Equilibrium Manifold Research Articles

Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation and order reduction

In this paper, we perform the numerical analysis of operator splitting techniques for nonlinear reaction-diffusion systems with an entropic structure in the presence of fast scales in the reaction term. We consider both linear diagonal and quasi-linear non-diagonal diffusion; the entropic structure implies the well-posedness and stability of the system as well as a Tikhonov normal form for the nonlinear reaction term [23]. It allows to perform a singular perturbation analysis and to obtain a reduced and well-posed system of equations on a partial equilibrium manifold as well as an asymptotic expansion of the solution. We then conduct an error analysis in this particular framework where the time scale associated to the fast part of the reaction term is much shorter that the splitting time step Δt thus leading to the failure of the usual splitting analysis techniques. We define the conditions on diffusion and reaction for the order of the local error associated with the time splitting to be reduced or to be preserved in the presence of fast scales. All the results obtained theoretically on local error estimates are then illustrated on a numerical test case where the global error clearly reproduces the scenarios foreseen at the local level. We finally investigate the discretization of the corresponding problems and its influence on the splitting error in terms of the previously conducted numerical analysis.

Properties of Diagrams of Equilibrium Open Evaporation Accompanied by Instantaneous Chemical Reactions

The rules of embedding various chemical equilibrium manifolds into a concentration simplex are established. An approach to the synthesis of equilibrium distillation diagrams for the instantaneous reaction A1 + A2 ⇄ A3 + A4 is suggested. The number of possible diagram types is determined, and diagram classification is reported.

Equilibrium behavior in markets and games: testable restrictions and identification

We provide a selective survey of the recent literature on the empirical implications of individually rational behavior in markets and games. We concentrate on work that develops empirical implications while making as few parametric assumptions as possible. We focus on two major themes: 1. the testable restrictions on the equilibrium manifold and the identification of economic fundamentals from the equilibrium manifold; and 2. the implications of the revealed preference theory of individual behavior for aggregated data.

Testable restrictions on the equilibrium manifold under random preferences

General equilibrium theory was criticized for its apparent irrefutability, as seemingly implied by the Sonnenschein–Mantel–Debreu theorem. This view was challenged by Brown and Matzkin [Econometrica, 64 (1996) 1249], who showed the existence of testable restrictions on the equilibrium manifold. Brown and Matzkin, however, maintain the assumption that individual preferences are invariant (against psychological evidence). I consider the Brown–Matzkin problem under random preferences: for each profile of endowments one observes a distribution of prices; does there exist a probability distribution of preferences that explains the observed distributions of prices via Walrasian equilibria? I argue that even under random utilities general equilibrium theory is falsifiable.

Feasible products in batch reactive distillation

AbstractA systematic feasibility evaluation for batch reactive distillation systems is addressed. In each batch configuration, feasibility criteria are obtained for ternary reactive systems by using residual curve maps (RCMs) and reaction equilibrium manifolds. If one of the reaction products is an unstable node (UN) and is reachable from all distillation regions, a rectifier can be used for producing pure products. Even if this unstable node is only reachable from one specific distillation region, a rectifier can still be used when a part of the reaction equilibrium curve lies within the same distillation region as the product. Symmetric results are conserved for a batch stripper with one stable node (SN) product. Finally, pure products are produced in middle‐vessel columns when the UN and SN products share the same distillation region with a part of the reaction equilibrium curve. Although these criteria are obtained for ternary systems, they can be extended to multicomponent reactive systems.

