This means that one needs to increase the number of samples by 100 to improve the accuracy by one decimal place. However, it is well known that MC methods suffer from slow convergence with convergence rate being inversely proportional to the square root of number of sample points. Monte Carlo (MC) realization can also be used to sample the uncertain space.
A uniform sampling of the uncertain space was used to solve the minimax problem. This, in conjunction with constraints on the terminal energy, resulted in a robust state and energy constrained controller. They formulated a linear programming (LP) problem with a cost function of minimizing the maximum excursion of the states over the duration of the maneuver called the minimax control. Vossler and Singh addressed the problem of design of the optimal control profiles for systems with uncertain parameters, for rest-to-rest maneuvers with state constraints. Deflection sampling is also used to constrain the permitted deflection in the design of the controller. where an analytical expression for the evolution of the deflection for a spring-mass system is derived and its gradient is forced to zero to identify time instants, which correspond to maximum or minimum deflections. In addition to the residual energy constraint, deflection constraints have been considered by Singhose et al. This is important in rest-to-rest class of problems.
The aforementioned problem forumulations have often considered minimizing the residual energy, i.e., the undesired energy in the system at the end of the maneuver. Numerical examples are used to illustrate the benefit of the proposed technique for the design of a rest-to-rest controller subject to deformation constraints and which are robust to uncertainties in the stiffness coefficient for the benchmark spring-mass-damper system. Linear programming (LP) is then used on the deterministic set of equations with constraints on the bounds of the states to determine the controller.
#CHAOS CONTROL PC PC#
A transformation of PC polynomial space to the Bernstein polynomial space permits determination of bounds on the evolving states of interest. The coefficients of the truncated PC expansion are determined using the Galerkin projection resulting in a set of deterministic equations. Polynomial chaos (PC) expansion, a spectral expansion, is used to parameterize the uncertain variables permitting the evolution of the uncertain states to be written as a polynomial function of the uncertain variables. This paper presents a methodology to design a state constrained controller, which is robust to time invariant uncertain variables.
Journal of Engineering for Gas Turbines and Power.Journal of Engineering and Science in Medical Diagnostics and Therapy.Journal of Electrochemical Energy Conversion and Storage.Journal of Dynamic Systems, Measurement, and Control.
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