A Comprehensive Survey on Advanced Control Techniques for T-S Fuzzy Systems Subject to Control Input and System Output Requirements

2. Control Methodology Subject to Input Constraint with Actuator Saturation

Input constraints in control engineering refer to the limitations imposed on the control inputs, such as voltage, torque, or force, due to physical, safety, or operational restrictions of actuators and system components. Actuators and other system components have finite capacities. If control inputs exceed their limits, actuator saturation can occur, where the actuator is unable to respond appropriately to control commands. Ignoring input constraints can cause the system to operate outside its designed safety margins, potentially leading to unsafe conditions. By considering input constraints, control engineers can design stable and safe systems under all operational conditions, even when faced with disturbances or uncertainties. In control systems, there are many types of input constraints. This paper will discuss the typical input constraint commonly discussed in the literature: actuator saturation for the T-S fuzzy system. First, let us consider the T-S fuzzy model represented as follows.

Plant Rule i:

Ψ_{1} (t)

M_{i 1}

and

Ψ_{2} (t)

M_{i 2}

and … and

Ψ_{q} (t)

M_{i q}

, then

$d x (t) = A_{i} x (t) + B_{u i} u (t) + B_{v i} v (t)$

(1)

where $Ψ (t) = [\begin{matrix} Ψ_{1} (t) & Ψ_{2} (t) & \dots & Ψ_{q} (t) \end{matrix}]$ are the premise variables, $i = 1, \dots, n$ and $n$ is the rules number, $M_{i q}$ are fuzzy sets, $q$ is the number of premise variables, $d x (t) \in ℜ^{m_{x}}$ is the state vector, $d x (t)$ denote $\dot{x} (t)$ for the continuous-time system, $d x (t)$ denote $x (t + 1)$ for the discrete-time system, $u (t) \in ℜ^{m_{u}}$ is the control input vector, $y (t) \in ℜ^{m_{y}}$ is the output vector, $v (t) \in ℜ^{m_{v}}$ is the external disturbance input vector which is a bounded unknown random signal, $A_{i} \in ℜ^{m_{x} \times m_{x}}$ , $B_{u i} \in ℜ^{m_{x} \times m_{u}}$ , $B_{v i} \in ℜ^{m_{x} \times m_{v}}$ and $C_{i} \in ℜ^{m_{y} \times m_{x}}$ are constant matrices, $m_{x}$ , $m_{u}$ , $m_{y}$ and $m_{v}$ are the dimensions of state, input, output and disturbance vectors, respectively.

Developing techniques to handle actuator saturation in T-S fuzzy systems emerged as fuzzy control methods gained popularity in the 1990s. Early fuzzy control systems, including T-S models, focused on approximating nonlinear systems using a set of local linear models. Still, they initially did not account for the actuator limits that could lead to degraded performance. The need to address actuator saturation became apparent as control systems began to be implemented in practical applications, such as robotics, aerospace, and industrial processes. In response, researchers started incorporating anti-windup compensators and saturation-aware control strategies into T-S fuzzy systems to prevent instability and performance degradation due to actuator saturation. These methods were designed to modify the controller output when saturation occurs, allowing the system to continue operating effectively even when the actuator limits are reached.

In general, the actuator saturation can be considered as

u (t) = s a t (\tilde{u} (t)) \in ℜ^{m_{u}}

, where the elements

{\tilde{u}}_{k} (t)

of control input vector

\tilde{u} (t)

is defined as follows:

${\tilde{u}}_{k} (t) = s a t ({\tilde{u}}_{k}) = \{\begin{matrix} {\tilde{u}}_{kL} \\ {\tilde{u}}_{k} \\ {\tilde{u}}_{kH} \end{matrix} \begin{matrix} if {\tilde{u}}_{k} < {\tilde{u}}_{kL} \\ if {\tilde{u}}_{kL} \leq {\tilde{u}}_{k} \leq {\tilde{u}}_{kH} \\ if {\tilde{u}}_{kH} < {\tilde{u}}_{k} \end{matrix}$

(3)

where ${\tilde{u}}_{kL} < 0 < {\tilde{u}}_{kH}$ and $k = 1, 2, \dots, m_{u}$ .

