Hyperbolic Partial Differential Equation Model Research Articles

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

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.

CNN-Based Temporal Video Segmentation Using a Nonlinear Hyperbolic PDE-Based Multi-Scale Analysis

An automatic temporal video segmentation framework is introduced in this article. The proposed cut detection technique performs a high-level feature extraction on the video frames, by applying a multi-scale image analysis approach combining nonlinear partial differential equations (PDE) to convolutional neural networks (CNN). A nonlinear second-order hyperbolic PDE model is proposed and its well-posedness is then investigated rigorously here. Its weak and unique solution is determined numerically applying a finite difference method-based numerical approximation algorithm that quickly converges to it. A scale-space representation is then created using that iterative discretization scheme. A CNN-based feature extraction is performed at each scale and the feature vectors obtained at multiple scales are concatenated into a final frame descriptor. The feature vector distance values between any two successive frames are then determined and the video transitions are identified next, by applying an automatic clustering scheme on these values. The proposed PDE model, its mathematical investigation and discretization, and the multi-scale analysis based on it represent the major contributions of this work. Some temporal segmentation experiments and method comparisons that illustrate the effectiveness of the proposed framework are finally described in this research paper.

Discrete output regulator design for linear distributed parameter systems

In this manuscript, we address discrete-time state and error feedback output regulator designs for a class of linear distributed parameter systems (DPS) with bounded input and output operators. By utilising the Cayley–Tustin bilinear transform, a linear infinite-dimensional discrete-time system is obtained without model spatial approximation or model order reduction. Based on the internal model principle, discrete state and error feedback regulators are designed. In particular, discrete Sylvester regulator equations are formulated, and their solvability is proved and linked to the continuous counterparts. To ensure the stability of the closed-loop system, the design of stabilising feedback gain and its dual problem of stabilising output injection gain design are provided in the discrete-time setting. Finally, three simulation examples including a first-order hyperbolic partial differential equation model and a 1-D heat equation with considerations of step-like, ramp-like and harmonic exogenous signals are shown to demonstrate the effectiveness of the proposed method.

A Novel Nonlinear Second Order Hyperbolic Partial Differential Equation-Based Image Restoration Algorithm With Directional Diffusion

Recently, variational and partial differential equation (PDE)-based algorithms have become very important for image restoration. In this study, we propose a new second order hyperbolic PDE model based on directional diffusion for image restoration. This hyperbolic PDE restoration model can simply diffuse along the edge's tangential direction in the observed image, thereby removing noise while preserving the image edges and fine details, which avoids the staircase effect in the restored image. An effective numerical scheme is proposed for handling the computation of our approach using the finite difference method. Successful image restoration experiments demonstrated that the proposed second order hyperbolic PDE-based model obtains superior performance compared with other models at preserving edges and it avoids the staircase effect.

A hyperbolic partial differential equation model for filtering turbulent flows

A two dimensional partial differential equation scheme that uses a second-order hyperbolic diffusion equation for image denoising is developed to a three dimensional model for filtering isotropic turbulence. The mathematical derivation of the model is introduced and a consistent finite difference numerical approximation scheme is proposed. The model is tested against the velocity fields of isotropic turbulence that are simulated by solving the lattice Boltzmann(LB) and the Navier–Stokes(NS) equations, respectively. The D3Q15 LB model and the Fourier spectral method are used to solve the LB equation and the Navier–Stokes vorticity equation at grid points of 2563, respectively. Also the filtering method with the choice of the same filtering parameters is applied against a synthetic turbulent field with the same number of points. The high rotation vortex identification method Q is used to visualize the total, coherent and incoherent fields in all cases of the study. Despite the different nature of the LB and NS direct numerical simulations and the synthetic turbulence, all input parameters for the hyperbolic filtering model are chosen the same in all cases. Comparisons between the LB and NS filtered energy spectra, coherent vortices, incoherent regions, skewness, flatness and the fourth order moments are also considered. The same features are calculated for the non-filtered and filtered synthetic turbulent fields. It is shown that the coherent part preserves all statistical features of turbulence. However, the isotropy is the only feature that has been preserved for the incoherent part and other statistical features are divergent in all cases of the study.

