Results on Transversal and Axial Motions of a System of Two Beams Coupled to a Joint through Two Legs

In recent years there has been renewed interest in inflatable-rigidizable space structures because of the efficiency they offer in packaging during boost-to-orbit. However, much research is still needed to better understand dynamic response characteristics, including inherent damping, of truss structures fabricated with these advanced material systems. We present results of an ongoing research related to a model consisting of an assembly of two beams with Kelvin-Voight damping, coupled to a simple joint through two legs. The beams are clamped at one end but at the other end they satisfy a boundary condition given in terms of an ODE coupling boundary terms of both beams, which reflects geometric compatibility conditions. The system is then written as a second order differential equation in an appropriate Hilbert space  in which well-posedness, exponential stability as well as other regularity properties of the solutions can be obtained. Two different finite dimensional approximation schemes for the solutions of the system are presented. Numerical results are presented and comparisons are made.Fil: Burns, J. A.. Interdisciplinary Center for Applied Mathematics; Estados UnidosFil: Cliff, E. M.. Interdisciplinary Center for Applied Mathematics; Estados UnidosFil: Liu, Z.. University of Minnesota at Duluth; Estados UnidosFil: Spies, Ruben Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentin

Cooling Capacity Control for Multi-Evaporator Vapor Compression Systems

Multi-evaporator vapor compression systems (ME-VCS) simultaneously provide cooling to multiple zones. The thermodynamic conditions in these zones are independent: the heat loads often differ, and the occupants of these spaces often have different desired room temperatures. Therefore, in order to regulate each zone to its desired setpoint temperature, the amount of thermal energy removed by each evaporator must be modulated independently. However, the common evaporating pressure within all evaporators introduces coupling that makes this objective difficult---the valve and piping arrangement imposes the constraint that all evaporators operate at the same temperature. (Systems considered here do not have valves at the outlet of each evaporator and therefore the individual evaporator pressures cannot be independently controlled.) In order to reduce the per-zone cooling, existing control strategies duty cycle the evaporator (alternate between a fully-open and fully-closed valve). However, duty cycling causes periodic disturbances to not only the local zone, but also to many machine temperatures and pressures, and these disturbances are often not transient but instead persist indefinitely. Fluctuations induced by the periodic disturbances can degrade the ability of the machine to regulate zone temperatures with zero steady state error, cause excessively high or low temperatures during peaks of the period, and couple into most machine signals of interest in ways that are difficult to describe with low order dynamical models. As an alternative to duty cycling, an observed behavior of refrigerant mass distribution in multi-path heat exchangers is exploited for control purposes. Multi-path heat exchangers are characterized by an inlet header pipe that splits refrigerant flow to two or more parallel paths through the heat exchanger and collects those paths into a common outlet header pipe. In the paper, we describe the following empirical phenomenon exploited for control: as the inlet valve is decreased, refrigerant mass flow rate entering the heat exchanger is reduced, and at some critical flow rate, refrigerant is shown to preferentially flow in some paths more than others, causing maldistribution. This uneven refrigerant distribution is repeatable and reduces the capacity in a continuous manner. The refrigerant distribution can be detected by temperature sensors along different paths of a multi-path heat exchanger. As some paths are starved for refrigerant they become superheated, and this uneven superheating process is unstable. A feedback controller is designed to provide stability and robustness to per-zone conditions. Finally, setpoints for this controller that relate per-path superheat temperature to overall evaporator capacity is created in such a way as to be robust to changes in local zone temperatures and the overall system evaporating temperature, which provides zone decoupling and ultimately creates a virtual control input for a supervisory controller such as a model predictive controller

Proportional-Integral Extremum Seeking for Optimizing Power of Vapor Compression Systems