Considering the T-S fuzzy system (1)–(2) with the actuator saturation defined in (3), some control methods such as robust control design [46,47,48,49,50,51,52], anti-windup compensators [53,54,55,56,57], and output feedback control [58,59,60,61,62] have been developed in the literature. In [46], the authors introduced a robust control design for uncertain nonlinear systems modeled by T-S fuzzy systems with actuator saturation. They derived robust control laws that consider actuator saturation, significantly advancing the application of T-S fuzzy systems in scenarios where uncertainties and actuator limits coexist. A robust fuzzy control strategy was introduced in [47] for handling actuator saturation in nonlinear systems represented by T-S fuzzy models. The authors employed LMIs to derive stability conditions that ensure the system remains stable and robust under varying actuator saturation levels. This paper provided a systematic approach to designing controllers that simultaneously handle nonlinearity and actuator constraints. Reference [48] addressed the stabilization of T-S fuzzy systems considering actuator saturation and time delay. It proposed a robust control design using LMIs to ensure system stability under these challenging conditions. The dual consideration of time delay and actuator saturation makes this work a valuable contribution to systems where time-sensitive control and physical input constraints are crucial. In [49,50], the authors used LMIs to design a robust controller, which guarantees the system’s performance under the given constraints. LMIs provide a systematic and computationally efficient way to derive sufficient conditions for stability and control performance. In [51], the control design was extended to handle multiplicative noises, which are disturbances that scale with the system’s state or control inputs. A fuzzy control strategy was proposed to ensure the system can operate effectively under these noise conditions, preventing performance degradation and instability. Furthermore, the researchers in [52] dealt with multiple constraints at the same time, including both time delays and actuator saturation. This makes it useful for complex real-time systems where delays and input limits are critical. The authors in [53] developed an adaptive anti-windup control method for T-S fuzzy systems with unmodeled dynamics and actuator saturation. The adaptive approach allows the controller to adjust dynamically in response to unknown system behaviors, ensuring stable control under saturation. A robust anti-windup dynamic output–feedback control strategy was presented in [54] for uncertain T-S fuzzy systems with actuator saturation. It used LMIs to design a controller that guarantees stability despite system uncertainties and input constraints. In [55], the authors addressed the problem of input delay and uncertain delays in T-S fuzzy systems with actuator saturation by developing an anti-windup compensator. The authors proposed a fuzzy control design using LMIs to ensure the system can handle both saturation and time delays without sacrificing performance. This kind of paper is significant for its focus on managing time delays and actuator limitations, making them applicable to various control systems.

To overcome disturbances from the external environment, an anti-windup control strategy was proposed based on the T-S fuzzy model [56,57]. These papers aimed to design a state feedback fuzzy controller that satisfies the physical limitations of controllers while simultaneously mitigating the effect of disturbances and uncertainties. Applying the robust fuzzy output feedback control strategy, the researchers in [58,59,60,61,62] have dealt with the control problem of actuator saturation and measurement noise in T-S fuzzy systems. The control strategy ensures the system remains stable and performs well, even with state and input limitations. These approaches use LMIs to ensure that the system remains stable even with noisy measurements and input constraints, providing a robust solution for noisy environments where actuator limitations are significant. In [61], the control problem of nonlinear active suspension systems was investigated by proposing a dynamic output–feedback interval type-2 fuzzy control strategy. The use of interval type-2 fuzzy logic enhances the system’s ability to handle uncertainties in road conditions and nonlinearities in vehicle dynamics, making it more robust than traditional fuzzy control approaches. By addressing actuator saturation and time delays, the authors ensure that the suspension system remains stable and high-performing even under constraints. Using LMIs to derive the control design guarantees computational efficiency and robustness, improving ride comfort and vehicle stability in modern vehicles. In [62], the authors deal with the fuzzy control problem of two-dimensional nonlinear systems using a fuzzy output feedback dynamic sliding mode controller. This paper introduced a dynamic sliding mode controller that integrates fuzzy logic with sliding mode control, a powerful technique known for its robustness in dealing with system uncertainties and external disturbances. The key innovation of [62] is the development of a fuzzy output feedback sliding mode controller for two-dimensional nonlinear systems. This is an interesting combination of fuzzy logic and sliding mode control, two individually powerful techniques that offer enhanced robustness and flexibility in controlling complex systems when combined. This adaptability is crucial for complex, evolving systems like those in industrial automation and image processing.

3. Control Methodology Subject to Output Constraint with Passivity Performance

The concept of passivity in control systems originates from the field of energy-based control theory. A system is said to be passive if it does not generate more energy than it consumes, ensuring stability and robustness. The introduction of passive constraints into T-S fuzzy systems was a natural evolution of passive control principles applied to nonlinear systems. Passivity-based control methods aim to ensure that the energy input to the system is always greater than or equal to the energy output, which is critical in systems like robotics, power electronics, and mechanical systems. The passive constraint in the design of T-S fuzzy controllers ensured the system’s energy-based stability, making it more resilient to external disturbances and uncertainties.

Considering the T-S fuzzy system (1)–(2) with corresponding membership functions, it can be represented as follows:

$d x (t) = \frac{\sum_{i = 1}^{n} ω_{i} (t) \{A_{i} x (t) + B_{u i} u (t) + B_{v i} v (t)\}}{\sum_{i = 1}^{n} ω_{i} (t)} = \sum_{i = 1}^{n} h_{i} (t) \{A_{i} x (t) + B_{u i} u (t) + B_{v i} v (t)\}$

(4)

where

$ω_{i} (t) = \prod_{q = 1}^{n} M_{i q} (x_{q} (t))$

(5)

$h_{i} (t) = ω_{i} (t) / \sum_{i = 1}^{n} ω_{i} (t), h_{i} (t) \geq 0 and \sum_{i = 1}^{n} h_{i} (t) = 1 .$

(6)

$ω_{i} (t)$ is the normalized weight of i-th rule, and it is calculated by membership values. The T-S fuzzy system (4) can usually be controlled by using the Parallel Distributed Compensation (PDC) method [63]. The PDC-based fuzzy controller can be written as

Controller Rule i:

Ψ_{1} (t)