A new approach to the zero-dynamics assignment problem for nonlinear discrete-time systems using functional equations

The present research work aims at the development of a systematic method to arbitrarily assign the zero dynamics of a nonlinear discrete-time real analytic system by constructing the requisite synthetic output maps. The problem under consideration is motivated by the need to adequately address the control problem of nonminimum-phase nonlinear discrete-time systems, since the latter represent a rather broad class of systems due to the well-known effect of sampling on the stability of zero-dynamics. In the proposed approach, the above control objective can be attained through: (i) a systematic computation of synthetic output maps that induce minimum-phase behavior while being statically equivalent to the original output maps (both vanish on the equilibrium manifold) and (ii) the subsequent integration into the methodological framework of currently available nonminimum-phase compensation schemes for nonlinear discrete-time systems that rely on output redefinition. The mathematical formulation of the zero-dynamics assignment problem is realized via a system of nonlinear functional equations, and a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of functional equations can be proven to be locally analytic, and this enables the development of a solution method that is easily programmable with the aid of a symbolic software package. The synthetic output maps that induce the prescribed zero dynamics for the original nonlinear discrete-time system can be explicitly computed on the basis of the solution to the aforementioned system of functional equations.

Polytopic gain scheduled H ∞ control for robotic manipulators

A new approach to the design of a polytopic gain scheduled H∞ controller with pole placement for n-joint rigid robotic manipulators is presented. With linearization around the equilibrium manifold, the robotic system is transformed into a continuous linear parameter-varying (LPV) system with respect to the equilibrium manifold. A filter is introduced to obtain an augmented system, which is apt to have the polytopic gain scheduled controller designed. This system is put into a polytopic expression by a convex decomposition. Based on the concepts of quadratic D-stability and quadratic H∞ performance, the polytopic features are used to simplify the controller design to be a vertices' controller design for the polytope. A state feedback gain, which satisfies H∞ performance and dynamic characteristics for each vertex of the polytope, is designed with a Linear Matrix Inequality (LMI) approach. A global continuous gain scheduled controller is then obtained by a synthesis of the vertex gains. Experiments demonstrate the feasibility of the designed controller.

Longitudinal and Steering Stabilization of an Autonomous Vehicle

Time-varying stabilizing controller solution for both the kinematic and dynamic model of an underactuated autonomous vehicle is proposed. The vehicle stabilization is considered with sideslip effects and the dynamic of the yaw angle. We prove that the vehicle cannot be stabilized to a point using continuous pure-state feedback law. The stabilization problem is solved with an explicit homogeneous time-varying control law. We show that the equilibrium manifold of the system is locally exponentially stable.

Deployment of tensegrity structures

In this paper we present a strategy for tensegrity structures deployment. The main idea is to use a certain set of equilibria to which the undeployed and deployed configurations belong. In the state space this set is represented by an equilibrium manifold. The deployment is conducted such that the deployment trajectory is close to this equilibrium manifold.

Regular economies with non-ordered preferences

We consider exchange economies with non-ordered preferences and externalities. Under continuity and boundary conditions, we prove an index formula, which implies the existence of equilibria for all initial endowments, and the upper semi-continuity of the Walras correspondence. We then posit a differentiability assumption on the preferences. It allows us to prove that the equilibrium manifold is a nonempty differentiable sub-manifold of an Euclidean space. We define regular economies as the regular values of the natural projection. We show that the set of regular economies is open, dense and of full Lebesgue measure. From the index formula, one deduces that a regular economy has a finite odd number of equilibria, and, for each of them, the implicit function theorem implies that there exists a local differentiable selection. So, even with only non-ordered preferences and externalities, exchange economies have nice properties for equilibrium analysis.

Parameter identification and control of uncertain unified chaotic system via adaptive extending equilibrium manifold approach

A novel adaptive control method is proposed for controlling newly introduced uncertain unified system. The observer technique is applied to identify the unknown parameter of the unified system. Based on this observer, a simple controller is designed via extending equilibrium manifolds of the original system, which can stabilize the chaotic irregular states not only to arbitrary desired equilibrium-alike points but also to any target periodic orbits as designated online. Numerical simulations are provided to show the effectiveness of the proposed control method.

Output regulation of a class of unstructured models of continuous bioreactors: steady-state approaches.

