored in paper [40], highlight the integration of observer-based techniques to manage communication burdens and improve networked control system (NCS) effectiveness [40]. Nevertheless, complexities in implementation and limited applicability to certain system dynamics remain notable concerns [40]. Future endeavors may delve into stochastic time-delayed NCSs to bolster system performance and stability, mitigating these potential limitations [40]. These collective efforts reflect the ongoing pursuit to advance control methodologies, addressing real-world complexities and pushing the boundaries of system robustness and efficiency.

A literature review highlights the limited discussion on adaptive estimation of uncertainty boundaries combined with optimal TSMC based on T-S fuzzy systems. Building on the discussed advantages, this paper investigates an adaptive approach to design optimal TSMC for T-S fuzzy-based nonlinear systems. The proposed method involves online adjustment of the nonlinear sliding surface gain and adaptive TSM control input. Fuzzy rule consequences allocate local subsystems to desired negative values based on partial eigenstructure assignment. The study assumes unknown actuator uncertainty bounds, deriving an adaptive formula to determine these bounds. The TSMC design is a two-step process: proposing a new TSM control input and adaptive formula to find uncertainty bounds, followed by calculating suboptimal gains for each rule consequence using -performance index and partial eigenstructure assignment likewise [41]. Convex combination identifies state feedback control associated with -performance criteria and generalized partial eigenstructure assignment. An optimization problem determines the online optimal nonlinear sliding matrix gain. Extending this approach, the study addresses output tracking for strict feedback form nonlinear systems using the T-S fuzzy model, demonstrating effectiveness of the presented method through comparative simulations.

This paper presents an innovative approach to tackle the complex challenge of achieving optimal output tracking control in strict feedback nonlinear systems with unknown actuator uncertainty bounds, modeled using the TSFM. It introduces the concept of extending eigenstructure assignment to nonlinear systems, enabling the optimization of transient response characteristics by defining sliding surfaces and control inputs aligned with desired eigenvalues corresponding to local subsystems within fuzzy rule consequences. Additionally, this paper proposes the novel idea of online TSMC design, where the level of control effort is considered during sliding surface and control input design. Suboptimal gains are computed for each fuzzy rule consequence based on system and actuator uncertainty, as well as external disturbance, and then combined to derive a customized control law tailored to the specific nonlinear system. Throughout the control design process, -optimization techniques are applied to enhance transient response characteristics.

This paper is structured as follows. Section 2 introduces nonlinear systems with matched uncertainty subjected to external disturbance. Section 3 outlines major design processes and formulation derivations. Section 4 presents a comparative case study to showcase the strategy’s effectiveness, followed by the conclusion in Section 5.

Hint , where is a square matrix and represents the conjugate transpose of ; denotes the transpose of . In addition, denotes the Euclidean norm .

2. Problem Statement and Preliminaries

Consider the following class of nonlinear systems with matched uncertainty that are subjected to external disturbance:where the state vector, the -performance output vector, the control input vector, the external disturbance input with bounded energy, and the actuator matched uncertainty with unknown bounded value are denoted by , , and , respectively. Additionally, the nonlinear functions , , , , and have suitable dimensions and undetermined parameter values. The fuzzy system can be used to represent the nonlinear system mentioned in (1) as follows.

2.1. Plant Rule i

If is and …and is , thenwhere and () are the premise variables and fuzzy sets, respectively; is standing the number of the fuzzy rules; and matrices ; ; ; ; and have appropriate dimensions. Furthermore, , , , , and .

The overall T-S fuzzy model of the nonlinear system given in (1) will be inferred using a singleton fuzzifier, product inference, and center-average defuzzifier as follows:where denotes the relative weighting value (normalized membership function), in which ≥ 0 and andwhere represents the grade of membership of in . Following that, the T-S fuzzy controller will be designed in the same way as (1). Parallel Distributed Compensator (PDC) [1] is used to design the fuzzy controller u(t) as follows.

2.2. Controller Rule i

If is and …and is , thenwhere indicates the feedback gain. The overall PDC controller can then be represented aswhere denotes the normalized fuzzy membership functions presented in (4).