M_{i 1}

and

Ψ_{2} (t)

M_{i 2}

and … and

Ψ_{q} (t)

M_{i q}

, then It can be represented as follows:

$u (t) = \frac{\sum_{i = 1}^{n} ω_{i} (t) \{F_{i} x (t)\}}{\sum_{i = 1}^{n} ω_{i} (t)} = \sum_{i = 1}^{n} h_{i} (t) F_{i} x (t)$

(8)

Substituting (8) into (4), one can obtain the corresponding closed-loop systems

$d x (t) = \sum_{i = 1}^{n} \sum_{j = 1}^{n} h_{i} (t) h_{j} (t) \{(\frac{G_{i j} + G_{j i}}{2}) x (t) + B_{v i} v (t)\}$

(9)

where $G_{i j} = A_{i} + B_{u i} F_{j}$ .

The following passive constraint is defined with the supplied function to achieve the attenuation ability from external disturbance energy for the fuzzy system (9).

Definition 1.

Considering the T-S fuzzy system (1)–(2) with external disturbance

v (t)

and output

y (t)

, if there exists a scalar

γ > 0

such that the following inequality is satisfied, then the controlled T-S fuzzy system (9) achieves the passivity constraints [64].

$\int_{0}^{t_{p}} y^{T} (t) S v (t) d t > γ \int_{0}^{t_{p}} v^{T} (t) v (t) d t$

(10)

for all $t_{p} \geq 0$ and $v (t) \neq 0$ . $t_{p}$ is the terminal time of control and it is positive and $S$ is a given constant matrix with compatible dimensions.

By comparing the energy defined by the supplied function (10) with Definition 1, it can be ensured that the consumed energy is always greater than the disturbance input energy. This ensures that the control system can effectively attenuate the disturbance energy. Passivity plays a significant role in dissipative systems such as mechanical systems, electrical grids, and robotic systems, where energy considerations are critical. By incorporating a passive constraint, the controller ensures that the system remains safe and stable despite uncertainties or external disturbances. Passivity inherently provides robustness in control systems. When designing controllers for T-S fuzzy systems, ensuring passivity means that the system can handle certain uncertainty or external disturbances without losing stability. This is particularly important for nonlinear systems like T-S fuzzy systems, where the interaction between subsystems or operating regions can lead to unpredictable behaviors. For example, in robotics, passivity ensures that a robot interacting with its environment, e.g., through contact or manipulation, behaves stably and predictably, regardless of the external forces it encounters.

According to the passivity constraint in Definition 1, some researchers have investigated many control methodologies, such as LMI-based control, output feedback control, and adaptive control, for the T-S fuzzy systems. In [65,66,67,68,69,70,71], the authors addressed time delays and disturbances in T-S fuzzy systems by developing a passivity-based control strategy using LMIs. The authors derived LMI conditions that ensure the system remains passive and stable even under external disturbances, making it a valuable method for time-delayed systems such as networked control systems. In [65], the design of an extended dissipative filter for T-S fuzzy systems with multiple time delays was addressed. The method of this paper ensured that performance measures, such as H_∞ control and passivity, are optimized, making it highly suitable for systems where robustness and reliability are critical, such as networked control systems, telecommunication networks, and industrial automation. One of the most important contributions of [66] is the use of passivity-based control to ensure the stability of stochastic time-delay systems. Passivity ensures that the system dissipates energy, contributing to its robustness, which is particularly valuable in systems affected by random fluctuations or time delays. References [67,68] presented the stabilization of fuzzy stochastic systems that are influenced by multiplicative noise. Such systems are inherently more challenging to control due to the stochastic nature of the disturbances, which depend on the system’s state. This study’s combination of multiplicative noise, time delays, and passivity constraints is critical for systems with inherent time delays, such as marine and mechanical systems. The novel stabilization technique advances the understanding of managing multiplicative noise in fuzzy systems and ensures robustness and passivity. This is particularly useful for industrial process control, robotics, and mechanical systems applications, where noise and delays can significantly affect system performance.

In [69], the authors contributed significantly by integrating H_∞ control with passive control in the context of nonlinear descriptor systems. This combination ensures that the system remains robust against disturbances and stable through energy dissipation, even in time-varying delays and sensor faults. The authors of [70] developed an observer-based mixed H_∞ and passive control strategy for T-S fuzzy semi-Markovian jump systems with time-varying delays. The combination of H_∞ control for robust disturbance rejection, passive control for energy-based stability, and sliding mode control for robustness to uncertainties provides a comprehensive solution for complex control systems. The observer-based controller design approach allows the system to operate without full-state measurements, making the strategy applicable to systems where some states are unmeasurable. Reference [71] provided a study to the field of control systems by proposing a finite-time mixed H_∞ and passivity control strategy for T-S fuzzy systems with time-varying delays and actuator faults. The combination of H_∞ control, which ensures robustness to disturbances, and passivity control, which guarantees energy-based stability, provides a powerful framework for achieving finite-time stability. The finite-time analysis ensures that the system reaches its desired state within a predefined time interval, making the control strategy highly relevant for systems requiring rapid responses. This paper makes a significant contribution to the field of power system control by proposing a functional observer-based T-S fuzzy control strategy to ensure the quadratic stability of power system synchronous generators. Using the T-S fuzzy control approach, a functional observer-based control strategy for power system synchronous generators was investigated in [72]. The functional observer-based approach ensures that the controller can access accurate information about the system’s internal states, enabling more precise control of the synchronous generator and ensuring quadratic stability. In [73,74,75], the authors developed passivity-based output feedback control and observer-based control methods for T-S fuzzy systems. The authors derive LMIs that ensure the passivity and stability of the system using output feedback and observer feedback technologies. Reference [74] proposed a dynamic output–feedback dissipative control strategy for T-S fuzzy systems with time-varying input delays and output constraints. The use of dissipative control ensures that the system remains stable by dissipating energy, while the output–feedback design allows the system to be controlled based solely on output measurements. The authors of [75] studied the control of nonlinear differential-algebraic interconnected systems by proposing a decentralized, performance-constrained state-estimated fuzzy control strategy. The decentralized approach ensures that each subsystem can be controlled independently, making the method scalable and applicable to large-scale systems. The controlled system not only remains stable but also meets key performance metrics, making the control method more relevant for practical applications.

