Model reduction and control of reactor–heat exchanger networks

This paper focuses on the dynamics and control of process networks consisting of a reactor connected with an external heat exchanger through a large material recycle stream that acts as an energy carrier. Using singular perturbation arguments, we show that such networks exhibit a dynamic behavior featuring two time scales: a fast one, in which the energy balance variables evolve, and a slow time scale that captures the evolution of the terms in the material balance equations. We present a procedure for deriving reduced-order, non-stiff models for the fast and slow dynamics, and a framework for rational control system design that accounts for the time scale separation exhibited by the system dynamics. The theoretical developments are illustrated with an example and numerical simulation results.     

Stochastic approximation with two time scales

Asymptotic behaviour of a two time scale stochastic approximation algorithm is analysed in terms of a related singular ordinary differential equation.     

Application of the G-JF discrete-time thermostat for fast and accurate molecular simulations

A new Langevin–Verlet thermostat that preserves the fluctuation–dissipation relationship for discrete time steps is applied to molecular modeling and tested against several popular suites (AMBER, GROMACS, LAMMPS) using a small molecule as an example that can be easily simulated by all three packages. Contrary to existing methods, the new thermostat exhibits no detectable changes in the sampling statistics as the time step is varied in the entire numerical stability range. The simple form of the method, which we express in the three common forms (Velocity-Explicit, Störmer–Verlet, and Leap-Frog), allows for easy implementation within existing molecular simulation packages to achieve faster and more accurate results with no cost in either computing time or programming complexity.     

Chaos expansion for the solutions of stochastic differential equations

In this note we generalize the Isobe–Sato formula for kernels of the Wiener–Ito chaos expansion to nonautonomous systems. Expansion of a transition density is obtained and some version of Wiener's famous “black-box” identification problem is solved.     

Singularly perturbed systems of diffusion type and feedback control

Asymptotic approximations describing the behaviour of linear systems of diffusion type (convective or non-convective) with a small diffusivity, to which a feedback control of distributed or boundary type based on point sensors is applied, are constructed and proven to be correct. As a consequence one can find a near-optimal feedback control for a cost minimization problem with a quadratic performance index measuring the deviation of the stationary state from an ideal state, under the restriction of a prescribed exponential degree of stability of the stationary state.     

Time-scale decomposition of the reachable set of constrained linear systems

We consider a linear control system with a multiparameter singular perturbation representing multiple time scales and with constraints for the control and the slow state. The Hausdorff limit of the reachable set when the small parameters tend to zero is found. The result provides a basis for a time-scale approximation of the reachable set.on leave from the Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria.     

Model reduction in the presence of uncertainty in model parameters

This short note describes how to extend a certain class of existing model reduction techniques to take into account uncertainty in model parameters. The key idea of this extension is that the reduced-order model should not only contain the model parameters, but that the reduction procedure itself has to be geared for dealing with parametric uncertainty. This goal is achieved by augmenting the vector of inputs to the system with the uncertain parameters and by performing model reduction on the augmented system. It is shown that error bounds for the reduced-order model can be computed if the underlying system is linear with respect to the states, parameters, and inputs. A comparison between the presented technique and a conventional approach is made via two examples.     

Systematic physics constrained parameter estimation of stochastic differential equations

A systematic Bayesian framework is developed for physics constrained parameter inference of stochastic differential equations (SDE) from partial observations. Physical constraints are derived for stochastic climate models but are applicable for many fluid systems. A condition is derived for global stability of stochastic climate models based on energy conservation. Stochastic climate models are globally stable when a quadratic form, which is related to the cubic nonlinear operator, is negative definite. A new algorithm for the efficient sampling of such negative definite matrices is developed and also for imputing unobserved data which improve the accuracy of the parameter estimates. The performance of this framework is evaluated on two conceptual climate models.     

Model reduction of systems with zeros interlacing the poles

The problem of reducing a system with zeros interlacing the poles (ZIP) on the real axis is considered. It is proved that many model reduction methods, such as the balanced truncation, balanced residualization, suboptimal and optimal Hankel approximations, inherit the ZIP property. Properties of the Hankel singular values of ZIP systems are also listed.     

