Adaptive Fuzzy Boundary Observer Design for Uncertain Linear Coupled Hyperbolic Partial Differential Equation Systems

This paper study the problem of designing a fuzzy adaptive controller based on H ∞ observer for a class of uncertain nonlinear system modeled by first-order hyperbolic partial differential equation (PDE). The purpose of this paper is to design a new adaptive fuzzy law based on the principle of observer such that the first-order hyperbolic PDE system is exponentially stable with a desired H ∞ performance for disturbance attenuation, while the control constraint is respected. Initially, a Takagi-Sugeno (T-S) hyperbolic PDE model is proposed to represent the nonlinear PDE systems. Then the fuzzy adaptive observer with H ∞ controller is designed through seeking the feasible solutions of existing linear matrix inequality (LMI) optimization techniques, so using adjusting observer parameters, the control effectiveness can be estimated by state observer. For uncertain system, by designing the estimate of state observer, and introduced approximately error compensation, then the requirement of square integrability for optimal approximation error is inessential. Finally, the proposed design method is proved that the controller based on fuzzy adaptive H ∞ observer will stabilize the first-order hyperbolic PDE system.

This article studies the problem of prescribed‐time constrained tracking control for a class of hyperbolic partial differential equation (PDE) systems with actuator dynamics, which are described by a set of nonlinear ordinary differential equations (ODEs). Since the control input only appears in the ODE subsystem rather than directly on the boundary of PDE subsystem, the control task becomes quite challenging. The most important is that for the control of such ODE controlled PDE systems we mainly make the following two contributions: (1) the controlled output of the PDE system tracks the reference signal within the prescribed time; (2) the controlled output and all the actuator states are constrained. It is the first time that such a prescribed‐time constrained tracking control problem is addressed for the PDE‐ODE coupled system considered in this article. Through rigorous theoretical proof, it is demonstrated that all the system states and control signals are bounded and sufficiently continuous by configuring appropriate design parameters. Finally, the performance is investigated via numerical simulation.

On stochasticity in the Von Foerster hyperbolic partial differential equation system: Further applications to the modeling of an asynchronously dividing cellular system

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

This article investigates the problem of fuzzy boundary (FB) control for nonlinear second-order hyperbolic partial differential equation systems (HPDESs) with time-varying delay (TVD). Considering the existence of TVD that widely appears in actual HPDESs, the presented FB control approach is more practical and effectively reduces the control cost, requiring few actuators and sensors in comparison with the fuzzy distributed control method. Initially, one can obtain a fuzzy model of the nonlinear delayed second-order HPDES by utilizing Takagi–Sugeno (T–S) fuzzy modeling technology. Subsequently, by using the Lyapunov direct approach, sufficient conditions for the exponential stability (ES) of closed-loop nonlinear delayed HPDES based on spatial linear matrix inequality (SLMI) are derived. Then, the FB control design issue of nonlinear delayed second-order HPDES is expressed as an LMI feasibility issue. Lastly, the provided FB control approach is validated by the simulation results.

Modelling and simulation of the thernomechanical pulping process

Adaptive boundary observer design for linear hyperbolic systems; Application to estimation in heat exchangers

[formula omitted] fault-tolerant fuzzy intermittent control for nonlinear hyperbolic PDE systems with multiple delays and actuator failures

