Equilibrium Manifold Research Articles

Controller design for inverted pendulum stabilizing on equilibrium manifold based on vector field

In general, mechanical systems are stabilized on their equilibrium point. Equilibrium point is often not unique and they are continuously connected, which is an equilibrium manifold. To stabilize the mechanical system on an equilibrium manifold will enable optimal control including the selection of the stabilizing position. In this paper, we propose a controller design method that stabilizes a mechanical system on an equilibrium manifold based on vector field. The equilibrium manifold is derived from dynamic equations, and by setting an appropriate evaluation function, (1) an optimal equilibrium point from arbitrary initial value is calculated, (2) a trajectory, input and vector field are derived based on linear control theory, (3) a controller is designed using functional approximation. Simulations show that different initial values are stabilized to different equilibrium points, and experimental results show the effectiveness of the proposed method.

An All Speed Second Order IMEX Relaxation Scheme for the Euler Equations

We present an implicit-explicit finite volume scheme for the Euler equations. We start from the non-dimensionalised Euler equations where we split the pressure in a slow and a fast acoustic part. We use a Suliciu type relaxation model which we split in an explicit part, solved using a Godunov-type scheme based on an approximate Riemann solver, and an implicit part where we solve an elliptic equation for the fast pressure. The relaxation source terms are treated projecting the solution on the equilibrium manifold. The proposed scheme is positivity preserving with respect to the density and internal energy and asymptotic preserving towards the incompressible Euler equations. For this first order scheme we give a second order extension which maintains the positivity property. We perform numerical experiments in 1D and 2D to show the applicability of the proposed splitting and give convergence results for the second order extension.

Equilibrium Manifolds in 2D Fluid Traffic Models

The main goal of this paper is to analytically find a steady-state in a large-scale urban traffic network with known and constant demand and supply on its boundaries. Traffic dynamics are given by a continuous two-dimensional macroscopic model, where state corresponds to the vehicle density evolving in a 2D plane. Thereby, the flux magnitude is given by the space-dependent fundamental diagram and the flux direction depends on the underlying network topology. In order to find a steady-state, we use the coordinate transformation such that the 2D equation can be rewritten as a parametrized set of 1D equations. This technique allows us to obtain the curves along which the traffic flow evolves, which are essentially the integral curves of the flux field constructed from the network geometry. The results are validated by comparing the obtained steady-state with the one estimated by using a microsimulator.

Stability Analysis of Quadrature-Based Moment Methods for Kinetic Equations

In this paper, we give a systematic stability analysis of the quadrature-based moment method (QBMM) for the one-dimensional Boltzmann equation with BGK or Shakhov models. As reported in recent literature, the method has revealed its potential for modeling nonequilibrium flows, while a thorough theoretical analysis is largely missing but desirable. We show that the method can yield nonhyperbolic moment systems if the distribution function is approximated by a linear combination of $\delta$-functions. On the other hand, if the $\delta$-functions are replaced by their Gaussian approximations with a common variance, we prove that the moment systems are strictly hyperbolic and preserve the dissipation property (or $H$-theorem) of the kinetic equation. In the proof, we also determine the equilibrium manifold that lies on the boundary of the state space. The proofs are quite technical and involve detailed analyses of the characteristic polynomials of the coefficient matrices.

QUESTIONS AND PROBLEMS OF MATHEMATICAL MODELING QUA NONEQUILIBRIUM OF COMBUSTION PROCESSES

On the basis of thermodynamic analysis, new mathematical models of the combus- tion process (thermal theory) and vibrational combustion are constructed. A global inhomo- geneity of the system can be described as an inhomogeneous distribution of the enthalpy over a two-component mixture. In this case, for the combustion process in the phase space of the variables (%, P, T, n, S, E), an increase in the enthalpy is not a total differential. An increase in the enthalpy is a total differential on the local equilibrium manifold (a laminar combustion process). These two assertions, which allow one to single out in the phase space the corre- sponding adiabatic of the combustion process (the Hugoniot adiabatic) and the equation for the entropy, close the classical mathematical model of the combustion process. The above numerical experiments show that two regimes of the combustion process (deflagration and detonation) depend on the structure of the standard chemical potential Moreover, a control of the passive component velocity at the inlet results in (depending on the structure of the standard chemical potential) high-frequency oscillations, which are responsible for a blow-up.

