Gap Metric Research Articles

Models’ Bank Selection of Nonlinear Systems by Integrating Gap Metric, Margin Stability, and MOPSO Algorithm

This paper deals with the control of nonlinear systems where the multimodel approach has been used to build global controller. In multimodel approaches, two problems could be generally encountered: (1) how to find the required number of models and (2) what are their locations in an operating space. The developed method integrates gap metric, margin stability and multi-objective particle swarm optimization algorithm (MOPSO) to get a reduced model bank that provides necessary information for controller design. For this, the gap metric and the margin stability are respectively used as a distance measuring tool and as a guideline for selecting the model bank. The controller design is handled as a multi-objective optimization problem. In this context, the MOPSO algorithm is used for tuning optimal PID controllers that give the shortest rise time with a lower overshoot percentage and good margin stability.

A novel plant-wide process monitoring framework based on distributed Gap-SVDD with adaptive radius

With the increasing complexity of modern process industries, plant-wide process monitoring has become a challenging issue. In this paper, a novel monitoring framework based on distributed gap support vector data description (Gap-SVDD) with adaptive radius is proposed for plant-wide processes. Firstly, the plant-wide processes are divided into different subblocks by the mixed similarity measure for handling the heavy coupling process variables. Afterwards, gap metric is introduced as a kind of data preprocessing method, which is combined with SVDD to set up the monitoring model. The feature extraction of Gap-SVDD is more accurate than conventional methods. Finally, an adaptive radius is developed for Gap-SVDD. It is calculated by a modified univariate statistical method with a limited window length. The purpose of this strategy is to enhance the performance of Gap-SVDD. The superiority of the proposed framework is demonstrated by the revised Tennessee Eastman (TE) benchmark. Comparisons with other conventional methods are also provided.

The gap distance to the set of singular matrix pencils

We study matrix pencils sE−A using the associated linear subspace ker⁡[A,−E]. The distance between subspaces is measured in terms of the gap metric. In particular, we investigate the gap distance of a regular matrix pencil to the set of singular pencils and provide upper and lower bounds for it. A relation to the distance to singularity in the Frobenius norm is provided.

UAV Remote Sensing for Biodiversity Monitoring: Are Forest Canopy Gaps Good Covariates?

Forest canopy gaps are important to ecosystem dynamics. Depending on tree species, small canopy openings may be associated with intra-crown porosity and with space among crowns. Yet, literature on the relationships between very fine-scaled patterns of canopy openings and biodiversity features is limited. This research explores the possibility of: (1) mapping forest canopy gaps from a very high spatial resolution orthomosaic (10 cm), processed from a versatile unmanned aerial vehicle (UAV) imaging platform, and (2) deriving patch metrics that can be tested as covariates of variables of interest for forest biodiversity monitoring. The orthomosaic was imaged from a test area of 240 ha of temperate deciduous forest types in Central Italy, containing 50 forest inventory plots each of 529 m2 in size. Correlation and linear regression techniques were used to explore relationships between patch metrics and understory (density, development, and species diversity) or forest habitat biodiversity variables (density of micro-habitat bearing trees, vertical species profile, and tree species diversity). The results revealed that small openings in the canopy cover (75% smaller than 7 m2) can be faithfully extracted from UAV red, green, and blue bands (RGB) imagery, using the red band and contrast split segmentation. The strongest correlations were observed in the mixed forests (beech and turkey oak) followed by intermediate correlations in turkey oak forests, followed by the weakest correlations in beech forests. Moderate to strong linear relationships were found between gap metrics and understory variables in mixed forest types, with adjusted R2 from linear regression ranging from 0.52 to 0.87. Equally strong correlations in the same forest types were observed for forest habitat biodiversity variables (with adjusted R2 ranging from 0.52 to 0.79), with highest values found for density of trees with microhabitats and vertical species profile. In conclusion, this research highlights that UAV remote sensing can potentially provide covariate surfaces of variables of interest for forest biodiversity monitoring, conventionally collected in forest inventory plots. By integrating the two sources of data, these variables can be mapped over small forest areas with satisfactory levels of accuracy, at a much higher spatial resolution than would be possible by field-based forest inventory solely.