3. Main Results

In this section, we employ the conventional methodology for designing control inputs in the context of nonlinear systems characterized by matched uncertainties and external disturbances to establish a sliding mode control input. The process entails the utilization of both a reaching control input and an equivalent control input, constituting the input strategy for the reaching and sliding phases, respectively. The standard control input design procedure itself is composed of two primary constituents. In cases where nonlinear systems are subject to matched uncertainty in the absence of external disturbances, we propose the adoption of the reaching control input as the initial step. Subsequently, in the second phase, the development of an equivalent control input is carried out for nonlinear systems exposed to external disturbances, employing state feedback control while ensuring alignment with the performance benchmarks of robust control design. These benchmarks encompass -performance and eigenstructure assignment. As a result, during the process of designing the state feedback control input, we are empowered to address the challenge of robust control design for the Takagi–Sugeno fuzzy model when confronted with the presence of external disturbances. This is achieved by conscientiously incorporating considerations related to -performance characterization and partial eigenstructure assignment. This section is methodically divided into three subsections. The first and second subsections delineate the formulation of a Terminal Sliding Mode (TSM) controller tailored to the Takagi–Sugeno fuzzy model. This formulation is rooted in the primary and subsequent stages of the conventional sliding mode control design methodology, respectively. The third subsection subsequently delves into the intricate matter of output tracking.

3.1. Terminal Sliding Mode Controller Design

In the initial stage of our proposed development approach, it is imperative to consider the system state (3) when there is no external disturbance. This step is pivotal to the subsequent methodology.

This subsection proposes a terminal sliding surface for the TSFM given in (7) and (8). Consider the following terminal sliding surface:where the matrix denotes the nonlinear sliding surface gain with the appropriate dimension to be designed. Also, are integer odd numbers, and .

Lemma 1. The reduced-order sliding mode dynamics surface (10) for control the T-S fuzzy model (7) and (8) can be reached through the following control input.where ; is infinitely stable and M is a non-zero arbitrary matrix of appropriate dimension such that becomes an invertible matrix. Similarly, is a scalar matrix of appropriate dimension such that .

Proof. Select a Lyapunov candidate function as follows:where the time derivative of the sliding surface (10) isNow, applying time derivation to the Lyapunov function (12) leads toThen, by definition of , (14) is rewritten asLet , . Then, the system state dynamics (7) and (8) and the time derivative of the Lyapunov function (15) are rewritten asNow, suppose that the Euclidean norm of the is bounded by a known function , i.e., , where is a known scalar value. Hence, substituting (11) in (16) yieldswhere (17) according to uncertainty upper-bound can be rewritten asTherefore, from (18), one can concludewhere is a matrix with a negative real part value; then reachability conditions are held and the proof is complete.
The uncertainty upper-bound can be simplified by supposing , where is a known small positive scalar value. In this paper, we suppose that the uncertainty bound is unknown and we should estimate the uncertainty bound, i.e., . Therefore, the proposed terminal sliding mode control input (11) can be rewritten asLet be the difference between the estimated and real values of the upper-bound parameter . Then, the estimated uncertainty bound and its parameter are obtained as follows:where is a known small positive constant.

3.1.1. Attention

According to the above explanations, the Lyapunov function has been chosen as a function of the delta sliding surface that first satisfies the above conditions; secondly, the Lyapunov function and then the sliding surface and then the first and second state variables converge to zero, which makes the system stable.

Lemma 2. The sliding surface described in (10) converges in a finite amount of time when the state trajectories in (7) and (8) are combined with the control input (20), the estimated bound of matched uncertainty, and the parameter updating rule (21).

Proof. we select a candidate Lyapunov functions as follows:Now, applying the time derivative to (22) yieldsSubstituting given in (20) in (22) yieldswhere the reachability condition is held and the proof is complete.
The TSFM (7) and (8) and the control input (20) must now be revised and represented in new form in order to achieve the optimal state feedback gain (25). To accomplish this, take into account the following representation for the entire TSFM with actuator uncertainty.where , , and are the appropriate dimension matrices. The strict feedback form T-S fuzzy model in (7) and (8) can be proven equivalent to the TSFM proposed in (25) by introducing the parameter-varying matrices as follows:whereFor the sake of simplicity, letwhere the overall matrices in (28) are as follows:The control input (20) can be recast by taking into account the reformulated parameters stated in (29), as follows:where . Therefore, the control input can be rewritten aswhere and stand for the equivalent and switching control inputs. Section 3.2 delves into how Generalized Partial Eigenstructure Assignment optimizes sliding gain, driven by factors discussed earlier. These include using the Takagi–Sugeno (T-S) fuzzy model to handle uncertain nonlinearities, integrating it with sliding mode control for robustness and wider use, using adaptive methods for uncertainties linked to actuator dynamics, employing Terminal Sliding Mode Control (TSMC) to swiftly guide system dynamics to the sliding surface with robustness, recognizing the effectiveness of eigenstructure assignment for rapid control responses, and exploring the combination of adaptive uncertainty estimation with optimal TSMC. This method adjusts sliding surface gain dynamically for better responsiveness. Additionally, Section 3.3 expands to address output tracking in strict feedback nonlinear systems through the T-S fuzzy model. The main goal is to enhance control robustness, adaptability, and performance in complex and uncertain system dynamics.

