Design and stability analysis of fuzzy model based nonlinear controller for nonlinear systems using genetic algorithm
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This paper presents the stability analysis of fuzzy model based nonlinear control systems, and the design of nonlinear gains and feedback gains of the nonlinear controller using genetic algorithm with arithmetic crossover and nonuniform mutation. A stability condition is derived based on the Lyapunov stability theory with a smaller number of Lyapunov conditions. The solution of the stability conditions are also determined using GA. An application example of stabilizing a cart-pole type inverted pendulum system is given to show the stabilizability of the nonlinear controller.Keywords:
Lyapunov stability
This paper introduces the optimization methods of Genetic Algorithm.Based on the Different Location Crossover and the Same Location Crossover,a new leading crossover is proposed.Then it is a self-adaptive manner judgment to choose which crossover is used before the crossover operator.At last,five different tests of the simulation function are given.The results show that the leading crossover is more efficient to improve convergence than other crossovers.And the new method is easy to find the optimal solution.
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In this paper, the well-known Lyapunov stability theory is further investigated and two new concepts on broad-sense Lyapunov function and Lyapunov distance are introduced. The broad-sense Lyapunov stability theory is then developed. It is shown that a broad-sense Lyapunov function V(X) may be positive or negative. If the proposed Lyapunov distance satisfies the specified condition in this paper, the system origin will be asymptotically stable. It is shown that the Lyapunov stability theory is the special case of the proposed broadsense Lyapunov stability theory. Example is given to explain and verify the concepts on the broad-sense Lyapunov stability theory. Key-Words: Lyapunov stability theory, nonlinear system, control system
Stability theory
Lyapunov optimization
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Lyapunov stability
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This work presents a control of switched nonlinear systems. The approach is based on stabilisation by using control Lyapunov functions. A necessary and sufficient condition for stabilisation based on the existence of a common Lyapunov control function for switching subsystems is presented for the first time. Secondly, a stabilisation control device is designed for nonlinear switching system using the theory of common Lyapunov function and state feedback control of Lin-Sontag. Finally, this proposed approach is illustrated on the example of a switched reactor.
Control-Lyapunov function
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The Lyapunov stability theorem has been proposed for more than 100 years, and it is still one of the most important theories in control science and other fields. In this paper, a new stability theorem (Extended Lyapunov stability theorem) is proposed and proved to be different from Lyapunov stability theorem. The Lyapunov stability theorem demands that the time derivative of Lyapunov function is negative. But according to Extended Lyapunov stability theorem, the system can still keep stable when the time derivative of Lyapunov function is positive in some horizon and even in infinite horizon . So the conditions of Extended Lyapunov stability theorem is widely relaxed compared with the Lyapunov stability theorem. Inputs of actual systems are always limited by energy, under this background, a control law is designed to make system stable according to Extended Lyapunov stability theorem. And it can be proved that no Lyapunov function can be found to make the system stable. So with the help of Extended Lyapunov stability theorem better results can be obtained than those by Lyapunov stability theorem. At last, the numerical simulation result shows that Extended Lyapunov stability theorem is greatly different from Lyapunov stability theorem, such as the time derivative of energy function can be positive in infinite horizon. So it can be concluded that Extended Lyapunov stability theorem contains Lyapunov stability theorem and also it can be used more widely in many fields.
Lyapunov optimization
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It is known that selection and crossover operators contribute to generating solutions in genetic programming (GP). Traditionally, crossover points are selected randomly by a normal (canonical) crossover. However, the traditional method has several difficulties, in that building blocks (i.e. effective partial programs) are broken because of blind application of the normal crossover. This paper proposes a depth-dependent crossover for GP, in which the depth selection ratio is varied according to the depth of a node. This proposed method accumulates building blocks via the encapsulation of the depth-dependent crossover. We compare the performance of GP with depth-dependent crossover with that with normal crossover. Our experimental results clarify that the superiority of the proposed crossover to the normal method.
Crossover study
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This paper investigates the use of more than one crossover operator to enhance the performance of genetic algorithms. Novel crossover operators are proposed such as the Collision crossover, which is based on the physical rules of elastic collision, in addition to proposing two selection strategies for the crossover operators, one of which is based on selecting the best crossover operator and the other randomly selects any operator. Several experiments on some Travelling Salesman Problems (TSP) have been conducted to evaluate the proposed methods, which are compared to the well-known Modified crossover operator and partially mapped Crossover (PMX) crossover. The results show the importance of some of the proposed methods, such as the collision crossover, in addition to the significant enhancement of the genetic algorithms performance, particularly when using more than one crossover operator.
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Lyapunov stability
Stability theory
Mode (computer interface)
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This paper develops a method that control Lyapunov functions of a class of nonlinear systems can be constructed systematically. By using the control Lyapunov function a feedback control is established to stabilize the nonlinear system. The simulation shows the effectiveness of the method.
Constructive
Control-Lyapunov function
Lyapunov optimization
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This work presents a control of switched nonlinear systems. The approach is based on stabilisation by using control Lyapunov functions. A necessary and sufficient condition for stabilisation based on the existence of a common Lyapunov control function for switching subsystems is presented for the first time. Secondly, a stabilisation control device is designed for nonlinear switching system using the theory of common Lyapunov function and state feedback control of Lin-Sontag. Finally, this proposed approach is illustrated on the example of a switched reactor.
Control-Lyapunov function
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