Fuzzy Descriptor Systems Research Articles

Admissibility Analysis and Controller Design Improvement for T-S Fuzzy Descriptor Systems

In this paper, a stability analysis and the controller improvement of T-S fuzzy Descriptor system are studied. Firstly, by making full use of the related theory of fuzzy affiliation function and combining the design method of fuzzy Lyapunov function with the method of inequality deflation, a stability condition with wider admissibility and less system conservatism is proposed. The advantage of this method is that it is not necessary to ensure that each fuzzy subsystem is progressively stable. We also maximise the boundary of the derivatives of the affiliation function mined. Secondly, a PDC controller and a Non-PDC controller are designed, and the deflation conditions for the linear matrix inequalities of the two controllers are constructed. Finally, some arithmetic simulations and practical examples are given to demonstrate the effectiveness of the method studied in this paper, and the results obtained are less conservative and have larger feasible domains than previous methods.

Robust-optimal control of rotary inverted pendulum control through fuzzy descriptor-based techniques

Expanding upon the well-established Takagi–Sugeno (T–S) fuzzy model, the T–S fuzzy descriptor model emerges as a robust and flexible framework. This article introduces the development of optimal and robust-optimal controllers grounded in the principles of stability control and fuzzy descriptor systems. By transforming complicated inequalities into linear matrix inequalities (LMI), we establish the essential conditions for controller construction, as delineated in theorems. To substantiate the utility of these controllers, we employ the rotary inverted pendulum as a testbed. Through diverse simulation scenarios, these controllers, rooted in fuzzy descriptor systems, demonstrate their practicality and effectiveness in ensuring the stable control of inverted pendulum systems, even in the presence of uncertainties within the model. This study highlights the adaptability and robustness of fuzzy descriptor-based controllers, paving the way for advanced control strategies in complex and uncertain environments.

Robust H∞ Fuzzy Approach Design via Takagi‐Sugeno Descriptor Model. Application for 2‐DOF Serial Manipulator Tracking Control

This paper focuses on trajectory tracking control for robot manipulators. While much research has been done on this issue, many other aspects of this field have not been fully addressed. Here, we present a new solution using feedforward controller to eliminate parametric uncertainties and unknown disturbances. The Takagi‐Sugeno fuzzy descriptor system (TSFDS) is chosen to describe the dynamic characteristics of the robot. The combination of this fuzzy system and the robust H∞ performance makes the system almost isolated from external factors. The linear matrix inequalities based on the theory of Lyapunov stability is considered for control design. The proposed method has proven its effectiveness through simulation results.

Robust [formula omitted]-dissipative event-triggered sliding mode control of discrete T-S fuzzy descriptor systems with partial unmeasurable premise variables

In this paper, the admissibility analysis and dissipative sliding mode control (SMC) based on event-triggered mechanism (ETM) are studied for discrete T-S fuzzy descriptor systems (D-TSFDS). The problem of partial unmeasurable premise variables in D-TSFDS is dealt with by nonlinear dynamic programming. Based on ETM, a sliding mode surface based on measurable premise variables is proposed for the first time. Since the delay between the sensor and the controller in ETM, a Lyapunov-Krasovskii function is used to acquire the sufficient conditions for the closed-loop system to be admissible with dissipative performance. A new SMC law is given to ensure the reachability conation. The simulation results demonstrate the effectiveness of the proposed method.

Fuzzy switching function-based sliding mode controller design for T-S fuzzy descriptor systems

The design of switching functions is extremely important in the design of sliding mode controllers in that it closely relates to the solvability of the sliding mode control problems. This paper is concerned with the design problems of switching functions and fuzzy sliding mode controllers for T-S fuzzy descriptor systems. First, a generalized regular form of T-S fuzzy descriptor systems is defined. Then the design method of the linear switching function-based fuzzy sliding mode controller is developed and the result obtained indicates a relationship between the design parameter in the switching function and the gain matrix in the controller of T-S fuzzy systems. From this relationship, a new framework of the switching function design for T-S fuzzy descriptor systems is formulated. Moreover, the design method of the fuzzy switching function-based sliding mode controllers is proposed to exemplify the proposed framework. It is shown that the proposed fuzzy switching function-based method is less conservative than the linear switching function-based method from the viewpoint of the solvability of the sliding mode control problems. Finally, the results obtained are verified through four illustrative examples.

Positivity and Stability Analysis of T-S Fuzzy Descriptor Systems With Bounded and Unbounded Time-Varying Delays.

