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Reinforcement learning (RL) agents can learn to control a nonlinear system without using a model of the system. However, having a model brings benefits, mainly in terms of a reduced number of unsuccessful trials before achieving acceptable control performance. Several modelling approaches have been used in the RL domain, such as neural networks, local linear regression, or Gaussian processes. In this article, we focus on techniques that have not been used much so far: symbolic regression (SR), based on genetic programming and local modelling. Using measured data, symbolic regression yields a nonlinear, continuous-time analytic model. We benchmark two state-of-the-art methods, SNGP (single-node genetic programming) and MGGP (multigene genetic programming), against a standard incremental local regression method called RFWR (receptive field weighted regression). We have introduced modifications to the RFWR algorithm to better suit the low-dimensional continuous-time systems we are mostly dealing with. The benchmark is a nonlinear, dynamic magnetic manipulation system. The results show that using the RL framework and a suitable approximation method, it is possible to design a stable controller of such a complex system without the necessity of any haphazard learning. While all of the approximation methods were successful, MGGP achieved the best results at the cost of higher computational complexity. Index Terms–AI-based methods, local linear regression, nonlinear systems, magnetic manipulation, model learning for control, optimal control, reinforcement learning, symbolic regression.

In this paper, in order to control a nonlinear dynamic system via multi-model controller, we propose a systematic approach to determine the nominal local linear models. These models are selected from the local models bank and results in a reduced nominal models set that provides enough information to design a multi-model controller. To determine the initial local models bank, gap metric is used so that the distance between two successive local models is smaller than a threshold value. Then, a systematic approach that aims to get a reduced nominal models bank is developed. Based on this approach, first, a binary gap matrix is defined by combining gap metric and stability information. Then, several rows of this matrix are selected such that the sum of them becomes a non-zero vector. The proposed approach along with a designed robust controller is validated on a pH neutralization regarding to its highly nonlinear behavior.

The dynamics of three local linear models of air sea-interation commonly employed in climate or ocean simulations is compared. The models differ by whether or not the ocean velocity is included in the shear calculation applied to the ocean and the atmosphere. Analytic calculations for the models with deteministic and random forcing (white and colored) are presented. The short term behavior is similar in all models, which only small quantitative differences, while the longterm behavior differs qualitatively between the models. The fluctuation-dissipation-relation, which connects the fast excitation to the slow dissipation, is establised for all models with random forcing. The fluctuation-dissipation-theorem, which compares the response to an external forcing to internal fluctuations is established for a white-noise forcing and a colored forcing when the phase space is augmented by the forcing variable. Using results from numerical integrations of stochastic differential equations shows that the fluctuation-theorem, which compares the probability of positive to negative fluxes of the same magnitude, averaged over time-intervals of varying length, holds for the energy gained by the ocean from the atmosphere.

This paper presents a cooperative fuzzy model predictive control (CFMPC) for heating and cooling of buildings. Because of different supply zones, the large time constants and the non-linear building dynamics with respect to the different seasons, a CFMPC concept is proposed. The overall non-linear building is split into different zones, which are consisting of input-coupled Takagi-Sugeno (TS)-fuzzy models. Each such TS-fuzzy model is constituted by a local linear model network (LLMN). The LLMN consists of local linear models (LLM), which are representing the different seasons: winter, transition season (fall and spring), and summer. The control of each LLM is realized by model predictive control (MPC). For each building zone the associated MPCs are output-blended by the fuzzy membership functions, which leads to fuzzy model predictive control (FMPC). In addition to the FMPCs a global MPC is controlling the thermally activated building systems, which affect all other zones. To coordinate the different controllers a cooperative iteration-loop is assumed, which leads to cooperative fuzzy model predictive control (CFMPC). The concept is developed for a specific demonstration building and can be easily adapted for other complex buildings. A simulation example demonstrates that the proposed CFMPC achieves a performance increase with less energy consumption, as compared to FMPC controllers and historical measured data of the demonstration building.