Supervisory Control of Multiple Switching Laws With Performance Guidance for Aeroengines

This brief is concerned with the problem of speed regulation control for a two-spool turbofan engine. Firstly, after analyzing the operation characteristics of the aeroengine, we give multiple switched equilibrium manifold expansion (EME) models on the basis of multiple operation modes. Meanwhile, the switching laws are based on the performance indexes that reflect the operation characteristics of the aeroengine in different operation modes. Then, a supervisor is proposed to orchestrate multiple switching laws through checking which operation mode the aeroengine operates in. Furthermore, to enhance the robustness of the aeroengine control system, an optimal control strategy is employed. However, as an acknowledged difficult problem, the analytic nonlinear optimal controller can be hardly obtained. Therefore, we present a modified linear quadratic regulator algorithm for the controller design and give a notion of an EME controller. Moreover, we offer a design process of the switching controllers and present a sufficient condition of the asymptotic stability of the closed-loop switched system for the aeroengine. Finally, a software simulation and a semiphysical simulation experiment for some aeroengine are provided to demonstrate the effectiveness of the proposed method.

Control chaos to different stable states for a piecewise linear circuit system by a simple linear control

Abstract In this paper, we mainly investigate chaos control of a piecewise linear circuit system. According to the characteristic of this system, we modify Hwang’s linear continuous controller and obtain a more simple controller consisting of two parts, by which we find from theory the extent of control parameter when chaotic motion is controlled to equilibrium manifold, equilibrium point, periodic orbit or limit cycle. Numerical simulation also verifies the method is effective.

Stable cycling in quasi-linkage equilibrium: Fluctuating dynamics under gene conversion and selection

Genetic systems with multiple loci can have complex dynamics. For example, mean fitness need not always increase and stable cycling is possible. Here, we study the dynamics of a genetic system inspired by the molecular biology of recognition-dependent double strand breaks and repair as it happens in recombination hotspots. The model shows slow-fast dynamics in which the system converges to the quasi-linkage equilibrium (QLE) manifold. On this manifold, sustained cycling is possible as the dynamics approach a heteroclinic cycle, in which allele frequencies alternate between near extinction and near fixation. We find a closed-form approximation for the QLE manifold and use it to simplify the model. For the simplified model, we can analytically calculate the stability of the heteroclinic cycle. In the discrete-time model the cycle is always stable; in a continuous-time approximation, the cycle is always unstable. This demonstrates that complex dynamics are possible under quasi-linkage equilibrium.

Slow invariant manifolds and its approximation in a multi-route reaction mechanism: A case study of iodized H2/O mechanism

To reduce the complexity of the reaction mechanism, we need to split down the complex mechanism in different available reaction routes by using Horiuti rule. However, the model reduction and refinement approaches for different reaction routes are highly dependent upon the slow invariant manifold approximations. The main objectives of the current article are to find the solution curves through the method of construction of invariant manifold (Spectral Quasi Equilibrium Manifold) taking single and multi-route reaction mechanisms. Further, Method of Invariant Grid (MIG) is applied to refine the approximated solution curve. A comparison between approximated and refined solution curves of different reaction routes for idealized H2 /O six-step reversible system is also reported graphically. The results are obtained through MATLAB.

Boundary-derivative direct method for computing saddle node bifurcation points in voltage stability analysis

The Saddle Node Bifurcation (SNB) is the location where equilibrium solutions of one parametric system distinguish when its parameter changes. Load distance between the SNB and the present operation point can reflect the voltage stability of the studied system. At present, Continuous Power Flow (CPF) and Point of Collapse (POC)/Direct Method are main techniques for tracking the point. However, CPF needs multiple power flow calculations, which is not suitable for fast and accurate applications, while POC has problems of large coefficient matrix and initialization of eigenvector. A novel direct method is proposed to calculate the SNB point, titled the Boundary-Derivative Direct Method (BDDM). A one-dimensional boundary condition, which reflects the slope of the equilibrium manifold, is configured to replace the multidimensional conditions of POC. As a result, the dimension of the coefficient matrix is drastically cut and the initialization of eigenvector is avoided. Moreover, we extend the algorithm, considering multi-load growth and general limitations. Several instances on IEEE 9-bus system show the convergence of the proposed algorithm and the cases in IEEE 118-bus system and 2000-Bus Texas System, which contain multi-load increments and reactive power limitations of generators, are used to analyze the effectiveness of the proposed algorithm.

