(S,N)-implications on bounded lattices

Since the birth of the fuzzy sets theory several extensions have been proposed. For these extensions, different sets of membership functions were considered. Since fuzzy connectives, such as conjunctions, negations and implications, play an important role in the theory and applications of fuzzy logics, these connectives have also been extended. An extension of fuzzy logic, which generalizes the ones considered up to the present, was proposed by Joseph Goguen in 1967. In this extension, the membership values are drawn from arbitrary bounded lattices. The simplest and best studied class of fuzzy implications is the class of (S,N)-implications, and in this chapter we provide an extension of (S,N)-implications in the context of bounded lattice valued fuzzy logic, and we show that several properties of this class are preserved in this more general framework.

Minimizing impact of bounded uncertainty on McNaughton's scheduling algorithm via interval programming

Uncertainty of data affects decision making process as it increases the risk and the costs of the decision. One of the challenges in minimizing the impact of the bounded uncertainty on any scheduling algorithm is the lack of information, as only the upper bound and the lower bound are provided without any known probability or membership function. On the contrary, probabilistic uncertainty can use probability distributions and fuzzy uncertainty can use the membership function. McNaughton's algorithm is used to find the optimum schedule that minimizes the makespan taking into consideration the preemption of tasks. The challenge here is the bounded inaccuracy of the input parameters for the algorithm, namely known as bounded uncertain data. This research uses interval programming to minimise the impact of bounded uncertainty of input parameters on McNaughton’s algorithm, it minimises the uncertainty of the cost function estimate and increase its optimality. This research is based on the hypothesis that doing the calculations on interval values then approximate the end result will produce more accurate results than approximating each interval input then doing numerical calculations.

Design of a norm-bounded LQG controller for power distribution networks with distributed generation

This paper presents a novel excitation control design to improve the voltage profile of power distribution networks with distributed generation and induction motor loads. The system is linearised by perturbation technique. Controller is designed using the linear-quadratic-Gaussian (LQG) controller synthesis method. The LQG controller is addressed with norm-bounded uncertainty. The approach considered in this paper is to find the smallest upper bound on the H∞ norm of the uncertain system and to design an optimal controller based on this bound. The design method requires the solution of a linear matrix inequality. The performance of the controller is tested on a benchmark power distribution system. Simulation results show that the proposed controller provides impressive oscillation damping compared to the conventional excitation controller.

ε-bounded state estimation for time-delay systems with bounded disturbances

A new problem on ε-bounded functional state estimation for time-delay systems with unknown bounded disturbances is studied in this paper. In the presence of unknown bounded disturbances, the common assumption regarding the observers matching condition is no longer required. In this regard, instead of achieving asymptotic convergence for the observer error, the error is now required to converge exponentially within a ball with a small radius ε > 0. This means that the estimate converges exponentially within an ε-bound of the true value. A general observer that utilises multiple-delayed output and input information is proposed. Sufficient conditions for the existence of the proposed observer are first given. We then employ an extended Lyapunov-Krasovskii functional which combines the delay-decomposition technique with a triple-integral term to study the ε-convergence problem of the observer error system. Moreover, the obtained results are shown to be more effective than the existing results for the cases with no disturbances and/or no time delay. Three numerical examples are given to illustrate the obtained results.

Extending scheduling algorithms to minimise the impact of bounded uncertainty using interval programming

 This research proposed a new methodology to extend algorithms to accept interval-based uncertain parameters. The methodology is applied on scheduling algorithms, including heuristic and meta-heuristic algorithms and produced optimal results with higher accuracy. The research outcomes are effective for decision making process using uncertain or predicted data.

Reachable sets bounding for switched systems with time-varying delay and bounded disturbances

In this paper, we address the problem of finding outer bound of forward reachable sets and inter-bound of backward reachable sets of switched systems with an interval time-varying delay and bounded disturbances. By constructing a flexible Lyapunov–Krasovskii functional combining with some recent refined integral-based inequalities, some sufficient conditions are derived for the existence of (1) the smallest possible outer bound of forwards reachable sets; and (2) the largest possible inter-bound of backward reachable sets. These conditions are delay dependent and in the form of linear matrix inequalities, which therefore can be efficiently solved by using existing convex algorithms. A constructive geometric design of switching laws is also presented. Two numerical examples with simulation results are provided to demonstrate the effectiveness of our results.

Reachable sets bounding for generalized neural networks with interval time-varying delay and bounded disturbances

This paper deals with the problem of finding outer bound of forwards reachable sets and interbound of backwards reachable sets of generalized neural network systems with interval nondifferentiable time-varying delay and bounded disturbances. Based on constructing a suitable Lyapunov–Krasovskii functional and utilizing some improved Jensen integral-based inequalities, two sufficient conditions are derived for the existence of: (1) the smallest possible outer bound of forwards reachable sets and (2) the largest possible interbound of backwards reachable sets. These conditions are delay dependent and in the form of matrix inequalities, which therefore can be efficiently solved by using existing convex algorithms. Three numerical examples with simulation results are provided to demonstrate the effectiveness of our results.