In this paper, a new PID controller design scheme is proposed for non-linear systems. According to this scheme, the system model is first generated by using local linear models. Then, it is described a design of internal model controllers(IMC) based on the local linear models. The internal model control has a simple structure and has a high robustness for system uncertainties. However, there are few studies of IMC schemes for non-linear systems. On the other hand, lots of controlled systems have non-linearities. Finally the effectiveness of the newly proposed control scheme is experiment examples in comparison with the conventional control methods for non-linear systems.

In this paper, a wavelet-based local linear neuro fuzzy model is developed for the multi-step ahead crude oil price prediction. In the proposed approach, the price series is first decomposed into the approximation and detail components by means of discrete wavelet transform. The approximation and detail components are then modeled by two individual neuro fuzzy local linear models to obtain final results. A comprehensive evaluation study is carried out using various series with different structures and features. The forecasting results have been compared to the recent results available in the literature. The results confirm the superiority of the proposed method in terms of forecasting performance and accuracy criteria.

© 2017 IEEE. This paper presents a comparative study of two widely accepted model predictive control schemes based on mixed logical dynamical (MLD) and nonlinear modeling approaches with application to a continuous stirred tank reactor (CSTR) system. Specifically, we approximate the nonlinear behavior of a CSTR system with multiple local linear models in a MLD framework. The main benefit of such a scheme is the significant improvement in model accuracy when compared with a single linearized model. The benefits and trade-offs associated with predictive control laws synthesized using MLD and nonlinear modeling approaches are also compared.

Objective: Previous studies have found both increased and decreased cortisol levels in depressed patients. These inconsistent findings may be explained by the fact that traditional group-based studies are not suitable to capture complex intra-individual dynamics between cortisol and affect, and inter-individual differences therein. The current study used a time-series approach to gain deeper insight into the nature of these complex dynamics and to investigate possible underlying nonlinear dynamical features. Method: Time-series data (90 measurements) were collected for cortisol and negative affect (NA) in depressed (n=15) and non-depressed (n=15) participants. The relationship between cortisol and NA in each individual was analyzed with SMAP, which estimates local linear vector autoregression (VAR) models with different degrees of nonlinearity in the prediction. The best-predicting model, and whether this model was linear or nonlinear, was determined by using the normalized root mean square error (NRMSE) between the models’ predicted values and the observed values. Univariate and multivariate models were compared to explore the connection between cortisol and NA. Results: Nonlinear cortisol predictions were best in 90% of the participants, whereas nonlinear NA predictions were best in 39% of the participants. Multivariate analyses showed that in 48% of the participants, cortisol was better predicted when NA was included in models that otherwise consisted of time delayed values of cortisol alone. Vice versa, in 39% of the participants, NA was better predicted when cortisol was included in models that otherwise consisted of time delayed values of NA alone. The connection between cortisol and NA was stronger in the depressed group, although the results showed considerable inter-individual heterogeneity within the diagnostic groups. Conclusion: In many individuals, cortisol and NA may be interacting parts of a common dynamical system and their con-nection may be stronger in depressed patients.

In this paper, we propose a neuro-fuzzy modeling approach for identification of a generalized first-order Takagi-Sugeno (TS) fuzzy models with considerably low number of local linear models (LLMs). For this purpose, the main idea is to identify uneven linear operating regimes of a process by LLMs localized in rather flexible fuzzy subspaces (FSs). To this end, a split-and-merge clustering approach is introduced. Through the split procedure, a generalized TS model is identified whose LLMs are localized in simple FSs. Then, by merging these simple FSs, so to keep the generality of the model, more flexible FSs are produced to localize the LLMs. As an illustrative example, the proposed modeling method is utilized to approximate a benchmark nonlinear function. Moreover, the performance of our approach is compared with that of several well-known state-of-the-art approaches in identification of a steam generator. It is demonstrated that our identified model generalizes well, with considerable lower number of LLMs. Robustness of the proposed modeling method against additive noise and outliers is studied as well, where good resilience against noise and drop of performance in the face of frequent outliers is observed in prediction of Mackey-Glass time-series.