This article deals with the output regulation of continuous bioreactors in the face of constant disturbances and inverse dynamics. Nonlinear controllers developed on the basis of approximate equilibrium manifolds can almost attenuate measurable or unmeasurable disturbances on the output. This nonlinear feed-forward/feedback control framework without any tuning parameters can be directly implemented to strictly nonlinear systems. Under dynamic actuator constraints and the availability of only output signals for use in the control law, closed-loop simulations demonstrate that the proposed control techniques are superior to a nonlinear PI control scheme based on the identified Hammerstein model.

Public Information, Private Information, and the Multiplicity of Equilibria in Coordination Games

I study coordination games with incomplete public and private information and relate equilibrium convergence to convergence of higher-order beliefs. As the players' signals become more and more precise, the equilibrium manifold converges to the correspondence of common knowledge equilibria, whenever the variance of the public signal converges to 0 at a rate faster than one half the rate of convergence of the variance of private signals. The same condition also determines the convergence of common p-belief to common knowledge, which leads to a simple intuition for its origin and an immediate generalization of the former results about equilibrium convergence. Journal of Economic Literature Classification Numbers: C72, D82.

Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure

We investigate the reduction of complex chemistry in gaseous mixtures. Weconsider an arbitrarily complex network of reversible reactions. We assume that their ratesof progress are given by the law of mass action and that their equilibrium constants arecompatible with thermodynamics; it thus provides an entropic structure [14] [23]. We studya homogeneous reactor at constant density and internal energy where the temperature canencounter strong variations. The entropic structure brings in a global convex Lyapounovfunction and the well-posedness of the associated finite dimensional dynamical system. Wethen assume that a subset of the reactions is constituted of 'Fast' reactions. The partialequilibrium constraint is linear in the entropic variable and thus identifies the 'Slow' and'Fast' variables uniquely in the concentration space through constant orthogonal projections.It is proved that there exists a convex compact polyhedron invariant by the dynamicalsystem which contains an affine foliation associated with a Tikhonov normal form. The reductionstep is then identified using the orthogonal projection onto the partial equilibriummanifold and proved to be compatible with the entropy production. We prove the globalexistence of a smooth solution and of an asymptotically stable equilibrium state for boththe reduced system and the complete one. A global in time singular perturbation analysisproves that the reduced system on the partial equilibrium manifold approximates the fullchemistry system. Asymptotic expansions are obtained.

Parameterization and Stabilization of the Equilibrium Points of Affine Non-linear Control Systems

The paper investigates the equilibrium points of affine non-linear control systems and constructs a scheduled control law composed of a feedforward control and a family of state feedbacks. When the design of a non-linear system is decomposed into the design of a family of linear time-invariant systems, it is required to generate parameterized linear models for the plant and to develop a scheduling scheme guaranteeing stability. To solve these difficulties, we parameterized the equilibrium points of the system by constructing a coordinate transformation into a new coordinate system with the parameter coordinates. Then, the system is represented by a parameterized family of linear models along its equilibrium manifold. With these parameterized linear families, we designed a scheduled control law with a feedforward control and local linear robust controllers so that the overall feedback stabilizes the plant about the equilibrium points. The approach is illustrated by applying it to the control of an arm-driven inverted pendulum.

Theory of stability regions for a class of nonhyperbolic dynamical systems and its application to constraint satisfaction problems

The concept of a stability region (region of attraction) of nonlinear dynamical systems is widely applied in many fields such as engineering and the sciences. In this paper, we develop a comprehensive theory of the stability region and its boundary for a class of nonhyperbolic dynamical systems. We then apply the theory to develop an effective method for solving constraint satisfaction problems. Several necessary and sufficient conditions for an equilibrium manifold (the generalized concept of an equilibrium point) to lie on the stability boundary of the class of nonhyperbolic dynamical systems are derived. The stability boundary of such nonhyperbolic dynamical systems is completely characterized and shown to consist of the union of the stable manifolds of the equilibrium manifolds on the stability boundary. An effective scheme to estimate the stability region of such a dynamical system by using an energy function is developed. These analytical results are then applied to the development of computational methods to systematically find feasible regions of constraint satisfaction problems. Several numerical examples are given to illustrate the effectiveness of the computational method.