Physical assessments on chemically reacting species and reduction schemes for the approximation of invariant manifolds

We have simplified the complex chemical reaction by reducing it from a high dimension to a low dimension to observe the behavior of participating species near the equilibrium point and obtained their solution curves without affecting the accuracy of the whole mechanism. For this purpose, we have applied three well-established techniques: Quasi Equilibrium Manifold (QEM), Spectral Quasi Equilibrium Manifold (SQEM) and Intrinsic Low Dimension Manifold (ILDM) over a Ternkin-Boudart Mechanism. However, the invariant solution curves and their comparison for the reduced species of Ternkin-Boudart reaction mechanism have never been reported so far. Therefore, the aim of this article is to study such facts. All the above-mentioned model-reduction techniques are applied to solve the high dimensional complex problem and comparison between invariant curves of the reduced species is presented graphically and leading towards the higher dimensional manifold. The efficiency of ILDM is compared with the QEM and SQEM graphically and in tabulated form by using MATLAB.

Slow invariant manifold assessments in multi-route reaction mechanism

A complex chemical scheme is need to reduce their complexity without losing their key features. A chemical reaction scheme has braked down in different available reaction routes. On matrix algebraic approach bases, we analyze key-components, key reactions and reaction routes. A model reduction technique Spectral Quasi Equilibrium Manifold (SQEM) has applied to reduce the dimension of reaction mechanism. However, for the case study the behaviour of different available routes of reaction mechanism presented graphically. The reaction routes and invariants for idealized H2/O six-step reversible system and comparison has not been reported so far. To overcome this difficulty a clue about reaction routes have addressed through analysis of stoichiometry, the route compression with respect to their slow invariant manifold (SIM), the physical behaviour of chemically reacting species near to equilibrium point and extension towards the high dimension are expressed graphically.

On the Existence of and Lower Bounds for the Number of Optimal Power Flow Solutions

An optimal power flow (OPF) problem can have one or multiple locally optimal solutions. It can also admit no solution. Few studies have addressed the existence of OPF solutions and derived a (tight) lower bound for the number of OPF solutions. The present paper is devoted to the analytical aspects of OPF solutions. Specifically, a necessary and sufficient condition for the existence of an OPF solution is presented and a tight lower bound for the number of OPF solutions is derived; in other words, an existence theorem is developed for the OPF solution and a computable lower bound for the number of solutions is derived by investigating a dynamical system called the quotient gradient system (QGS), which is built from the set of constraints. It is shown that an OPF solution exists if and only if the QGS has a regular stable equilibrium manifold (SEM) and the number of OPF solutions is not less than the number of regular SEMs for the QGS. Two test systems with 9 and 118 buses, respectively, are evaluated, and the numerical results are summarized to validate the presented analytical results and illustrate that multiple OPF solutions can lie in the same SEM.

Robust Fault Identification of Turbofan Engines Sensors Based on Fractional-Order Integral Sliding Mode Observer

Abstract This paper presents a robust fault identification scheme based on fractional-order integral sliding mode observer (FOISMO) for turbofan engine sensors with uncertainties. The equilibrium manifold expansion (EME) model is introduced due to its simplicity and accuracy for nonlinear system. A fractional-order integral sliding mode observer is designed to reconstruct faults on sensors, in which the fractional-order integral sliding surface guarantees the fast convergence of reconstruction. The observer parameters is selected according to L2 gain theory in order to minimize the effect of uncertainties on the fault reconstruction signal. Simulations in Matlab/Simulink show high reconstruction accuracy of the proposed method despite the present of uncertainties.

Asymptotic preserving discretization of a Jin–Xin model with implicit equilibrium manifold on a bounded domain

Abstract In this paper, we design and analyze a numerical scheme that approximates a Jin–Xin linear system with implicit equilibrium on a bounded domain. This scheme relaxes toward the asymptotic limit of the linear system. The main properties of the limiting scheme are that it does not require to invert the implicit function defining the manifold and that it provides an accurate discretization of the boundary conditions.