Output feedback control plays a critical role in designing controllers for T-S fuzzy systems, as it addresses the practical limitations of control systems where full-state measurements may not be available. Output feedback control techniques for T-S fuzzy systems became more sophisticated, incorporating robust control, adaptive control, and handling time delays and input constraints. These developments were critical in making output feedback control applicable to a broader range of industrial and practical systems, where full-state measurements are often impractical or impossible to obtain. The use of LMIs became standard for deriving stability and performance guarantees, and researchers increasingly focused on methods that could handle real-time changes in system behavior. The papers investigated in [49,67,76,77,78] introduced a robust output feedback control approach for T-S fuzzy systems with uncertainties. It ensures that the system remains passive and stable despite uncertainties in the system’s dynamics. The designed controller can guarantee passivity and stability despite uncertainties, making it valuable for applications with unknown or varying system parameters. An output–feedback control strategy for fuzzy singularly perturbed systems was developed in [78] by using a nonhomogeneous stochastic communication protocol. The combination of output–feedback control and fuzzy logic allows the system to be controlled effectively using only output measurements, while the introduction of the stochastic communication protocol ensures that the system remains robust in the face of random communication delays and packet loss. In [79], the authors addressed the challenge of controlling uncertain singular fuzzy systems. Singular systems (also known as descriptor systems) are a class of systems that can model more complex dynamic behavior than regular state-space systems. These systems are often harder to observe and control due to their inherent structural complexity and potential for irregular behavior. This paper makes significant contributions to controlling uncertain singular T-S fuzzy systems by introducing an observer-based robust fuzzy controller that guarantees passivity and stability using LMIs. By integrating the observer design theory, the fault diagnosis under the H_∞ criterion was developed for nonlinear descriptor systems with stability analysis formulated in the LMI framework [80]. In addition, Reference [81] significantly contributed to the field of control systems by proposing a decentralized, estimated-state feedback fuzzy compensator for nonlinear interconnected descriptor systems with unmeasured states. By addressing the challenge of controlling large-scale interconnected systems in the presence of unmeasured states and system uncertainties, the paper provides a practical and scalable solution that is highly relevant for applications.

Adaptive fuzzy control methods were seen as powerful tools to cope with parameter uncertainties and nonlinear behaviors in many applications. As T-S fuzzy systems became more widely used for modeling complex systems, adaptive control began to evolve to handle various challenges, such as input constraints, external disturbances, and time delays. The goal was to ensure the system could adapt to real-time changes, improving robustness and stability. Researchers started integrating robust control techniques with adaptive control in T-S fuzzy systems. LMI-based approaches became popular due to their computational efficiency in solving the control design problem. As LMI constraints define feasible regions in optimization problems, which can be represented as convex hulls encompassing all possible solutions, these methods are closely related to convex hulls. From a practical application perspective, the convex hull represents the smallest convex boundary enclosing a set of points and helps define the region of feasible solutions or potential configurations in physical systems. References [82,83] addressed nonlinear T-S fuzzy systems with time delays using an adaptive control framework. The authors proposed a fuzzy-based control approach that adjusts to variations in system delays, ensuring closed-loop stability in dynamic environments. The control strategy is adapted in real-time to maintain stability, even when the system faces time delays, making it suitable for complex applications. In [84,85,86,87,88], the authors introduced an adaptive passivity control method for uncertain T-S fuzzy systems. The approach allows for real-time adjustments to the controller to maintain passivity and stability in the face of system uncertainties. This adaptive feature makes the controller suitable for systems with unknown or time-varying parameters. The authors of [86] contributed to controlling T-S fuzzy switched stochastic nonlinear systems by proposing a dissipativity-based composite anti-disturbance control strategy. The ability to handle multisource disturbances ensures that the system remains stable and performs well, even in the presence of a variety of disturbances from different sources. The dissipativity framework guarantees robust stability by ensuring that the system dissipates energy. In contrast, the use of LMIs ensures that the control design is computationally efficient and suitable for complex applications. A passivity-based adaptive fuzzy control strategy was developed in [87] for stochastic nonlinear switched systems. The incorporation of passivity guarantees system stability, even in the presence of stochastic disturbances and mode switching. The adaptive nature of the control system allows it to adjust in real time, ensuring that the system remains stable across different modes of operation and under varying conditions.