3.2. Optimal Sliding Gain Design Using Generalized Partial Eigenstructure Assignment

In the second step of the ideal TSMC design, we set and and then assume that the TSM controller (30) only contains the equivalent control part, which corresponds to the control law (31), to demonstrate the disturbance attenuation/rejection property of the suggested approach for the nonlinear T-S fuzzy model (3). The system dynamics (3) is therefore revised as follows.

The core challenge pertains to the adaptive identification of the optimal nonlinear sliding surface coefficient, denoted as S, with the dual objectives of ensuring system stability and fulfilling the performance criteria of characterization. This challenge remains pertinent even within the constraints of motion governed by considerations. To effectively address this intricate challenge, the approach involves addressing a multichannel state-feedback predicament, aimed at judiciously selecting the most suitable state feedback control input. This selection holds significant influence over the subsequent design of the sliding matrix gain, S. The formulation of a -centric sliding mode control strategy is predicated on the delineation of two pivotal steps: Step 1. The amalgamation of the multichannel optimization problem with the generalized partial eigenstructure assignment technique takes precedence. This entails the derivation of a state feedback matrix, denoted as K, which serves to fulfill not only the multichannel constraints but also the nuanced requirement of situating the m poles of individual rule consequence subsystems at precisely predefined negative eigenvalues. Step 2. The explicit determination of the sliding matrix gain, S, is contingent upon the state feedback outcomes gleaned from the preceding Step 1.

The subsequent sections undertake a comprehensive exegesis of these critical steps, elucidating their intricacies and implications in depth.

3.2.1. -LMI Characterization

This subsection presents a new extended LMI characterization of performance for the TSFM (32) from disturbance input to output .

Lemma 3. The following claims are identical for the nonlinear system (32):(a), such that is a stable matrix and the performance specification from the disturbance input to the output is less than .(b) and such that where .(c), and nonsingular matrix such that for where .

Proof. It should be noted that (a) and (b) are equivalent versions of the conventional -state-feedback synthesis for the T-S fuzzy system proposed by Lemma (1) in [42]. Schur complement lemma allows for the rewriting of (36) asin which . Applying the congruence transformation on (39) results inwhere , and . Then, (40) can be rewritten asBy using the Projection lemma [37], inequality (41) holds if the following inequalities are satisfied:Inequality (42) indicates and inequality (43) is equivalent toBy multiplying both sides of (44) by , one obtainswhere . From the Schur complement, (45) is equivalent to (33). The proof is completed.

Note that by using the Projection lemma, one can realize that (37) is an extension of (34), and the proof is omitted for brevity.

3.2.2. Multichannel- State-Feedback Gain Design Using LMI Characterization

The state feedback gain in this subsection is designed to conform to the following performance categorization.where and are weighted vectors used to select a specific input/output channel and represents a performance from to the output vector [43]. The number of channels is specified by . By replacing , and in (32) with , and , respectively, where , it is possible to derive . By minimizing the norm, a specific performance associated with signals and can be ensured (46). Now, by recasting (36)–(38) with different Lyapunov variables of channel and global variables for all channels, the multichannel LMI characterization can be derived. Therefore, by using Lemma 3, the LMI characterization for the ψ th channel can be expressed as follows:where , and are global LMI decision variables and (). The optimization problem (46) is then modified to

3.2.3. Generalized Partial Eigenstructure Assignment

The LMI characterizations (47)–(49) can be utilized to assign the m poles of each rule consequence linear subsystem (27) to a set of predetermined negative values (51) utilizing the PDC controller. Assignments of partial eigenstructures, which combine performance, are suggested in this subsection. The assignment of the following partial eigenstructures is done by state feedback.This issue can be broken down into the following two steps.(1)Calculate the base vector in the null space for .(2)The state feedback can be calculated as with arbitrary wheresuch that , and . It should be noted that some and arrays are dependent on the assignment of eigenstructure to a predetermined value . In other words, other arrays that are not used for eigenvalue assignment can be used to achieve additional constraints. As a result, the first step in designing a -based SMC is to recast (50) by the LMI characterizations (47)–(49) with , , and as shown below.

