# CTDE 2018 Abstracts

Full Papers

Paper Nr: | 1 |

Title: | ## State- and Uncertainty-observers-based Controller for a Class of T-S Fuzzy Models |

Authors: | ## Hugang Han, Daisuke Hamasaki and Jiabao Fu |

Abstract: | A control system design based on the T-S fuzzy model with uncertainty is considered in this paper. At first, state observer is developed to estimate the state properly despite the existence of the uncertainty. Then, uncertainty observer is derived using the estimated state. Finally, a controller based on the observers is proposed in an effort to counteract the influence of the uncertainty whatever possible. In addition, the Nussbaum-type function and its relevant properties are used in the controller design to cover the observers’ error and the part of estimated uncertainty that is not possibly used through the control matrices. As a result, the closed-loop control system becomes asymptotically stable. |

Paper Nr: | 5 |

Title: | ## Construction of a Complete Lyapunov Function using Quadratic Programming |

Authors: | ## Peter Giesl, Carlos Argáez, Sigurdur Hafstein and Holger Wendland |

Abstract: | A complete Lyapunov function characterizes the behaviour of a general dynamical system. In particular, the state space is split into the chain-recurrent set, where the function is constant, and the part characterizing the gradient-like flow, where the function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov function provide information about the stability of connected components of the chain-recurrent set and the basin of attraction of attractors therein. In a previous method, a complete Lyapunov function was constructed by approximating the solution of the PDE V0(x) = −1, where 0 denotes the orbital derivative, by meshfree collocation. We propose a new method to compute a complete Lyapunov function: we only fix the orbital derivative V0(x0) = −1 at one point, impose the constraints V0(x) ≤ 0 for all other collocation points and minimize the corresponding reproducing kernel Hilbert space norm. We show that the problem has a unique solution which can be computed as the solution of a quadratic programming problem. The new method is applied to examples which show an improvement compared to previous methods. |

Paper Nr: | 6 |

Title: | ## Combination of Refinement and Verification for the Construction of Lyapunov Functions using Radial Basis Functions |

Authors: | ## Peter Giesl and Najla Mohammed |

Abstract: | Lyapunov functions are an important tool for the determination of the domain of attraction of an equilibrium point of a given ordinary differential equation. The Radial Basis Functions collocation method is one of the numerical methods to construct Lyapunov functions. This method has been improved by combining it with a refinement algorithm to reduce the number of collocation points required in the construction process, as well as a verification that the constructed function is a Lyapunov function. In this paper, we propose a combination of both methods in one, called the combination method. This method constructs a Lyapunov function with the refinement algorithm and then verifies its properties rigorously. The method is illustrated with examples. |

Paper Nr: | 7 |

Title: | ## Local Lyapunov Functions for Nonlinear Stochastic Differential Equations by Linearization |

Authors: | ## Hjörtur Björnsson, Peter Giesl, Skuli Gudmundsson and Sigurdur Hafstein |

Abstract: | We present a rigid estimate of the domain, on which a Lyapunov function for the linearization of a nonlinear stochastic differential equation is a Lyapunov function for the original system. By using this estimate the demanding task of computing a lower bound on the γ-basin of attraction for a nonlinear stochastic systems is greatly simplified and the application of a resent numerical method for the same purpose facilitated. |

Paper Nr: | 8 |

Title: | ## Verification of a Numerical Solution to a Collocation Problem |

Authors: | ## Hjortur Bjornsson and Sigurdur Hafstein |

Abstract: | In a recent method to compute Lyapunov functions for nonlinear stochastic differential equations a subsequent verification of the results is needed. The theory has been developed but there are several practical difficulties in its implementation because of the huge amount of function evaluations needed during verification. We study several different methods and compare their accuracy and efficiency. |