The investigation of a memory-based adaptive integral sliding-mode controller for fractional-order T-S fuzzy systems was studied in [88]. By using fractional calculus combined with T-S fuzzy modeling, the authors provided a more accurate control strategy for systems where past states influence current dynamics. The integral sliding-mode approach developed in [88] guarantees fast convergence to the desired trajectory, making the control strategy suitable for real-time applications in robotics, biomechanical systems, and energy management.

4. Control Methodology Subject to Input and Output Constraints with Optimal Scheme

The H₂ constraint has been a crucial part of control theory, particularly in simultaneously designing controllers for systems subject to input and output constraints. In the context of T-S fuzzy systems, the development of H₂ control emerged from the need to optimize the performance of nonlinear systems that can be approximated by a set of local linear models. The application of H₂ control in T-S fuzzy systems dates back to the late 1990s and early 2000s when researchers began applying classical linear control techniques to nonlinear fuzzy models. The early use of H₂ control in T-S fuzzy systems focused on minimizing the energy of the disturbances affecting the system. The H₂ norm measures the sensitivity of the system’s output to external inputs, and minimizing it reduces the overall influence of disturbances on the system’s behavior. This approach allowed for smoother and more predictable control, making it ideal for systems with small, random disturbances.

Based on the above PDC-based fuzzy controller (8), the H₂ constraint can usually be defined by the following cost function.

$J = \int_{0}^{\infty} \{x^{T} (t) Q x (t) + u^{T} (t) R u (t)\} d t$

(11)

By designing a PDC-based fuzzy controller (8) subject to minimizing the above cost function (11), the H₂ constraint for the closed-loop T-S fuzzy system (9) can be satisfied. As the complexity of systems modeled by T-S fuzzy logic increased, so did the need for more refined control methods. Early techniques involved solving H₂ optimization problems using LMIs, which allowed for a systematic way to design controllers that could handle the nonlinear dynamics of T-S fuzzy systems while maintaining the desired level of disturbance attenuation. The H₂ constraint plays an essential role in ensuring that T-S fuzzy systems can effectively handle disturbances and input constraints. Below are three main control methods used to address H₂ constraints in T-S fuzzy systems: LMI-based H₂ control, H₂/H_∞ mixed control, and MPC with H₂ optimization.

LMI-based H₂ control uses LMIs to formulate the H₂ control problem in T-S fuzzy systems. The system’s performance is optimized by minimizing the H₂ norm, which represents the system’s sensitivity to external disturbances. LMIs allow the H₂ control problem to be efficiently solved using convex optimization techniques. The method provides a systematic way to ensure system robustness by addressing uncertainty and disturbances. However, LMI-based methods may lead to conservative control designs, meaning the controller is more robust than necessary, possibly at the cost of performance. For large-scale systems, the LMI formulation can become computationally demanding. A multiple Lyapunov function (MLF) approach was introduced in [89] to achieve H₂ guaranteed cost control for the uncertain T-S fuzzy system. Lyapunov functions are commonly used to assess system stability. Using multiple Lyapunov functions allows for a more flexible and powerful method to handle systems with switching dynamics or time-varying uncertainties. In order to control the uncertain discrete-time fuzzy systems, a robust H₂ guaranteed cost control approach was developed [90]. The key contribution of this paper is the use of poly-quadratic Lyapunov functions to ensure the stability of the uncertain fuzzy system. The use of poly-quadratic Lyapunov functions allows the control strategy to handle time-varying uncertainties and switching dynamics more effectively than traditional methods. Reference [91] addressed the problem of H₂ control for T-S fuzzy systems with hard constraints. The rigid constraints refer to strict limits on the system’s inputs and outputs, which must be maintained to ensure safety, stability, or performance requirements in practical applications. Using LMI-based techniques allows for the derivation of stability and performance conditions in a computationally efficient way. In [92], the authors studied the problem of resilient guaranteed cost control for uncertain T-S fuzzy systems with time-varying delays and Markov jump parameters. The inclusion of time-varying delays and Markov jump parameters introduces additional complexity, as the system may experience random jumps between different modes of operation, with the system parameters changing according to a Markov process. The main contribution of [92] is to develop a methodology for the control of uncertain T-S fuzzy systems by proposing a resilient guaranteed cost control strategy that handles time-varying delays and Markov jump parameters.