Each local subsystem (2) with PDC controller (5) contain eigenvalues (51), if (52) be used to calculate state feedback, which is proved by the following lemma.

Lemma 4. The set of eigenvalues for each fuzzy rule with the state feedback , where and are reported in (52), contains the subset (51) for each fuzzy rule.

Proof. The following brief proof demonstrates that a set of (51) is the subset of eigenvalues .where , and stands for the canonical or standard basis of [44–47].

3.2.4. Adaptive Sliding Gain Matrix Design

The fact that (6) equals (11) can lead to one approach for determining the sliding gain matrix, which proceeds as follows:

If is used as the state feedback control input, then (56) can be expressed as follows:

Flowing minimization is suggested to resolve (57):where is the state feedback gain, derived from (56), and is the scalar variable. This ensures that the m poles of each rule consequence subsystem with PDC controller are precisely situated at λ [48–50].

By employing a straightforward relaxation procedure, one can analyze (58) and obtain the positive definite matrix as shown below:

Using LMI optimization, the minimization problem stated in (59) is reformulated as follows:

As a result, the multichannel--based terminal SMC problem is summarised in (60) to find the global solution for the optimization problem, yielding the terminal sliding mode gain [51, 52].

The following theorem offers a concise statement of the suggested design strategy for the multichannel -based Adaptive Dynamic TSMC for (25) [53–55].

Theorem 5. Assume that the state feedback is a solution to the optimization problem (56) for some , . The multichannel -performance constraints are then guaranteed, and the resulting reduced-order sliding mode dynamics obtained by the control law are demonstrated aswhere represents the switching control component of the proposed controller in (20), which is asymptotically stable.

Proof. Select a candidate Lyapunov function as follows:Using the time derivative to analyze the Lyapunov function (62) now results inand substituting from (25) in (63) yieldsAfter that, (64) is revised as follows:Substituting from (61) into (65) yieldsIn this study, it is claimed that are Hurwitz matrices; hence, (66) leads towhere the reachability condition is satisfied, and the maximum eigenvalues of the Hurwitz matrices , i.e., , are located on the left side of the s-plane. Hint 3.1. When the trajectories in this investigation are on the sliding surface, that is, when , the system tracking dynamic isThe real components of the eigenvalues are entirely negative, proving the infinite stability of the TSMC system. We can also rewrite (68) asWe now integrate both sides of (69). Then, using the closed interval (), we derive the resulting equation as follows:We can deduce from (70) that when the system with the initial condition arrives at the terminal sliding mode at , the system state converges to in finite time and remains there for In other words, for , the state trajectory will belong to the sliding line. However, the convergence time, given in (70), is affected by the parameters and .

3.3. Application to Output Tracking

The suggested method in this paper is expanded to handle the challenge of output tracking in nonlinear systems characterized by strict feedback structure, employing the T-S fuzzy model. The output to be measured is represented as. The subsequent steps are recommended to achieve this objective. Step 1. The first state tracking error can be expressed as , where and stand for the desired output vector and desired first state vector, respectively. This is because the measured output vector is an available output vector. The first and second time derivatives of are then calculated as shown below. Define , and the system tracking error dynamics (8) is rewritten as The output tracking problem (72), which is now expressed in terms of the tracking error and time-derivative of the tracking error , is to be developed with a new control input . This equation must be true for the new control input: Then, system (72) is rewritten as Now, the terminal sliding surface for the state tracking error (74) can be modified as where the appropriate dimension matrix stands for the nonlinear sliding surface gain. The new control input of the TSMC given in (75) can be proposed as where and denote the equivalent and switching new control input, respectively. Step 2. In this case, assuming that , we define the -performance output tracking error as , where represents the desired -performance output vector. The output tracking error dynamics is then changed as (77) to evaluate the disturbance attenuation supplied for the system dynamic given in (32). According to the PDC form, the new control input is now designed for the tracking error dynamic (77) as such that the new control input gains in (78) are acquired by solving the optimization problem (56). Step 3. For the output tracking problem, the terminal sliding mode gain must be obtained. As a result of (77) and (78), the minimization problem (60) is modified accordingly in this case. Step 4. The sliding surface gain S obtained in the optimization problem (79) and the new control input condition given in (76) are used to calculate the practical control input .

