Modelling framework for artificial hybrid dynamical systems

Many current industry branches use hybrid approaches to solve complex application problems. Over the last decades, different tools for the simulation of such hybrid systems (e.g. Hysdel and YAMLIP) as well as the identification of hybrid systems (e.g. HIT, MLP and OAF NN) have been developed. The framework presented in this work facilitates the integration of artificial feed-forward neural networks in the modelling process of hybrid dynamical systems (HDS). Additionally, the framework provides a structured language for characterising these feed-forward networks itself. Therefore, an interdisciplinary exchange in the field of neural networks and its integration into hybrid dynamical systems is enabled. Focusing on hybrid systems with autonomous events, two different approaches, namely the artificial hybrid model and the artificial hybrid dynamics, are introduced. Challenges of the modelling process of HDS are reflected and advantages as well as disadvantages are discussed. The case study includes two common examples of HDS and analyses the simulation results and examines limitations of the modelling framework.     

Model predictive control of nonlinear hybrid systems with discrete inputs employing a hybrid fuzzy model

The paper deals with model predictive control (MPC) of nonlinear hybrid systems with discrete inputs based on reachability analysis. In order to implement a MPC algorithm, a model of the process that we are dealing with is needed. In the paper, a hybrid fuzzy modelling approach is proposed. The hybrid system hierarchy is explained and the Takagi–Sugeno fuzzy formulation for hybrid fuzzy modelling purposes is tackled. An efficient method of identification of the hybrid fuzzy model is also discussed.

An algorithm that is–due to its MPC nature–suitable for controlling a wide spectrum of systems (provided that they have discrete inputs only) is presented.

The benefits of the algorithm employing a hybrid fuzzy model are verified on a batch reactor example. The results suggest that by suitably determining the cost function, satisfactory control can be attained, even when dealing with complex hybrid–nonlinear–stiff systems such as the batch reactor.

Finally, a comparison between MPC employing a hybrid linear model and a hybrid fuzzy model is carried out. It has been established that the latter approach clearly outperforms the approach where a linear model is used.     

Rigid–flexible interactive dynamics modelling approach

Dynamics modelling of multi-body systems composed of rigid and flexible elements is elaborated in this article. The control of such systems is highly complicated due to severe underactuated conditions caused by flexible elements and an inherent uneven non-linear dynamics. Therefore, developing a compact dynamics model with the requirement of limited computations is extremely useful for controller design, simulation studies for design improvement and also practical implementations. In this article, the rigid–flexible interactive dynamics modelling (RFIM) approach is proposed as a combination of Lagrange and Newton–Euler methods, in which the motion equations of rigid and flexible members are separately developed in an explicit closed form. These equations are then assembled and solved simultaneously at each time step by considering the mutual interaction and constraint forces. The proposed approach yields a compact model rather than a common accumulation approach that leads to a massive set of equations in which the dynamics of flexible elements is united with the dynamics equations of rigid members. The proposed RFIM approach is first detailed for multi-body systems with flexible joints, and then with flexible members. Then, to reveal the merits of this new approach, few case studies are presented. A flexible inverted pendulum is studied first as a simple template for lucid comparisons, and next a space free-flying robotic system that contains a rigid main body equipped with two manipulating arms and two flexible solar panels is considered. Modelling verification of this complicated system is vigorously performed using ANSYS and ADAMS programs. The obtained results reveal the outcome accuracy of the new proposed approach for explicit dynamics modelling of rigid–flexible multi-body systems such as mobile robotic systems, while its limited computations provide an efficient tool for controller design, simulation studies and also practical implementations of model-based algorithms.     

Dynamic systems identification with Gaussian processes

This paper describes the identification of nonlinear dynamic systems with a Gaussian process (GP) prior model. This model is an example of the use of a probabilistic non-parametric modelling approach. GPs are flexible models capable of modelling complex nonlinear systems. Also, an attractive feature of this model is that the variance associated with the model response is readily obtained, and it can be used to highlight areas of the input space where prediction quality is poor, owing to the lack of data or complexity (high variance). We illustrate the GP modelling technique on a simulated example of a nonlinear system.     

Probabilistic verification of a biodiesel production system using statistical model checking

Biochemical system designers are increasingly using formal modelling, simulation, and verification methods to improve the understanding of complex systems. Probabilistic models can incorporate realistic stochastic dynamics, but creating and analysing probabilistic models in a formal way is challenging. In this work, we present a stochastic model of biodiesel production that incorporates an inexpensive test of fuel quality, and we validate the model using statistical model checking, which can be used to evaluate simple or complex temporal properties efficiently. We also describe probabilistic simulation and analysis techniques for stochastic hybrid system (SHS) models to demonstrate the properties of our model. We introduce a variety of properties for various configurations of the reactor as well as results of testing our model against the properties.     

