Zsófia Lendek

Observer design for Takagi-Sugeno descriptor models

This paper considers observer design for nonlinear descriptor systems. We propose approaches based on Takagi-Sugeno (TS) models. An extended estimation vector is used in order to keep the descriptor structure of the observer. The design conditions are expressed as linear matrix inequalities. The proposed observer structure, via an intermediate variable as estimated variable, is able to recover the previous observer results for TS descriptors. Moreover, through a direct extension via so-called Finslers Lemma, relaxed conditions are obtained. Numerical examples show the effectiveness of the proposed approaches.

Stability of Cascaded Fuzzy Systems and Observers

A large class of nonlinear systems can be well approximated by Takagi-Sugeno (TS) fuzzy models with linear or affine consequents. It is well known that the stability of these consequent models does not ensure the stability of the overall fuzzy system. Therefore, several stability conditions have been developed for TS fuzzy systems. We study a special class of nonlinear dynamic systems that can be decomposed into cascaded subsystems, which are represented as TS fuzzy models. We analyze the stability of the overall TS system based on the stability of the subsystems and prove that the stability of the subsystems implies the stability of the overall system. The main benefit of this approach is that it relaxes the conditions imposed when the system is globally analyzed, thereby solving some of the feasibility problems. Another benefit is that by using this approach, the dimension of the associated linear matrix inequality (LMI) problem can be reduced. For naturally distributed applications, such as multiagent systems, the construction and tuning of a centralized observer may not be feasible. Therefore, we also extend the cascaded approach to the observer design and use fuzzy observers to individually estimate the states of these subsystems. A theoretical proof of stability and simulation examples are presented. The results show that the distributed observer achieves the same performance as the centralized one, while leading to increased modularity, reduced complexity, lower computational costs, and easier tuning. Applications of such cascaded systems include multiagent systems, distributed process control, and hierarchical large-scale systems.

Controller Design for TS Models Using Delayed Nonquadratic Lyapunov Functions

In the last few years, nonquadratic Lyapunov functions have been more and more frequently used in the analysis and controller design for Takagi-Sugeno fuzzy models. In this paper, we developed relaxed conditions for controller design using nonquadratic Lyapunov functions and delayed controllers and give a general framework for the use of such Lyapunov functions. The two controller design methods developed in this framework outperform and generalize current state-of-the-art methods. The proposed methods are extended to robust and H∞ control and α-sample variation.

Parametric Bayesian Filters for Nonlinear Stochastic Dynamical Systems: A Survey

Nonlinear stochastic dynamical systems are commonly used to model physical processes. For linear and Gaussian systems, the Kalman filter is optimal in minimum mean squared error sense. However, for nonlinear or non-Gaussian systems, the estimation of states or parameters is a challenging problem. Furthermore, it is often required to process data online. Therefore, apart from being accurate, the feasible estimation algorithm also needs to be fast. In this paper, we review Bayesian filters that possess the aforementioned properties. Each filter is presented in an easy way to implement algorithmic form. We focus on parametric methods, among which we distinguish three types of filters: filters based on analytical approximations (extended Kalman filter, iterated extended Kalman filter), filters based on statistical approximations (unscented Kalman filter, central difference filter, Gauss-Hermite filter), and filters based on the Gaussian sum approximation (Gaussian sum filter). We discuss each of these filters, and compare them with illustrative examples.

Distributed Kalman filtering for cascaded systems

The Kalman filter provides an efficient means to estimate the state of a linear process, so that it minimizes the mean of the squared estimation error. However, for naturally distributed applications, the construction and tuning of a centralized observer may present difficulties. Therefore, we propose the decomposition of a linear process model into a cascade of simpler subsystems and the use of a Kalman filter to individually estimate the states of these subsystems. Both a theoretical comparison and simulation examples are presented. The theoretical results show that the distributed observers, except for special cases, do not minimize the overall error covariance, and the distributed observer system is therefore suboptimal. However, in practice, the performance achieved by the cascaded observers is comparable and in certain cases even better than the performance of the centralized observer. A distributed observer system also leads to increased modularity, reduced complexity, and lower computational costs.

Application of Takagi-Sugeno observers for state estimation in a quadrotor

In this paper, the validity of Takagi-Sugeno observers to estimate the angular positions and speeds in the experimental platform of a quadrotor will be assessed. Takagi-Sugeno observers are compared to observers based on the linearized model designed with the same optimization criteria and design parameters. Experimental results confirm that Takagi-Sugeno models and observers behave similarly to linear ones around the linearization point, and have a better performance over a larger operating range.

Online self-evolving fuzzy controller with global learning capabilities

This paper presents an online self-evolving fuzzy controller with global learning capabilities. Starting from very simple or even empty configurations, the controller learns from its own actions while controlling the plant. It applies learning techniques based on the input/output data collected during normal operation to modify online the fuzzy controller’s structure and parameters. The controller does not need any information about the differential equations that govern the plant, nor any offline training. It consists of two main blocks: a parameter learning block that learns proper values for the rule consequents applying a local and a global strategy, and a self-evolving block that modifies the controller’s structure online. The modification of the topology is based on the analysis of the error surface and the determination of the input variables which are most responsible for the error. Simulation and experimental results are presented to show the controller’s capabilities.

On non-PDC local observers for TS fuzzy systems

In this paper we propose a method to design local observers for Takagi-Sugeno fuzzy models obtained from nonlinear systems by the sector nonlinearity approach. When a global observer cannot be designed, using our method it is still possible to design observers that are valid in a well-defined region of the state-space. The design is based on a nonquadratic Lyapunov function. Depending on whether or not the scheduling vector is a function of the states to be estimated, the conditions are formulated as an LMI or a BMI problem, respectively. The results are illustrated on simulation examples, for which classical observer design conditions are unfeasible.

Improvements on non-quadratic stabilization of continuous-time Takagi-Sugeno descriptor models

This paper presents a relaxed approach for stabilization and H∞ disturbance rejection of continuous-time Takagi-Sugeno models in descriptor form. Based on Finslers Lemma, the control law can be conveniently decoupled from a non-quadratic Lyapunov function. These developments include and outperform previous results on the same subject while preserving the advantage of being expressed as linear matrix inequalities. Two examples are presented to illustrate the improvements.

Periodic Lyapunov functions for periodic TS systems

Abstract In this paper we consider stability analysis and controller design for periodic Takagi–Sugeno fuzzy models. To develop the conditions, we use a switching nonquadratic Lyapunov function defined at the time instants when the subsystems switch. Using the proposed conditions we are able to handle periodic Takagi–Sugeno systems where the local models or even the subsystems are unstable or cannot be stabilized. The application of the conditions is illustrated on numerical examples.