Unconstrained Degrees Of Freedom Research Articles

Decentralized control using selectors for optimal steady-state operation with changing active constraints

We study the optimal steady-state operation of processes where the active constraints change. The aim of this work is to eliminate or reduce the need for a real-time optimization layer, moving the optimization into the control layer by switching between appropriately selected controlled variables (CVs) in a simple way. The challenge is that the best CVs, or more precisely the reduced cost gradients associated with the unconstrained degrees of freedom, change with the active constraints. This work proposes a framework based on decentralized control that operates optimally in all active constraint regions, with region switching mediated by selectors. A key point is that the nullspace associated with the unconstrained cost gradient needs to be selected in accordance with the constraint directions so that selectors can be used. A main benefit is that the number of SISO controllers that need to be designed is only equal to the number of process inputs plus constraints. The main assumptions are that the unconstrained cost gradient is available online and that the number of constraints does not exceed the number of process inputs. The optimality and ease of implementation are illustrated in a simulated toy example with linear constraints and a quadratic cost function. In addition, the proposed framework is successfully applied to the nonlinear Williams–Otto reactor case study.

Optimal operation of a natural gas sweetening plant

The most common method of removing acid gases from natural gas is through the use of solvent-based gas sweetening plants, which are energy-intensive. The present study aims to establish a plant-wide structure to operate natural gas sweetening plants optimally. Thus, a steady-state simulation of the Khangiran plant was carried out, and a 17% reduction in energy usage was achieved as a result of the optimization by the application of surrogate models. The self-optimizing method was applied to find the best set of controlled variables (CVs). There was found one unconstrained degree of freedom (DOF). The best CV for the remaining DOF was found to be the top stage temperature of the regenerator. Aspen Plus Dynamics was used to examine the performance of the proposed control structure utilizing simple traditional PIDs and compare it to modern feedback and feedforward MPCs.

An approach to analyze the position and orientation between two parts assembled by non-ideal planes

Performance analysis, which plays a key role in the design stage, is employed to estimate whether product performance can satisfy design requirements. In general, product performance is gained after parts are assembled; product performance is influenced by the position and orientation deviations (PODs) that occur in directions of the constrained degrees of freedom (DOFs) due to the surface deviations of mating-surfaces. Furthermore, PODs are uncertain because the surface deviations as well as positions in the unconstrained DOF directions can vary randomly. Thus, predicting the consequences of uncertain PODs on product performance is key for performance analysis. Considering that planes are extensively used in assemblies, this study aims to propose a statistical approach to analyze the uncertain PODs of non-ideal planes. A modeling method from the perspective of manufacturing errors is employed to describe the uncertain surface deviations. A method for computing the uncertain PODs based on the progressive adjustment of coordinate systems is proposed. The maximum PODs that characterize the most unfavorable assembly situation are determined as evaluation indicators. Finally, the effectiveness of the presented approach is verified by a case study. Because both the effects of uncertain surface deviations and uncertain positions on PODs can be considered, the approach is expected to help predict the practical effects of uncertain PODs on product performance accurately during the design stage.

Invariants for optimal operation of a reactor-separator-recycle process

Globally optimal invariants for the one unconstrained degree of freedom in an A + B → C reactor-separator-recycle process are derived for four alternative reaction kinetic expressions from an analysis of the overall plant material balance. The derivation is performed for the simplified economic objective of minimizing recycle rate (Mode I) at given production and maximizing production (Mode II) with maximum recycle rate as the capacity bottleneck. For power law kinetics, holding the reactor A/B ratio constant is obtained as the invariant while a more complex non-linear expression is obtained for Langmuir-Hinshelwood kinetics. For the more practical economic objective of minimizing separation column boilup or maximum column boilup as the capacity bottleneck, the invariant as the controlled variable (CV) achieves near optimal operation with the loss being <0.1% over the envisaged operating space in all cases. The invariants are thus good self-optimizing CVs (SOCVs). The work provides a physical basis for using reactant ratio as an SOCV in plantwide control.

