Optimal feedback gains of a delayed proportional-derivative (PD) control for balancing an inverted pendulum

In the dynamics analysis and synthesis of a con-trolled system, it is important to know for what feedback gains can the controlled system decay to the demanded steady state as fast as possible. This article presents a sys-tematic method for finding the optimal feedback gains by taking the stability of an inverted pendulum system with a delayed proportional-derivative controller as an example. First, the condition for the existence and uniqueness of the stable region in the gain plane is obtained by using the D-subdivision method and the method of stability switch. Then the same procedure is used repeatedly to shrink the stable region by decreasing the real part of the rightmost charac-teristic root. Finally, the optimal feedback gains within the stable region that minimizes the real part of the rightmost root are expressed by an explicit formula. With the optimal feedback gains, the controlled inverted pendulum decays to its trivial equilibrium at the fastest speed when the initial val-ues around the origin are fixed. The main results are checked by numerical simulation.     

On constructing a Green’s function for a semi-infinite beam with boundary damping

The main aim of this paper is to contribute to the construction of Green’s functions for initial boundary value problems for fourth order partial differential equations. In this paper, we consider a transversely vibrating homogeneous semi-infinite beam with classical boundary conditions such as pinned, sliding, clamped or with a non-classical boundary conditions such as dampers. This problem is of important interest in the context of the foundation of exact solutions for semi-infinite beams with boundary damping. The Green’s functions are explicitly given by using the method of Laplace transforms. The analytical results are validated by references and numerical methods. It is shown how the general solution for a semi-infinite beam equation with boundary damping can be constructed by the Green’s function method, and how damping properties can be obtained.     

Impact of Strong Magnetic Fields on Collision Mechanism for Transport of Charged Particles

One of the main applications in plasma physics concerns the energy production through thermo-nuclear fusion. The controlled fusion is achieved by magnetic confinement i.e., the plasma is confined into a toroidal domain (tokamak) under the action of huge magnetic fields. Several models exist for describing the evolution of strongly magnetized plasmas, most of them by neglecting the collisions between particles. The subject matter of this paper is to investigate the effect of large magnetic fields with respect to a collision mechanism. We consider here linear collision Boltzmann operators and derive, by averaging with respect to the fast cyclotronic motion due to strong magnetic forces, their effective collision kernels.     

Preparation and characterization of (Pebax 1657 + silica nanoparticle)/PVC mixed matrix composite membrane for CO<Subscript>2</Subscript>/N<Subscript>2</Subscript> separation

In this work, the films of poly(ether-block-amide) (Pebax 1657) and hydrophilic/hydrophobic silica nanoparticles (0–10 wt%) were coated on a poly(vinyl chloride) (PVC) ultrafiltration membrane to form new mixed matrix composite membranes (MMCMs) for CO₂/N₂ separation. The membranes were characterized by SEM, FTIR, DSC and XRD. Successful formation of a non-porous defect-free dense top layer with ~4 μm of thickness and also uniform dispersion of silica nanoparticles up to 8 wt% loading in Pebax matrix were confirmed by SEM images. The gas permeation results showed an increase in the permeance of all gases and an increase in ideal CO₂/N₂ selectivity with the increase in silica nanoparticle contents. Comparison between the incorporation of hydrophilic and hydrophobic silica nanoparticle into Pebax matrix revealed that the great enhancement of CO₂ solubility is the key factor for the performance improvement of Pebax + silica nanoparticle membranes. The best separation performance of the hydrophilic silica nanoparticle-incorporated Pebax/PVC membrane for pure gases (at 1 bar and 25 °C) was obtained with a CO₂ permeability of 124 barrer and an ideal CO₂/N₂ selectivity of 76, i.e., 63 and 35% higher than those of neat Pebax membrane, respectively. The corresponding values for hydrophobic silica nanoparticle-incorporated Pebax/PVC membrane were 107 barrer for CO₂ permeability and 61 for ideal CO₂/N₂ selectivity. Also the performances of MMCMs improved upon pressure increase (1–10 bar) owing to the shift in plasticizing effect of CO₂ towards the higher pressures. In addition, an increase in permeabilities with a decrease in ideal selectivity was observed upon temperature increase (25–50 °C) due to the intensification of chain mobility.     

Ergodicity of a Single Particle Confined in a Nanopore

We analyze the dynamics of a gas particle moving through a nanopore of adjustable width with particular emphasis on ergodicity. We give a measure of the portion of phase space that is characterized by quasiperiodic trajectories which break ergodicity. The interactions between particle and wall atoms are mediated by a Lennard-Jones potential, so that an analytical treatment of the dynamics is not feasible, but making the system more physically realistic. In view of recent studies, which proved non-ergodicity for systems with scatterers interacting via smooth potentials, we find that the non-ergodic component of the phase space for energy levels typical of experiments, is surprisingly small, i.e. we conclude that the ergodic hypothesis is a reasonable approximation even for a single particle trapped in a nanopore. Due to the numerical scope of this work, our focus will be the onset of ergodic behavior which is evident on time scales accessible to simulations and experimental observations rather than ergodicity in the infinite time limit.     

Discontinuous alkylation of diphenylamine with nonene for maximum catalyst utilization

This study examines the alkylation of diphenylamine (DPA) with nonene (NON) in a liquid phase catalyzed by acid-treated clay-based catalysts from commercial suppliers (Fulcat 22B, Nobelin MM, and Jeltar 300). Alkylations were conducted to achieve the highest possible selectivity of diisononyldiphenylamine (DNDPA), low selectivity of monoisononyldiphenylamine, and a maximum triisononyldiphenylamine yield of 4%. This study also examines the reaction conditions to selectively form dialkylated diphenylamine from DPA and NON in a batch reactor. Repeated use of the catalyst during the alkylation of DPA with NON was also investigated. Catalyst deactivation takes place during the alkylation of each batch and intensifies with repeated catalyst use, resulting in low DNDPA selectivity. The regenerated catalyst was sufficiently active only until the regeneration of the first and second batches. After the third batch, the catalyst’s selectivity for DNDPA was very low, and its reuse in the alkylation of DPA with NON was not efficient. Therefore, to achieve the maximum length of catalyst activity, the fresh catalyst was gradually added to the used catalyst from a previous batch, thus maintaining a high activity of eight batches. The reduction in catalyst activity was probably caused by the irreversible adsorption of substances on the surface, a loss of microporous structure, and a loss of surface acidity. DPA or alkylated products are adsorbed on the surface oxygen of the catalyst through nitrogen and form nitro formations. The fresh and regenerated catalysts were characterized by their surface area, surface acidity, pore size distribution, and pore volume.