Boundary Observer for Congested Freeway Traffic State Estimation via Aw-Rascle-Zhang model

Abstract This paper develops boundary observer for estimation of congested freeway traffic states based on Aw-Rascle-Zhang(ARZ) partial differential equations (PDE) model. Traffic state estimation refers to acquisition of traffic state information from partially observed traffic data. This problem is relevant for freeway due to its limited accessibility to real-time traffic information. We propose a boundary observer design so that estimates of aggregated traffic states in a freeway segment are obtained simply from boundary measurement of flow and velocity. The macroscopic traffic dynamics is represented by the ARZ model, consisting of 2 × 2 coupled nonlinear hyperbolic PDEs for traffic density and velocity. Analysis of the linearized ARZ model leads to the study of a hetero-directional hyperbolic PDE model for congested traffic regime. Using spatial transformation and PDE backstepping method, we construct a boundary observer with a copy of the nonlinear plant and output injection of boundary measurement errors. The output injection gains are designed for the error system of the linearized ARZ model so that the exponential stability of error system in the L2 norm and finite-time convergence to zero are guaranteed. Simulations are conducted to validate the boundary observer design for nonlinear ARZ model without knowledge of initial conditions.

Traffic congestion control for Aw–Rascle–Zhang model

This paper develops boundary feedback control laws to reduce stop-and-go oscillations in congested traffic. The macroscopic traffic dynamics are governed by Aw–Rascle–Zhang (ARZ) model, consisting of second-order nonlinear partial differential equations (PDEs). A criterion to distinguish free and congested regimes for the ARZ traffic model leads to the study of hetero-directional hyperbolic PDE model of congested traffic regime. To stabilize the oscillations of traffic density and speed in a freeway segment, a boundary input through ramp metering is considered. We discuss the stabilization problem for freeway segments respectively, upstream and downstream of the ramp. For the more challenging upstream control problem, our full-state feedback control law employs a backstepping transformation. Both collocated and anti-collocated boundary observers are designed. The exponential stability in L2 sense and finite time convergence to equilibrium are achieved and validated with simulation. In the absence of relaxation time and boundary parameters’ knowledge, we propose adaptive output feedback control design. Control is applied at outlet and the measurement is taken from inlet of the freeway segment. We use the backstepping method to obtain an observer canonical form in which unknown parameters multiply with measured output. A parametric model based on this form is derived and gradient-based parameter estimators are designed. An explicit state observer involving the delayed values of the input and the output is introduced for state estimation. Using the parameter and state estimates, we develop an adaptive output feedback control law which achieves convergence to the steady regulation in the L2 sense.

Modeling and Analysis of a Nonlinear Age-Structured Model for Tumor Cell Populations with Quiescence

We present a nonlinear first-order hyperbolic partial differential equation model to describe age-structured tumor cell populations with proliferating and quiescent phases at the avascular stage in vitro. The division rate of the proliferating cells is assumed to be nonlinear due to the limitation of the nutrient and space. The model includes a proportion of newborn cells that enter directly the quiescent phase with age zero. This proportion can reflect the effect of treatment by drugs such as erlotinib. The existence and uniqueness of solutions are established. The local and global stabilities of the trivial steady state are investigated. The existence and local stability of the positive steady state are also analyzed. Numerical simulations are performed to verify the results and to examine the impacts of parameters on the nonlinear dynamics of the model.

Robust Control of a Cable From a Hyperbolic Partial Differential Equation Model

This article presents a detailed study of the robust control of a cable’s vibrations, with emphasis on considering a model of infinite dimension. Indeed, using a partial differential equation model of the vibrations of an inclined cable with sag, we are interested in studying the application of $\mathcal H_{\infty }$ -robust feedback control to this infinite dimensional system. The approach relies on Riccati equations to stabilize the system under measurement feedback when it is subjected to external disturbances. Henceforth, this article focuses on the construction of a standard linear infinite dimensional state space description of the cable under consideration before writing its approximation of finite dimension and studying the $\mathcal H_{\infty }$ feedback control of vibrations with partial observation of the state in both cases. The closed-loop system is numerically simulated to illustrate the effectiveness of the resulting control law.