Enhanced-efficiency operating variables selection for vapor compression refrigeration cycle system

In this paper, a novel enhanced-efficiency selection of operating variables based on self-optimizing control (SOC) method for the vapor compression refrigeration cycle (VCC) system is proposed. An objective function is proposed to maximize the energy efficiency of the VCC system while meeting with the demand of indoor thermal comfort. With the detailed analysis of operating variables, three unconstrained degrees of freedom are selected among all the candidate operating variables. Then two SOC methods are applied to determine the optimal individual controlled variables (CVs) and measurement combinations as CVs. The model predictive control (MPC) method based controllers and PID controllers are designed for different sets of CVs, and the experimental results indicate that the proposed selection of CVs can achieve a good trade-off between optimal (or near optimal) stable operation and enhanced-efficiency of the synthesized control structure.

Economic Plantwide Control of the Cumene Process

Economic plantwide control of the cumene process over a large throughput range (design to maximum achievable throughput) is studied. The process has 12 steady state operating degrees of freedom (DOFs), which are optimized for maximum hourly profit. At maximum throughput, the highest number of constraints (eight) is active leaving four unconstrained degrees of freedom. Reasonable controlled variables (CVs) corresponding to these are chosen with implementation of a constant setpoint policy at lower throughputs providing near optimal operation. For economic plantwide control, the best pairings for tight control of the maximum throughput active constraints and self-optimizing CVs are first implemented followed by inventory control loop pairings using the remaining valves. The resulting control structure consists of long level loops that manipulate the two fresh feeds to maintain the bottom sump level of the first two distillation columns. Simulation results show that the unconventional control structure provides smooth operation over the wide throughput range with tight control of the active constraints and the self-optimizing CVs. Comparison with a conventional plantwide control system (throughput manipulator at fresh propylene feed) shows that the synthesized control structure is simpler in requiring no overrides for handling constraints and achieving superior economic performance.

Selection of Controlled Variables for a Natural Gas to Liquids Process

The aim of this work is to select the best individual or combined controlled variables (CVs) for a natural gas to hydrocarbon liquids (GTL) process based on the idea of self-optimizing control. The objective function is to maximize the variable income of the plant, and two modes of operation are studied. In mode I, where the natural gas flow rate is given, there are three unconstrained degrees of freedom (DOFs) and the corresponding individual self-optimizing CVs are selected as (i) CO2 removal in fresh synthesis gas (syngas), (ii) CO mole fraction in fresh syngas, and (iii) CO mole fraction in recycle tail gas from the Fischer−Tropsch (FT) reactor. This set of CVs gives a worst-case loss of 1,393 USD/h. Adding one, two, and three measurements and controlling measurement combinations decrease the worst-case loss significantly, to 184, 161, and 53 USD/h, respectively. In mode II, the natural gas flow rate is a degree of freedom and it is optimal to increase it as much as possible to maximize profit. The variable income increases almost linearly until the oxygen flow rate becomes active. Practically, this is the maximum achievable income. Theoretically, it is possible to increase the natural gas flow rate to improve the objective function, but this results in large recycle flow rates to the FT reactor (similar to snowballing) because its volume is the limitation.

Economic back-off selection based on optimal multivariable controller

This paper discusses the minimum backed off operating point selection problem based on process economics. In this work, we consider the case where the nominal operating point is not completely constrained, i.e., there are some unconstrained degrees of freedom or manipulations available. In this regard, we propose a stochastic formulation that ensures feasible dynamic operating region within the prescribed confidence limit. Furthermore, the formulation also finds a suitable multivariable controller to achieve economic benefits. The problem is nonlinear and non-convex and hence an iterative solution procedure is proposed such that at each step in the iteration, a convex problem is solved. Finally, the approach is illustrated using an evaporation process.

Economically efficient operation of CO 2 capturing process part I: Self-optimizing procedure for selecting the best controlled variables

We study optimal operation of a post-combustion CO 2 capturing process where optimality is defined in terms of a cost function that includes energy consumption and penalty for released CO 2 to the air. Three different operational regions are considered. In region I, with a nominal flue gas flowrate, there are two optimally unconstrained degrees of freedom (DOFs) and the corresponding best self-optimizing controlled variables (CVs) are found to be the CO 2 recovery in the absorber and the temperature at tray no. 16 in the stripper. In the region II, with an increased flue gas flowrate, the heat input is saturated and there is one unconstrained DOF left. The best CV is temperature at tray no. 13 in the stripper. In region III, when the flue gas flowrate is further increased, the process reaches the minimum allowable CO 2 recovery of 80% and there is no unconstrained DOF. We have then reached the bottleneck and a controller is needed to adjust the feed flowrate to avoid violating this minimum. The exact local method and the maximum gain rule are applied to find the best CVs in each region.