Finite-time stabilization of nonlinear impulsive dynamical systems

Finite-time stability involves dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time. For continuous-time dynamical systems finite-time convergence implies nonuniqueness of system solutions in reverse time, and hence, such systems possess non-Lipschitzian dynamics. For impulsive dynamical systems, however, it may be possible to reset the system states to an equilibrium state achieving finite-time convergence without requiring non-Lipschitzian system dynamics. In this paper, we develop sufficient conditions for finite-time stability of impulsive dynamical systems using both scalar and vector Lyapunov functions. Furthermore, we design hybrid finite-time stabilizing controllers for impulsive dynamical systems that are robust against full modelling uncertainty. Finally, we present a numerical example for finite-time stabilization of large-scale impulsive dynamical systems.     

First steps toward formal controller synthesis for bipedal robots with experimental implementation

Bipedal robots are prime examples of complex cyber–physical systems (CPSs). They exhibit many of the features that make the design and verification of CPS so difficult: hybrid dynamics, large continuous dynamics in each mode (e.g., 10 or more state variables), and nontrivial specifications involving nonlinear constraints on the state variables. In this paper, we propose a two-step approach to formally synthesize controllers for bipedal robots so as to enforce specifications by design and thereby generate physically realizable stable walking. In the first step, we design outputs and classical controllers driving these outputs to zero. The resulting controlled system evolves on a lower dimensional manifold and is described by the hybrid zero dynamics governing the remaining degrees of freedom. In the second step, we construct an abstraction of the hybrid zero dynamics that is used to synthesize a controller enforcing the desired specifications to be satisfied on the full order model. Our two step approach is a systematic way to mitigate the curse of dimensionality that hampers the applicability of formal synthesis techniques to complex CPS. Our results are illustrated with simulations showing how the synthesized controller enforces all the desired specifications and offers improved performance with respect to a classical controller. The practical relevance of the results is illustrated experimentally on the bipedal robot AMBER 3.     

Hybrid decentralized maximum entropy control for large-scale dynamical systems

In the analysis of complex, large-scale dynamical systems it is often essential to decompose the overall dynamical system into a collection of interacting subsystems. Because of implementation constraints, cost, and reliability considerations, a decentralized controller architecture is often required for controlling large-scale interconnected dynamical systems. In this paper, a novel class of fixed-order, energy-based hybrid decentralized controllers is proposed as a means for achieving enhanced energy dissipation in large-scale lossless and dissipative dynamical systems. These dynamic decentralized controllers combine a logical switching architecture with continuous dynamics to guarantee that the system plant energy is strictly decreasing across switchings. The general framework leads to hybrid closed-loop systems described by impulsive differential equations. In addition, we construct hybrid dynamic controllers that guarantee that each subsystem–subcontroller pair of the hybrid closed-loop system is consistent with basic thermodynamic principles. Special cases of energy-based hybrid controllers involving state-dependent switching are described, and an illustrative combustion control example is given to demonstrate the efficacy of the proposed approach.     

Stability analysis for networked control systems with sampling,transmission protocols and input delays

In this paper, the stability problem is investigated for networked control systems. Input delays and multiple communication imperfections containing time-varying transmission intervals and transmission protocols are considered. A unified framework based on the hybrid systems with memory is proposed to model the whole networked control system. Hybrid systems with memory are used to model hybrid systems affected by delays and permit multiple jumps at a jumping instant. The stability analysis depends on the Lyapunov–Krasovskii functional approaches for hybrid systems with memory and the proposed stability theorem does not need strict decrease of the Lyapunov–Krasovskii functional during jumps. Based on the developed stability theorems, stability conditions for networked control systems are established. An explicit formula is given to compute the maximal allowable transmission interval. In the special case that the networked control system contains linear dynamics, an explicit Lyapunov functional is constructed and stability conditions in terms of linear matrix inequalities (LMI) are proposed. Finally, an example of a chemical batch reactor is given to illustrate the effectiveness of the proposed results.     