A new and systematic method to design digital controllers for uncertain chaotic systems with structured uncertainties is presented in this paper. Takagi–Sugeno (TS) fuzzy model is used to model the chaotic dynamic system, while the uncertainties are decomposed such that the uncertain chaotic system can be rewritten as a set of local linear models with an additional disturbed input. Conventional control techniques are utilized to develop the continuous-time controllers first. Then, the digital controllers are obtained as the digital redesign of the continuous-time controllers using the state-matching approach. The performance of the proposed controller design is illustrated through numerical examples.

This paper introduces a variant of k Bipartite Neighbors (k-BN), called k-BN2, for use in function prediction. Like k-BN, k-BN2 selects k instances surrounding the query, i.e., the novel instance, and keeps them bipartitely. However, in order to improve the prediction precision, based on the bipartite neighborhood, k-BN2 combines local linear models and the global nonlinear model to predict the value of the novel instance. Applied to two real measured datasets, k-BN2 outperforms the typical k-BN and those methods in which kBN or a related approximate physical model alone is exploited.

In diesem Beitrag wird im ersten Teil ein systematisches Vorgehen zur Identifikation des Luftpfades (Luft- und Abgastrakt) von Dieselmotoren auf Basis einer dynamischen Motorvermessung vorgestellt. Dafür werden lokal lineare Modelle mit Ausgangsfehlerschätzung verwendet. Mit dem identifiziertenModell erfolgt im zweiten Teil eine automatisierte Parametrisierung der Luftmassenstrom- und Ladedruckregelung. A systematic procedure is introduced for the identification of the air path (air- and exhaust manifold) of diesel engines, by means of dynamic engine measurements. Local linear models in output error configuration are used. Furthermore, an automated calibration of the feedback-controllers is presented for the air mass flow and the charging pressure.

A new method for identifying Hammerstein/Wiener models is proposed, where the parameters of a piecewise linear characteristic and a linear transfer function are estimated by recursive least-squares (RLS). Using smooth validity functions and an adaptive algorithm for optimal partitioning of the local linear models an efficient and accurate model can be identified.

Takagi-Sugeno (TS) modeling formalism has been widely used to build up multi-model representations for nonlinear systems. In this paper, TS approach is applied to get a multi-model description of Ozone generation process in a specific geographical zone. This is done in two steps: firstly, an adequate structure of the desired multi-model is designed. This structure involves a set of local linear models (each one is valid for a certain range of operating conditions) and an interpolative mechanism that combines the outputs of the local models into a continuous global output. The second step consists in identifying the parameters of the local models using the parametric identification approach. The identification scheme developed is applied to model ozone generation in the Basse-Normandie region (France). The model thus obtained turned out to be satisfactory and currently used to build-up a predictor for this region.

Application of the data driven modelling framework entitled Grid of Linear Models to derive a predictive model for a fed-batch fermentation process is presented. The modelling framework utilises a large number of local linear models to approximate the behaviour of the non-linear process. Industrial data from a fed-batch fermentation has been supplied by Novozymes A/S. A model for the process has been identified based on these data. The model possesses good predictive capabilities and it is intended to be implemented in a model predictive control framework.

In this paper, the possibilities of possibilistic fuzzy clustering in identification of local linear models for control will be discussed. A generalisation of possibilistic algorithms in the bibliography is presented. It will provide a trade-off between an “expected shape” of the membership functions and the model fit. If low antecedent overlap occurs, undermodelling must be suspected.

The aim of the present study is to derive nonlinear instrument variable methods by using local linear models. Two algorithms to estimate consistent local ARX-models of the system order are presented. A local ARX-model with a regressor of higher order than the system is simulated to estimate an approximately noise-free data set. In the first algorithm this approximately noise-free data is used as estimation data to a local ARX-model of the system order. The second algorithm uses the simulated data as instrument in a local instrument variable method. The algorithms are demonstrated on both simulated and laboratory data.

The multi-objective identification of nonlinear dynamic models consisting of local linear models is considered. The tradeoff between global model accuracy and local model interpretability is explicitly considered by introducing weights on the criteria for local model accuracy. A strategy is proposed to tune the local weights in order to achieve similar tradeoff for each local model. In this way, better generalization is achieved. The multi-objective identification algorithm has been applied to predict the engine load of an off-road vehicle operating under varying working load conditions. The analysis tools have proven useful for the construction of an accurate and robust engine load prediction model. The resulting model can directly be used in model-based control algorithms in automatic tuning systems that explicitly deal with constraints on the working region.

