Tuning of PID Controllers Based on Bode’s Ideal Transfer Function

This paper presents a new strategy for tuning PID controllers based on a fractional reference model. The model is represented as an ideal closed-loop system whose open-loop is given by the Bode’s ideal transfer function. The PID controller parameters are determined by the minimization of the integral square error (ISE) between the time responses of the desired fractional reference model and of the system with the PID controller. The resulting closed-loop system (with the PID controller) has the desirable feature of being robust to gain variations with step responses exhibiting an iso-damping property. Several examples are presented that demonstrate the effectiveness and validity of the proposed methodology.     

On the Propagation of Long‐Wave Perturbations in a Two‐Layer Free‐Boundary Rotational Fluid

A mathematical model for the propagation of longwave perturbations in a freeboundary shear flow of an ideal stratified twolayer fluid is considered. The characteristic equation defining the velocity of perturbation propagation in the fluid is obtained and studied. The necessary hyperbolicity conditions for the equations of motion are formulated for flows with a monotonic velocity profile over depth, and the characteristic form of the system is calculated. It is shown that the problem of deriving the sufficient hyperbolicity conditions is equivalent to solving a system of singular integral equations. The limiting cases of weak and strong stratification are studied. For these models, the necessary and sufficient hyperbolicity conditions are formulated, and the equations of motion are reduced to the Riemann integral invariants conserved along the characteristics.     

Hyperbolicity of Steady‐State Equations of Gas Shear Flows in a Thin Layer

The steadystate threedimensional motion of an ideal gas in a thin layer of variable height is considered. In the longwave approximation, the equations of gas dynamics reduce to a system of integrodifferential equations. The generalized characteristics and hyperbolicity conditions of the obtained system are found.     

“Thermal layer” effect in MHD: Interaction of a parallel shock with a layer of decreased density

Interaction of a parallel fast MHD shock with a layer of decreased density is discussed using ideal MHD approach. This is an extrapolation of gas dynamic thermal layer effect on ideal MHD. Computer simulations show that a magnetic field of a moderate intensity ( 1) may change the character of the flow for intermediate Mach numbers (M 5) and a new raking regime may occur which is not observed in the absence of a magnetic field. Self similar precursor analogous to that in gas dynamics may develop in the case of highM and low density in the layer but magnetic forces essentially decrease its growth rate. This problem appears in connection with cosmical shock propagation where planetary magnetic tails play the role of the thermal layer, and it may also be observed in the laboratory when the shock is strong enough to heat the walls ahead of it.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

A novel framework to design and compare limit cycle minimizing controllers: demonstration with integer and fractional-order controllers

In this paper, a constrained optimization problem is formulated to tune the limit cycle minimizing controllers meeting additional loop-shaping performances such as phase margin and gain crossover frequency. A graphical approach is proposed so as to determine the superior controller in terms of better limit-cycle suppression. The framework is illustrated with a suitable case of elementary servo plant which has separable static backlash nonlinearity in its model. For this plant, integer-order controllers and their fractional counterparts (PI and \( PI ^\alpha , [ PI ]^\alpha \) ; PID and \( PI ^\alpha D^\beta \) ) are designed and compared. Interestingly, it is found that the fractional controllers produce better limit-cycle responses than their integer counterparts while both meeting the rest of the specifications. Correspondingly, the better sustained oscillations in the plant output response are obtained with fractional controllers. Such a ‘fractional superiority’ is further verified with the closed-loop nonlinear simulation.     

Disruption of Shaped‐Charge Jets by a Pulsed Current

This paper gives results from experimental studies of the effect of electricpulse parameters on the development of current instability and disruption of shapedcharge jets. A simple physical model for the development of current instability and the decrease in the cavern depth in the target is proposed. Notations are introduced and analytical expressions are obtained for the critical current (critical linear current density) and the ideal shape of the current pulse required for the disruption of shapedcharge jets. It is shown that the estimate of the final cavern depths in steel target for the proposed model is in fairly good agreement with both experimental values of cavern depths in targets and with the results of numerical calculations using quasitwodimensional unsteady models of MHD instability and volume disruption.     

Fractional Dispersion,Lévy Motion,and the MADE Tracer Tests

The macrodispersion experiments (MADE) at the Columbus Air Force Base in Mississippi were conducted in a highly heterogeneous aquifer that violates the basic assumptions of local second-order theories. A governing equation that describes particles that undergo Lévy motion, rather than Brownian motion, readily describes the highly skewed and heavy-tailed plume development at the MADE site. The new governing equation is based on a fractional, rather than integer, order of differentiation. This order (), based on MADE plume measurements, is approximately 1.1. The hydraulic conductivity (K) increments also follow a power law of order =1.1. We conjecture that the heavy-tailed K distribution gives rise to a heavy-tailed velocity field that directly implies the fractional-order governing equation derived herein. Simple arguments lead to accurate estimates of the velocity and dispersion constants based only on the aquifer hydraulic properties. This supports the idea that the correct governing equation can be accurately determined before, or after, a contamination event. While the traditional ADE fails to model a conservative tracer in the MADE aquifer, the fractional equation predicts tritium concentration profiles with remarkable accuracy over all spatial and temporal scales.     

Seismic-vibration mitigation of a nonlinear structural system with an ATMD through a fuzzy PID controller

This paper discusses the design of fuzzy PID type controllers (FPIDC) to improve seismic control performance of a nonlinear structural system with an active tuned mass damper (ATMD) against earthquakes. Since structural systems have nonlinearities and uncertainties, fuzzy-based controllers are adequate because of their robust character and satisfactory performance in active structural control. The main advantages of this controller are the ability to handle nonlinearities and uncertainties effectively. In the literature, various structures for fuzzy PID (including PI and PD) controllers have been proposed. In order to obtain proportional, integral and derivative control actions altogether, it is intuitive and convenient to combine PI and PD actions to form a fuzzy PID controller. The simulated system has fifteen degrees of freedom and is modeled using nonlinear behavior of the base–structure interaction. The system is then simulated against the ground motion of the Northridge earthquake (M _w =6.7) in USA on 17 January, 1994. Finally, the time history of the storey displacements, accelerations, ATMD displacements, control voltage and frequency responses of both the uncontrolled and controlled cases are presented. The ground motion recorded of the El-Centro and Kocaeli earthquakes has been used to evaluate the effectiveness of the proposed control algorithm. The robustness of the controller has been checked through the uncertainty in stiffness of the structure. Simulation results exhibit that superior vibration suppression is achieved by the use of designed fuzzy PID type controllers.     

Evolution of the Diffusion Mixing Layer of Two Gases upon Interaction with Shock Waves

A mathematical model of mechanics of a twovelocity twotemperature mixture of gases is developed. Based on this model, evolution of the mixing layer of two gases with different densities under the action of shock and compression waves is considered by methods of mathematical simulation in the onedimensional unsteady approximation. In the asymptotic approximation of the full model, a solution of an initialboundary problem is obtained, which describes the formation of a diffusion layer between two gases. Problems of interaction of shock and compression waves with the diffusion layer are solved numerically in the full formulation. It is shown that the layer is compressed as the shock wave traverses it; the magnitude of compression depends on shockwave intensity. As the shock wave passes from the heavy gas to the light gas, the mixing layer becomes overcompressed and expands after shockwave transition. The wave pattern of the flow is described in detail. The calculated evolution of the mixinglayer width is in good agreement with experimental data.