In this paper we shall derive energy integral inequalities for the linear hyperbolic partial differential equation of any order in several variables with multiple characteristics and constant coefficients. In [1] Garding uses an idea by Leray [6] to develop energy integrals for totally hyperbolic equations (including variable coefficients). Totally hyperbolic equations are subject to the severe restriction that all characteristics remain distinct(2). In his method G'arding multiplies the given operator of order n by another operator of order n 1 whose characteristics separate those of the given operator. Using the Fourier transform and Parseval's formula he arrives at the energy inequality, which is an a priori estimate of the square integral of the solution of the given equation and all the derivatives up to order n -1. In our case of an equation with multiple characteristics we use a variant of G'arding's method. Our energy inequality is of a different form in as much as it provides an a priori estimate of the integral of the square of the solution and certain non-negative integral forms of its derivatives up to order n -1. However, we also show that for totally hyperbolic equations our energy inequalities reduce to the standard form. In hyperbolic equations with multiple characteristics the influence of the lower order terms must be taken into account and hence the form of these lower order terms is suitably restricted. In totally hyperbolic equations no restrictions are imposed on the lower order terms. Our definition of hyperbolicity is therefore such that in the case of distinct characteristics no restrictions are imposed on the lower order terms. In [2] Gairding has given a criterion for hyperbolicity of equations in several variables with multiple characteristics and constant coefficients. A. Lax [4] extended this criterion, in the case of two independent variables, to include variable coefficients. We shall not be concerned here with

ABSTRACTThis article focuses on the boundary output feedback exponential stabilization and disturbance rejection for a hyperbolic partial differential equation (PDE) system with nonlinear boundary condition and unknown control‐matched disturbance. An infinite‐dimensional disturbance estimator is designed by utilizing three boundary measurement outputs to estimate the disturbance. Subsequently, a boundary output feedback stabilizing controller is designed based on the disturbance estimate, effectively compensating for the disturbance in the feedback loop. Besides conventional operator semigroup theory and Lyapunov direct approach, novel theoretical analysis methods including the techniques of boundary reflection and iterated sequence are developed to prove rigorously the well‐posedness and exponential stability of the resulting closed‐loop system. Finally, some numerical simulations are carried out to validate the theoretical conclusions and the effectiveness of the proposed methods.

This work proposes a methodology for the synthesis of distributed output feedback controllers for systems of two-time-scale first-order hyperbolic partial differential equations (PDEs) for which the manipulated inputs, the controlled and the measured outputs are distributed in space. The proposed methodology is applied to a typical convection-reaction process with time-scale multiplicity.

This paper presents a model for a two-dimensional pedestrian movement flow. The model is derived by extending a one-dimensional vehicle traffic flow model that uses two coupled partial differential equations (PDEs) to govern vehicular motion. This model modifies the vehicular traffic model so that bidirectional flow is possible, and also the pedestrian movement can be controlled to model different behaviors. The model satisfies the conservation principle and is classified as a hyperbolic PDE system. Analysis of the model is drawn based on the theoretical aspects as well as the numerical simulation results achieved using the finite volume method. The original vehicular macroscopic model was derived by extending a corresponding microscopic model. This paper follows the same strategy and shows how microscopic pedestrian behavior can be used to derive the macroscopic behavior

In this paper, a hybrid fuzzy adaptive output feedback control design approach is proposed for a class of multiinput and multioutput strict-feedback nonlinear systems with unknown time-varying delays, unmeasured states, and input saturation. First, fuzzy logic systems are employed to approximate unknown nonlinear functions in the system. Next, a smooth function is used to approximate the input saturation and an adaptive fuzzy state observer is constructed to solve the problem of unmeasured states. Based on the designed adaptive fuzzy state observer, a serial-parallel estimation model is established. By applying adaptive fuzzy dynamic surface control technique and utilizing the prediction error between the system states observer model and the serial-parallel estimation model, a new fuzzy controller with the composite parameters adaptive laws is developed based on Lyapunov-Krasovskii functional. It is proved that all variables of the closed-loop system are bounded and the system outputs can follow the given bounded reference signals as close as possible. A simulation example is provided to further show the effectiveness of this novel control scheme.

Output-feedback control of uncertain nonlinear systems using a self-structuring adaptive fuzzy observer

In this paper, we consider the controllability and the observability of a networked system of linear hyperbolic partial differential equations with coupled boundary conditions. Using the method of characteristics, we characterize them by the controllability/observability of a discrete-time system which is defined on the boundaries of the PDE system, has low-dimension, and hence the analysis is easy for. We also show the effectiveness of our approach using an simple example.