The structure identification problem when estimating local linear models can be eased by using Analysis of Variance (ANOVA) as a prior step in the estimation procedure. The information gained from using ANOVA on the input/output data is what regressors that should be used to partition the input space and what regressors are needed only for ihe linear models in each part. Also the complexity of the partitioning can be restricted due to the extra information.

A simple gain-scheduling approach to the position control of electrostatically levitated materials is presented. An electrostatic levitator is currently being constructed at the German Aerospace Center (DLR). It can be used for containerless processing of metals and nonconducting materials. Time-varying sample charges and the fact that the plant is open-loop unstable make this a difficult and challenging control problem. The nonlinear parameter-dependent system is approximated by two local linear models. Based on these models, two local linear controllers are designed. LMI-based pole region assignment is used to reduce the design procedure to the tuning of three parameters with intuitive meaning. The sample charge is estimated with a discrete-time extended Kalman filter and used as scheduling parameter. Experimental results demonstrate that gain-scheduled control achieves a significantly better performance than fixed-gain, observer-based state feedback control.

Concentrated solar power (CSP) is an efficient means of renewable energy that makes use of solar radiation to produce electricity instead of making use of conventional fossil fuel techniques such as burning coal. The aim of this study is the simulation and characterisation of a CSP plant in order to gain a better understanding of the dominant plant dynamics. Due to the nature of the study, the dissertation is divided into two main parts namely the simulation of a CSP plant model and the characterisation of the plant model. Modelling the CSP plant takes the form of developing an accurate Flownex® model of a 40 MW combined cycle CSP plant. The model includes thermal energy storage as well as making use of a duct burner. The Flownex® model is based on an existing TRNSYS model of the same plant. The Flownex® model is verified and validated, by making use of a bottom-up approach, to ensure that the developed model is in fact correct. The characterisation part of this dissertation involves evaluating the dynamic responses unique to that of a CSP plant as stated in the literature. This involves evaluating the dominant dynamic behaviour, the presence of resonant and anti-resonant modes found within the control bandwidth, and the change in the dynamics of the plant as the plants’ operating points change throughout the day. Once the developed model is validated, characterisation in the form of evaluating the open loop local linear models of the plant is implemented. In order to do so, these models are developed based on model identification processes, which include the use of system identification software such as Matlab® SID Toolbox®. The dominant dynamic behaviour of the plant model, obtained from the developed local linear models, represents that of an over damped second order system that changes as the operating points of the plant change; with the models’ time responses and the bandwidth decreasing and increasing respectively as the thermal energy inputs to the plant increases. The frequency response of the developed local linear models also illustrates the presence of resonant and antiresonant modes found within the control bandwidth of the solar collector field’s temperature response. These modes however are not found to be present in the mechanical power output response of the plant. The use of adaptive control, such as feedforward and gain-scheduled controllers, for the plant should be developed to compensate for the dynamic behaviours associated with that of a CSP plant. MIng (Computer and Electronic Engineering), North-West University, Potchefstroom Campus, 2015

