A gap metric based multiple model approach for nonlinear switched systems

Many applications in chemical engineering often exhibit a switching character due to the presence of discrete modes in the course of their operation. First principles models of such systems constructed using process simulators are far too complex for use in online applications, especially in model-based control. For such systems, numerous control-relevant modeling approaches have been reported in the literature such as mixed logic dynamical (MLD) models [1] and piece wise affine (PWA) [2] models among others. These models describe the evolution of states in each discrete mode using linear equations. Fewer control-relevant models have been reported that address the nonlinear behavior of switched systems. To model nonlinear hybrid systems, Nandola and Bhartiya [3] proposed a multiple linear model approach wherein multiple linear models are used to describe the dynamic behavior in each mode of the hybrid system. However, no guidelines were provided to select the number of models necessary in each mode and their region of validity. In this work, we address these lacunae by presenting a systematic multiple model approach to describe nonlinear switched systems. The method involves a trajectory based linearization and employs a model bank with a set of local linear models for each discrete operational mode. The model bank is generated by linearizing the first principles model across a carefully designed trajectory based on accuracy of multi-step ahead predictions. The numerous models thus obtained are clustered using the gap metric as the distance measure and representative models are selected. The selected linear models are aggregated using Bayesian or Fuzzy approaches to obtain the global model for the nonlinear switched system. A simulation case study of spherical two-tank system and an experimental case study of a benchmark problem consisting of three tanks are used to validate the proposed modeling strategy.     

Multiple model reduction approach using gap metric and stability margin for control nonlinear systems

This paper deals with the control of nonlinear systems using multimodel approach. The main idea of this work consists on the association of the gap metric and the stability margin tools to reduce the number of models constituting the multimodel bank. In fact, the self-organisation map (SOM) algorithm is used, firstly, to develop a preliminary multimodel bank. Then, the gap metric and the stability margin are computed to determine the redundancy of the initial multimodel bank. So, the multimodel controller is elaborated based on the reduced model bank. Simulations confirm the method for selecting the appropriate number of local models which should be used in the controller design.

     

Multiple model bank selection based on nonlinearity measure and H-gap metric

This paper provides a systematic method for model bank selection in multi-linear model analysis for nonlinear systems by presenting a new algorithm which incorporates a nonlinearity measure and a modified gap based metric. This algorithm is developed for off-line use, but can be implemented for on-line usage. Initially, the nonlinearity measure analysis based on the higher order statistic (HOS) and the linear cross correlation methods are used for decomposing the total operating space into several regions with linear models. The resulting linear models are then used to construct the primary model bank. In order to avoid unnecessary linear local models in the primary model bank, a gap based metric is introduced and applied in order to merge similar linear local models. In order to illustrate the usefulness of the proposed algorithm, two simulation examples are presented: a pH neutralization plant and a continuous stirred tank reactor (CSTR).     

Conservatism of the gap metric

It is shown by way of simple examples that the gap metric may be inappropriate for quantifying the uncertainty for lightly damped systems, e.g. flexible structures such as spacecraft and aircraft. Two plants having otherwise identical transfer functions, except that their zeros on the imaginary axis are located slightly apart due to perturbations, are usually considered very close, i.e. their closed-loop characteristics under most stabilizing feedbacks are very similar. However, it is demonstrated that when the gap is used to quantify the plant perturbations, the computed gap may be large. Consequently, the stability of the perturbed plant with a stabilizing controller designed for the nominal plant cannot be guaranteed via the existing stability theorems using the gap metric     

Optimal robustness in the gap metric

The application of the gap metric to robust stabilization of feedback systems is considered. In particular, a solution to the problem of robustness optimization in the gap metric is presented. The problem of robust stabilization under simultaneous plant-controller perturbations is addressed, and the least amount of combined plant-controller uncertainty, measured by the gap metric, that can cause instability of a nominally stable feedback system is determined. Included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the T-gap metric. A key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations. This settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well-defined metric     

System-theoretical approach to model reduction and system-order determination

This paper is concerned with (1) the problem of the construction of lower-order models and (2) the Telated problem of the order determination of a real system based upon an estimated model with an overestimated order. Methods of the construction of stable lower-order models and the system-order determination are proposed. The approach adopted is to obtain a minimal realization of the original system by taking the principal components of the predictors of the outputs as the state and then to construct reduced models based upon a measure of reducibility defined in connection with the minimal realization algorithm. The measure of reducibility is useful to get a priori information about how small the order of the reduced model can be without much deterioration. Simulation studies are also carried out to demonstrate the effectiveness of the measure of reducibility and the proposed methods.     

A complexity metric approach to software quality measurement

Traditional statistical quality control for dealing with quality variation during mass production is not appropriate for software quality assessment. Therefore determination and measurement of user desirable software attributes become a subject of research interest. In this paper, software attributes are classified into attributes of subjective judgment, attributes based on complexity metrics, and attributes with rigorous mathematical definition. Methods of measuring those attributes are proposed in the paper. To measure the quality of software as a whole, model that integrates qualitative software attributes and quantitative complexity metrics is also given. Finally, distribution of software life cycle costs in relation to the weighting of software quality attributes is discussed.     

