SET-MEMBERSHIP IDENTIFICATION OF CONTINUOUS-TIME SYSTEMS

An alternative method was proposed for identification of linear continuous time system with noisy output. Using Gaussian modulating filters, by numerical integration, a discrete equivalent identification model that was parametrized directly in continuous time model parameters was obtained, and the LS estimation algorithm was developed. The LS estimator is biased even with white measurement output noise, and the bias is determined by variance of output noise. On the basis of obtained output measurement noise variance, the bias was compensated and the consistent estimates of system parameters were given. Simulation which compares the standard LS with BCLS algorithm indicates that the proposed algorithm is an efficient method for noisy output continuous time system identification.

Chapter 9 - Discrete-Time Signals and Systems

<div class="htmlview paragraph">Many methods have been suggested for the solution of parabolic partial differential equations. Prior efforts have heavily focused on explicit difference equations, implicit difference equations, and a combination of these two methods. Some work has been done in exponential methods and other obscure methods. This paper introduces an advanced exponential method for obtaining solutions of parabolic partial differential equations.</div> <div class="htmlview paragraph">The explicit, implicit, Crank-Nicolson, DuFort-Frankel, modified DuFort-Frankel, original exponential method, and the advanced exponential method are briefly described and compared. The advanced exponential method is unconditionally stable with whatever time step is taken. Accuracy and speed of solution for this method is compared to other solution methods for various example test cases. For the example test cases, the method gave excellent results when compared to the analytical solution and to other methods. The method has been introduced into the Systems Improved Numerical Differencing Analyzer and Fluid Integrator (SINDA/FLUINT) code as a solver. The advanced exponential method gives excellent speed, accuracy, and stability for solving parabolic partial differential equations.</div>

We show the continuous dependence of solutions of linear nonautonomous second-order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-* topology of delay coefficients is required. The results are important in the applications of the theory of Lyapunov exponents to the investigation of PDEs with delay.

A criterion is introduced for determining the optimum sampling interval for the identification of system parameters. It increases the accuracy and stability in identification procedures. This is done by employing a bilinear transformation from the z-plane to the w-plane. Some results of simulations are also included. These reveal the usefulness of the suggested algorithm even in the presence of measurement noise.

In this paper we present the analytical solution of a one-dimensional (1D) parabolic differential equation (PDE), encountered in many engineering studies under the name of the heat equation. To solve the equation, we use the method of separation of variables together with Fourier series. After applying the separation of variables, the initial equation decomposes into two ordinary differential equations (ODE). The equations obtained correspond to the spatial (x) and temporal (t) components. The spatial solution, for the first equation, is expressed by sinusoidal functions corresponding to the eigenfunctions, and the temporal component, for the second equation, has a characteristic exponential dependence. The determination of the Fourier series coefficients is based on the initial condition. Thus, we constructed an exact solution in the form of convergent series. The method used in solving the equation, together with the boundary conditions and initial conditions, can be used as a reference for validating numerical methods or practical experiments performed in the laboratory.

Well-posedness of parabolic differential and difference equations

We consider a class of problems in which traces of the first spacial derivative of the solution appear in the coefficients of a parabolic partial differential equation. Such problems arise in free boundary problems and the determination of unknown coefficients in various parabolic differential equations. Existence, uniqueness and continuous dependence on the data are demonstrated. The analysis involves a priori estimates and the Schauder fixed point theorem. Overall, the article generalizes some of the results of Cannon and Yin (1989, Journal of Differential Equations, 79, 266–288).

Relying on the classical connection between backward stochastic differential equations and nonlinear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional semilinear parabolic PDEs. This scheme is inspired by the approach coming from machine learning and developed using deep neural networks in Han et al. (2018, Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci., 115, 8505–8510. Our algorithm is based on a Picard iteration scheme in which a sequence of linear-quadratic optimization problem is solved by means of stochastic gradient descent algorithm. In the framework of a linear specification of the approximation space, we manage to prove a convergence result for our scheme, under some smallness condition. In practice, in order to be able to treat high-dimensional examples, we employ sparse-grid approximation spaces. In the case of periodic coefficients and using pre-wavelet basis functions, we obtain an upper bound on the global complexity of our method. It shows, in particular, that the curse of dimensionality is tamed in the sense that in order to achieve a root mean squared error of order $\varepsilon $, for a prescribed precision $\varepsilon $, the complexity of the Picard algorithm grows polynomially in $\varepsilon ^{-1}$ up to some logarithmic factor $|\!\log (\varepsilon )|$, whose exponent grows linearly with respect to the PDE dimension. Various numerical results are presented to validate the performance of our method, and to compare them with some recent machine learning schemes proposed in E et al. (2017, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat., 5, 349–380) and Huré et al. (2020, Deep backward schemes for high-dimensional nonlinear PDEs. Math. Comput., 89, 1547–1579).

In this article, we study the adaptive stability for parabolic partial differential equation (PDE)-ordinary differential equation (ODE) cascade systems with actuator dynamics, where the actuator dynamics are nonlinear subject to unknown parameters. Compared with a class of PDE–ODE coupled systems that the control input only acts on the PDE boundary and the linear sandwiched system without uncertainty, the structure of such systems is more complex. First of all, infinite-dimensional backstepping transformation is adopted. The original PDE-ODE cascade system is changed to a new system that is easier to design. On this basis, finite-dimensional backstepping transformation and adaptive compensation technology are combined to develop a state-feedback controller. Then, the boundedness of all the signals in the closed-loop system is proved by the Lyapunov functional analysis. Furthermore, the control law and the original system states eventually converge to zero. Finally, different simulation data are presented to illustrate the validity of the theoretical results.

Convergence analysis of constraint energy minimizing generalized multiscale finite element method for a linear stochastic parabolic partial differential equation driven by additive noises

Expected exit time for time-periodic stochastic differential equations and applications to stochastic resonance

PDE backstepping control of one-dimensional heat equation with time-varying domain

In this paper we consider a control problem for the convection diffusion equation and investigate the impact of including actuator dynamics with delays. The problem is motivated by applications to control of energy efficient buildings where actuation is provided by a HVAC system. The basic model is governed by a parabolic partial differential equation (PDE) with boundary inputs. The boundary inputs are assumed to be the output of an actuator governed by a delay differential equation. Thus, one augments the PDE with a delay equation model of an actuator with delays. The combined system is described by a coupled delay partial differential equation. We show that under suitable conditions, the coupled delay PDE system is well posed in a standard Hilbert space and we use this corresponding abstract formulation to construct numerical methods for control design. We apply these results to a simple 1D boundary control system to illustrate the ideas and numerical methods.

In this paper, the extension of the generalized finite difference method to the solution of anisotropic elliptical and parabolic partial differential equations is given in the case of considering two space dimensions. The explicit finite difference formulae and the criterion of stability for the anisotropic parabolic equations are given. The stability condition has been expressed in function of the coefficients of the star equation for irregular clouds of nodes. Various anisotropic cases of simulation of cardiac tissue, including elliptic and parabolic equations, have been solved, and the results show the accuracy of the method.