4. Simulation Results

In this section, a real-world illustration involving a mass-spring mechanical system is examined to evaluate the effectiveness and practicality of the suggested optimal adaptive terminal sliding mode control system design.

Example 1. Consider a mass-spring mechanical system depicted in Figure 1, where , , , , , and denote the displacement from the reference points, the viscous damping forces, the restoring forces of the springs, the kinetic friction forces, the masses, and the external input, respectively. Newton’s law states that the motion equations can be represented aswhere the actuator uncertainty () and the external disturbance () are denoted correspondingly. The viscous damping force and the restoring force of the spring are both described by the variables , , , and as and , respectively. The nonlinear system (80) is represented as the following state-space form by the definitions , , , and .Using the sector nonlinearity approaches [1], we select and as the premise variables with the corresponding membership functions as and , and , where and denote the upper and lower bounds of the premise variable, respectively. For the system dynamic (81), one derives the following desired state variable asTherefore, one obtains the practical input from (71) and (81) as . Here, the system parameters are set as , , , . Also, and are selected as the desired output vector. Then, the upper and the lower bounds of the premise variables are obtained as , and , respectively. The fuzzy rule matrices are obtained asand ().
In this study, we select the predetermined eigenvalue as , the nonlinear sliding fractional power as , the initial state condition as , the actuator uncertainty as , and the arbitrary uncertainty parameters as , and the external disturbance is as follows:We devote different Lyapunov matrix variables for each channel in (51). Also, the predetermined value is considered for the -performance from the external disturbance vector to the first output vector, i.e., . Then, the results show that the -performance is obtained for the first output vector, i.e., . In this case, the state feedback gains and the optimal sliding gains are obtained asA standard TSMC based on T-S fuzzy model (81) and (82) combined with performance is analyzed in order to compare the proposed approach’s effectiveness with that of similar efforts. This approach demonstrates that the performance is assured. Figures 2–13 contain the illustrated comparative simulations.Figures 2–4 illustrate the superiority of the adaptive optimal TSMC approach over the conventional method with respect to reference tracking. In Figures 6 and 7, the trajectories of the first and second vectors on the surface demonstrate that the adaptive optimal TSMC approach reaches the origin point more rapidly and with minimal chattering compared to the conventional method. Figures 8 and 9 visualize the new control inputs proposed in (76), computed by integrating the characterization and the generalized eigenstructure assignment method through the adaptive optimal TSMC minimization problem (79) alongside the calculation using the conventional method. Figures 6–9 indicate that the method proposed in this paper offers numerous advantages, including a significant reduction in chattering phenomena and its adverse effects, decreased control effort, and notably reduced reaching time.
To demonstrate the effectiveness of the proposed approach in reducing reaching time and chattering phenomena, refer to Figures 10 and 11. Furthermore, according to (81) and Figure 1, the first output vector, , denotes the difference between the displacements of the first and second masses. It is preferable for these functions to reach zero in finite time with minimal chattering to ensure smooth displacement. On the other hand, the second output vector, , solely represents the displacement of the second mass. Figure 12 shows that reaches zero faster than that with the conventional method. Figures 12 and 13 demonstrate that although the difference in the second mass displacement () is not substantial using both approaches, the disparity in finding the displacement of the first mass () between the two approaches is remarkable.

(a)

(b)

5. Conclusion and Future Works

This study presents an innovative strategy aimed at devising the optimal switching surface for implementing sliding mode control in a nonlinear system defined by TSFM. To achieve this objective, the approach involves the generation of a state feedback gain for each localized subsystem within the closed-loop setup. This gain is computed using a convex optimization strategy, which ensures the allocation of predetermined complex values with negative real components to a predefined number of eigenvalues, while simultaneously adhering to the -norm criteria. Following the acquisition of the state feedback during the initial phase, the associated convex optimization problem is subsequently solved to ascertain the sliding surface. Notably, the suggested method offers several notable benefits. It takes into account the level of control efforts during the design of SMC and facilitates the adaptive determination of the nonlinear sliding surface gain based on trajectory conditions. The efficacy of this proposed approach is substantiated through simulation outcomes. In the realm of future research, potential avenues could encompass the development of effective techniques to address matched/unmatched uncertainty and the judicious selection of optimal TSMC parameters to ensure the attainment of the desired outcomes.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2024 Farzad Soltanian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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