Parameter identification of aggregated thermostatically controlled loads for smart grids using PDE techniques

This paper develops methods for model identification of aggregated thermostatically controlled loads (TCLs) in smart grids, via partial differential equation (PDE) techniques. Control of aggregated TCLs provides a promising opportunity to mitigate the mismatch between power generation and demand, thus enhancing grid reliability and enabling renewable energy penetration. To this end, this paper focuses on developing parameter identification algorithms for a PDE-based model of aggregated TCLs. First, a two-state boundary-coupled hyperbolic PDE model for homogenous TCL populations is derived. This model is extended to heterogeneous populations by including a diffusive term, which provides an elegant control-oriented model. Next, a passive parameter identification scheme and a swapping-based identification scheme are derived for the PDE model structure. Simulation results demonstrate the efficacy of each method under various autonomous and non-autonomous scenarios. The proposed models can subsequently be employed to provide system critical information for power system monitoring and control.

Fuzzy output tracking control of semi-linear first-order hyperbolic PDE systems with matched perturbations

This paper considers the design problem of a fuzzy output tracking controller via a finite number of actuators for a class of nonlinear distributed parameter systems with unknown matched perturbations described by semi-linear first-order hyperbolic partial differential equations (PDEs). Firstly, a T–S fuzzy hyperbolic PDE model is assumed to be used to accurately represent the semi-linear first-order hyperbolic PDE system. Subsequently, based on the T–S fuzzy PDE model, the classic variable structure control technique is adopted to develop a fuzzy output tracking controller such that the output of the semi-linear PDE system exponentially tracks a desired reference and the state of the system is uniformly bounded in the Filippov sense. By constructing a strict Lyapunov function, the sufficient condition on the existence of the fuzzy output tracking controller is derived and presented in terms of algebraic linear matrix inequalities (LMIs) in space with an equality constraint. It has been shown that when these algebraic LMIs with an equality constraint are feasible, fuzzy tracking control design can be obtained via the existing convex optimization techniques. Finally, the developed design methodology is successfully applied to the control of a semi-linear Lotka–Volterra PDE system and the achieved results illustrate its effectiveness.

Guaranteed cost distributed fuzzy observer‐based control for a class of nonlinear spatially distributed processes

The guaranteed cost distributed fuzzy (GCDF) observer‐based control design is proposed for a class of nonlinear spatially distributed processes described by first‐order hyperbolic partial differential equations (PDEs). Initially, a T–S fuzzy hyperbolic PDE model is proposed to accurately represent the nonlinear PDE system. Then, based on the fuzzy PDE model, the GCDF observer‐based control design is developed in terms of a set of space‐dependent linear matrix inequalities. In the proposed control scheme, a distributed fuzzy observer is used to estimate the state of the PDE system. The designed fuzzy controller can not only ensure the exponential stability of the closed‐loop PDE system but also provide an upper bound of quadratic cost function. Moreover, a suboptimal fuzzy control design is addressed in the sense of minimizing an upper bound of the cost function. The finite difference method in space and the existing linear matrix inequality optimization techniques are used to approximately solve the suboptimal control design problem. Finally, the proposed design method is applied to the control of a nonisothermal plug‐flow reactor. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2366–2378, 2013

Design of distributed [formula omitted] fuzzy controllers with constraint for nonlinear hyperbolic PDE systems

This paper investigates the problem of designing a distributed H∞ fuzzy controller with constraint for a class of nonlinear spatially distributed processes modeled by first-order hyperbolic partial differential equations (PDEs). The purpose of this paper is to design a distributed fuzzy state feedback controller such that the closed-loop PDE system is exponentially stable with a prescribed H∞ performance of disturbance attenuation, while the control constraint is respected. Initially, a Takagi–Sugeno (T–S) hyperbolic PDE model is proposed to accurately represent the nonlinear PDE system. Then, based on the T–S fuzzy PDE model, a distributed H∞ fuzzy controller design with constraint is developed in terms of a set of coupled differential/algebraic linear matrix inequalities (D/ALMIs) in space. Furthermore, a suboptimal distributed H∞ fuzzy controller with constraint is proposed to minimize the level of attenuation. The finite difference method in space and the existing linear matrix inequality (LMI) optimization techniques are employed to approximately solve the suboptimal fuzzy control design problem. Finally, the proposed design method is applied to the distributed control of a nonlinear system described by two coupled first-order hyperbolic PDEs to illustrate its effectiveness.