Nonlinearity Analysis and Multi-Model Modeling of an MEA-Based Post-Combustion CO2 Capture Process for Advanced Control Design

The monoethanolamine (MEA)-based post-combustion CO2 capture plant must operate flexibly under the variation of the power plant load and the desired CO2 capture rate. However, in the presence of process nonlinearity, conventional linear control strategy cannot achieve the best performance under a wide operation range. Considering this problem, this paper systematically studies the multi-model modeling of the MEA-based CO2 capture process for the purpose of (1) implementing well-developed linear control techniques to the design of an advanced controller and (2) achieving a wide-range flexible operation of the CO2 capture process. The local linear models of the CO2 capture process are firstly established at given operating points using the method of subspace identification. Then the nonlinearity distribution at different loads of an upstream power plant and different CO2 capture rates is investigated via the gap metric. Finally, based on the nonlinearity investigation results, the suitable linear models are selected and combined together to form the multi-model system. The proposed model is validated using the measurement data, which is generated from a post-combustion CO2 capture model developed in the go-carbon capture and storage (gCCS) simulation platform. As the proposed multi-linear model has a simple mathematical expression and high prediction accuracy, it can be directly employed as the control model of a practical advanced control strategy to achieve a wide operating range control of the CO2 capture process.

Wind Turbine Torque Oscillation Reduction Using Soft Switching Multiple Model Predictive Control Based on the Gap Metric and Kalman Filter Estimator

A new multiple model predictive control (MMPC) is reported to regulate the output power of the National Renewable Energy Laboratory (NREL) 1.5 MW baseline wind turbine (WT). The maximum power point tracking in the wide range of the partial load regime without losing any control performance is the goal of the proposed algorithm. To do this, the whole area of the partial load regime is divided into adequate linear models using a mathematical tool called gap metric. Next, we set different MPCs (controller bank) to each model. The controller selection is conducted based on wind speed value estimated using a neural network estimator. The sudden change between controllers applies a hazardous chattering to control signal. To tackle this problem and reduce the chattering, a new soft switching MMPC (SSMMPC) is introduced. Furthermore, the stability of the closed-loop system is guaranteed using Lyapunov theory creating novelty in this work. Moreover, to validate the proposed algorithm, SSMMPC and classical MMPC are implemented on the NREL 1.5 MW baseline WT using fatigue aerodynamic structure and turbulence simulator. Finally, the simulation results and comparisons demonstrate the effectiveness of the proposed controller in the tracking error, oscillation in control signal, and mechanical power.

Gap metric–based model bank construction for wind turbine predictive control

SummaryThis paper presents a novel model bank construction method for the multiple model predictive control of wind turbine system. The gap metric is used to measure the dynamic difference between the linearized models of the wind turbine system at different wind speed. Two algorithms are then proposed to divide the wind speed range in different operating regions. Meanwhile, a complete and nonredundant linear model bank is established to approximate the wind turbine system in the whole operating region. We take the robust model predictive control algorithm to design the local controller and utilize the wind speed as the switching criterion to combine the submodels. The simulation study on a 5‐MW wind turbine verifies the efficiency of the proposed method.

Multimodel Control of Nonlinear Systems: An Improved Gap Metric and Stability Margin-Based Method

This paper presents a new multimodel controller design approach incorporating stability and performance criteria. The gap metric is employed to measure the distance between local models. An efficient method based on state feedback strategy is introduced to improve the maximum stability margin of the local models. The proposed method avoids local model redundancy, simplifies the multimodel controller structure, and supports employing of many linear control techniques, while does not rely on a priori experience to choose the gridding threshold value. To evaluate the proposed method, three benchmark nonlinear systems are studied. Simulation results demonstrate that the method provides the closed-loop stability and performance via a simple multimodel structure in comparison with the opponents.

Optimal direct adaptive soft switching multi-model predictive control using the gap metric for spacecraft attitude control in a wide range of operating points

In this paper, a new Multi-Input Multi-Output (MIMO) Multi-model Predictive Control (MMPC) with direct adaptive structure is used for spacecraft attitude control in a wide range of operating points and maneuvers. Because of the highly nonlinear dynamics, the linearized model characteristics are extremely depended to the operating points. In such cases, an expected performance, in the wide range of the operating points, never can be achieved using a single controller and single model (even instability may be anticipated). To handle this problem, in this paper, we divide the whole operating range of the spacecraft to construct sub-models (model bank) using a mathematical tool called as gap metric. Next, an adaptive MPC based on sub-models is designed. In this procedure, there are two problems: stability when switching among models and the minimum number of sub-models. Hard switching among models to update the controller's model causes extreme chattering on the control signal and reference tracking. The motivation of this paper is to present a new solution for the mentioned problems. To solve the first problem (remove chattering), an adaptive soft switching law to tune the controller parameters, when selecting new model, based on the Lyapunov theory is introduced as the main novelty of this paper. This guarantees the stability of the closed loop control system. For solution of the second problem, the number of optimal sub-models is obtained using different simulations. Finally, the effectiveness of the suggested method is proven via simulations.