Fuzzy-hybrid modelling of an Ackerman steered electric vehicle

Physical system modelling with known parameters together with 2-D or high order look-up tables (obtained from experimental data), have been the preferred method for simulating electric vehicles. The non-linear phenomena which are present at the vehicle tyre patch and ground interface have resulted in a quantitative understanding of this phenomena. However, nowadays, there is a requirement for a deeper understanding of the vehicle sub-models which previously used look-up tables. In this paper the hybrid modelling methodology used for electric vehicle systems offers a two-stage advantage: firstly, the vehicle model retains a comprehensive analytical formulation and secondly, the ‘fuzzy’ element offers, in addition to the quantitative results, a qualitative understanding of specific vehicle sub-models. In the literature several hybrid topologies are reported, sequential, auxiliary, and embedded.In this paper, the hybrid model topology selected is auxiliary and within the same hybrid model, the first paradigm used is the vehicle dynamics together with the actuator/gearbox system. The second paradigm is the non-linear fuzzy tyre model for each wheel. In particular, conventional physical system dynamic modelling has been combined with the fuzzy logic type-II or type-III methodology. The resulting hybrid-fuzzy tyre models were estimated for a-priori number of rules from experimental data. The physical system modelling required the available vehicle parameters such as the overall mass, wheel radius and chassis dimensions. The suggested synergetic fusion of the two methods, (hybrid-fuzzy), allowed the vehicle planar trajectories to be obtained prior to the hardware development of the entire vehicle. The strength of this methodology is that it requires localised system experimental data rather than global system data. The disadvantage in obtaining global experimental data is the requirement for comprehensive testing of a vehicle prototype which is both time consuming process and requires extensive resources. In this paper the authors have proposed the use of existing experimental rigs which are available from the leading automotive manufacturers. Hence, for the ‘hybrid’ modelling, localised data sets were used. In particular, wheel-tyre experimental data were obtained from the University tyre rig experimental facilities. Tyre forces acting on the tyre patch are mainly responsible for the overall electric vehicle motion. In addition, tyre measurement rigs are a well known method for obtaining localised data thus allowing the effective simulation of more detailed mathematical models. These include, firstly, physical system modelling (conventional vehicle dynamics), secondly, fuzzy type II or III modelling (for the tyre characteristics), and thirdly, electric drive modelling within the context of electric vehicles. The proposed hybrid model synthesis has resulted in simulation results which are similar to piece-wise ‘look-up’ table solutions. In addition, the strength of the ‘hybrid’ synthesis is that the analyst has a set of rules which clearly show the reasoning behind the complex development of the vehicle tyre forces. This is due to the inherent transparency of the type II and type III methodologies. Finally, the authors discussed the reasons for selecting a type-III framework. The paper concludes with a plethora of simulation results.     

Preconditioned Dynamic Iteration for Coupled Differential-Algebraic Systems

The network approach to the modelling of complex technical systems results frequently in a set of differential-algebraic systems that are connected by coupling conditions. A common approach to the numerical solution of such coupled problems is based on the coupling of standard time integration methods for the subsystems. As a unified framework for the convergence analysis of such multi-rate, multi-method or dynamic iteration approaches we study in the present paper the convergence of a dynamic iteration method with a (small) finite number of iteration steps in each window. Preconditioning is used to guarantee stability of the coupled numerical methods. The theoretical results are applied to quasilinear problems from electrical circuit simulation and to index-3 systems arising in multibody dynamics.     

A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

In this work, the issue of estimation of reachable sets in continuous bimodal piecewise affine systems is studied. A new method is proposed, in the framework of ellipsoidal bounding, using piecewise quadratic Lyapunov functions. Although bimodal piecewise affine systems can be seen as a special class of affine hybrid systems, reachability methods developed for affine hybrid systems might be inappropriately complex for bimodal dynamics. This work goes in the direction of exploiting the dynamical structure of the system to propose a simpler approach. More specifically, because of the piecewise nature of the Lyapunov function, we first derive conditions to ensure that a given quadratic function is positive on half spaces. Then, we exploit the property of bimodal piecewise quadratic functions being continuous on a given hyperplane. Finally, linear matrix characterizations of the estimate of the reachable set are derived.     

Simulation in manufacturing and business: A review

This paper reports the results of a review of simulation applications published within peer-reviewed literature between 1997 and 2006 to provide an up-to-date picture of the role of simulation techniques within manufacturing and business. The review is characterised by three factors: wide coverage, broad scope of the simulation techniques, and a focus on real-world applications. A structured methodology was followed to narrow down the search from around 20,000 papers to 281. Results include interesting trends and patterns. For instance, although discrete event simulation is the most popular technique, it has lower stakeholder engagement than other techniques, such as system dynamics or gaming. This is highly correlated with modelling lead time and purpose. Considering application areas, modelling is mostly used in scheduling. Finally, this review shows an increasing interest in hybrid modelling as an approach to cope with complex enterprise-wide systems.     

