Nonlinear observer design for two-time-scale systems

A two-time-scale system involves both fast and slow dynamics. This article studies observer design for general nonlinear two-time-scale systems and presents two alternative nonlinear observer design approaches, one full-order and one reduced-order. The full-order observer is designed by following a scheme to systematically select design parameters, so that the fast and slow observer dynamics are assigned to estimate the corresponding system modes. The reduced-order observer is derived based on a lower dimensional model to reconstruct the slow states, along with the algebraic slow-motion invariant manifold function to reconstruct the fast states. Through an error analysis, it is shown that the reduced-order observer is capable of providing accurate estimation of the states for the detailed system with an exponentially decaying estimation error. In the last part of the article, the two proposed observers are designed for an anaerobic digestion process, as an illustrative example to evaluate their performance and convergence properties.     

Invariant manifolds and the calculation of the long-term asymptotic response of nonlinear processes using singular PDEs

The present research work proposes a new systematic approach to the problem of finding invariant manifolds and computing the long-term asymptotic behavior of nonlinear dynamical systems. In particular, nonlinear processes are considered whose dynamics is driven by an external time-varying ‘forcing’ or input/disturbance term, or by a set of time-varying process parameters, or by the autonomous dynamics of an upstream process. The formulation of the problem is conveniently realized through a system of singular first-order quasi-linear PDEs, and a rather general set of conditions for solvability is derived. In particular, within the class of analytic solutions the aforementioned set of conditions guarantees the existence and uniqueness of a locally analytic solution. The solution of the system of singular PDEs is then proven to be a locally analytic invariant manifold of the nonlinear dynamical system. The local analyticity property of the invariant manifold enables the development of a series solution method, which can be easily implemented with the aid of a symbolic software package such as maple. Under a certain set of conditions, it is shown that the invariant manifold computed exponentially attracts all system trajectories, and therefore, the long-term asymptotic process response is described and calculated through the restriction of the process dynamics on the invariant manifold. Finally, in order to illustrate the proposed approach and method, a representative enzymatic bioreactor example is considered.     

Dynamic modeling and reaction invariant control of pH

A systematic treatment of the dynamics and control of acid-base reactions is presented. A state model consisting of mass balances for the species present is formed. Through a linear transformation of the concentration vector a new model is set up. The state vector of this model is partitioned into a reaction invariant and a reaction variant part. The model is thus split into two parts, one describing the physical properties of the reactor system independent of chemical reactions, the other describing the chemical reactions.For fast acid-base reactions the reaction invariant part of the model is sufficient to define the thermodynamic state of the system. The reaction rate vector is thus eliminated from the model. In this case the reaction variant part of the model consists of a static equation, relating pH to the reaction invariant state variables.The model obtained provides a sound basis for design of control loops for feedback and feedforward control of pH. As a direct application a new scheme for adaptive control of pH is proposed.     

A model-based characterization of the long-term asymptotic behavior of nonlinear discrete-time processes using invariance functional equations

The present research work proposes a new approach to the problem of quantitatively characterizing the long-term dynamic behavior of nonlinear discrete-time processes. It is assumed that in order to analyze the process dynamic behavior and digitally simulate it for performance monitoring purposes, the discrete-time dynamic process model considered can be obtained: (i) either through the employment of efficient and accurate discretization methods for the original continuous-time process which is mathematically described by a system of nonlinear ordinary (ODEs) or partial differential equations (PDEs) or (ii) through direct identification methods. In particular, nonlinear processes are considered whose dynamics can be viewed as driven: (i) either by an external time-varying “forcing” input/disturbance term, (ii) by a set of time-varying process parameters or (iii) by the autonomous dynamics of an upstream process. The formulation of the problem of interest can be naturally realized through a system of nonlinear functional equations (NFEs), for which a rather general set of conditions for the existence and uniqueness of a solution is derived. The solution to the aforementioned system of NFEs is then proven to represent a locally analytic invariant manifold of the nonlinear discrete-time process under consideration. The local analyticity property of the invariant manifold map enables the development of a series solution method for the above system of NFEs, which can be easily implemented with the aid of a symbolic software package such as MAPLE. Under a certain set of conditions, it is shown that the invariant manifold computed attracts all system trajectories, and therefore, the asymptotic process response and long-term dynamic behavior are determined through the restriction of the discrete-time process dynamics on the invariant manifold.     

Wavelet-like collocation method for finite-dimensional reduction of distributed systems

A wavelet-like collocation method is proposed to approach the reduction of dissipative distributed systems, expressed by means of partial differential equations, applying the methods of inertial manifold theory. The collocation method proposed, based on localized trial functions, provides a convenient numerical framework to develop approximate inertial manifolds in the case of partial differential problems (e.g. reaction/diffusion models) containing nonpolynomial nonlinearities. The collocation method is based on the interpolation of concentration/temperature fields by means of Gaussian–sinc functions. As model systems, we consider reaction diffusion schemes such as the non-isothermal model for stockpile ignition and the Elezgaray–Arneodo diffusion model.     