A Unified Approach to Kinematic and Tolerance Analysis of Locating Fixtures

A workholding fixture should ensure a stable and precise positioning of the workpiece with respect to the machine tool. This requirement is even more important when modular fixtures are used for the sake of efficiency and reconfigurability. They include standard locating elements, which set the part in a predefined spatial orientation by contacting its datum surfaces. In the computer-based design of a fixture, the layout of locators must be tested against two main sources of problems. Kinematic analysis verifies that any relative motion between the part and the worktable is constrained. Tolerance analysis evaluates the robustness of part orientation with respect to manufacturing errors on datum surfaces. We propose a method to carry out both tests through a common set of geometric parameters of the fixture configuration. These derive from the singular value decomposition of the matrix that represents positioning constraints in screw coordinates. For a poorly designed fixture, the decomposition allows us to find out either unconstrained degrees of freedom of the part or a possible violation of tolerance specifications on machined features due to geometric errors on datum surfaces. In such cases, the analysis provides suggestions to plan the needed corrections to the locating scheme. This paper describes the procedure for kinematic and tolerance analysis and demonstrates its significance on a sample case of fixture design.

Null Space Method for Selecting Optimal Measurement Combinations as Controlled Variables

The issue in this paper is to select controlled variables c as combinations of the measurements y. The objective is to obtain self-optimizing control, which is when we can achieve near-optimal steady-state operation with constant setpoints for the controlled variables, without the need to reoptimize when new disturbances perturb the plant. The null space method yields locally optimal controlled variables c = Hy that are linear combinations of measurements y. The requirement is that we at least have as many measurements as there are unconstrained degrees of freedom, including disturbances, and that the implementation error is neglected. The method is surprisingly simple. From a steady-state model of the plant, the first step is to obtain the optimal sensitivity matrix F, with respect to the disturbances. The optimal matrix H satisfies HF = 0; therefore, the next step is to obtain H in the left null space of F. As an illustration, the method is used to obtain temperature combinations for control of a Petlyu...

DYNAMIC SIMULATION OF A SPATIAL 3-DOF TENSEGRITY MECHANISM

Tensegrity mechanisms have the advantage of being relatively lightweight due to. their extensive use of cables and springs. As such, they have the potential of being an attractive alternative to conventional mechanisms in certain application environments. However, the presence of unconstrained degrees of freedom in tensegrity mechanisms leads to a dynamic behaviour that cannot be directly controlled with the actuators. In this work, the dynamic model of a novel spatial three-degree-of-freedom (3-DOF) tensegrity mechanism is developed using the Lagrangian formulation. The resulting equations of motion are then solved to simulate the mechanism's motion between equilibrium configurations. Since the mechanism is subjected to holonomic nonlinear geometrical constraints, these must be considered during the solution of its forward dynamic problem. It is seen that the use of damping in the springs is not very efficient in dissipating the mechanism's energy during motion.

Variable-Gain Constraint Stabilization for General Multibody Systems with Applications

This paper presents a general and efficient formulation applicable to a vast variety of rigid and flexible multibody systems. It is based on a variable-gain error correction with scaling and adaptive control of the convergence parameter. The methodology has the following distinctive features. (i) All types of holonomic and non-holonomic equality constraints as well as a class of inequalities can be treated in a plain and unified manner. (ii) Stability of the constraints is assured. (iii) The formulation has an order Ncomputational cost in terms of both the constrained and unconstrained degrees of freedom, regardless of the system topology. (iv) Unlike the traditional recursive order Nalgorithms, it is quite amenable to parallel computation. (v) Because virtually no matrix operations are involved, it can be implemented to very simple general-purpose simulation programs. Noting the advantages, the algorithm has been realized as a C++ code supporting distributed processing through the Message-Passing Interface (MPI). Versatility, dynamical validity and efficiency of the approach are demonstrated through numerical studies of several particular systems including a crawler and a flexible space structure.

A Parallel O(N) Formulation for General Multibody Dynamics

A new methodology for dynamical analyses applicable to a very large class of rigid and flexible multibody systems is presented. It is based on a variable-gain error correction method with scaling, and has the following distinctive features: (i)All kinds of holonomic and nonholonomic equality constraints can be treated in a plain and unified manner; (ii)Stability of the constraints is always attained; (iii)The formulation has an order N computational cost in terms of both the constrained and unconstrained degrees of freedom, regardless of the system topology; (iv)Unlike the traditional recursive order N algorithms, it is quite amenable to parallel computation; (v)Since no matrix operations are involved, it can be implemented to very simple general-purpose simulation programs. Versatility, dynamical validity and efficiency of the approach are checked through numerical studies of several particular systems.

