Non-quadratic Lyapunov Function Research Articles

Semiglobal exponential stability of the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm for constrained optimization

We consider the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm, also known as the first-order Lagrangian method, for constrained optimization problems involving a smooth strongly convex cost and smooth convex constraints. We prove that, for every given compact set of initial conditions, there always exists a sufficiently small stepsize guaranteeing exponential stability of the optimal primal-dual solution of the problem with a domain of attraction including the initialization set. Inspired by the analysis of nonlinear oscillators, the stability proof is based on a non-quadratic Lyapunov function including a nonlinear cross term.

Proportional multi-integral observer design for Takagi–Sugeno systems with unmeasurable premise variables: Conservatism reduction via polyquadratic Lyapunov function

In this paper, we present the design of a new unknown-input observer for Takagi–Sugeno fuzzy systems with unmeasurable premise variables. The work proposes an improvement of the simultaneous state and unknown-input observer. The main idea is to consider a non-quadratic Lyapunov function in order to reduce the conservatism of its quadratic counterpart. Moreover, the mean value theorem is used to express the error dynamics in a way that reduces the conservatism of the bounded terms assumptions. The stability conditions are formulated as Bilinear Matrix Inequalities. To address these, we propose an iterative algorithm based on linear matrix inequalities, which transforms the Bilinear form into a sets of linear matrix inequalities. Finally, two examples are given to highlight the performance of the proposed observer.

Relaxed local nonquadratic stabilization for nonhomogeneous hidden Markov jump fuzzy systems over non-closed system operating range

This paper is dedicated to investigating the problem of relaxed local nonquadratic stabilization of nonhomogeneous hidden Markov jump fuzzy systems (HMJFSs) over a non-closed system operating range in the continuous-time domain. Within the framework of the mode-dependent nonquadratic Lyapunov function, the proposed method formulates the local stabilization conditions in a less conservative manner. Different from other recent studies, our formulation process is carried out without assumptions on the additional state restrictions required to establish a closed operating range, which opens the possibility to further increase the size of the domain of attraction. Furthermore, this method provides a systematic way to solve the nonconvex problem arising from the asynchronism between hidden and observed modes through augmented state-based congruent transformations. Not only that, significant progress is made in further reducing conventional conservatism and computational complexity by removing unnecessary auxiliary conditions. Finally, two numerical examples are given to show the validity of the proposed method.

Observer design for Takagi–Sugeno fuzzy systems with unmeasured premise variables: Conservatism reduction using line integral Lyapunov function

In this paper, a non-quadratic Lyapunov function is employed to reduce conservatism in a nonlinear observer designed for a class of continuous-time Takagi–Sugeno fuzzy systems with unmeasurable premise variables. This structure presents greater challenges compared to systems with measured variables. To overcome this issue, we utilize the mean value theorem and the sector nonlinearity transformation to convert the nonlinear error dynamics into a linear parameter-varying system. Moreover, we introduce the line integral Lyapunov function, which based on the integral of the membership functions, in order to ensure the global stability of the fuzzy systems under consideration. The use of this function offers several notable advantages over conventional quadratic forms, including a reduction in conservatism. Additionally, this type of functions constructed in a manner that eliminates the need for generating time derivatives of the membership functions, thereby simplifying calculations and analysis in comparison to other nonquadratic functions. Furthermore, it also enables capturing the system’s behavior along a trajectory. The stability conditions are more relaxed and expressed as linear matrix inequalities, which can be solved using a linear programming approach through specialized software tools. To validate the effectiveness of the proposed methodology, we conducted a hardware-in-the-loop test using a flexible joint robot. The obtained results clearly underscore the success of the proposed approach.

Relaxed nonquadratic H∞ robust constrained event-triggered fuzzy controller design for discrete-time nonlinear systems

This paper introduces an [Formula: see text] robust constrained event-triggered fuzzy controller for discrete-time nonlinear systems in the presence of system parameter uncertainties, the parameter uncertainties of the output matrices and the disturbances, simultaneously. To reduce the communication resources and improve the robust performance, the event-triggered mechanism is incorporated with the [Formula: see text] robust fuzzy controller design. Moreover, by employing the fuzzy modeling of the actuator saturation and a nonquadratic Lyapunov function (NQLF), the ultimate bounded stability (UBS) of the closed-loop system is guaranteed subject to the amplitude limitations of the control signal. This means, a new linear matrix inequality (LMI) problem is derived to ensure the robustness and constraints in an offline scheme based on the event-triggered strategy. Besides, to meet the practical problem to further improve the controlled system performance, the control input rate constraint is also considered and a new LMI condition is provided with no online computational burden. Therefore, the computational burden of the proposed controller is greatly reduced with less conservatism. Two simulation examples including a numerical dynamic system and a truck–trailer system are employed to illustrate the less conservatism and computational complexity of the proposed method compared with the existing approaches.

