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This brief presents the design and implementation of observer design strategies for experimental non-smooth continuous and discontinuous systems. First, a piece-wise linear observer is implemented for an experimental setup consisting of a harmonically excited flexible steel beam with a one-sided support which can be considered as a benchmark for a class of flexible mechanical systems with one-sided restoring characteristics. Second, an observer is developed for an experimental setup that describes a dynamic rotor system which is a benchmark for motion systems with friction and flexibility. In both cases, the implemented observers guarantee global asymptotic stability of the estimation error dynamic in theory. Simulation and experimental results are presented to demonstrate the performance of the observers in practice. These results support the use of (switched) observers to achieve state reconstruction for such non-smooth and discontinuous mechanical systems.

This tutorial paper is concerned with the identification of hybrid models, i.e. dynamical models whose behavior is determined by interacting continuous and discrete dynamics. Methods specifically aimed at the identification of models with a hybrid structure are of very recent date. After discussing the main issues and difficulties connected with hybrid system identification, and giving an overview of the related literature, this paper focuses on four different approaches for the identification of switched affine and piecewise affine models, namely an algebraic procedure, a Bayesian procedure, a clustering-based procedure, and a bounded-error procedure. The main features of the selected procedures are presented, and possible interactions to still enhance their effectiveness are suggested.

Disentangling the complexity of biochemical networks requires knowledge about the quantitative relationships between the individual components. However, the nonlinear dynamics of biochemical processes are difficult to identify with traditional identification methods. We therefore propose to model the complex nonlinear biochemical processes with several simpler systems (modes), together with a switching rule that determines which mode is active, i.e. with a hybrid system. We consider the example of a nonlinear biochemical oscillator, and propose a simple piecewise affine (PWA) approximation. Qualitative analysis shows that the PWA model is able to capture the dynamics of the nonlinear oscillator. Hybrid identification procedure is subsequently applied to identify the parameters of the PWA model.

Book abstract: Setting out core theory and reviewing a range of new methods, theoretical problems and applications, this handbook shows how hybrid dynamical systems can be modelled and understood. 60 expert authors involved in the recent research activities and industrial application studies provide practical insights on topics ranging from the theoretical investigations over computer-aided design to applications in energy management and the process industry. Structured into three parts, the book opens with a thorough introduction to hybrid systems theory, illustrating new dynamical phenomena through numerous examples. Part II then provides a survey of key tools and tool integration activities. Finally, Part III is dedicated to applications, implementation issues and system integration, considering different domains such as industrial control, automotive systems and digital networks. Three running examples are referred to throughout the book, together with numerous illustrations, helping both researchers and industry professionals to understand complex theory, recognise problems and find appropriate solutions.

So far, the processes involved in regulation of intracellular calcium (Ca/sub i//sup 2+/) in cardiomyocytes have been mainly studied through biochemical and isolated cell analysis. Here, we present a novel technique to model and identify cardiac Ca/sub i//sup 2+/-cycling under physiologically relevant conditions in the intact beating heart. Ca/sub i//sup 2+/ was measured using fluorescence techniques in ex vivo perfused rat hearts. For analysis, we developed a parametric mathematical model, switching between active and inactive calcium release. The kinetic parameters of the two submodes of the model were computed using a recently developed technique from hybrid system identification. Application of the method to control and isoproterenol-stimulated hearts resulted in parameter values within a physiologically reliable range.

In this paper we present an output feedback controller design method for a class of bi-modal piecewise linear systems. The proposed output feedback controller consists of a switching state observer and a static state feedback. The observer and the controller are designed separately using the concept of input-to-state stability (ISS). A sufficient small gain condition is derived that guarantees global asymptotic stability of the system in closed loop with the designed observer and state feedback. The derived theory is illustrated by an example.

This paper proves the formal equivalence between the class of switched affine systems and the class of switched autoregressive systems with exogenous inputs. These classes of hybrid dynamical systems have been studied recently for various purposes. It is shown that any observable switched affine system admits a representation as a switched autoregressive system and every switched autoregressive system admits a representation as a switched affine system.

In this paper, we present a novel procedure for the identification of hybrid systems in the class of piecewise ARX systems. The presented method facilitates the use of available a priori knowledge on the system to be identified, but can also be used as a black-box method. We treat the unknown parameters as random variables, described by their probability density functions. The identification problem is posed as the problem of computing the a posteriori probability density function of the model parameters, and subsequently relaxed until a practically implementable method is obtained. A particle filtering method is used for a numerical implementation of the proposed procedure. A modified version of the multicategory robust linear programming classification procedure, which uses the information derived in the previous steps of the identification algorithm, is used for estimating the partition of the piecewise ARX map. The proposed procedure is applied for the identification of a component placement process in pick-and-place machines.

In many practical situations, the outputs of a plant are not measured exactly, but are corrupted by quantization errors. Often the effect of the quantization error is neglected in the control design phase, which can lead to undesirable effects like limit cycles and even chaotic behavior once the controller has been implemented. In this paper we present a method based on l/sub 1/ optimal control that minimizes the amplitude of the oscillations in the to-be-controlled variables. Analytical and numerical examples illustrate the elegance of the l/sub 1/-theory in this setting.

This paper deals with a radar track before detect application in a multi target setting. Track before detect is a method to track weak objects (targets) on the basis raw radar measurements, e.g. the reflected target power. In classical target tracking, the tracking process is performed on the basis of pre-processed measurements, that are constructed from the original measurement data every time step. In this way no integration over time takes place and information is lost. In this paper we will give a modelling setup and a particle filter based algorithm to deal with a multiple target track before detect situation. In simulations we show that, using this method, it is possible to track multiple, closely spaced, (weak) targets.

Identification of hybrid systems is complex because it is not a priori known which data is generated by which discrete mode of the system. A novel procedure is presented for the simultaneous determination of the model parameters and classification of the data. Initial parameter estimates are obtained based on a priori knowledge of the modes of the system. Next, the data points are subsequently classified and with each new classified data point the parameter estimates are refined. The proposed procedure is applied in the estimation of parameters in a hybrid description of calcium cycling in the intact heart.

—In this paper we present an experimental observer design for a rotor system with friction. The model of the system exhibits set-valued static friction law with the Stribeck effect. As the model of the setup is non-smooth and non-Lipschitz, observer has to be designed using mathematical tools from convex analysis and the theory of differential inclusions. The designed observer guarantees that there exists a unique solutionto the observer dynamics and that the estimated state converges to the true state of the system. Simulation and experimental results illustrate the design and performance of the observer in practice.