The reaction routes comparison with respect to slow invariant manifold and equilibrium points

Mathematically, complex chemical reactions can be simplified by “model reduction,” that is, the rigorous way of approximating and representing a complex model in simplified form. Furthermore, to reduce the dimension of the reaction mechanism there are different available model reduction techniques (MRT). Two MRT Spectral Quasi Equilibrium Manifold (SQEM) and Intrinsic Low Dimensional Manifold (ILDM) are applied here. Both techniques are good approximation techniques but not completely analytical. While for a case study, the complex behavior of the two-route reaction mechanism allied graphically. The characteristic properties of nodes and trees of reaction mechanism are addressed with the graph theory. The invariant solution curves and their comparison for two-route reaction mechanism have never been reported so far. Therefore, the aim of this article is to study such facts. The possible approaches to their solution and key problems are based on the Horiuti’s rules and their slow invariants. Both model-reduction techniques are applied to solve the high dimensional complex problem and comparison between invariant curves of both routes are presented graphically. The efficiency of ILDM with respect to time is compared with the SQEM graphically and in tabulated form by using MATLAB.

Dynamic balance control based on an adaptive gain-scheduled backstepping scheme for power-line inspection robots

This paper presents an adaptive gain-scheduled backstepping control &#x0028 AGSBC &#x0029 scheme for the balance control of an underactuated mechanical power-line inspection &#x0028 PLI &#x0029 robotic system with two degrees of freedom and a single control input. First, a nonlinear dynamic model of the balance adjustment process of the PLI robot is constructed, and then the model is linearized at a nominal equilibrium point to overcome the computational infeasibility of the conventional backstepping technique. Second, to solve generalized stabilization control issue for underactuated systems with multiple equilibrium points, an equilibrium manifold linearized model is developed using a scheduling variable, and then a gain-scheduled backstepping control &#x0028 GSBC &#x0029 scheme for expanding the operational area of the controlled system is constructed. Finally, an adaptive mechanism is proposed to counteract the impact of external disturbances. The robust stability of the closed-loop system is ensured by Lyapunov theorem. Simulation results demonstrate the effectiveness and high performance of the proposed scheme compared with other control schemes.

Nonlinear System Modeling based on System Equilibrium Manifold

Abstract The main objective of the present paper is to provide an approximate method for nonlinear system with a family of equilibrium points (EPs). By investigating the system’s equilibrium manifold (EM) and its expansion form, a class of nonlinear system is modeled analytically. The property of the EM expansion form and the effect of mapping design and parameterizing method to the model have been discussed. Then an approximate nonlinear model for aircraft engine was applied, followed by an identification procedure for aircraft engine. It is shown that the modeling method based on system’s EM can not only comply with the system physical characteristics, but also satisfy the accuracy of modeling requirement. Numerical simulations are given to demonstrate the good precision in capturing the nonlinear behavior of nonlinearities and a simpler structure.

Geometric control of 3D needle steering in soft-tissue

In this paper, a 3D automated needle steering system is presented that can enhance the performance of needle-based procedures. The system comprises a nonholonomic needle steering model and a nonlinear controller for 3D needle steering. First, a reduced-order needle steering model is presented. Next, a geometric reduction procedure is carried out to present the nonlinear control system in a transformed format. Finally, the transformed model is used to design a two-step controller. The controller first stabilizes the system on an equilibrium manifold of the system and later employs a switching law to stabilize it on an equilibrium point in the manifold. The former performs insertion of the needle up to a desired depth and the latter performs retraction/insertion motion that guides the needle toward a desired point at the given depth. Validation experiments are performed on a phantom and ex-vivo animal tissues and the results are compared with manual needle insertions performed by skilled surgeons. The mean error of our 3D needle steering system is 60% less than manual needle insertions.

Routes to Multiple Equilibria for Mass-Action Kinetic Systems

In this work we explore two potential mechanisms inducing multiple equilibria for weakly reversible networks with mass-action kinetics. The study is performed on a class of polynomial dynamic systems that, under some mild assumptions, are able to accommodate in their state-space form weakly reversible mass-action kinetic networks. The contribution is twofold. We provide an explicit representation of the set of all positive equilibria attained by the system class in terms of a set of (positive parameter dependent) algebraic relations. With this in hand, we prove that deficiency-one networks can only admit multiple equilibria via folding of the equilibrium manifold, whereas a bifurcation leading to multiple branches is only possible in networks with deficiencies larger than one. Interestingly, some kinetic networks within this latter class are capable of sustaining multiple equilibria for any reaction simplex, as we illustrate with one example.