Polynomial Fuzzy Observer-Based Feedback Control for Nonlinear Hyperbolic PDEs 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.

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

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.

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.

The work presents the analysis of the free boundary value problem related to the one-dimensional invasion model of new species in biofilm reactors. In the framework of continuum approach to mathematical modelling of biofilm growth, the problem consists of a system of non-linear hyperbolic partial differential equations governing the microbial species growth and a system of semi-linear elliptic partial differential equations describing the substrate trends. The model is completed with a system of elliptic partial differential equations governing the diffusion and reaction of planktonic cells, which are able to switch their mode of growth from planktonic to sessile when specific environmental conditions are found. Two systems of non-linear differential equations for the substrate and planktonic cells mass balance within the bulk liquid are also considered. The free boundary evolution is governed by a differential equation that accounts for detachment. The qualitative analysis is performed and a uniqueness and existence result is presented. Furthermore, two special models of biological and engineering interest are discussed numerically. The invasion of Anammox bacteria in a constituted biofilm inhabiting the deammonification units of the wastewater treatment plants is simulated. Numerical simulations are run to evaluate the influence of the colonization process on biofilm structure and activity.

A design method of finite dimensional nonlinear disturbance observer based control (NLDOBC) is proposed for a class of nonlinear dissipative partial differential equation (PDE) systems, where the disturbance is modeled by an exosystem of nonlinear ordinary differential equations (ODEs). Motivated by the fact that the dominant structure of the dissipative PDE is usually characterized by a finite number of degrees of freedom, the modal decomposition technique is initially applied to the PDE system to give a slow subsystem of low dimensional nonlinear ODEs. By using the nonlinear slow subsystem and exosystem, a nonlinear disturbance observer (NLDO) is constructed to estimate the disturbance. Then, on the basis of NLDO, an NLDOBC method is proposed in terms of linear matrix inequality (LMI) to ensure the exponential stability of the closed-loop PDE system in the presence of the modeled disturbance. Finally, the simulation results on control of one dimensional Fisher diffusion-reaction system show the effectiveness of the proposed method.

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.

: Complex physical phenomena involving chemically reacting systems or the transport of heat or fluids are often modeled by coupled systems of time-dependent, nonlinear partial differential equations. The difficulties in understanding the stability of the differential equation systems and in designing efficient, accurate numerical methods for their solution are widely recognized and were the focus of this research. We have worked on four general aspects of the analysis and numerical approximation of systems of partial differential equations. These areas of research are: (1) modeling aspects and stability analysis for nonlinear time-dependent partial differential equations; (2) use and analysis of finite element or finite difference methods to discretize coupled systems of nonlinear differential equations; (3) development of adaptive or local grid refinement capabilities to resolve local phenomena in large-scale applications; and (4) development of data structures, preconditioners, and efficient solution algorithms for large-scale problems on new computer architectures. Emphasis has been placed upon multiphase or multicomponent, transport-dominated flow processes with dynamic local phenomena. The research also involved a mix of analysis, algorithm development, and large-scale computation using newer computer architectures.

Identification of parameters in partial differential equations

Joint uncertainties and state estimation of a class of linear coupled hyperbolic partial differential equation systems in the presence of unstructured and structured uncertainties are studied in this paper. For unstructured uncertainties which are completely unknown, by employing Takagi-Sugeno fuzzy logic system to approximate the unstructured uncertainties, a novel adaptive fuzzy boundary observer is developed to estimate both unknown system states as well as unknown weights in the fuzzy logic system, and the estimation errors are ultimately bounded. Therein, in the design of the proposed observer, a set of swapping filters and infinite dimensional backstepping technique are combined. On the other hand, for structured uncertainties that can be described in a concrete parameterized form, the proposed method can easily achieve the exact estimation of weights and states to their true values. The rigorous proof is provided to show that the ultimately bounded estimation errors for the case of unstructured uncertainties and the exponential convergent estimation errors for the case of structured uncertainties can be realized. Finally, three illustrative simulations are carried out to show the feasibility and effectiveness of the developed methods in this paper.

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

Summary There is no standard framework for solving optimization problems for systems described by agent-based models (ABMs). We present a method for constructing individual-level controls that steer the population-level dynamics of an ABM towards a desired state. Our method uses a system of partial differential equations (PDEs) with control functions to approximate the dynamics of the ABM with control. An optimal control problem is formulated in terms of the PDE model to mimic the optimization goal of the ABM. Mathematical theory is used to derive optimal controls for the PDE model, which are numerically approximated and transformed for use in the ABM. We use the Sugarscape ABM, a prototype ABM that includes agent and environmental heterogeneity and accumulation of agent resources over time. We present a PDE model that approximates well the spatial, temporal, and resource dynamics of the Sugarscape ABM. In both models, control represents taxation of agent wealth with the goal to maximize total taxes collected while minimizing the impact of taxation on the population over a finite time. Solutions to the optimal control problem yield taxation rates specific to an agent's location and current wealth. The use of optimal controls (generated by the PDE model) within the ABM performed better than other controls we evaluated, even though some error was introduced between the ABM and PDE models. Our results demonstrate the feasibility of using a PDE to approximate an ABM for control purposes and illustrate challenges that can arise in applying this technique to sophisticated ABMs. Copyright © 2016 John Wiley & Sons, Ltd.

Deep neural network modeling of unknown partial differential equations in nodal space

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.

The initial attached cell layer in multispecies biofilm growth is studied in this paper. The corresponding mathematical model leads to discuss a free boundary problem for a system of nonlinear hyperbolic partial differential equations, where the initial biofilm thickness is equal to zero. No assumptions on initial conditions for biomass concentrations and biofilm thickness are required. The data that the problem needs are the concentration of biomass in the bulk liquid and biomass flux from the bulk liquid. The differential equations are converted into an equivalent system of Volterra integral equations. We use Newton-Raphson method to solve the nonlinear system of Volterra integral equations (SVIEs) of the second kind. This method converts the nonlinear system of integral equations into a linear integral equation at each step.