On the existence and computation of reaction invariants

For model reduction of chemically reacting systems with fast reversible reactions, reaction invariant compositions, as introduced by Doherty and co-workers for reactive distillation processes, can be used. Reaction invariant compositions are obtained from a nonlinear transformation of the original composition variables. The transformation requires the choice of a suitable set of reference components. In the present paper it is proven, that such a set of suitable reference components always exists. Furthermore, a systematic procedure for computing such a feasible set is given.     

Fours rotatifs: Modèle dynamique du mouvement du lit

The motions of the granular bed inside a rotary furnace are governed by complex dynamics consisting of two modes. The axial velocity of the active layer on top of the bed represents the fast mode, while the overall velocity in the bed follows a slower mode. The proposed nonlinear model focuses on the dynamics of the slow mode. The difficulties of an analytic solution are avoided by a numerical method that employs several correlations. The model can reproduce the experimental transient responses. It is concluded that rotary furnaces respond better to small amplitude excitations than to large ones; consequently, closed-loop control is well suited for this type of furnace.     

Non-equilibrium effects on fast reactions in a heterogeneous batch reactor

Recent studies by the authors of the heterogeneous catalysis of fast binary reactions have taken a dynamical systems approach, assuming that fast enough reactions are confined to a manifold upon which surface equilibrium holds. This approximation makes substantial simplification possible, for instance in the case of a batch reactor, it allows a naturally sixth order system to be approximated by a two dimensional manifold for the dynamics of two modified Thiele moduli. Nevertheless, a proper assessment of how much faster must the velocity of surface reaction be than the velocity of mass transfer to the catalytic surface before the quasi-equilibrium on the surface holds should be made. In this paper, a theory for the systematic correction to infinitely fast reactions is made for large but finite velocity reactions. It is compared to full numerical solutions to the model equations. Recommendations about the regime of applicability of the quasi-equilibrium approximation are made. In general, the predictions of the quasi-equilibrium theory hold for ratios of mass transfer coefficients to reaction velocity ξ of less than 1/1000, with qualitative agreement in regimes of less than 1/100. The general trend, however, is that the stronger the kinetic asymmetry between the mass transfer coefficients of the reactants, the slower the reaction rate can be and still have the quasi-equilibrium theory hold. A perturbation analysis demonstrates that the quasi-equilibrium theory is a regular limit of the fast non-equilibrium theory. In the irreversible case, a matched asymptotic analysis gives the same prediction for the switch time from effective surface depletion of one reagent to the other as the quasi-equilibrium theory. Furthermore, it gives an estimate of the smoothing out of the transition zone with a temporal width of ξ^1/2. It should be noted that the continual drive for improved catalyst activities inevitably leads to mass transfer limited reactions, and thus this regime is not uncommon.     

基于LTSA的FS-SVDD方法及其在化工过程监控中的应用

基于支持向量数据描述(SVDD)方法的非高斯过程监控和故障诊断具有众多优点。然而在对SVDD离线建模时需要在整个训练样本集上操作,对大样本集计算量相当大,也不利于在线操作时模型的更新。对此提出一种基于特征样本的SVDD(FS-SVDD),采用特征样本提取方法用少数几个特征样本代替原始数据集进行训练,显著降低了建模复杂度。同时,针对传统的线性降维算法如主成分分析(PCA)存在的提取过程数据非线性结构能力不足的缺点,首先用局部切空间排列(LTSA)方法提取出低维子流形,进行有效的维数约减;接着在这个低维子流形上执行SVDD算法;最后,利用相应统计指标进行过程监控。在TE过程上的仿真表明上述方法的有效性。     

Modeling of an industrial scale hydrodynamic cavitation multiphase reactor for Prileschajew epoxidation

The hydrodynamic cavitation multiphase reactor (HCMR) is emerging as a promising alternative for the intensification of liquid–liquid heterogeneous reactions, but research on HCMR modeling is lacking. In this article, an HCMR model was developed using Prileschajew epoxidation as the model system. First, based on experimental measurements of oil/water two-phase flow downstream of hydrodynamic cavitation devices, semiempirical correlations were proposed to describe the droplet size and droplet size distribution (DSD) as functions of flow conditions and geometry parameters. Then, with boundary conditions calculated by the DSD correlation, a droplet dynamics simulation in a reaction tank was performed by computational fluid dynamics coupled with population balance model to obtain the two-phase interfacial area. Finally, the acquired reactor model was substituted into an overall kinetic model, to simulate the epoxidation reaction in HCMR. Model predictions were verified by experimental results measured on an industrial scale HCMR.     

