Sub-optimal control of fuzzy linear dynamical systems under granular differentiability concept

In this study, a new fuzzy system structure that reduces the number of inputs is proposed for dynamic system identification applications. Algebraic fuzzy systems have some disadvantages due to many inputs. As the number of inputs increase, the number of parameters in the training process increase and hence the classical fuzzy system becomes more complex. In the conventional fuzzy system structure, the past information of both inputs and outputs are also regarded as inputs for dynamic systems, therefore the number of inputs may not be manageable even for “single input and single output” systems. The new dynamic fuzzy system module (DFM) proposed here has only a single input and a single output. We have carried out identification simulations to test the proposed approach and shown that the DFM can successfully identify non-linear dynamic systems and it performs better than the classical fuzzy system.

Fuzzy cognitive maps (FCMs) can structure virtual worlds that change with time. A FCM links causal events, actors, values, goals, and trends in a fuzzy feedback dynamical system. A fuzzy rule defines a fuzzy patch in the input -output state-space of a system. It links commonsense knowledge with state-space geometry. A FCM connects the fuzzy rules or causal flow paths that relate events. It can guide actors in a virtual world as the actors move through a web of cause and effect and react to events and to other actors. Experts draw FCM causal pictures of the virtual world. Complex FCMs can give virtual worlds with ?>new ?> or chaotic equilibrium behavior. Simple FCMs give virtual worlds with periodic behavior. They map input states to limit-cycle equilibria. A FCM limit cycle repeats a sequence of events or a chain of actions and responses. Limit cycles can control the steady-state rhythms and patterns in a virtual world. In nested FCMs each causal concept can control its own FCM or fuzzy function approximator. Appendix A shows how an additive fuzzy system can uniformly approximate any continuous (or bounded measurable) function on a compact domain to any degree of accuracy. This gives levels of fuzzy systems that can choose goals and causal webs as well as move objects and guide actors in the webs. FCM matrices sum to give a combined FCM virtual world for any number of knowledge sources. Adaptive FCMs change their fuzzy causal web as causal patterns change and as actors act and experts state their causal knowledge. Neural learning laws change the causal rules and the limit cycles. Actors learn new patterns and reinforce old ones. In complex FCMs the user can choose the dynamical structure of the virtual world from a spectrum that ranges from mildly to wildly nonlinear. We use an adaptive FCM to model an undersea virtual world of dolphins, fish, and sharks.

System of simultaneous linear equations plays a vital role in mathematics, Operations Research, Statistics, Physics, Engineering and Social Sciences etc. In many applications at least some of the system's parameters and measurements are represented by fuzzy numbers rather than crisp numbers. Therefore it is imperative to develop mathematical models and numerical procedures to solve such a fuzzy linear system. The general model of a fuzzy linear system whose coefficient matrix is crisp and the right hand side column is an arbitrary fuzzy vector. In the fully fuzzy linear system all the parameters are considered to be fuzzy numbers. Since triangular fuzzy numbers is a special case of trapezoidal fuzzy numbers, hence in this paper we considered fully fuzzy linear system with trapezoidal fuzzy numbers. LU decomposition method for a crisp matrix is well known in solving linear system of equations. We discuss LU decomposition of the coefficient matrix of the fully fuzzy linear system, in which the coefficients are trapezoidal fuzzy numbers.

A linear equation is an equation in which each term contains a constant with a variable of degree one or single and can be described as a straight line in a Cartesian coordinate system. A Linear equations system is a collection of several linear equations. A system of linear equations whose coefficients and variables are fuzzy numbers is called a fully fuzzy linear equation system. This study aims to apply a metaheuristic algorithm to solve a system of fully fuzzy linear equations. The objective function used is the minimization objective function. At the same time, the metaheuristic algorithms used in this research are Particle Swarm Optimization (PSO), Firefly Algorithm (FA), and Cuckoo Search (CS). The input in this research is a fully fuzzy linear equation system matrix and parameters of the PSO, FA, and CS algorithms. The resulting output is the best objective function and the variable value of the fully fuzzy linear equations system. The work was compared for accuracy with the Gauss-Jordan elimination method from previous studies with the help of the Matlab programming language. The results obtained indicate that the Particle Swarm Optimization (PSO) algorithm is better at solving fully fuzzy linear equation systems than the Firefly Algorithm (FA) and Cuckoo Search (CS). This case can be seen from the value of the resulting objective function close to the value of the Gauss-Jordan elimination methodKeywords: Mathematics, investation

This paper investigates the robust control for dynamical systems subject to uncertainty. The uncertainty is assumed to be (possibly fast) time varying and bounded. The bound is unknown but lies within a prescribed fuzzy set (hence the fuzzy dynamical system). We propose an approach for the robust control design which is implemented in two steps. First, a class of robust controls is proposed based on tunable parameters. The proposed controls are deterministic and are not conventionally IF-THEN rules based. By the Lyapunov minimax approach, we prove that the proposed controls are able to guarantee deterministic system performance, namely, uniform boundedness and ultimate uniform boundedness. Second, optima seeking from the proposed controls is considered to improve fuzzy system performance. We formulate the optima-seeking problem as a two-player (one leader and one follower) Stackelberg game by developing two cost functions, each of which is in charge of one tunable parameter (i.e., the player). Each cost function consists of an average fuzzy system performance index and the associated player's control effort. We show that the solution of the optimal design problem (i.e., the optima of the tunable parameters), which is called the Stackelberg strategy, always exists and how to obtain the backwards-induction outcome is provided. Simulation results on the walking control of a biped robot model are presented for demonstration.