An experimental search-based approach to cohesion metric evaluation

In spite of several decades of software metrics research and practice, there is little understanding of how software metrics relate to one another, nor is there any established methodology for comparing them. We propose a novel experimental technique, based on search-based refactoring, to ‘animate’ metrics and observe their behaviour in a practical setting. Our aim is to promote metrics to the level of active, opinionated objects that can be compared experimentally to uncover where they conflict, and to understand better the underlying cause of the conflict. Our experimental approaches include semi-random refactoring, refactoring for increased metric agreement/disagreement, refactoring to increase/decrease the gap between a pair of metrics, and targeted hypothesis testing. We apply our approach to five popular cohesion metrics using ten real-world Java systems, involving 330,000 lines of code and the application of over 78,000 refactorings. Our results demonstrate that cohesion metrics disagree with each other in a remarkable 55 % of cases, that Low-level Similarity-based Class Cohesion (LSCC) is the best representative of the set of metrics we investigate while Sensitive Class Cohesion (SCOM) is the least representative, and we discover several hitherto unknown differences between the examined metrics. We also use our approach to investigate the impact of including inheritance in a cohesion metric definition and find that doing so dramatically changes the metric.     

Equivalence of a behavioral distance and the gap metric

In this article it is shown that for the class of stable linear state space systems with a fixed MacMillan degree, the topology of the gap between the graphs of the systems is equivalent to the topology of the gap between the extended graphs of the systems.     

Upper and lower bounds for approximation in the gap metric

Upper and lower bounds for the closest approximant of degree k <n in the gap metric to a plant of degree n are obtained. The bounds are expressed in terms of the singular values of two Hankel operators defined in terms of the symbol of the graph of the plant. The question of robust stability and performance of feedback systems is examined in the context of approximation of plant and controller in the gap metric     

The gap metric: Robustness of stabilization of feedback systems

In this paper we introduce the gap metric to study the robustness of the stability of feedback systems which may employ not necessarily stable open-loop systems. We elaborate on the computational aspects of the gap metric and provide upper bounds to the gap in cases where the exact formulas do not apply, By admissible uncertainties we mean those which preserve closed-loop stability and a specified small tolerance on the I/O behavior of a feedback system. We show that admissible uncertainties are precisely those which are constrained in the gap. Finally, we conclude that any metric which preserves a continuous relationship between open-loop systems and the corresponding stable feedback interconnections must have the topology of the gap metric.     

Feedback stability under simultaneous gap metric uncertainties in plant and controller

The stability robustness of a feedback system is studied in this paper by assuming that the plant and the controller are subject to independent uncertainties and that the uncertainties are measured by the gap metric. A fairly complete solution is obtained by exploring the trigonometric structure of the graphs of the plant and the controller.     

On the computation of the gap metric for LTV systems

In this paper, we consider the robust stabilization problem for linear discrete time-varying (LTV) systems using the gap metric. In particular, we show that the time-varying (TV) directed gap reduces to an operator with a TV Hankel plus Toeplitz structure. Computation of the norm of such an operator can be carried out using an iterative scheme involving a TV Hankel operator defined on a space of Hilbert–Schmidt causal operators. The “infimization” in the TV directed gap formula is shown to be, in fact, a minimum by using duality theory. The latter holds as well in the time-invariant case.     

The robustness of a Nash equilibrium simulation model: Game-theoretic approach using variable metric projection method

This paper proposes a Nash equilibrium model that applies continuous time replicator dynamics to the analysis of oligopoly markets. The robustness of the proposed simple Nash equilibrium model under the simultaneous constraints of allocation of product and market share using a simulation method to derive an optimal solution for production decisions by rival firms in oligopoly markets is tested by changing profit and cost function parameters, as well as the initial production values and market shares of the firms examined in this study. The effects of differences in conjectural variation and initial allocation of market share on the convergent values are considered, particularly in the case of corner solutions. This approach facilitates the understanding of the robustness of attaining equilibrium in an oligopoly market.     

Application of fuzzy goal programming approach to multi-objective linear fractional inventory model

In this paper, we propose a model and solution approach for a multi-item inventory problem without shortages. The proposed model is formulated as a fractional multi-objective optimisation problem along with three constraints: budget constraint, space constraint and budgetary constraint on ordering cost of each item. The proposed inventory model becomes a multiple criteria decision-making (MCDM) problem in fuzzy environment. This model is solved by multi-objective fuzzy goal programming (MOFGP) approach. A numerical example is given to illustrate the proposed model.     

A relationship-based approach to model integration

A key problem in model-based development is integrating a collection of models into a single, larger, specification as a way to construct a functional system, to develop a unified understanding, or to enable automated reasoning about properties of the resulting system. In this article, we suggest that the choice of a particular model integration operator depends on the inter-model relationships that hold between individual models. Based on this observation, we distinguish three key integration operators studied in the literature—merge, composition and weaving—and describe each operator along with the notion of relationship that underlies it. We then focus on the merge activity and provide a detailed look at the factors that one must consider in defining a merge operator, particularly the way in which the relationships should be captured during merge. We illustrate these factors using two merge operators that we have developed in our earlier work for combining models that originate from distributed teams.