Dynamics modelling and Hybrid Suppression Control of space robots performing cooperative object manipulation

Dynamics modelling and control of multi-body space robotic systems composed of rigid and flexible elements is elaborated here. Control of such systems is highly complicated due to severe under-actuated condition caused by flexible elements, and an inherent uneven nonlinear dynamics. Therefore, developing a compact dynamics model with the requirement of limited computations is extremely useful for controller design, also to develop simulation studies in support of design improvement, and finally for practical implementations. In this paper, the Rigid–Flexible Interactive dynamics Modelling (RFIM) approach is introduced as a combination of Lagrange and Newton–Euler methods, in which the motion equations of rigid and flexible members are separately developed in an explicit closed form. These equations are then assembled and solved simultaneously at each time step by considering the mutual interaction and constraint forces. The proposed approach yields a compact model rather than common accumulation approach that leads to a massive set of equations in which the dynamics of flexible elements is united with the dynamics equations of rigid members. To reveal such merits of this new approach, a Hybrid Suppression Control (HSC) for a cooperative object manipulation task will be proposed, and applied to usual space systems. A Wheeled Mobile Robotic (WMR) system with flexible appendages as a typical space rover is considered which contains a rigid main body equipped with two manipulating arms and two flexible solar panels, and next a Space Free Flying Robotic system (SFFR) with flexible members is studied. Modelling verification of these complicated systems is vigorously performed using ANSYS and ADAMS programs, while the limited computations of RFIM approach provides an efficient tool for the proposed controller design. Furthermore, it will be shown that the vibrations of the flexible solar panels results in disturbing forces on the base which may produce undesirable errors and perturb the object manipulation task. So, it is shown that these effects can be significantly eliminated by the proposed Hybrid Suppression Control algorithm.     

ISSUE INFORMATION

Modeling social‐ecological systems is difficult due to the complexity of ecosystems and of individual and collective human behavior. Key components of the social‐ecological system are often over‐simplified or omitted. Generalized modeling is a dynamical systems approach that can overcome some of these challenges. It can rigorously analyze qualitative system dynamics such as regime shifts despite incomplete knowledge of the model's constituent processes. Here, we review generalized modeling and use a recent study on the Baltic Sea cod fishery's boom and collapse to demonstrate its application to modeling the dynamics of empirical social‐ecological systems. These empirical applications demand new methods of analysis suited to larger, more complicated generalized models. Generalized modeling is a promising tool for rapidly developing mathematically rigorous, process‐based understanding of a social‐ecological system's dynamics despite limited knowledge of the system.     

Predicting high-codimension critical transitions in dynamical systems using active learning

Complex dynamical systems, from those appearing in physiology and ecology to Earth system modelling, often experience critical transitions in their behaviour due to potentially minute changes in their parameters. While the focus of much recent work, predicting such bifurcations is still notoriously difficult. We propose an active learning approach to the classification of parameter space of dynamical systems for which the codimension of bifurcations is high. Using elementary notions regarding the dynamics, in combination with the nearest-neighbour algorithm and Conley index theory to classify the dynamics at a predefined scale, we are able to predict with high accuracy the boundaries between regions in parameter space that produce critical transitions.     

Interaction of intelligent and autonomous systems – part II: realization of cognitive technical systems

In this study, the application of a meta-modelling technique to structure complex environments is demonstrated with two examples. A special situation-operator model developed to model the human-machine-interaction is applied to automated supervision within the HMI context and to realize flexible and situated interaction of autonomous systems. A concept to automated monitoring of human operators is introduced to illustrate the structuring of complex environments as the first example. In the second example, this modelling technique is applied to a mobile robot to autonomously build and update a mental model of the interaction with the environment as an example of a cognitive technical system.     

A method for reduction of human ventricular action potential model

ABSTRACT

Mathematical modelling and computer simulations are important tools in the field of cardiac electrophysiology. High computational costs of complex models make them difficult to apply in large-scale simulations like tissue. Therefore, model reduction are of particular importance in heart studies. In this paper, we introduce a technique for simplification of ventricular cell(VC) complex models. By using this technique, starting with a complex model of human VC including 17state variables, we reduce the number of state variables to two. Our simplified model is compared with the original one via several electrophysiological features and computational efficiency. Results show that the reduced model has acceptable behaviours in single cell and one-dimensional simulation, moreover, is 55 times faster than the original one. As the presented method does not depend on the reference model, it may be applied to every cardiac cell models or each complex excitable dynamical systems with the same dynamics as VC.     

Local expansion concepts for detecting transport barriers in dynamical systems

In the last two decades, the mathematical analysis of material transport has received considerable interest in many scientific fields such as ocean dynamics and astrodynamics. In this contribution we focus on the numerical detection and approximation of transport barriers in dynamical systems. Starting from a set-oriented approximation of the dynamics we combine discrete concepts from graph theory with established geometric ideas from dynamical systems theory. We derive the global transport barriers by computing the local expansion properties of the system. For the demonstration of our results we consider two different systems. First we explore a simple flow map inspired by the dynamics of the global ocean. The second example is the planar circular restricted three body problem with Sun and Jupiter as primaries, which allows us to analyze particle transport in the solar system.