Objective.In mechanically ventilated patients the expiratorytime constant provides information about respiratory mechanics. In thepresent study a new method, fuzzy clustering, is proposed to determineexpiratory time constants. Fuzzy clustering differs from other methodssince it neither interferes with expiration nor presumes any functionalrelationship between the variables analysed. Furthermore, time constantbehaviour during expiration can be assessed, instead of an average timeconstant. The time constants obtained with fuzzy clustering are comparedto time constants conventionally calculated from the same expirations.Methods.20 mechanically ventilated patients, including 10patients with COPD, were studied. The data of flow, volume and pressurewere sampled. From these data, four local linear models were detected byfuzzy clustering. The time constants (τ) of the local linear models(clusters) were calculated by a least-squares technique. Time constantbehaviour was analysed. Time constants obtained with fuzzy clusteringwere compared to time constants calculated from flow-volume curves usinga conventional method. Results.Fuzzy clustering revealed twopatterns of expiratory time constant behaviour. In the patients withCOPD an initial low time constant was found (mean τ 1: 0.33 s, SD0.21) followed by higher time constants; mean τ 2: 2.00 s (SD0.91s), mean τ 3: 3.45 s (SD 1.44) and mean τ 4: 5.47 s (SD2.93). In the other patients only minor changes in time constants werefound; mean τ 1: 0.74 s (SD 0.30), mean τ 2: 0.90 s (SD 0.23),mean τ 3: 1.04 s (SD 0.42) and mean τ 4: 1.74 s (SD 0.78). Boththe pattern of expiratory time constants, as well as the time constantscalculated from the separate clusters, were significantly differentbetween the patients with and without COPD. Time constants obtained withfuzzy clustering for cluster 2, 3 and 4 correlated well with timeconstants obtained from the flow-volume curves. Conclusions.Inmechanically ventilated patients, expiratory time constant behaviour canbe accurately assessed by fuzzy clustering. A good correlation was foundbetween time constants obtained with fuzzy clustering and time constantsobtained by conventional analysis. On the basis of the time constantsobtained with fuzzy clustering, a clear distinction was made betweenpatients with and without COPD.

Extracellular protein concentrations and gradients initiate a wide range of cellular responses, such as cell motility, growth, proliferation and death. Understanding inter-cellular communication requires spatio-temporal knowledge of these secreted factors and their causal relationship with cell phenotype. Techniques which can detect cellular secretions in real time are becoming more common but generalizable data analysis methodologies which can quantify concentration from these measurements are still lacking. Here we introduce a probabilistic approach in which local-linear models and the law of mass action are applied to obtain time-varying secreted concentrations from affinity-based biosensor data. We first highlight the general features of this approach using simulated data which contains both static and time-varying concentration profiles. Next we apply the technique to determine concentration of secreted antibodies from 9E10 hybridoma cells as detected using nanoplasmonic biosensors. A broad range of time-dependent concentrations was observed: from steady-state secretions of 230 pM near the cell surface to large transients which reached as high as 56 nM over several minutes and then dissipated.

A method for combining multiple local models to describe a nonlinear system. The local model weights are based on the linear interpolation of the current operating point from the closest local model operating points defined as the filtered value of the current process input. This local modeling method can be used to describe systems with changing system gains and dynamics, as well as input multiplicity. A case study for a benchmark CSTR reactor is presented. Local linear models are used to synthesize IMC based PID controllers.

Divide-and-conquer methods, in the context of nonlinear dynamic systems modelling and control design, break the problem down into a number of simpler, frequently linear modelling and control design problems, each associated with a restricted operating region. The most widespread divide-and-conquer nonlinear control design method is probably gain-scheduling. The gain-scheduling method has been successfully applied in many fields, ranging from process control to aerospace engineering. The basic idea behind the method is to divide the nonlinear system to be controlled into local subsystems described by linear dynamic models. A linear control problem is then solved for each of these subsystems. The global control solution—called gain-scheduling control—is obtained by merging partial local solutions. The idea of divide-and-conquer seems very attractive, but its application is not straightforward. The following issues arising during system modelling and control design are treated in this chapter: the properties of the global nonlinear model vis a vis the properties of the local linear models; controller design from first-principles nonlinear continuous-time models; and control design based on local models identified from measured data. The problems are elaborated on simulated examples that include vehicle dynamics and a continuous stirred tank reactor. How to properly use these methods is illustrated by pressure control design on a semi-industrial gas–liquid separator unit.

International audience; This paper presents a new air traffic complexity metric based on dynamical systems. Based on a set of radar observations (position and speed) a vector field interpolating these data is constructed. Once the field has been obtained, the Lyapunov spectrum of the associated dynamical system is computed on points evenly spaced on a spatial grid. The results of the computations are summarized on complexity maps, with high values indicating areas to avoid or to carefully monitor. A first approach based on linear dynamical system enable to compute an aggregate complexity metric. In order to produce complexity maps, two extensions of the previous approach have been developed (one in space and another in space and time). Finally, an approximation is proposed in order to localize the computation of the vector field by the mean of Local Linear Models.