The H₂/H_∞ mixed control approach combines both H₂ and H_∞ control objectives. While the H₂ control focuses on minimizing the energy of the disturbance, the H_∞ control aims to minimize the worst-case disturbance, enhancing robustness further. The H₂/H_∞ mixed control provides a balance between minimizing the disturbance’s energy and guaranteeing robustness against worst-case scenarios. This method allows designers to tailor the control strategy to specific system requirements by weighting the H₂ and H_∞ objectives. However, finding the right balance between the H₂ and H_∞ norms can be challenging and may require trial and error or advanced tuning methods. A robust adaptive mixed H₂/H_∞ interval type-2 fuzzy control method was developed in [93] for nonlinear uncertain systems. The combination of mixed H₂/H_∞ control with interval type-2 fuzzy logic provides a flexible and robust framework for managing uncertainties and nonlinearities. The adaptive nature of the control method [93] allows the system to adjust dynamically to changing conditions where system dynamics are time-varying or unpredictable. In [94], a fuzzy control approach was investigated to control the nonlinear dynamic systems by proposing a fuzzy mixed H₂/H_∞ sampled-data control strategy. The combination of H₂ control for optimizing system performance and H_∞ control for ensuring robustness to disturbances and uncertainties provides a balanced approach to controlling nonlinear systems. The use of sampled-data control makes the method highly applicable to digital control systems, where system measurements and control actions are taken at discrete intervals. A multi-objective H₂/H_∞ fuzzy control strategy for nonlinear mean-field stochastic jump-diffusion systems was developed in [95]. This approach ensures the system maintains robust performance (H_∞) while optimizing its response to random disturbances (H₂). The multi-objective approach allows for a balance between performance and robustness.

Reference [95] addressed mean-field stochastic effects, where individual components of the system interact with the overall system state. This is highly relevant for multi-agent networks and social dynamics applications, where collective behaviors impact individual decision-making and overall system performance. The authors of [96] proposed a robust mixed H₂/H_∞ fuzzy tracking control methodology for photovoltaic systems subject to asymmetric actuator saturation. Photovoltaic systems, used to convert solar energy into electricity, are nonlinear and can be influenced by environmental factors such as temperature and sunlight variability. This research is particularly relevant for photovoltaic systems and other renewable energy applications, where maintaining robust performance under uncertain environmental conditions and physical limitations is critical for long-term reliability and efficiency. Reference [97] deals with the problem of mixed H₂/H_∞ control for nonlinear stochastic systems using an event-triggered mechanism and a closed-loop Stackelberg game framework. An innovation of this paper is the introduction of an event-triggered mechanism for control-ling nonlinear stochastic systems. In event-triggered control, the system updates control actions only when certain predefined conditions are met rather than at regular time intervals, as in traditional control methods. In [98], the authors deal with the control of uncertain networked systems by proposing an adaptive event-triggered H₂/H_∞ control method that addresses both cyber-attacks and network imperfections. The integration of hybrid cyber-attack resilience, including DoS and deception attacks, ensures that the system remains robust even in the face of sophisticated cyber threats. The adaptive event-triggered mechanism optimizes communication and computational efficiency, making the system suitable for practical applications in IoT, smart grids, and industrial control systems.

MPC is a popular method that can be extended to include H₂ constraints for optimal control of T-S fuzzy systems. MPC optimizes the control input over a finite horizon, taking into account future disturbances and minimizing the H₂ norm. MPC anticipates future disturbances and optimizes the control input accordingly, leading to better performance in dynamic environments. It can simultaneously handle multiple constraints, e.g., input, state, and H₂ norm, making it suitable for complex systems. However, it should be noted that designing an MPC controller with H₂ constraints involves careful tuning of prediction and control horizons, which can be difficult for systems with high uncertainties. Integrating fuzzy logic into MPC [99] provides a robust approach for controlling nonlinear systems. The proposed fuzzy MPC is particularly useful for systems where traditional linear models are inadequate, and this optimization method makes it feasible to apply the approach in control applications. The balance between control accuracy, computational efficiency, and real-time feasibility makes the proposed method a valuable tool for improving the performance of fuzzy MPC in complex and nonlinear systems. Reference [100] addressed the problem of fuzzy-constrained predictive optimal control for high-speed trains with actuator dynamics. The control of high-speed trains is a complex task due to the nonlinear behavior of the system, safety constraints, and the need for precise speed and position control. The control design aims to optimize the H₂ performance of the high-speed train by minimizing deviations from the desired speed and position while ensuring safety. The predictive control approach allows the system to anticipate future changes and proactively adjust control actions, improving safety and efficiency. In [101], the authors introduced an approach that integrates Binary Particle Swarm Optimization (BPSO) with T-S fuzzy predictive control. This integration optimizes the controller parameters, improving the system’s ability to handle nonlinear dynamics and ensuring better performance in automotive systems. The proposed control design method combines BPSO, an optimization algorithm inspired by the social behavior of birds and fish, with T-S fuzzy predictive control. Its key contribution is the use of BPSO to optimize the parameters of the fuzzy predictive controller, improving both H₂ performance and robustness. The authors of [102] developed an approximation method for optimal MPC solutions using T-S fuzzy models. This method reduces the computational complexity of solving the MPC problem, making it feasible for real-time applications requiring fast decision-making. The proposed T-S fuzzy MPC strategy ensures that the control system optimizes performance by predicting future behavior and adjusting control actions accordingly. This approach minimizes deviations from the desired output and reduces control effort, ensuring optimal H₂ performance under both linear and nonlinear conditions. Reference [103] used a hierarchical scheme for optimization to manage the complexity of discrete-time large-scale systems by decomposing them into subsystems. It employs the Lyapunov-Razumikhin function, which simplifies the analysis compared to the Krasovskii method, particularly for systems with significant delays and disturbances. The system dynamics introduced in [103] are represented using fuzzy T-S models, which are well-suited for handling nonlinearities and complex interconnections within large-scale systems. The proposed model explicitly considers persistent disturbances and ensures robust performance under such conditions using LMIs. This research advanced the field by integrating hierarchical optimization, fuzzy modeling, and innovative stability analysis to address challenges in discrete-time large-scale system control.

