Stefan Gering

Synthesis of local state feedback for continuous-time recurrent fuzzy systems

A synthesis method for global stabilization of equilibria in continuous-time recurrent fuzzy systems is presented. By exploiting the fact that in recurrent fuzzy systems, the state derivative is an interpolation of gradients at corners of hypersquares in the input state space, our approach is based on directly influencing these gradients such that the controlled system is asymptotically stable. In addition, it is shown that after obtaining a feasible solution, the resulting controller can be optimized towards higher performance. Besides the clarity of the synthesis process, the approach offers the advantage of an interpretation of the resulting controller as static fuzzy system. Thus, the resulting controller can be linguistically interpreted. The synthesis method is illustrated by two examples.

Sum of squares approaches for control of continuous-time recurrent fuzzy systems

In recent years, a concise theory of recurrent fuzzy systems has emerged and methods for utilizing these fuzzy systems with dynamics for modeling and fault detection were developed. At the same time, sum of squares decompositions in conjunction with semidefinite programming were successfully applied for the synthesis of controllers for polynomial systems. In this paper, we combine both approaches and present sum of squares based control strategies for continuous-time recurrent fuzzy systems. The system dynamics under consideration is defined by gradients at discrete points in the input-state space. An alternative description as piecewise polynomial system is possible. This motivates to utilize a controller switching between local polynomial control laws. We propose three different approaches for controller synthesis based on this idea and demonstrate this new synthesis method by means of an example. In addition, advantages and drawbacks of the approaches are discussed.

Fuzzy control of continuous-time recurrent fuzzy systems

Abstract This article presents a synthesis method for stabilization of equilibria in continuous-time recurrent fuzzy system. It is first shown how to obtain a hybrid system representation from the given linguistic differential equations defining the recurrent fuzzy system. Based on this, the controller synthesis algorithm presented is then a two step procedure. By means of the synthesis step, a fuzzy controller is obtained by solving a single optimization problem, which guarantees that no chattering effects will occur in the controlled recurrent fuzzy system. Stability of the equilibrium is then proven by a verification procedure utilizing sum of squares methods. A performance optimization by means of additional polynomial controllers is discussed. The main advantage of our method relies in the computational efficiency of the synthesis problem as well as in the linguistic interpretability of the resulting controller. In addition, chattering effects are proven not to occur.

A Piecewise Approximation Approach to Nonlinear Systems: Stability and Region of Attraction

This paper discusses piecewise bilinear models and recurrent fuzzy systems as particular classes of dynamic fuzzy systems, which can be dealt with in the same framework due to their structural similarity. As universal approximators, they are capable of representing any continuous nonlinear system dynamics with arbitrary accuracy by means of a rule base. First, basic definitions of both systems are revisited here, and unifying canonical forms are given. Then, a stability criterion is derived for the system classes, which is based on approximating piecewise quadratic Lyapunov functions and formulated in terms of linear matrix inequalities. The main contribution is the demonstration of the stability analysis and estimation of region of attraction by means of these models, outperforming methods reported in the literature.

Robust stabilization of recurrent fuzzy systems via switching control

A method for stabilization of known equilibria in recurrent fuzzy systems is presented, which particularly accounts for model uncertainties. Since the dynamics of recurrent fuzzy systems are defined over a rectangular grid, it is first observed that for stability analysis, only gradient conditions at grid points have to be considered given that the inputs are piecewise constant. Therefore, a robust structure variable controller is proposed, switching between constant inputs. In order to prevent the system from deadlock phenomena due to the switching of the system, the structure variable control is augmented by a piecewise polynomial controller, guaranteeing asymptotic stability. The proposed method is applied to the example of an inverted pendulum.

Decentralized Bayesian consensus over networks

This paper deals with networked, dynamical multi-agent systems (MAS) trying to reach consensus about their states subject to uncertain data transfer and noisy measurements. For this, an analogy between the deterministic consensus protocol and a Gaussian process is established. First, the consensus problem is modeled as a stochastic process to consider uncertain initial states and noisy information flow over the network. Next, necessary conditions for decentral inference are derived, two decentral approximative inference protocols are developed and the dependency between communication density and approximation error is presented. Furthermore, a provably convergent and computationally efficient Gaussian consensus protocol is realized. Finally, it is shown that taking measurement noise into account the Gaussian consensus protocol naturally extends to a decentralized Kalman filter for consensus systems.

Rule predictive control and model predictive control strategies for Recurrent Fuzzy Systems

Recurrent Fuzzy Systems allow for an approximate modeling of system dynamics based on expert knowledge or measured data. In this paper, the applicability of model predictive control strategies for control of these dynamic fuzzy systems is considered. It is shown that each of the different forms for representation of the system dynamics leads to a specific model predictive control strategy. The main result is the proposition of an explicit model predictive control strategy based on the rule base representation, outperforming existing control strategies in terms of online computation time. The performance of all control strategies is also illustrated and compared by means of a bio reactor example.

Feedforward Tracking Control of Flat Recurrent Fuzzy Systems

Flatness based feedforward control has proven to be a feasible solution for the problem of tracking control, which may be applied to a broad class of nonlinear systems. If a flat output of the system is known, the control is often based on a feedforward controller generating a nominal input in combination with a linear controller stabilizing the linearized error dynamics around the trajectory. We show in this paper that the very same idea may be incorporated for tracking control of MIMO recurrent fuzzy systems. Their dynamics is given by means of linguistic differential equations but may be converted into a hybrid system representation, which then serves as the basis for controller synthesis.

Stabilization of Recurrent Fuzzy Systems via Sum of Squares-based Hybrid Control

Abstract This paper presents an approach for stabilization of equilibria in recurrent fuzzy systems. This type of dynamic fuzzy systems being defined via linguistic rules can be interpreted as interpolation between constant gradients, and therefore as hybrid dynamical system. It is shown that the latter viewpoint allows for a precise description of the system dynamics, but on the other hand lacks transparency. In order to render a given equilibrium of the recurrent fuzzy system globally asymptotically stable, local polynomial controllers are computed via sum of squares optimization to allow only for deterministic mode transitions on a micro level. In addition, the controlled recurrent fuzzy system can then be interpreted as finite deterministic automaton, thus allowing for analysis of system properties on a more abstract macro level. Relaxations are proposed in cases where recurrent fuzzy systems may not be rendered deterministic and the method is applied to two examples.

Synthese von Zustands- und Ausgangsrückführungen für rekurrente Fuzzy-Systeme

Zusammenfassung Als spezielle Klasse dynamischer Fuzzy-Systeme bieten rekurrente Fuzzy-Systeme die Möglichkeit, dynamische Prozesse anhand von Expertenwissen oder Messdaten zu modellieren. Dabei zeichnet sich die Regelbasis durch die Möglichkeit der linguistischen Interpretierbarkeit und somit der Transparenz aus. Dieser Artikel stellt Ansätze vor, mit denen sowohl (beobachterbasierte) Zustands- als auch Ausgangsrückführungen zur Stabilisierung bekannter Ruhelagen mit Hilfe bilinearer Matrixungleichungen ausgelegt werden können. Die sich ergebenden Regler sind dabei strukturell äquivalent zu Fuzzy-Reglern und können somit ebenfalls linguistisch interpretiert werden. Hinsichtlich der Ausgangsrückführungen wird gezeigt, dass sich bekannte Syntheseansätze aus der linearen Systemtheorie ebenfalls auf Ausgangsrückführungen für rekurrente Fuzzy-Systeme übertragen lassen.