Robust Synchronization in Multi-Agent Networks With Unstable Dynamics

The problem of synchronization in heterogenous linear time-invariant networks is investigated. An approach blending the theory of integral quadratic constraints (IQCs) with the gap metric is proposed to study the problem within a unifying framework, where both the agents and communication channels can be dynamic, infinite-dimensional, unstable, and uncertain. Structural properties of the uncertainty are described by IQCs and exploited in the analysis to reduce conservatism. The homotopy employed in IQC analysis is defined with respect to the graph topology induced by the gap metric, whereby open-loop unstable dynamics are accommodated. Sufficient conditions for synchronism are provided, extending recent developments which have been shown to unify several existing synchronization analysis results in the literature.

Gap Metrics for Linear Time-Varying Systems

The relationship between two gap metrics is investigated for a class of time-varying linear systems. Specifically, systems are considered to be causal linear maps between finite-energy continuous-time signals of nonuniformly lower-bounded support, and by definition, the corresponding input-output graphs admit normalized so-called strong-right and strong-left representations. First, a time-varying generalization of Vinnicombe's $\nu$-gap is revisited to rectify an omission in the original developments and to verify compliance with the axioms of a metric. Subsequently, it is shown that this generalized $\nu$-gap metric induces the same topology as an apposite adaptation of Feintuch's time-varying gap metric for discrete-time systems defined over finite-energy signals of uniformly lower-bounded support. The two generalized metrics are therefore qualitatively equivalent in robust stability analysis for linear time-varying feedback interconnections. Quantitatively, the $\nu$-gap between a given pair of systems is never larger in value than the Feintuch gap.

Fault Diagnosis Method Based on Gap Metric Data Preprocessing and Principal Component Analysis

Principal component analysis (PCA) is widely used in fault diagnosis. Because the traditional data preprocessing method ignores the correlation between different variables in the system, the feature extraction is not accurate. In order to solve it, this paper proposes a kind of data preprocessing method based on the Gap metric to improve the performance of PCA in fault diagnosis. For different types of faults, the original dataset transformation through Gap metric can reflect the correlation of different variables of the system in high-dimensional space, so as to model more accurately. Finally, the feasibility and effectiveness of the proposed method are verified through simulation.

Selection of glucose metabolism models with an observer error metric

Abstract We present a novel approach to the selection of dynamic system models with regard to stabilisability in terms of their observer error dynamics. For that, we introduce the gap metric for the observer error dynamics as a distance measure. Models are selected for different subsets in such a way, that the gap metric is minimised. The procedure is subsequently applied to classify parametrised Gottingen Minipig models, which were obtained from animal experimental data, into different subsets. These obtained classes are then set up as the basis for an optimal experimental design procedure and result in improved convergence properties, as well as, in reduced implementation effort.

Bifurcation of critical points along gap-continuous families of subspaces

We consider the restriction of twice differentiable functionals on a Hilbert space to families of subspaces that vary continuously with respect to the gap metric. We study bifurcation of branches of critical points along these families and apply our results to semilinear systems of ordinary differential equations.

Optimal tuning of cascaded control architectures for nonlinear HVAC systems

Cascaded controllers have been shown in recent literature to significantly reduce hunting behavior and improve the performance of a variety of HVAC systems. The cascaded architecture uses multiple proportional-integral-derivative loops, standard in industry and building automation controls, to minimize the effects of load dependent nonlinearities. This article builds on previous work by defining and demonstrating the properties of two nonlinear gap metrics that quantify the closed-loop linearization of the cascaded control structure and enable proper tuning of the inner-loop proportional gain. This is followed by the development of a linear quadratic framework that enables simultaneous tuning of inner- and outer-loop control gains. The proposed gap metrics and tuning procedures are demonstrated with two case studies. A simple single input, two output example plant is used to demonstrate the benefits of the proposed tuning method compared to standard proportional-integral-derivative control and successive loop closure tuning techniques. The tuning approach is also applied to the control of a two-ton, residential heat pump system. The proposed metrics and tuning techniques showed significant improvement in control over standard proportional-integral-derivative control tuning methods in both simulation and experiment.