This work focuses on the stability analysis of positive Takagi-Sugeno (T-S) fuzzy descriptor systems with time-varying delays. An equivalent augmented system is constructed to investigate the positivity and stability of T-S fuzzy descriptor time-delay systems. By using this transformed system, a necessary and sufficient positivity condition is first derived for systems, which can be verified by linear programming (LP). Then, based on the positivity of T-S fuzzy descriptor systems, a sufficient condition is put forward for the asymptotic stability of systems with bounded and unbounded time-varying delays. Finally, several examples are provided to show the effectiveness of the obtained results.

Dissipativity‐based adaptive integral sliding mode control of Takagi‐Sugeno fuzzy descriptor systems

This paper focuses on the problem of adaptive integral sliding mode control (ISMC) with dissipativity for a type of complex non-linear system represented by Takagi-Sugeno (T-S) fuzzy descriptor models. First, T-S fuzzy descriptor systems with different derivative matrices are transformed into augmented systems that have the same properties as the original systems, and an integral sliding mode surface function is designed considering the descriptor matrices. Then, based on the dissipativity theory and the non-parallel distributed compensation (non-PDC) method, ISMC strategies and dissipativity-based ISMC strategies are proposed, and the system uncertainties and external disturbances with unknown upper bounds are addressed with an adaptive controller. According to the Lyapunov theory and the dissipativity theory, a fuzzy Lyapunov function is designed, and the obtained criteria are presented in the form of strict linear matrix inequalities (LMIs); this approach guarantees that the closed-loop system is asymptotically stable and strictly ( Q , R , S ) − α $(\mathcal{Q},\mathcal{R},\mathcal{S}) - {{\alpha}}$ dissipative. Moreover, the controller gain matrices and Lyapunov matrices are decoupled by introducing the corresponding auxiliary variables and employing Finsler's lemma. Finally, two examples are provided to illustrate the effectiveness and merit of the proposed method.

Admissibility and admissibilization for T–S fuzzy descriptor systems with partially unknown membership functions

In this paper, the admissibility analysis and controller design issues for a class of nonlinear descriptor systems described by the T–S descriptor fuzzy model with partially unknown membership functions are discussed. New sufficient conditions are first given to ensure the admissibility of such fuzzy descriptor systems. To fully utilise the information of membership functions, the known and unknown parts of membership functions are handled by staircase functions and the bounds of the unknown part, respectively. Different from the traditional parallel distribution compensation (PDC) method, the state-feedback controller design condition is given under an imperfect premise matching method via linear matrix inequalities (LMIs). Some examples are given to show the validity of the proposed methods.

Non-Fragile Observer-Based Adaptive Integral Sliding Mode Control for a Class of T-S Fuzzy Descriptor Systems With Unmeasurable Premise Variables.

The issue of non-fragile observer-based adaptive integral sliding mode control for a class of Takagi–Sugeno (T-S) fuzzy descriptor systems with uncertainties and unmeasurable premise variables is investigated. General nonlinear systems are represented by nonlinear T-S fuzzy descriptor models, because premise variables depend on unmeasurable system states and fuzzy models have different derivative matrices. By introducing the system state derivative as an auxiliary state vector, original fuzzy descriptor systems are transformed into augmented systems for which system properties remain the same. First, a novel integral sliding surface, which includes estimated states of the sliding mode observer and controller gain matrices, is designed to obtain estimation error dynamics and sliding mode dynamics. Then, some sufficient linear matrix inequality (LMI) conditions for designing the observer and the controller gains are derived using the appropriate fuzzy Lyapunov functions and Lyapunov theory. This approach yields asymptotically stable sliding mode dynamics. Corresponding auxiliary variables are introduced, and the Finsler's lemma is employed to eliminate coupling of controller gain matrices, observer gain matrices, Lyapunov function matrices, and/or observer gain perturbations. An observer-based integral sliding mode control strategy is obtained to assure that reachability conditions are satisfied. Moreover, a non-fragile observer and a non-fragile adaptive controller are developed to compensate for system uncertainties and perturbations in both the observer and the controller. Finally, example results are presented to illustrate the effectiveness and merits of the proposed method.

Admissibility and Design Issues for T-S Fuzzy Descriptor Systems With Perturbed Derivative Matrices in the Rules

This article mainly investigates the admissibility analysis and the parallel distributed compensator (PDC) for fuzzy descriptor systems with perturbed derivative matrices in the rules. The standard Takagi–Sugeno (T–S) fuzzy model is generalized into a fuzzy descriptor model with the perturbed derivative matrices existing in each rule, which can be used to represent a larger class of nonlinear and/or uncertain systems. Based on linear matrix inequalities (LMIs) technique, a new admissibility analysis condition for the considered fuzzy descriptor system is first presented, where the new results involve slack matrices for lessening the conservatism. Furthermore, we further cope with the fuzzy PDC synthesis associated with admissibility and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ρ</i> -admissibility assurance for the resulting closed-loop fuzzy descriptor systems. Since all the proposed criteria can be explicitly formulated in terms of LMIs or parametric LMIs, we can handily verify them via current LMI solvers. Finally, illustrative examples are given to demonstrate that the proposed approach is valid and effective.