一类由偏微分方程描述时滞随机系统的变结构控制

研究了一类由偏微分方程描述的Ito型时变时滞随机系统的变结构控制问题．首先构造了系统的滑动流形,设计了变结构控制律;然后证明T系统的滑动模具有次可达性,并且利用Halanay不等式的方法给出了系统滑动模运动为均方稳定运动的一个充分条件．     

Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay

We regard the stochastic functional differential equation with infinite delay as the result of the effects of stochastic perturbation to the deterministic functional differential equation , where is defined by x_t(θ)=x(t+θ),θ(−∞,0]. We assume that the deterministic system with infinite delay is exponentially stable. In this paper, we shall characterize how much the stochastic perturbation can bear such that the corresponding stochastic functional differential system still remains exponentially stable.     

Model reduction and model predictive control of energy-efficient buildings for electrical heating load shifting

In France, buildings account for a significant portion of the electricity consumption (around 68%), due to an important use of electrical heating systems. This results in high peak load in winter and causes tensions on the production-consumption balance. In view of reducing such fluctuations, advanced control systems (including the Model Predictive Control framework) have been developed to shift heating load while maintaining indoor comfort and taking advantage of the building thermal mass. In this paper, a framework for developing optimisation-based control strategies to shift the heating load in buildings is introduced. The balanced truncation method and a time-continuous optimisation method were used to develop a real-time control of the heating power. These two methods are well suited for control problems and yield precise results. The novelty of the approach is to use reduced models derived from advanced building simulation software. A simulation case study demonstrates the controller performance in the synthesis of a predictive model-based optimal energy management strategy for a single-zone test building of the “INCAS” platform built in Le Bourget-du-Lac, France, by the National Solar Energy Institute (INES). The controller exhibits excellent performance, reaching between 6 and 13% cost reduction, and can easily be applied in real-time.     

Stability of weak numerical schemes for stochastic differential equations

This paper considers numerical stability and convergence of weak schemes solving stochastic differential equations. A relatively strong notion of stability for a special type of test equations is proposed. These are stochastic differential equations with multiplicative noise. For different explicit and implicit schemes, the regions of stability are also examined.     

On the asymptotic stability of coupled delay differential and continuous time difference equations

A two-step Liapunov-Krasovskii methodology for checking the asymptotic stability of nonlinear coupled delay differential and continuous time difference equations is proposed here. The feasibility of such methodology is shown by means of Liapunov-Krasovskii functionals with nonconstant kernels in the integrals, for instance discretized Liapunov-Krasovskii ones. An illustrative example taken from the literature, showing the effectiveness of the proposed method, is reported.     

Model reduction of uncertain systems retaining the uncertainty structure

Model reduction of high order linear-in-parameters discrete-time systems is considered. The main novelty of the paper is that the coefficients of the original system model are assumed to be known only within given intervals, and the coefficients of the derived reduced order model are also obtained in intervals, such that the complex value sets of the uncertain original and reduced models will be optimally close to each other on the unit circle. The issue of inclusion of one value set in another is also addressed in the paper. The meaning of model reduction is defined for linear-in-parameters systems. The algorithm for obtaining the value sets of such systems is derived in the paper. Then, applying a novel approach, the infinity norm of “distance” between two polygons representing the original and the reduced uncertain systems is minimized. A noteworthy point is that by a special definition of this distance the problem is formulated as a linear semi-infinite programming problem with linear constraints, thus reducing significantly the computational complexity. Numerical example is provided.     

On stability in distribution of stochastic differential delay equations with Markovian switching

This paper provides a new sufficient condition for stability in distribution of stochastic differential delay equations with Markovian switching (SDDEs). It can be considered as an improvement to the result given by Yuan C. et al. in [6].     

State-space truncation methods for parallel model reduction of large-scale systems

We discuss a parallel library of efficient algorithms for model reduction of large-scale systems with state-space dimension up to (10⁴). We survey the numerical algorithms underlying the implementation of the chosen model reduction methods. The approach considered here is based on state-space truncation of the system matrices and includes absolute and relative error methods for both stable and unstable systems. In contrast to serial implementations of these methods, we employ Newton-type iterative algorithms for the solution of the major computational tasks. Experimental results report the numerical accuracy and the parallel performance of our approach on a cluster of Intel Pentium II processors.