Observer-Based Controller Using Line Integral Lyapunov Fuzzy Function for TS Fuzzy Systems: Application to Induction Motors

This paper deals with the stabilization problem of a nonlinear system described by a Takagi–Sugeno fuzzy (TSF) model with unmeasurable premise variables via a robust controller. Applying the sector nonlinearity techniques, the nonlinear system is represented by a decoupled fuzzy model. Then, we design a robust observer-based controller for the obtained fuzzy system by utilizing the differential mean value approach. The observer and controller gains are obtained by the separation principle, in which the problem is solved in the sum of linear matrix inequalities (LMIs). The paper presents two main contributions: A state feedback controller is designed using differential mean value (DMVT) which ensures robust stabilization of the nonlinear system. Additionally, the Luenberger observer is extended to the Takagi–Sugeno fuzzy models. The second contribution is to reduce conservatism in the obtained conditions, a non-quadratic Lyapunov function (known as the line integral Lyapunov fuzzy candidate (LILF)) is employed. Two examples are provided to further illustrate the efficiency and robustness of the proposed approach; specifically, the Takagi–Sugeno fuzzy descriptor of an induction motor is derived and a robust observer-based controller applied to the original nonlinear system.

𝑯∞ reliable fuzzy control for vehicle dynamics stability

This article introduces a reliable control scheme for a four-wheel vehicle. This scheme assumes that actuators fail and external disturbances occur to the system. In contrast to existing results, this study assumes the actuator fault model includes linear and nonlinear terms, and an output feedback controller is designed to improve vehicle stability and handling when actuators fail. Using Takagi-Sugeno (T-S) fuzzy models, a reliable fuzzy static output feedback (SOF) controller is designed to address the nonlinear aspect of the system. Based on the non-quadratic Lyapunov function with auxiliary matrices, less conservative sufficient conditions are established such that the closed-loop system is stable with a γ level of H_∞ performance against external disturbances. Furthermore, using an appropriate model transformation, a set of linear matrix inequalities (LMIs) is formulated to synthesize the controller gains. The proposed scheme is then tested using numerical experiments to demonstrate potential applications and validate its effectiveness.

Robust fault-tolerant tracking control for a class of T-S fuzzy systems subject to actuator failure and external disturbance

Based on the Takagi-Sugeno (T-S) fuzzy model approach, this study discusses the robust fault-tolerant tracking controller design for non-linear systems affected by external disturbances, uncertainties, and actuator failures. In contrast to existing results, this study assumes the actuator fault model includes linear and nonlinear terms, and a state feedback controller is designed to improve the tracking and stability of the system when actuators fail. Using a non-quadratic Lyapunov function, new sufficient conditions for L_2-gain tracking performance analysis are derived to determine simultaneously the minimal level of the L_2-gain and controller gains. The robustness of the proposed approach is also investigated. An illustration of the theoretical developments is provided by a Duffing forced oscillation system.

Optimal constrained integral sliding mode control design for fuzzy-based nonlinear systems

This study introduces a novel H∞ optimal constrained integral sliding mode control (OCISMC) for nonlinear systems due to the matched/unmatched external disturbances based on Takagi–Sugeno (TS) fuzzy models. Based on the Lyapunov function, the appropriate conditions are provided to reach the sliding mode from any initial condition and robustness against the matched disturbances is guaranteed. The optimal nominal part of the proposed OCISMC is designed based on an online optimization problem. To reach robustness against the unmatched disturbances and constrained controller, the H∞ performance and the limitations of the control signal are included in the design procedure of the suggested OCISMC. Furthermore, by considering a non-quadratic Lyapunov function (NQLF), a new linear matrix inequality problem with less conservatism is derived. Finally, two examples are simulated and compared with the existing works to illustrate the capability of the proposed OCISMC.

Output Feedback Control of TS Fuzzy Systems with Unmeasured Nonlinearities via a Local Separation Principle

This paper presents a fuzzy observer-based control framework for TS fuzzy models with nonlinear consequents (N-TS). The N-TS fuzzy model is constructed such that only measured nonlinearities are considered to obtain the fuzzy membership functions, and all unmeasured nonlinearities are modeled as local consequent parts. Based on the differential mean value theorem, the closed-loop system is rewritten in an adequate format, which allows proving the separation principle for asymptotic TS fuzzy observer-based control design. Sufficient design conditions, expressed in terms of linear matrix inequalities, are derived via delayed nonquadratic Lyapunov functions for the synthesis of both stabilizing fuzzy controller and Luenberger-like fuzzy observer. Using Lyapunov-based arguments of a local separation principle, a constructive procedure is provided to maximize the closed-loop domain of attraction (DoA) estimation, which is guaranteed to be included inside the predefined fuzzy modeling region. Numerical examples are given to illustrate the effectiveness of the proposed approach.