Unknown input observer based robust control for fuzzy descriptor systems subject to actuator saturation

For the stabilization problem of non linear system, a new control based on unknown input observer (UIO) is proposed to consider unknown inputs and actuators saturation. The proposed control method consists of designing an H∞ control for nonlinear systems based on TS fuzzy systems subject to unknown inputs and actuators saturation using the idea of UIO. To estimate state variables and unknown inputs, the unknown input observer is investigated. The saturation effect is considered through the polytopic representation. The descriptor representation is used to derive the new robust stabilization conditions of closed-loop system with unknown and constraints inputs. Both the UIO and controller gains are formulated as an optimization problem, which is expressed under Linear Matrix Inequalities (LMI). Simulations results are given to show the effectiveness of the proposed method.

Sliding mode observer based control for T-S fuzzy descriptor systems

&lt;p style='text-indent:20px;'&gt;In this paper, the problem of sliding mode observer (SMO) based sliding mode control (SMC) for nonlinear descriptor delay systems is studied. First, based on the T-S fuzzy dynamic modeling technique, the nonlinear descriptor system is transformed into a combination of local linear models. Then, a integral-type sliding surface (ITSS) based SMO is constructed for the error system. In the sequel, sufficient linear matrix inequality (LMI) conditions are established to ensure the admissibility of the sliding motions and obtain the observer gain matrix. Furthermore, two novel SMC laws are developed to ensure the reachability conditions and stabilize the descriptor systems. Finally, simulations are provided to show the effectiveness of the method.&lt;/p&gt;

Admissibility analysis of Takagi–Sugeno fuzzy descriptor systems with time-varying delay via nonorthogonal free-matrix-based inequality

This paper studies the admissibility of Takagi–Sugeno (T-S) fuzzy descriptor systems with time-varying delay. To analyze the admissibility, various free-matrix-based inequalities (FMIs) are used in the existing works, since they directly produce bounds for integral terms without time-varying delay in denominator. However, the slack matrices in these inequalities are incomplete with some zero components, which may lead to incomplete coupling system information. In this case, conservatism is unavoidable. To remove the issue, a nonorthogonal FMI (NFMI) is proposed by constructing nonorthogonal polynomials with auxiliary scalars and auxiliary vectors, which includes some existing FMIs as its special cases. Based on the NFMI, a less conservative admissibility criterion is obtained for T-S fuzzy descriptor system. Two examples are illustrated to demonstrate the effectiveness of the obtained results.

H∞ and Passive Fuzzy Control for Non-Linear Descriptor Systems with Time-Varying Delay and Sensor Faults

In this paper, the problem of reliable control design with mixed H∞ /passive performance is discussed for a class of Takagi–Sugeno TS fuzzy descriptor systems with time-varying delay, sensor failure, and randomly occurred non-linearity. Based on the Lyapunov theory, firstly, a less conservative admissible criterion is established by combining the delay decomposition and reciprocally convex approaches. Then, the attention is focused on the design of a reliable static output feedback (SOF) controller with mixed H∞ /passive performance requirements. The key merit of the paper is to propose a simple method to design such a controller since the system output is subject to probabilistic missing data and noise. Using the output vector as a state component, an augmented model is introduced, and sufficient conditions are derived to achieve the desired performance of the closed-loop system. In addition, the cone complementarity linearization (CCL) algorithm is provided to calculate the controller gains. At last, three numerical examples, including computer-simulated truck-trailer and ball and beam systems are given to show the efficacy of our proposed approach, compared with existing ones in the literature.

Sliding-Mode Observer-Based Fault Reconstruction for T-S Fuzzy Descriptor Systems

This article discusses the problem of sliding-mode observer (SMO)-based fault reconstruction for T-S fuzzy descriptor systems by using the RBF neural network. First, a descriptor learning SMO, without the observability condition of fast subsystem, is developed such that the reconstruction of fault and states is achieved, simultaneously. Then, a sufficient linear matrix inequality (LMI) condition is presented to guarantee the admissibility of the sliding motion. Meanwhile, the LMI condition ensures the boundedness of the state and fault estimation errors. By utilizing robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> strategy, the impact of disturbance on the system can be reduced. Moreover, an RBF neural network-based sliding-mode control (SMC) strategy is adopted to estimate the nonlinearity, unknown positive constants, and ensure the reachability condition, simultaneously. Finally, an example is presented to verify the efficacy of our approach.