In this paper, we propose an approach for recognizing online Persian isolated characters using LLNF model. Local Linear Neuro Fuzzy (LLNF) Model is a powerful approach for classification tasks. It uses divide-and-conquer strategy to partition the problem space into sub-problems and construct Local Linear Models (LLMs). In order to classify the characters, at first, we extract some generic features of Persian character and build a features vector. Then we construct a LLNF model by the features vector as input data. The constructed LLNF model will be later used to recognize the written letters. Our experimental results for 100 different people show recognition rate of 99.15%.

This paper presents a case study of model predictive control (MPC) applied to a continuous stirred tank reactor (CSTR). It is proposed to approximate nonlinear behavior of a plant by several local linear models, enabling a piecewise affine (PWA) description of the model used to predict and optimize future evolution of the reactor behavior. Main advantage of the PWA model over traditional approaches based on single linearization is a significant increase of model accuracy which leads to a better control quality. It is also illustrated that, by adopting the PWA modeling framework, MPC strategy can be implemented using significantly less computational power compared to nonlinear MPC setups.

Danielsson, R. and Malmquist, G., 1992. Multi-dimensional simplex interpolation. An approach to local models for prediction. Chemometrics and Intelligent Laboratory Systems , 14: 115–128. The use of local linear models for prediction is investigated. A number of calibration samples, constituting a prediction set, in the neighbourhood of the unknown sample is selected from a given set. The extreme case of local models, where the prediction set forms a simplex, is given particular consideration. Two methods for selection of the prediction set in this case are introduced, ensuring interpolation within the simplex. These methods are compared to multiple linear regression, using a larger prediction set. The prediction error is divided into two contributions: (i) the data error emanating from random and systematic errors in both the dependent and the independent variables; (ii) the interpolation error due to lack-of-fit for the linear model. The properties of both these contributions as a function of the size of the prediction set are studied by simulations. The results are compared to theoretical relations. Simulated experiments in size exclusion chromatography are used as an application. A refinement step in the prediction, using a recalculated distance measure for the selection of the prediction set, is introduced leading to a substantial decrease in the prediction error.

6 pages; International audience; This work aims at evaluating the use of TS MIMO model for the Detection and Isolation of faults of sensors on a nonlinear process. TS model involves a set of local linear models whose contribution on the output signals is expressed by a weighting function. In a previous work, an identification scheme was proposed for TS MIMO model. It doesn't need for the outputs of the process to be separable, as it is usually the case for the identification methods based on MISO model. A model-based method, for faults detection is proposed from the identified model. This new approach differs from the observers and parity space based approaches, as it only uses a partial knowledge of the model. Outputs are projected into a sub-space that is orthogonal to the sequences of input-output weighted signals to provide a residuals set. The process of fault isolation exploits the structure of the residuals set in order to recognize the acting fault. The sensitivity of the residuals to the occurrence of faults is proved while the incidence matrix of the residuals is deduced from simulation. It shows the ability of the generated set of residuals to isolate the fault of sensors. © 2009 IFAC.

This paper presents a new method for control of complex MIMO plants based on a quantised state description. The control algorithm solves a minimisation problem for a set of user defined convex control objective functions with the plant dynamics as a constraint. The solution makes use of local linear models that are effectively calculated by propagating the state on-line using a quantised state description. The method is demonstrated on a model of an autonomous underwater vehicle.

Motion capture is a prevalent technique for capturing and analyzing human articulations. A common problem encountered in motion capture is that some marker positions are often missing due to occlusions or ambiguities. Most methods for completing missing markers may quickly become ineffective and produce unsatisfactory results when a significant portion of the markers are missing for extended periods of time. We propose a data-driven, piecewise linear modeling approach to missing marker estimation that is especially beneficial in this scenario. We model motion sequences of a training set with a hierarchy of low-dimensional local linear models characterized by the principal components. For a new sequence with missing markers, we use a pre-trained classifier to identify the most appropriate local linear model for each frame and then recover the missing markers by finding the least squares solutions based on the available marker positions and the principal components of the associated model. Our experimental results demonstrate that our method is efficient in recovering the full-body motion and is robust to heterogeneous motion data.