Distributed Fuzzy Control Design of Nonlinear Hyperbolic PDE Systems With Application to Nonisothermal Plug-Flow Reactor

This paper considers the problem of fuzzy control design for a class of nonlinear distributed parameter systems that is described by first-order hyperbolic partial differential equations (PDEs), where the control actuators are continuously distributed in space. The goal of this paper is to develop a fuzzy state-feedback control design methodology for these systems by employing a combination of PDE theory and concepts from Takagi-Sugeno (T-S) fuzzy control. First, the T-S fuzzy hyperbolic PDE model is proposed to accurately represent the nonlinear first-order hyperbolic PDE system. Subsequently, based on the T-S fuzzy-PDE model, a Lyapunov technique is used to analyze the closed-loop exponential stability with a given decay rate. Then, a fuzzy state-feedback control design procedure is developed in terms of a set of spatial differential linear matrix inequalities (SDLMIs) from the resulting stability conditions. Furthermore, utilizing the finite-difference approximation method (with a backward difference for the spatial derivative), a recursive algorithm is presented to solve the SDLMIs via the existing LMI optimization techniques. Finally, the developed design methodology is successfully applied to the control of a nonisothermal plug-flow reactor.

Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems

The Linear-Quadratic (LQ) optimal control problem is studied for a class of first-order hyperbolic partial differential equation models by using a nonlinear in finite-dimensional (distributed parameter) Hilbert state-space description. First the dynamical properties of the linearized model around some equilibrium pro. le are studied. Next the LQ-feedback operator is computed by using the corresponding operator Riccati algebraic equation whose solution is obtained via a related matrix Riccati differential equation in the space variable. Then the latter is applied to the nonlinear model, and the resulting closed-loop system dynamical performances are analyzed.

Size Structured Competition Model with Periodic Coefficients

A model of Diekmann et al. [14] for two size-structured populations competing for a single resource is extended in this paper to allow for periodic variation in parameter values (having a common period). The hyperbolic partial differential equation model is reduced to systems of ordinary differential equations describing the dynamic of two size-structured species competing for a single unstructured resource. The existence of nontrivial periodic solutions of those systems is considered. Under fairly general conditions, global bifurcation techniques as used by Gushing are used to show the existence of a continuum of solutions that bifurcate from a noncritical periodic solution of the reduced single species system. The positivity and the stability of the bifurcating branch solutions are studied. The stability is shown to depend on the stability of the solutions of the reduced system and the direction of bifurcation. The results and techniques can be generalized to the case of n, n ≥ 1, size-structured species competing for a single resource. It is also shown that the average size of individuals of each species is governed by a nonautonomous logistic equation whose solution under fairly general conditions approaches the unique periodic solution of the classical periodic logistic equation.

Recursive estimation for a noisy image represented by a hyperbolic PDE model

Abstract A recursive estimation technique is applied to the noisy image which is represented by a hyperbolic partial differential equation (PDE). Approximating a PDE model by a finite-difference approximation leads to an autoregressive (AR) model representation which Jain pointed out. In this paper, we use a new PDE model for the image representation. To apply a recursive estimation scheme to the image which is degraded by white noise, we propose the transformation of the.AR model into a state-space representation. To this representation, we apply a Kalman strip processor to reduce the order of the computation and storage, and the strip smoother (the optimal fixed-interval smoother) is also applied to obtain a better estimated imago. Three numerical examples are illustrated to show the effectiveness of the proposed algorithm.