이산시간 T-S 퍼지 시스템에 대한 정적 출력궤환 제어: k-샘플 차분법

This study deals with the problem of static output feedback control for discrete-time T-S fuzzy systems via a nonquadratic Lyapunov function and its k-sample variation, i.e., △SUBk/SUBV(χSUBt/SUB)=V(χSUBt+k/SUB)-V(χSUBt/SUB). A sufficient condition for the existence of the static output feedback controller is presented in terms of linear matrix inequalities (LMIs). The proposed sufficient stabilizing condition does not require any transformation matrices and equality constraints. Finally, two numerical examples are illustrated to verify the relaxation achieved with the proposed method.

Robust fuzzy observer‐based fault‐tolerant control: A homogeneous polynomial Lyapunov function approach

In this paper, a homogeneous polynomial Lyapunov function (HPLF) is employed in robust observer-based Integrated Fault-Tolerant Control (IFTC) of nonlinear systems modelled by the polynomial fuzzy model (PFM). This makes the design problem benefit from the conservative reduction property of polynomial Lyapunov functions (PLFs) and simultaneously avoid the emergence of non-convex terms due to differentiation of the polynomial Lyapunov matrix. As a result, the less conservative sum of square (SOS) conditions are obtained in the form of polynomial matrix inequalities (PMI), which are solved via SOSTOOLS. For the first time, a non-quadratic Lyapunov function is used to design a polynomial fuzzy unknown-input observer that estimates the system's states and actuator faults in the presence of model uncertainties and external disturbances. The effectiveness of the proposed approach in providing more relaxed and less conservative results along with the better fault tolerance is illustrated through three simulating examples and compared with the quadratic Lyapunov function (QLF) approach.

Stability and Stabilization of TS Fuzzy Systems via Line Integral Lyapunov Fuzzy Function

This paper is concerned with the stability and stabilization problem of a Takagi-Sugeno fuzzy (TSF) system. Using a non-quadratic function (well-known integral Lyapunov fuzzy candidate (ILF)) and some lemmas, new sufficient conditions are established as linear matrix inequalities (LMIs), which are solved with a stochastic fractal search (SFS). The main advantage of the technique used is its small conservatives. Motivated by the mean value theorem, a state feedback controller based on a non-quadratic Lyapunov function is designed. Unlike other approaches based on poly-quadratic Lyapunov candidates, stability conditions of the closed loop are obtained in LMI regions. It is important to highlight that the time derivatives of membership functions do not appear in the used line integral Lyapunov function, which is the well-known problem of poly-quadratic Lyapunov functions. A numerical example is given to show the advantages and the utility of the integral Lyapunov fuzzy candidate, which provides a wider feasibility region than other Lyapunov functions.

Local H∞ Control for Continuous-Time T-S Fuzzy Systems via Generalized Non-Quadratic Lyapunov Functions

This paper further develops a relaxed method to reduce conservatism in H∞ feedback control for continuous-time T-S fuzzy systems via a generalized non-quadratic Lyapunov function. Different from the results of some exisiting works, the generalized H∞ state feedback controller is designed. The relaxed stabilization conditions are obtained by applying Finsler’s lemma with the homogenous polynomial multipliers, and the H∞ performance is acquired by solving an optimization problem. In addition, the proposed method could be expanded to handle other control problems for fuzzy systems. Two examples are given to show the validity of the proposed results.

Membership-function-dependent dynamic output feedback [formula omitted] controller design of continuous-time T–S fuzzy systems via non-quadratic Lyapunov function

The dynamic output feedback H∞ (DOFH) control under imperfect premise matching (IPM) is studied in this paper for continuous-time Takagi–Sugeno (T–S) fuzzy systems. Different from the existing results, the DOFH switching controller, which enjoys membership functions (MFs) distinct from the fuzzy systems, is designed. First, the non-quadratic Lyapunov function (NQLF) based on MFs is utilized to design the controller. The time derivatives of MFs are addressed by the switching strategy. Second, a method based on linear matrix inequality (LMI) is given to make the controller gains solvable. Third, an improved method is developed to incorporate the more boundary information of MFs into the stability conditions to reduce conservatism. Finally, three examples are used to certify the advantage of the approach.

