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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.

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.

© 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 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.

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.

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.

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.

The presented work studies the multi-input multi-output (MIMO) control of Heating Ventilating and Air Conditioning (HVAC) systems. When using such HVAC systems in special applications, e.g., the conditioning of the intake air of combustion engine test benches, stringent requirements on the control accuracy have to be met. However, in industry the control of HVAC systems is typically based on single-input single-output (SISO) standard PI(D)-control strategies. Due to the simple structure of the PI(D)-concepts, the potential of the considered plants is not fully exploited. To overcome this problem, in this work, a model based control technique consisting of a linear model predictive control (LMPC) method and local linear neuro fuzzy models (LLNFM) is proposed. Thereby, for the LMPC, at each time instant a linear state space model is generated based on a dynamic LLNFM of the plant. In this way, simple linear control methods can be used, by still taking into account the nonlinearities of the plant. For the computation of the LLNFMs the so called Local Linear Model Tree (LoLiMoT) algorithm is used. Since the real world system cannot be excited arbitrarily (disturbances acting on the system cannot be influenced), for the generation of the identification data, a detailed mathematical model of the HVAC system is used. The modeling of the core components (heating and cooling coil) is based on thermodynamic relations. Within this thesis, extensions of the mathematical model are described. One extension enables the simulation of a variable air mass flow. Furthermore, the cooling coil model is extended in order to capture the drying of the coils after a dehumidification period. Based on these extensions the quality of the plant model and subsequently the performance of the proposed control concept can be improved. Regarding the model complexity, based on the LLNFM approach an essential reduction of the model order (from e.g., n =172 to n = 4) can be achieved to apply it for model predictive control strategies. All the control and modeling methods used in this work are validated in simulation as well as on a real world system. The available test bench allows an air transport of max. 5000m 3 /h. All the components used within this plant are typically installed in commercial industrial HVAC systems. The presented work provides a tool which allows systematic model based controller design for a large class of HVAC systems., Die vorliegende Arbeit beschäftigt sich mit der Mehrgrößenregelung von raumlufttechnischen (RLT)-Anlagen. Beim Einsatz solcher RLT-Anlagen in speziellen Anwendungen, wie z.B. der Konditionierung der Ansaugluft von Verbrennungsmotorprüfständen, sind die Anforderungen hinsichtlich der Regelgenauigkeit sehr hoch. Bei den in der Praxis üblicherweise eingesetzten Regelungsstrategien handelt es sich um konventionelle PI(D)-Regler, die als Eingrößensystem implementiert werden. Durch die einfache Struktur der PI(D)-Regler und die zusätzliche Einschränkung auf Eingrößensysteme kann das Potential der zu regelnden Anlagen nicht hinreichend ausgenutzt werden. Um diese Schwierigkeiten zu überwinden, wird in dieser Arbeit eine Regelungsstrategie vorgeschlagen, die sich aus einer modellbasierten prädiktiven Regelung und einem Netzwerk aus lokal linearen Modellen zusammensetzt. Für die modellprädiktive Regelung wird zu jedem Zeitschritt ein neues lineares Zustandsraummodell aus den lokal linearen Modellen generiert. Durch diese Vorgehensweise können einfache lineare Regelungsmethoden eingesetzt werden, und dadurch wird das nichtlineare Verhalten solcher Anlagen geeignet berücksichtigt. Zur Identifikation der lokal linearen Modelle wird der Local Linear Model Tree (LoLiMoT) Algorithmus verwendet. Da das reale System, zur Erzeugung der Identifikationsdaten, nicht ausreichend gut angeregt werden kann (auf das System einwirkende Störungen können nicht beeinflusst werden), wird dafür ein detailliertes mathematisches Modell der RLT-Anlage verwendet. Die Modellierung der wesentlichen Komponenten (Heiz- und Kühlregister) basiert auf thermo-dynamischen Zusammenhängen. Im Rahmen dieser Arbeit werden diese Modelle erweitert. Eine Erweiterung ermöglicht die Simulation eines variablen Luftmassenstromes. Zusätzlich wird das Modell des Kühlregisters erweitert, um das Abtrocknen der Rohre nach einer Entfeuchtungsphase abzubilden. Durch diese Erweiterungen kann die Modellqualität erhöht und das Verhalten der darauf basierten Regelungsstrategie verbessert werden. Bezüglich der Modellkomplexität wird durch den Einsatz von lokal linearen Modellen eine wesentliche Ordnungsreduktion (von z.B. n =172 auf n = 4) erreicht, die den Einsatz in modellprädiktiven Reglern ermöglicht. Alle in dieser Arbeit beschriebenen Modellierungs- und Regelungsmethoden werden in Simulation, sowie an einer realen Anlage validiert. Die zur Verfügung stehende Testanlage ermöglicht einen Lufttransport von max. 5000m 3 /h. Alle in dieser Anlage verwendeten Komponenten werden typischerweise in kommerziellen industriellen RLT-Anlagen verbaut. Die vorgestellte Arbeit liefert ein Werkzeug zum systematischen Entwurf modellbasierter Regler für eine große Klasse von klimatechnischen Anlagen., Dipl.-Ing. Daniel Schwingshackl, Zusammenfassung in deutscher Sprache, Dissertation Alpen Adria Universität Klagenfurt 2016