A general fuzzy linear system of equations is investigated using embedding approach. In the literature of fuzzy systems two types of linear systems are more important: 1) Fuzzy Linear Systems, 2) Fully Fuzzy Linear Systems. In both class of these systems usually the authors considered triangular type of fuzzy numbers. In this paper, we introduce a linear system of equations with trapezoidal fuzzy numbers as an extension of the fuzzy linear systems which was first introduced by Friedman et al. [6]. Conditions for the existence of a trapezoidal fuzzy solution to n×n linear system are derived and a numerical procedure for calculating the solution is illustrated with some examples.

<abstract> <p>The asymptotic behavior of solutions of fuzzy control systems is a component of the study of fuzzy control theory. The study of stability for T-S (Takagi-Sugeno) fuzzy systems, which process qualitative data through linguistic expressions, is the subject of this paper. Asymptotic stability is conservative in many real-world applications due to measurement noise and other disruptions. The ultimate limit, which indicates that the mistakes stay in a specific area close to the origin after a long enough amount of time, is a crucial characteristic that is frequently defined for such systems. We are interested with the problem of the state feedback controller for T-S fuzzy models with uncertainties where the global exponential ultimate boundedness of solutions is studied for certain fuzzy control systems. We use common quadratic Lyapunov function and parallel distributed compensation controller techniques to study the asymptotic behavior of the solutions of fuzzy control system in presence of perturbations. An example demonstrating the validity of the main result is discussed.</p> </abstract>

As a key theoretical basis of fuzzy analysis, a new representation of the two-dimension fuzzy numbers is proposed in this paper. We also present a new method for solving the fuzzy system of linear equations with the two-dimension fuzzy data based on the new representation of the two-dimension fuzzy numbers, which may translate a fuzzy system of linear equations into two real systems of linear equations, one is an n × n real system of linear equations and the other is a (2 n ) × (2 n ) real system of linear equations, where the coefficients, the right hand term and the unknown term of the (2 n ) × (2 n ) real systems are all positive real numbers. Accordingly, we compare the classical methods with the technique on the basis of the study of the two-dimensional fuzzy linear system proposed in this paper for an one-dimensional fuzzy linear system, and the results shows that they have the same solutions. Finally, the conditions of the existence of the solutions for a fuzzy system of linear equations are discussed, and the examples are given to show the efficiency and effectiveness of the method investigated in this article.

Nonlinear Optimal and Near-Optimal Controls of a Space Robotic System via Parallel Computation

The current research attempts to offer a new method for solving fuzzy linear Volterra integral equations system. This method converts the given fuzzy system into a linear system in crisp case by using the Taylor expansion method. Now the solution of this system yields the unknown Taylor coefficients of the solution functions. The proposed method is illustrated by an example and also results are compared with the exact solution by using computer simulations.

Extrapolation parameters in Refinement of Jacobi over Relation method in solving fuzzy linear systems were hypothesized to influence the rate of convergence. This paper aims to investigate the relationship between extrapolation parameters and approximate solutions of fuzzy linear systems when solved using Refinement of Jacobi over Relaxation method. The Algorithms of Refinement of Jacobi over Relaxation method are proposed based on the Jacobi method that specifically used in solving the system. A five by five fuzzy linear system is given to investigate the convergence to exact solution with three different values of extrapolation parameters. The numerical results show that there is a positive correlation between extrapolation parameter and convergence to exact solution. The three extrapolation parameters suggest that convergence to exact solution can be increased in line with the increase in the values of extrapolation parameters.

A method for reducing the order of a linear time-invariant dynamic system is presented. It is shown that it is possible to retain the predominant eigenvalues (or any other set of eigenvalues) of the exact system in the lower order model that possesses the property that its state is an aggregation of the state variables of the original system. Also it is shown that the output of the reduced order model can be constrained to contain all the modes of the exact output and to be close to the actual output of the original system within a specified tolerance. The performance of the original system is investigated for an optimal output regulator problem, when it is controlled on the assumption that its behavior is governed by that of the lower order model. Relations are obtained for the performance degradation that results with the above suboptimal control policy. Numerical examples show that the suboptimal control can be used in practice to lessen the computational complexity required for the higher order optimal control. The stability of the suboptimal control is not guaranteed; however, it is reasonable to expect it to be asymptotically stable when the order of reduction is not excessively high, because the outputs of the exact and lower order models are tolerably close.

Dimensional properties of optimal and sub-optimal nonlinear control systems

This paper focuses on investigating the cross-border financial contagion based on a fuzzy dynamical system scenario simulation from a perspective of analyzing the volatility of international capital flows for a panel of 50 countries in emerging markets and advanced economies from 1980 to 2011. Increasing evidence has shown that financial globalization has developed into a complex nonlinear dynamical system made up of economic subsystems with extensive financial connections and linkages. The contagion effects of the spread of bonanzas in the 50 countries are identified and analyzed. The Hodrick–Prescott filter is employed to address the long-term net capital inflow trend. The comovement of financial contagion between the source country of financial turbulence and the volatility-affected country is described as a fuzzy dynamical system in which the driving and response systems are coupled. A fuzzy dynamical system scenario simulation model under a liberal economy is established by employing nonlinear differential equations to describe the contagion mechanism and the international capital flow volatility effects. The model is then extended to a dynamical system model with macroeconomic control. The coupling strength uncertainty is addressed by employing an interval type-2 fuzzy theory method. The properties of the volatility equilibrium point for the two models are discussed, and the volatility contagion principles based on locally asymptotic stability analysis are derived to explain the different volatility transmission patterns. Policy suggestions are given in three situations for providing managerial insights for policymakers and the explorations of response strategies are also presented. The global financial crisis in 2008 is used as an experimental study to demonstrate the validity and effectiveness of the simulation and modeling method.

On Atangana–Baleanu fractional granular calculus and its applications to fuzzy economic models in market equilibrium