In fuzzy control problems with both input and output constraints, the advantages and limitations of the three important methods are summarized in Table 1.

Since the H₂ performance criterion provides an efficient optimization approach with the LMI technique, an increasing number of authors have integrated the criterion to resolve the control problem of practical nonlinear systems [89,90,91,92,104,105,106]. Based on the T-S fuzzy model, stabilization was achieved for the nonlinear linked robot arm under the H₂ index by solving the optimization problem of LMI conditions [90,91]. The advantage of the optimization is to obtain a better balance between the system variables and control efforts. This criterion combination can develop a fuzzy controller to stabilize the control systems at a certain performance level while preserving the control costs. Therefore, the authors of [104] extended the H₂ fuzzy control approach to improve the energy conversion issue of wind turbines by minimizing the quadratic cost function (11). In [105], the control force was successfully reduced by incorporating the optimal H₂ performance into the fuzzy backstepping sliding mode control for helicopter flight control. The authors proposed a fuzzy controller design for spacecraft orbit tracking control with LMI-based H₂ performance to minimize fuel usage and deal with uncertainties [106]. It is worth pointing out that the authors also indicated the H₂/H_∞ mixed fuzzy control as a promising direction for future application research.

Simultaneously considering robustness and optimal control performance, the authors developed H₂/H_∞ mixed fuzzy controller designs based on the T-S fuzzy model for various applications of nonlinear systems [93,94,95,96,97,98,107,108,109]. In [93], the authors considered various types of disturbances as system noise and proposed a fuzzy control approach with H₂/H_∞ mixed optimization performance to simulate a 3-prismatic-spherical-prismatic parallel robot. Satisfying the H₂/H_∞ mixed performance was very helpful for control applications significantly affected by environmental disturbances, especially in the cases of truck–trailer systems [94] and autonomous vehicles [108] on uneven ground. For other engineering applications, authors solved the Maximum Power Point Tracking (MPPT) control problem of photovoltaic systems with rapid climatic changes by the H₂/H_∞ mixed fuzzy control method. Furthermore, some researchers extended the H₂/H_∞ mixed fuzzy control to various application fields [95,109]. In [95], the authors developed the fuzzy controller for financial dynamic systems to mitigate disturbances due to changes in national and international situations and oil prices with minimal funds. The authors applied the H₂/H_∞ mixed fuzzy control approach to the COVID-19 spread model, which helped to overcome the disturbance caused by the sudden increase in the number of infected people and minimize control effort [109].

The MPC method has been demonstrated to offer an efficient control strategy in various practical scenarios and to deal with performance constraints [2] effectively. Recently, many researchers have continued to promote the development of MPC methods. Owing to the advantages of fuzzy control theory in decision-making and robustness, MPC-based fuzzy control has provided a new direction for improving control performance over traditional MPC methods [27,29,99,100,101,102,103,110,111,112,113]. In [27], the authors developed a fuzzy model predictive control approach combined with sliding mode control to improve the performance of robot and buck converter systems under uncertainties and disturbances. Other authors also integrated MPC-based fuzzy control with BPSO to optimize fuel injection performance in automotive applications by minimizing the given cost function [101]. Notably, the MPC method can serve as a powerful approach in tracking control problems of time-varying trajectories to enhance the tracking performance by the proper design of the predictive mechanism. Many researchers have advanced the development of MPC-based fuzzy control to investigate the tracking control problem under performance constraints for a wide range of applications, including but not limited to endpoint tracking of truck trailers [29], velocity tracking of high-speed trains [100], MPPT of wind turbines [110], path-tracking of ground vehicles [111], and position tracking of semi-submersible platform [112].

From the above statements, it can be observed that LMI-based H₂ control is widely applied in systems requiring optimal performance with manageable computational demands. A common application is in aerospace systems, where H₂ control is used for flight path tracking and stability augmentation. For instance, aircraft control systems often rely on H₂ control to minimize deviations from desired flight paths while handling moderate disturbances. The H₂/H_∞ mixed control method is particularly valuable in systems requiring both performance optimization and high robustness to disturbances. This approach is used in automotive systems for active suspension systems and vehicle stability control, where road comfort and safety are prioritized. The method allows suspension systems to maintain smooth rides over varying terrains while ensuring stability under harsh driving conditions. MPC is frequently used in applications with complex constraints, such as chemical process control and oil refining. In these industries, MPC enables precise control over variables like temperature, pressure, and flow rates, optimizing efficiency while staying within strict operational constraints. Moreover, MPC can serve as a powerful method for time-varying tracking problems by predicting possible future motions. Table 1 shows that LMI-based H₂ control is computationally efficient, H₂/H_∞ mixed control offers a balanced approach to performance and robustness, and MPC excels at managing complex constraints. The choice among them depends on specific application requirements, such as robustness, computational resources, and constraint-handling capabilities.

