New developments in mathematical control and information for fuzzy systems

Hamid Reza Karimi, Mohammed Chadli and Peng Sh

Parallel Distributed Compensation for Piecewise Bilinear Models and Recurrent Fuzzy Systems Based on Piecewise Quadratic Lyapunov Functions

Piecewise Bilinear Models and Recurrent Fuzzy Systems are universal approximators for any smooth nonlinear dynamics. One of their advantage is the efficient representation of the modeled system dynamics by means of rule-bases or look-up-tables. In this paper, it is shown how to obtain provably stabilizing controllers by means of piecewise quadratic Lyapunov functions. Interpolating controllers with affine local controllers are considered for interpolation, akin to the concept of parallel distributed compensation widely used for control of Takagi-Sugeno systems

Stability Constraints of Markov State Kinetic Models Based on Routh- Hurwitz Criterion

In computational neuroscience, receptors, channels and more generally signaling pathways are often modeled with Markov state models to represent biochemical reactions, which are then implemented with bilinear equations. One of the goals of these models, once calibrated with experimental results is to predict the dynamics of the biological system they represent in response to molecular perturbations and therefore facilitate and enhance the success rate of drug discovery and development. To model receptors under both pathological and physiological conditions, modelers usually modify the ligand association and dissociation parameters in the kinetic model during the optimization phase of model development. However, some parameter values may lead to unstable models, making calibration very difficult, time-consuming and inefficient before performing predictive in silico studies. In order to guarantee model stability during the parameter optimization phase, we propose to linearize bilinear kinetic models around an operating point. Considering the model input as piecewise constant, we propose an algorithm based on the Routh-Hurwitz criterion to generate stability constraints on model parameters. As an example, we apply this algorithm to the gamma-aminobutyric acid (GABA) receptor subtype A (GABAA receptor) model, as developed by Pugh and Raman (2005). The results obtained with the Routh-Hurwitz criterion provide constraint equations. These equations, once integrated into the parameter optimization process, guarantee the stability of the model and thus the success of the optimization process. An additional benefit is that the constraint equations allow determining the boundaries of the stability domain of the model. In the example provided, the Routh-Hurwitz criterion indicates that the model with the chosen parameters becomes unstable if GABA concentration rises above 6.54 mM. The proposed algorithm has also the advantage of being fast and easy to implement

Robust Control

The need to be tolerant to changes in the control systems or in the operational environment of systems subject to unknown disturbances has generated new control methods that are able to deal with the non-parametrized disturbances of systems, without adapting itself to the system uncertainty but rather providing stability in the presence of errors bound in a model. With this approach in mind and with the intention to exemplify robust control applications, this book includes selected chapters that describe models of H-infinity loop, robust stability and uncertainty, among others. Each robust control method and model discussed in this book is illustrated by a relevant example that serves as an overview of the theoretical and practical method in robust control

A review of convex approaches for control, observation and safety of linear parameter varying and Takagi-Sugeno systems

This paper provides a review about the concept of convex systems based on Takagi-Sugeno, linear parameter varying (LPV) and quasi-LPV modeling. These paradigms are capable of hiding the nonlinearities by means of an equivalent description which uses a set of linear models interpolated by appropriately defined weighing functions. Convex systems have become very popular since they allow applying extended linear techniques based on linear matrix inequalities (LMIs) to complex nonlinear systems. This survey aims at providing the reader with a significant overview of the existing LMI-based techniques for convex systems in the fields of control, observation and safety. Firstly, a detailed review of stability, feedback, tracking and model predictive control (MPC) convex controllers is considered. Secondly, the problem of state estimation is addressed through the design of proportional, proportional-integral, unknown input and descriptor observers. Finally, safety of convex systems is discussed by describing popular techniques for fault diagnosis and fault tolerant control (FTC).Peer ReviewedPostprint (published version