1,721,784 research outputs found
Model order reduction for a linearized robust PDE constrained optimization
No full textWe investigate an optimal control problem governed by a time dependent partial differential equations and due to manufacturing, there are uncertainties which are deterministic in the present work. The resulting optimization problem considers the worst-case scenario over the uncertain domain. The main idea of the method is to linearize the considered cost functional with respect to the uncertain parameters, then to consider the inner maximization of the obtained linear approximation, which can be rewritten as a standard optimization problem. The solution of this problem involves the solution of several PDEs that is computationally expensive, and for this reason, we propose a model reduction approach in order to reduce the complexity of the problem. Numerical results are presented to show the effectiveness of the presented approach
A time-adaptive POD method for optimal control problems
No full textIn this paper we present a time adaptive technique for the solution of optimal control problems where the dynamic is given by an evolutive semi linear PDE. The method is based on a model reduction using a POD approximation coupled with a Hamilton-Jacobi equation characterizing the value function of the corresponding control problem for the reduced system. The choice of the POD basis is updated according to the evaluation of a numerical indicator in order to guarantee a global accurate solution. This is obtained via a sub-division of the time horizon into sub-intervals where the residual is below a given threshold. Some numerical tests illustrate the main features of this approach. © IFAC
HJB-POD feeback control of advection-diffusion equation with a model predictive control snapshot sampling
No full textIn this paper we present the approximation of an infinite horizon optimal control problem for evolutive advection-diffusion equations. The method is based on a model reduction technique, using a Proper Orthogonal Decomposition (POD) approximation, coupled with a Hamilton-Jacobi-Bellman (HJB) equation which characterizes the value function of the corresponding control problem for the reduced system. We show that it is possible to improve the surrogate model by means of a Model Predictive Control (MPC) solver. Finally, we present numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches
Randomized model order reduction
Full text linkThe singular value decomposition (SVD) has a crucial role in model order reduction. It is often utilized in the offline stage to compute basis functions that project the high-dimensional nonlinear problem into a low-dimensional model which is then evaluated cheaply. It constitutes a building block for many techniques such as the proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). The aim of this work is to provide an efficient computation of low-rank POD and/or DMD modes via randomized matrix decompositions. This is possible due to the randomized singular value decomposition (rSVD) which is a fast and accurate alternative of the SVD. Although this is considered an offline stage, this computation may be extremely expensive; therefore, the use of compressed techniques drastically reduce its cost. Numerical examples show the effectiveness of the method for both POD and DMD
Nonlinear Model Order Reduction via Dynamic Mode Decomposition
Full text linkWe propose a new technique for obtaining reduced order models for nonlinear dynamical systems. Specically, we advocate the use of the recently developed dynamic mode decomposition (DMD), an equation-free method, to approximate the nonlinear term. DMD is a spatio-temporal matrix decomposition of a data matrix that correlates spatial features while simul-taneously associating the activity with periodic temporal behavior. With this decomposition, one can obtain a fully reduced dimensional surrogate model and avoid the evaluation of the nonlinear term in the online stage. This allows for a reduction in the computational cost and, at the same time, accurate approximations of the problem. We present a suite of numerical tests to illustrate our approach and to show the e ectiveness of the method in comparison to existing approaches
Feedback control of parametrized PDEs via model order reduction and dynamic programming principle
Full text linkIn this paper, we investigate infinite horizon optimal control problems for parametrized partial differential equations. We are interested in feedback control via dynamic programming equations which is well-known to suffer from the curse of dimensionality. Thus, we apply parametric model order reduction techniques to construct low-dimensional subspaces with suitable information on the control problem, where the dynamic programming equations can be approximated. To guarantee a low number of basis functions, we combine recent basis generation methods and parameter partitioning techniques. Furthermore, we present a novel technique to construct non-uniform grids in the reduced domain, which is based on statistical information. Finally, we discuss numerical examples to illustrate the effectiveness of the proposed methods for PDEs in two space dimensions
Asymptotic Stability of POD based Model Predictive Control for a semilinear parabolic PDE
Full text linkIn this article a stabilizing feedback control is computed for a semilinear parabolic partial differential equation utilizing a nonlinear model predictive (NMPC) method. In each level of the NMPC algorithm the finite time horizon open loop problem is solved by a reduced-order strategy based on proper orthogonal decomposition (POD). A stability analysis is derived for the combined POD-NMPC algorithm so that the lengths of the finite time horizons are chosen in order to ensure the asymptotic stability of the computed feedback controls. The proposed method is successfully tested by numerical examples
Model order reduction approaches for infinite horizon optimal control problems via the HJB equation
No full textWe investigate feedback control for infinite horizon optimal control problems for partial differential equations. The method is based on the coupling between Hamilton-Jacobi-Bellman (HJB) equations and model reduction techniques. It is well-known that HJB equations suffer the so called curse of dimensionality and, therefore, a reduction of the dimension of the system is mandatory. In this report we focus on the infinite horizon optimal control problem with quadratic cost functionals. We compare several model reduction methods such as Proper Orthogonal Decomposition, Balanced Truncation and a new algebraic Riccati equation based approach. Finally, we present numerical examples and discuss several features of the different methods analyzing advantages and disadvantages of the reduction methods
Model order reduction approaches for the optimal design of permanent magnets in electro-magnetic machines
No full textIn an electromagnetic machine with permanent magnets the excitation field is provided by a permanent magnet instead of a coil. The center of the generator, the rotor, contains the magnet. Our optimization goal consists in finding the minimum volume of the magnet which gives a desired electromotive force. This results in an optimization problem for a parametrized partial differential equation (PDE). We propose a goal-oriented model order reduction approach to provide a reduced order surrogate model for the parametrized PDE which then is utilized in the numerical optimization. Numerical tests will be provided in order to show the effectiveness of the proposed method
A residual based snapshot location strategy for POD in distributed optimal control of linear parabolic equations
No full textIn this paper we study the approximation of a distributed optimal control problem for linear parabolic PDEs with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the basis functions are obtained upon information contained in time snapshots of the parabolic PDE related to given input data. In the present work we show that for POD-MOR in optimal control of parabolic equations it is important to have knowledge about the controlled system at the right time instances. For the determination of the time instances (snapshot locations) we propose an a-posteriori error control concept which is based on a reformulation of the optimality system of the underlying optimal control problem as a second order in time and fourth order in space elliptic system. Finally, we present numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches
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