Simultaneous Sensor and Actuator Fault Reconstruction by Using a Sliding Mode Observer, Fuzzy Stability Analysis, and a Nonlinear Optimization Tool

This paper proposes a Takagi–Sugeno (TS) fuzzy sliding mode observer (SMO) for simultaneous actuator and sensor fault reconstruction in a class of nonlinear systems subjected to unknown disturbances. First, the nonlinear system is represented by a TS fuzzy model with immeasurable premise variables. By filtering the output of the TS fuzzy model, an augmented system whose actuator fault is a combination of the original actuator and sensor faults is constructed. An performance criteria is considered to minimize the effect of the disturbance on the state estimations. Then, by using two further transformation matrices, a non-quadratic Lyapunov function (NQLF), and fmincon in MATLAB as a nonlinear optimization tool, the gains of the SMO are designed through the stability analysis of the observer. The main advantages of the proposed approach in comparison to the existing methods are using nonlinear optimization tools instead of linear matrix inequalities (LMIs), utilizing NQLF instead of simple quadratic Lyapunov functions (QLF), choosing SMO as the observer, which is robust to the uncertainties, and assuming that the premise variables are immeasurable. Finally, a practical continuous stirred tank reactor (CSTR) is considered as a nonlinear dynamic, and the numerical simulation results illustrate the superiority of the proposed approach compared to the existing methods.

Relaxed observer-based stabilization and dissipativity conditions of T-S fuzzy systems with nonhomogeneous Markov jumps via non-PDC scheme

This paper aims to design a robust observer-based dissipative controller for discrete-time Takagi–Sugeno (T-S) fuzzy systems with nonhomogeneous Markov jumps through a non-parallel distributed compensation (non-PDC) scheme. Based on a mode-dependent nonquadratic Lyapunov function, the final form of the stabilization conditions is expressed as linear matrix inequalities in a less conservative manner. To be specific, this paper proposes a decoupling technique that can address the inherent nonconvex terms by extracting them from the stabilization conditions, where all slack variables are set to be fuzzy-basis-dependent for less conservative performance. Furthermore, the proposed stabilization method adopts a one-step design strategy that simultaneously designs the fuzzy observer and control gains without any iteration procedures by employing a positive tuning parameter. In particular, the time-varying transition probabilities included in the stabilization conditions are effectively removed using a modified relaxation technique that can avoid excessive use of free weighting matrices. Finally, based on four examples, the validity of the proposed method is verified through comparison with other studies.

Zonotopic observer designs for uncertain Takagi–Sugeno fuzzy systems

This paper addresses the zonotopic observer design for nonlinear systems affected by uncertainties, i.e., state disturbances and measurement noises using Takagi–Sugeno (TS) fuzzy technique. The system uncertainties are considered as unknown but bounded, which are handled via a set-membership framework. For state estimation purposes, we develop an algorithm to recursively compute the zonotope containing the mismatching nonlinear term caused by unmeasured nonlinearities. Then, two methods are proposed to design the zonotopic observer gains. The first method is based on the minimization of the F−radius of zonotopes, for which the membership-function-dependent observer gain must be completely computed online. For the second method, an ℋ∞ approach is used together with a nonquadratic Lyapunov function to determine the observer gain. Then, the zonotopic observer design is reformulated a convex optimization problem under linear matrix inequalities (LMIs), which can be effectively solved with numerical solvers. An autonomous vehicle application is provided to demonstrate and analyze the effectiveness of both proposed methods.

Trajectory tracking for a class of switched nonlinear systems: Application to PMSM

This paper proposes a state-dependent switching control for a class of switched nonlinear systems, whose model describes a permanent magnet synchronous machine (PMSM) fed by a three-phase voltage source inverter. Due to its high torque density, high efficiency and wide velocity range, this electrical drive is widely used for traction and several applications in robotics, aerospace, electric vehicles among others. The proposed design conditions are based on a non-quadratic Lyapunov function, dependent on the machine shaft displacement, and assure asymptotic tracking of a pre-specified time-varying rotational velocity profile with guaranteed performance. Properties of the nonlinear system under consideration are used to derive design conditions expressed in terms of linear matrix inequalities that can be solved efficiently. Special cases involving asymptotic stability toward step and ramp velocity profiles are presented. Experimental results are used to validate the proposed technique.

Static Linear Anti-Windup Design With Sign-Indefinite Quadratic Forms

We design static anti-windup gains to mitigate the effect of input saturation in linear output feedback closed loops. The design is conducted with the help of a non-quadratic Lyapunov function involving sign-indefinite quadratic forms, which allows for additional degrees of freedom to be exploited, for the anti-windup gain design. Synthesis conditions, combining the use of sign-indefinite quadratic forms and several sector bound properties are stated in the form of bilinear matrix inequalities ensuring global exponential stability of the closed-loop system. An iterative design algorithm is then devised, based on the resolution of a sequence of semidefinite programs. The superiority of the proposed technique over classical quadratic constructions is illustrated on an example borrowed from the literature, where quadratic positive definite functions are ineffective.