Conventionally, online methods for minimizing power consumption of vapor compression systems rely on the use of physical models. These model-based approaches attempt to describe the influence of commanded inputs, disturbances and setpoints on the thermodynamic behavior of the system and the resultant consumed electrical power. These models are then used online to predict the combination of inputs for a measured set of thermodynamic conditions that both meets the heat load and minimizes power consumption. However, these models of vapor compression systems must contain nonlinear terms of sufficient complexity in order to accurately describe the region near the optimum operating point(s), but also must rely on simplifying assumptions in order to produce a mathematically tractable representation. For these reasons, model-based online optimization of vapor compression machines have not gained traction in application, and have created an opportunity for model-free techniques such as extremum seeking control, which is gradient descent optimization implemented as a feedback controller. While traditional perturbation-based extremum seeking controllers for vapor compression systems have proven effective at minimizing power without requiring a process model, the algorithm\u27s requirement for multiple distinct timescales has limited the applicability of this method to laboratory tests where boundary conditions can be carefully controlled, or simulation studies with unrealistic convergence times. Perturbation-based extremum seeking requires that the control input be manipulated with a time constant approximately two orders of magnitude slower than the slowest vapor compression system dynamics, otherwise instabilities in the closed loop system occur. As a result, convergence to the optimum for slow processes such as thermal systems is restrictive due to inefficient estimation of the gradient, and slow (integral-action dominated) adaptation in the extremum seeking control law. In order to address this timescale separation issue, we have previously developed an algorithm called ``time-varying extremum seeking that more efficiently estimates the gradient of the performance metric and applied this algorithm to the problem of setpoint optimization for compressor temperatures. That algorithm improved the convergence rate to one timescale slower than the vapor compression machine dynamics. In this paper, we optimize power consumption through the application of a newly-developed proportional--integral extremum seeking controller (PI-ESC) that converges at the same timescale as the process. This method uses the improved gradient estimation routines of time-varying extremum seeking but also modifies the control law to include terms proportional to the estimated gradient. This modification of the control law, in turn, requires a revision to the gradient estimator in order to avoid bias. PI-ESC is applied to the problem of compressor discharge temperature selection for a vapor compression system so that power consumption is minimized. Because of the improved convergence properties of PI-ESC, we show that optimum values of discharge temperature can be tracked in the presence of realistic disturbances such as variation in the outdoor air temperature---enabling application of extremum seeking control to vapor compression systems in environments where previous methods have failed. The method is demonstrated experimentally on a 2.8 kW split ductless room air conditioner and in simulation using a custom-developed Modelica model

Direct Synthesis of 5-Aryl Barbituric Acids by Rhodium(II)-Catalyzed Reactions of Arenes with Diazo Compounds

A commercially available rhodium(II) complex catalyzes the direct arylation of 5‐diazobarbituric acids with arenes, allowing straightforward access to 5‐aryl barbituric acids. Free N--H groups are tolerated on the barbituric acid, with no complications arising from N--H insertion processes. This method was applied to the concise synthesis of a potent matrix metalloproteinase (MMP) inhibitor

Realtime Optimization of MPC Setpoints using Time-Varying Extremum Seeking Control for Vapor Compression Machines

Recently, model predictive control (MPC) has received increased attention in the HVAC community, largely due to its ability to systematically manage constraints while optimally regulating signals of interest to setpoints. For example, in a common formulation of an MPC control problem for variable compressor speed vapor compression machines, the setpoints often include the zone temperature and the evaporator superheat temperature. However, the energy consumption of vapor compression systems has been shown to be sensitive to these setpoints. Further, while superheat temperature is often preferred because it can be easily correlated to heat exchanger efficiency (and therefore cycle efficiency), direct measurement of superheat is not always available. Therefore, identifying alternate signals in the control of vapor compression machines that correlate to efficiency is desired. Conventionally, methods for maximizing the energy efficiency rely on the use of mathematical models of the physics of vapor compression systems. These model-based approaches attempt to describe the influence of commanded inputs on the thermodynamic behavior of the system and the consumed electrical energy, and they are used to predict the combination of inputs that both meet the heat load requirements and minimize energy consumption. However, these models of vapor compression systems rely on simplifying assumptions in order to produce a mathematically tractable representation. Further, they are difficult to derive and calibrate, and often do not describe variations over long time scales, such as those due to refrigerant losses or accumulation of debris on the heat exchangers. In this paper, we consider a model-free extremum seeking algorithm that adjusts setpoints provided to a model predictive controller. While perturbation-based extremum seeking methods have been known for some time, they suffer from slow convergence rates---a problem emphasized by the long time constants associated with thermal systems. Our method uses a new algorithm (time-varying extremum seeking), which has dramatically faster and more reliable convergence properties. In particular, we regulate the compressor discharge temperature using an MPC controller with setpoints selected from a model-free time-varying extremum seeking algorithm. We show that the relationship between compressor discharge temperature and power consumption is convex (a requirement for this class of realtime optimization), and use time-varying extremum seeking to drive these setpoints to values that minimize power. The results are compared to the traditional perturbation-based extremum seeking approach. Further, because the required cooling capacity (and therefore compressor speed) is a function of measured and unmeasured disturbances, the optimal compressor discharge temperature setpoint must vary according to these conditions. We show that the energy optimal discharge temperature is tracked with the time-varying extremum seeking algorithm in the presence of disturbances