The effect of perturbations of linear operators on their polar decomposition

The effect of matrix perturbations on the polar decomposition has been studied by several authors and various results are known. However, for operators between infinite-dimensional spaces the problem has not been considered so far. Here, we prove in particular that the partial isometry in the polar decomposition of an operator is stable under perturbations, given that kernel and range of original and perturbed operator satisfy a certain condition. In the matrix case, this condition is weaker than the usually imposed equal-rank condition. It includes the case of semi-Fredholm operators with agreeing nullities and deficiencies, respectively. In addition, we prove a similar perturbation result where the ranges or the kernels of the two operators are assumed to be sufficiently close to each other in the gap metric.

Feedback control of combustion instabilities from within limit cycle oscillations using [Formula: see text] loop-shaping and the ν-gap metric.

Combustion instabilities arise owing to a two-way coupling between acoustic waves and unsteady heat release. Oscillation amplitudes successively grow, until nonlinear effects cause saturation into limit cycle oscillations. Feedback control, in which an actuator modifies some combustor input in response to a sensor measurement, can suppress combustion instabilities. Linear feedback controllers are typically designed, using linear combustor models. However, when activated from within limit cycle, the linear model is invalid, and such controllers are not guaranteed to stabilize. This work develops a feedback control strategy guaranteed to stabilize from within limit cycle oscillations. A low-order model of a simple combustor, exhibiting the essential features of more complex systems, is presented. Linear plane acoustic wave modelling is combined with a weakly nonlinear describing function for the flame. The latter is determined numerically using a level set approach. Its implication is that the open-loop transfer function (OLTF) needed for controller design varies with oscillation level. The difference between the mean and the rest of the OLTFs is characterized using the ν-gap metric, providing the minimum required ‘robustness margin’ for an loop-shaping controller. Such controllers are designed and achieve stability both for linear fluctuations and from within limit cycle oscillations.

A Systematic Determination Approach of Model's Base Using Gap Metric for Nonlinear Systems

This paper proposes a novel gap metric based fuzzy decomposition approach resulting in a reduced model bank that provides enough information to design controllers. It requires, first, the determination of the model base. For this, the number of initial models is obtained via fuzzy c-means (FCM) algorithm. Then, a gap metric based method which aims to get a reduced model bank is developed. Based on the linear models bank, a set of linear controllers are designed and combined into a global controller for setpoint tracking control.

Modeling and Simulation for Nonlinear Pressurized Water Reactor Cores Using Gap Metric and Fuzzy Model with Transfer Function

This investigation is to deal with the modeling issue for nonlinear pressurized water reactor cores. Nonlinearity measure with gap metric and T-S fuzzy modeling are exploited to build the fuzzy model of a nonlinear core which approximates this nonlinear core model. The gap metric of the core is proposed to quantify the core nonlinearity. The curve in the whole range of core power level that is solved adopting the gap metric is the core nonlinearity measure. In terms of the measure, six linearized core models at six power levels are selected as local models of the nonlinear core model. Based on the local models and the introduction of the triangle membership function, the core fuzzy model is achieved. The core fuzzy model and nonlinear core model are simulated. Simulation results show that the core fuzzy model can approximate and substitute the nonlinear core model.

Gap formation following climatic events in spatially structured plant communities.

Gaps play a crucial role in maintaining species diversity, yet how community structure and composition influence gap formation is still poorly understood. We apply a spatially structured community model to predict how species diversity and intraspecific aggregation shape gap patterns emerging after climatic events, based on species-specific mortality responses. In multispecies communities, average gap size and gap-size diversity increased rapidly with increasing mean mortality once a mortality threshold was exceeded, greatly promoting gap recolonization opportunity. This result was observed at all levels of species richness. Increasing interspecific difference likewise enhanced these metrics, which may promote not only diversity maintenance but also community invasibility, since more diverse niches for both local and exotic species are provided. The richness effects on gap size and gap-size diversity were positive, but only expressed when species were sufficiently different. Surprisingly, while intraspecific clumping strongly promoted gap-size diversity, it hardly influenced average gap size. Species evenness generally reduced gap metrics induced by climatic events, so the typical assumption of maximum evenness in many experiments and models may underestimate community diversity and invasibility. Overall, understanding the factors driving gap formation in spatially structured assemblages can help predict community secondary succession after climatic events.