Adaptive pole removal control for a batch polymerization reactor

A systematic method of controller design is introduced to determine the overall charcteristic behaviour developed on process pole removal. The time delay compensation is automatically incorporated in the proposed control law. The implemented tuning parameters in the law are confined to the range between 0 and 1 to guarantee the stability of the overall control system. Subsequently, the fixed and adaptive control strategies are implemented to simulate a batch PVC reaction system. The adaptive control scheme provides good, roubust control of this simulated reactor notwithstanding the wide range of operating conditions and non-linear dynamics of the system. However, the fixed control scheme performs well only for a noise-free system. In addition, two limiting control laws, derived from the proposed method, are also used to simulate the reactor. The results indicate that these laws are not suitable for this non-linear reaction system.     

A dynamic model for rectangular capillary suction apparatus

A model for the dynamic behaviour of wet front distance and liquid saturation in rectangular capillary suction apparatus (RCSA) is developed. Governing equations based on the mass/momentum balance are derived and solved numerically. The calculation shows that the effects of the intial conditions on system dynamics vanish rapidly, and the system will then evolve along a slow manifold independent of the initial conditions. When time is large, the dimensionless wet front distance with time is a straight line on a log–log plot with slope 1/2, and the liquid saturation under the inner cell will approach to a constant which depends only on the product of solid concentration and the averaged specific resistance if the RCSA is fixed. Optimum experimental conditions are suggested. A rapid method based on the model for estimating the averaged specific resistance of cake is proposed and compared with experimental findings.     

Neural network‐based optimal iterative controller for nonlinear processes

A new optimal iterative neural network‐based control (OINNC) strategy with simple computation and fast convergence is proposed for the control of processes with nonlinear dynamics. The process dynamics is captured by a forward neural network, and the control is determined by a simple iterative optimization during each sampling interval based on a linearized neural network model. In addition, a feedback control is incorporated into the system to compensate for any model mismatches and to reject disturbances. With the proposed system, the tracking error is shown to be confined to the origin. An application of the proposed OINNC scheme to a nonlinear process results in superior performance when compared with a well‐tuned conventional PID controller.     

Regularized maximum likelihood estimation of sparse stochastic monomolecular biochemical reaction networks

A sparse parameter matrix estimation method is proposed for identifying a stochastic monomolecular biochemical reaction network system. Identification of a reaction network can be achieved by estimating a sparse parameter matrix containing the reaction network structure and kinetics information. Stochastic dynamics of a biochemical reaction network system is usually modeled by a chemical master equation (CME) describing the time evolution of probability distributions for all possible states. This paper considers closed monomolecular reaction systems for which an exact analytical solution of the corresponding chemical master equation can be derived. The estimation method presented in this paper incorporates the closed-form solution into a regularized maximum likelihood estimation (MLE) for which model complexity is penalized. A simulation result is provided to verify performance improvement of regularized MLE over least-square estimation (LSE), which is based on a deterministic mass-average model, in the case of a small population size.     

Dynamics of polymer molecules using neutron spin—echo

The dynamics of single polymer chains have been investigated in dilute solution and in the melt using the neutron spin—echo technique. In dilute solution the intramolecular motion is described to a first approximation by that for a Rouse chain incorporating hydrodynamic interactions (Zimm). It is characterised by an inverse correlation time which may be normalised by temperature, solvent viscosity and segment size, and which for long chains varies as Q³ at small scattering angles (Q is the change in wavevector on scattering). At very low Q vectors the predicted ‘universal’ regime is observed, but over most of the accessible range chemical structure also becomes important. For short chains, deviations from this Q³ behaviour are associated with overall molecular diffusion. In the melt, chain entanglements come into play and modify the simple Rouse-type motion. The correlation functions are described by a combination of Rouse motion over short distances and times and a long-time slow-motion predicted by the reptation model of melt dynamics.     

Bifurcation Analysis on Pt and Ir for the Reduction of NO by CO

Due to the importance of the NO‐CO reaction in current catalytic converters, reduction of NO by CO on Pt group catalysts is important to study. Various reaction mechanisms have been proposed for the NO‐CO reaction on Pt(100), which shows bifurcations, kinetic oscillations and multiple steady states under ultra high vacuum (UHV) conditions due to complex surface dynamics. Some experiments on supported Pt group catalysts reported in literature show oscillations and bistability under atmospheric conditions as well. Industrially relevant conditions require the modelling and detailed analysis of the system at atmospheric pressure. We have proposed a reaction mechanism for the NO‐CO system on Pt group catalysts and coupled it with an isothermal PSR model to obtain solutions at atmospheric conditions with the continuation software CONTENT 1.5, at different operating conditions. Simulation results suggest that Pt(111) shows bifurcations at certain operating conditions while Ir(111) shows stable solutions at all the operating conditions studied here.