Improvements on stability conditions and control design of Takagi–Sugeno fuzzy descriptor systems

Several methods to improve stabilization conditions of Takagi–Sugeno fuzzy descriptor systems (TSFDS) are presented. First, we prove that with a modified non–quadratic fuzzy Lyapunov function and a PDC controller, stabilization problems of TSFDS are reformulated as checking negativity of ”triple” fuzzy summations, and then relaxed methods of T–S fuzzy systems can be directly applied to descriptor systems. In the sequel, two relaxed methods are extended to TSFDS based on slack decision variables and Polya’s Theorem, respectively, but these conditions are only sufficient. Second, we design a non–quadratic fuzzy Lyapunov function which simultaneously consists of membership functions of derivative matrices and state matrices, and it generalizes previous related Lyapunov functions. Then with a non–PDC controller, not only sufficient but asymptotically necessary conditions for TSFDS are presented based on Polya’s theorem. All conditions are cast into LMIs, and simulation examples illustrate improvements and effectiveness of these methods.

Robust [formula omitted] observer-based sliding mode control for uncertain Takagi–Sugeno fuzzy descriptor systems with unmeasurable premise variables and time-varying delay

This work focuses on the observer-based sliding mode control (OBSMC) synthesis for non-linear T-S fuzzy descriptor time-varying delay systems, which subject to uncertainties and unmeasurable premise variables (UPVs). Due to the state variables are difficult to completely accessible, a sliding mode observer (SMO) is designed to estimate the unmeasurable variables, and the nonlinear systems are represented to an equivalent multiple model (MM) forms under the MM formulation with UPVs. Then two sliding surfaces are proposed for the observer system and error system respectively to ensure stabilization, which can effectively deal with the nonlinearity and uncertainties as well. By the utilization of the linear matrix inequality (LMI) technique, we present the stability criterions such that the sliding mode dynamics with H∞ performance are proved to be admissible. Moreover, the SMC is obtained so we can drive the system trajectories to the predefined sliding surface in a finite time. At last, three simulation examples are provided to certificate the availability of our approach.

Disturbance estimation and compensation for discrete-time large-scale IT-2 T–S fuzzy descriptor systems

This paper studies disturbance estimation and compensation for discrete-time large-scale nonlinear descriptor systems with unknown measurement noises. Interval type-2 (IT-2) Takagi–Sugeno (T–S) fuzzy model is used to represent nonlinear dynamics, where fuzzy representation is assumed to be appearing not only in both the state and input matrices but also in the matrix of derivative state. First, by using a novel model transformation, the fuzzy representation in the derivative matrix is formulated into the linear one. Then, an augmented fuzzy observer is introduced to implement a synchronous estimation for the system state and the unknown measurement noise, which guarantees that reachable sets of the closed-loop system are bounded by a given ellipsoid. Moreover, a compensation-based controller is employed to remove or relieve the uncertainties induced by measurement noises. Finally, the validity of the obtained theories is testified by a numerical example.

Reachable set bounding of output-feedback control for discrete-time large-scale IT-2 fuzzy descriptor systems using distributed event-based broadcasts

This paper studies the distributed event-based broadcasting of output-feedback control for discrete-time large-scale fuzzy descriptor systems subject to reachable set bounding. First, each nonlinear subsystem is represented by the interval type-2 fuzzy model, where fuzzy representation is assumed to be appearing not only in both the state and input matrices but also in the derivative matrix. By using a descriptor redundancy approach, the fuzzy representation in the derivative matrix is reformulated into the linear one. Then, we introduce an output-feedback fuzzy controller with distributed event-based broadcasting, where all subsystems communicate with each other via a distributed network, and each subsystem broadcasts its decision-making of transmission based on a prescribed event. Two event-based mechanisms (EBMs) are respectively proposed to examine when the system output and fuzzy premise variable should be transmitted to the controller. Moreover, by further employing a descriptor redundancy representation, combined with reachable set analysis method, it will be shown that the proposed fuzzy controller guarantees that the estimation error is bounded within the ellipsoidal boundary while communication data are transmitted to the controller as little as possible, and sufficient condition for designing the desired controller is derived in terms of linear matrix inequalities (LMIs). Finally, a simulation study is given to show the effectiveness of the proposed method.

Finite frequency fault estimation observer design for T-S Fuzzy discrete-time descriptor systems

This paper deals with the problem of fault estimation observer design for T-S fuzzy nonlinear discrete-time descriptor systems in the finite frequency domain. First, a novel fault estimation observer for the augmented descriptor system is given. Based on Parseval's theorem and Lyapunov theory, the performance in the finite frequency domain is derived via time-domain method. Then, new sufficient conditions of admissibility and robust performance index for the dynamic error system are obtained as bilinear matrix inequalities. Meanwhile, an iterative linear matrix inequality algorithm is given to obtain the optimal solution. Finally, an example is given to demonstrate the effectiveness of our method.