Testable Implications of General Equilibrium Theory: A Differentiable Approach

We characterize the equilibrium manifold of an economy when initial endowments can be individually observed. We adopt a local, differentiable perspective. We derive necessary and sufficient conditions that characterize the equilibrium manifold, and we show that, if these conditions are satisfied, then the economy can be identified. We show, further, that, if only aggregate data are available, observable restrictions vanish. We conclude that the availability of individual data is essential for the derivation of testable consequences of the general equilibrium construct.

BIFURCATION PLACEMENT OF HOPF POINTS FOR STABILIZATION OF EQUILIBRIA

Applications of bifurcation control to chemical engineering processes have aimed at rendering subcritical bifurcations supercritical. Beyond affecting the criticality of bifurcation points for control purposes, it is possible to deliberately introduce bifurcation points to stabilize parts of the equilibrium manifold of an ODE process model. This can be done without affecting the equilibrium manifold of the uncontrolled process. For brevity this approach is called bifurcation placement. The present paper focuses on the introduction of Hopf bifurcations, though the approach is more general. Conditions are stated for the introduction of a Hopf bifurcation at a given location on the equilibrium manifold of a system of ODEs. The proof is constructive in the sense that it allows to calculate the controller needed for the introduction of the desired bifurcation point. A simple motivating example and an application to a model of an industrial continuous polymerization process are presented.

Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure

In this Note, we investigate the reduction of complex chemistry in gaseous mixtures. We consider an arbitrarily complex network of reversible reactions, the equilibrium constant of which are compatible with thermodynamics, thus providing an entropic structure. We assume that a subset of the reactions is consituted of fast reactions and define a constant and linear projection onto the partial equilibrium manifold compatible with the entropy production. This reduction step is used for the study of a homogeneous reactor at constant density and internal energy where the temperature can encounter strong variations. We prove the global existence of a smooth solution and of an asymptotically stable equilibrium state for both the reduced system and the complete one. A global in time singular perturbation analysis proves that the reduced system on the partial equilibrium manifold approximates the full chemistry system. To cite this article: M. Massot, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 93–98.

Implementation of combustion chemistry by in situ adaptive tabulation of rate-controlled constrained equilibrium manifolds

A general methodology called in situ adaptive tabulation-rate-controlled constrained equilibrium (ISAT-RACCE) is developed to treat detailed chemistry in turbulent combustion calculations. This method combines dimension reduction and tabulation and can be implemented in a single computer program, which is independent of the detailed mechanism and of the level of reduction selected. The dimension reduction of a detailed mechanism is achieved by the RCCE approach, which is based on the maximum entropy principle of thermodynamics: and the tabulation is performed in situ during the combustion calculations and is made in the low-dimensional constraint subspace. This method is particularly attractive for very large kinetic mechanisms (e.g., including thousands of species) because the constrained equilibrium state depends on only a small number of constraints (or constrait potentials) which form the constraint subspace. Moreover, the unique and continuous maximum-entropy manifold determined by the RCCE assumption is well suited to the ISAT algorithm—an efficient computational technique for the implementation of combustion chemistry. Test calculations are performed for non-premixed methane/air combustion in a statistically homogeneous turbulent reactor, using a detailed kinetic mechanism with 31 species and 175 reactions. A direct approach of numerically integrating the full reaction equations (32-dimensional) is performed and the result is taken as the exact solution. The ISAT-RCCE calculations based on 10- to 16-dimensional constraint subspaces show good agreement with the accurate solution, and the accuracy of the results from the 16-dimensional ISAT-RCCE calculation is comparable to that of a reference calculation using ISAT and a 17-dimensional augmented reduced mechanism (derived from the same detailed mechanism). A speed-up factor of about 500 is obtained for the ISAT-RCCE calculation compared to the direct integration approach, which demonstrates the high efficiency of the new method.