Future research in this field holds significant promise, with ongoing integration into various emerging developments while handling input and output constraints via the T-S fuzzy model. The potential directions can be categorized into the following aspects.

(i) Artificial Intelligence (AI): Some researchers have developed a fuzzy control approach based on the T-S fuzzy model from the perspective of AI and the learning process [114,115,116]. However, these papers only provide an initial investigation, and the issue of input and output constraints has not been explored. For AI control applications that are destined to evolve, the combinations of input and output constraints are essential for complying with physical limitations and handling disturbance problems in practical control scenarios.

(ii) Network Systems: Due to the rapid progression of the internet, network control systems have emerged over the past decades. In [117,118,119], the researchers have, respectively, developed the fuzzy controller designs based on the H_∞, H₂/H_∞ mixed constraint and MPC based on the T-S fuzzy model. It can be observed that more advanced fuzzy control approaches for solving more complex control problems can still be explored in the future by interested researchers.

(iii) Large-Scale Systems: Nowadays, practical applications often involve more than one unit of equipment. In addition, control systems are required to have the ability to handle big data. Recently, T-S fuzzy-model-based constrained control methods have continuously emerged for large-scale systems [120], multi-agent systems [121], and big data systems [122]. Since these systems are highly related to the future development of unmanned vehicles, input and output constraints have become more crucial in controller design in practical working environments.

(i) Interval Type-2 (IT2) Fuzzy Model: Superior to the typical type-1 fuzzy model, the IT2 fuzzy model offers better capabilities to express uncertain factors in practical applications. Based on the IT2 fuzzy model, some researchers have developed performance-constrained fuzzy control methods [51,61]. Moreover, passivity, H₂/H_∞ mixed performance, and MPC have been considered in the IT2 fuzzy control approach of [93,123,124], and [125], respectively. As the control precision requirements have increased nowadays, the effect of uncertainties has inevitably become more important to handle.

(ii) Fractional Order Fuzzy Model: In recent years, fractional order systems have been widely investigated due to the more accurate representation of some physical phenomena in engineering applications, such as damping, diffusion, delayed responses, and so on. Several authors have integrated the H₂ and H_∞ performance into the fuzzy controller design for this type of non-integer-order systems [126,127]. However, this area remains an open issue that can still be further improved.

(iii) Singular Fuzzy Model: Singular systems provide an ability to describe the singular behavior in engineering applications for controller design. Notice that the singular phenomenon will occur in many control situations such as the unobservable variables and infinitesimal resistance in electronic systems. Based on the T-S fuzzy singular model, the control issues of nonlinear singular systems have been investigated subject to multiple performance constraints [30,31,52,69,77,79,85]. Nevertheless, the controller design process can still be further improved in efficiently handling these constraints under singularity.

(i) Sliding Mode Control: Throughout the history of the control field, sliding mode control has provided an efficient strategy that is insensitive to uncertainties and disturbances, since system dynamics can achieve convergence by following a predesigned surface. Therefore, many researchers have developed fuzzy sliding mode control by integrating performance requirements and MPC in [27,70,84,85,88,128,129,130]. Owing to the design flexibility of sliding mode control theory, more effective fuzzy sliding mode control methods are expected to be further explored.

(ii) Event-Triggered: Accompanied by network systems and multi-agent systems, the bandwidth usage for signal transmission has significantly increased. The event-triggered control strategy has recently been widely developed to reduce transmitted signals and conserve bandwidth. By designing an event-triggered mechanism, control systems can regulate the transmission of control commands. Based on the T-S fuzzy model, some researchers have enhanced control performance through constrained fuzzy event-triggered control methods [85,97,98,131,132]. In the future, the engineering applications may become more complex, which will also lead to increased requirements for adjusting and designing the constrained event-triggered mechanism.

(iii) Cyber-Attack: In addition to bandwidth usage, cyber-attacks are also a critical issue in practical control scenarios for systems whose control relies entirely on networks, such as multi-agent systems and interconnected systems. Denial of Service (DoS) and deception attacks are the two most common cyber-attacks. Against DoS attacks, researchers often apply the switching controller or event-triggered controller design method to handle the different cases caused by the attacks [133,134]. Researchers typically design compensations within the controller for deception attacks to counteract irregular changes [135]. However, both control issues require additional design modifications in the controller structure, making the design process more complex when combined with performance constraints. The development of more efficient, constrained, and anti-attack fuzzy control methods continues to be a valuable issue.

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Wen-Jer Chang www.mdpi.com

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A Comprehensive Survey on Advanced Control Techniques for T-S Fuzzy Systems Subject to Control Input and System Output Requirements

2. Control Methodology Subject to Input Constraint with Actuator Saturation

3. Control Methodology Subject to Output Constraint with Passivity Performance

4. Control Methodology Subject to Input and Output Constraints with Optimal Scheme

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2. Control Methodology Subject to Input Constraint with Actuator Saturation

3. Control Methodology Subject to Output Constraint with Passivity Performance

4. Control Methodology Subject to Input and Output Constraints with Optimal Scheme

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