Approximation of reachable sets using optimal control algorithms

To appearInternational audienceNumerical experiences with a method for the approximation of reachable sets of nonlinear control systems are reported. The method is based on the formulation of suitable optimal control problems with varying objective functions, whose discretization by Euler's method lead to finite dimensional non-convex nonlinear programs. These are solved by a sequential quadratic programming method. An efficient adjoint method for gradient computation is used to reduce the computational costs. The discretization of the state space is more efficiently than by usual sequential realization of Euler's method and allows adaptive calculations or refinements. The method is illustrated for two test examples. Both examples have non-linear dynamics, the first one has a convex reachable set, whereas the second one has a non-convex reachable set

Approximation of reachable sets using optimal control and support vector machines

We propose and discuss a new computational method for the numerical approximation of reachable sets for nonlinear control systems. It is based on the support vector machine algorithm and represents the set approximation as a sublevel set of a function chosen in a reproducing kernel Hilbert space. In some sense, the method can be considered as an extension to the optimal control algorithm approach recently developed by Baier, Gerdts and Xausa. The convergence of the method is illustrated numerically for selected examples

Grid approximation of singularly perturbed parabolic equations with moving boundary layers

A grid approximation of a boundary value problem is considered for a singularly perturbed parabolic reaction‐diffusion equation in a domain with boundaries moving along the x‐axis in the positive direction. For small values of the parameter ϵ (that is the coefficient of the highest‐order derivative in the equation, ϵ ∈ (0,1]), a moving boundary layer appears in a neighbourhood of the left lateral boundary SL 1. It turns out that, in the class of difference schemes on rectangular grids condensing in a neighbourhood of SL 1 with respect to x and t, there do not exist schemes that converge even under the condition P 0 −1 Â ϵ1/2, where P 0 is the total number of nodes in the meshes used, that is, P 0 Â N N 0, where the values N and N 0 define the numbers of mesh points in x and t. On such meshes, convergence under the condition N −1 + N 0 −1 ≤ ϵ1/4 cannot be achieved. Examination of widths similar to Kolmogorov's widths allows us to establish necessary and sufficient conditions for the ϵ‐uniform convergence of approximations to the solution of the boundary value problem. Using these conditions, a scheme is constructed on a mesh being piece‐wise uniform in a coordinate system adapted to the moving boundary. This scheme converges ϵ‐uniformly at the rate O(N −1 ln N + N0 −1). First Published Online: 14 Oct 201

Design Of Feedback Control For Active Mass Dampers Of Excited Structures

Annually, our world experiences thousands of seismic events that are the cause of hundreds of structural disasters and human fatalities. The objective of the presented research is to contribute to the world’s social, economic, and environmental needs by designing an optimized feedback control for active mass dampers (AMDs) by reducing oscillations. The optimal design will meet the required specifications and maintain a structure’s quasi-ideal, static position throughout a seismic event. The system’s equation of motion (EOM) is derived by using the Lagrangian Method and the free-body diagram. All the simulated and experimental responses of the AMD-1 system are obtained using MATLAB and Simulink. The experimental data is collected from various tests performed on a single-story building model. The techniques utilized for improvement of the AMD’s feedback control include parameter estimation, eigenvalue assignment, and linear quadratic regulation (LQR). As success is achieved with the AMD feedback control, future research can focus on idealizing the AMD’s performance in a system with multiple degrees of freedom

Design Of Feedback Control For Active Mass Dampers Of Excited Structures

Annually, our world experiences thousands of seismic events that are the cause of hundreds of structural disasters and human fatalities. The objective of the presented research is to contribute to the world’s social, economic, and environmental needs by designing an optimized feedback control for active mass dampers (AMDs) by reducing oscillations. The optimal design will meet the required specifications and maintain a structure’s quasi-ideal, static position throughout a seismic event. The system’s equation of motion (EOM) is derived by using the Lagrangian Method and the free-body diagram. All the simulated and experimental responses of the AMD-1 system are obtained using MATLAB and Simulink. The experimental data is collected from various tests performed on a single-story building model. The techniques utilized for improvement of the AMD’s feedback control include parameter estimation, eigenvalue assignment, and linear quadratic regulation (LQR). As success is achieved with the AMD feedback control, future research can focus on idealizing the AMD’s performance in a system with multiple degrees of freedom