Dynamic Modelling of an Off-Road Vehicle for the Design of a Semi-Active, Hydropneumatic Spring-Damper System

SUMMARYRide and handling simulation of a heavy off-road military vehicle was simultaneously performed using DADS software. A non-linear vehicle simulation model, complete with a total of 66 degrees of freedom, was developed. After adding constraints, joints and a driver model, 14 unconstrained degrees of freedom remained. Ride comfort and handling/stability evaluations were combined in a single simulation run by driving the vehicle over Belgian paving whilst simultaneously performing a double lane change handling manoeuvre. The aim of the study was to determine the “best” spring and damper characteristics for ride comfort and handling/stability respectively. These characteristics will be used to design a semi-active, hydropneumatic spring-damper, able to switch both spring and damper characteristics electronically between two discrete values. The simulation results indicated that, for best ride comfort, the suspension natural frequency should be in the region of 0.6 Hz while the damper characteristic should be reduced to a value between 20 and 50% of the baseline damper value. For best handling and stability, a suspension natural frequency of 2 Hz is required while the baseline damping should be doubled.

Identifying the characteristics of finite element analysis convergence curves

Given the mathematical idealization of a structure, modeling introduces all the approximations which vanish as the number of unconstrained degrees of freedom of the finite element grid approaches infinity. As a practical matter, the computer code selected to support the analysis limits the definition of the idealized model and hence, the accuracy of the analysis. Sensitivity analysis can furnish information on the effect of these limitations on structural response. However, by definition, the judgement exercised by the analyst in idealization is unassailable by the computer program. That is, we assume that during a finite element analysis the computer has no access to alternative unprogrammed idealizations. A very important example of idealization is the choice of the constitutive equations and the choice of the coefficients of its particularization. Choices affecting the relevance of the element model include use of Euclidean geometry to define the length, volume, and deformations of the structure and the coefficients chosen to particularize the geometry. The assumption of frictionless pins or spring supports, the limitation of the distribution of applied loadings, and the mathematical model of nodal displacement constraints are other examples of idealization choices. Examples of modeling approximations include element and/or node approximations. Use of straight lines to represent curves, polygons or hyperbolas to model circular edges, and stepped surface thicknesses typify approximations to the original geometry. Use of polynomials with few terms to represent deformations is the most important modeling approximation of the deformed geometry. Modeling applied loading and settlements by linear functions of element local coordinates illustrates modeling approximations of boundary conditions. By definition, when the modeling is optimum, the required computer solution accuracy is developed with a minimum of computer resources. The parameters of this optimization problem are the modeling choices, the analysis strategy and the efficiency of the computer configuration in implementing the choices. In this study, we assume that the constraint on the optimum is evaluation of the external work in the structure to a prespecified number of significant digits. We measure the efficiency of the computer configuration by the relative number of degrees of freedom in the analysis. Melosh [1] emphasizes that conventional analysis with element models satisfying minimum convergence requirements can be much less efficient than analysis using curve fitting. Examples suggest that use of fitting can reduce the number of calculations by more than two orders of magnitude and the storage requirements by more than three orders. Furthermore, the previous study indicates that using rational polynomials or hyperbolic estimating functions for curve fitting offers no advantage over polynomials.

First Cumulant for Chains with Constraints

AbstractThe Fixman-Kovac formulation of chain dynamics with constraints is used to calculate the first cumulant Q(q) of the dynamic scattering function. This general formalism is applied to the case of freely jointed chains. It is shown that the large q limit (q being the scattering wavenumber) of Q(q) for a chain of N bonds in the absence of hydrodynamic interaction is proportional to the ratio (2N+3)/3(N+l) representing the fraction of unconstrained degrees of freedom of the chain. The inclusion of hydrodynamic interaction seems to enhance the apparent segmental diffusion. The use of constrained chain dynamics has no appreciable effects, however, on the behavior of Q(q) in the small and intermediate q regions for long enough chains. This formalism can be used to interpret neutron (spin echo) scattering experiments from semiflexible polymers in solution.