离散动力系统反馈混沌化与控制算法的研究
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摘要
本学位论文主要研究离散系统的反馈混沌化与控制问题。混沌控制和反控制是一个新的研究课题。对离散系统反馈混沌化与控制的研究具有重要的理论意义和实际应用价值。
离散系统反馈混沌化与控制有几种不同的目标,包括混沌化一个离散系统,超混沌化一个离散系统,增强混沌,抑制或消除混沌等。Lyapunov指数是混沌系统的重要的特征量,可以用于判定混沌的存在性和强弱。解决离散系统反馈混沌化与控制问题的一种方法是将Lyapunov指数配置为不同的符号和数值。这方面的结果比较少,其中Chen-Lai算法具有代表性。Chen-Lai算法的主要结果是将受控系统的Lyapunov指数全部配置为大于一个给定的正常数c。Chen-Lai算法等在一定的程度上成功地解决了此类问题。然而Chen-Lai算法等的结果与目前学术界所接受的关于利用Lyapunov指数判定混沌存在性的判据并不完全吻合,在实用上也有一定的局限性。
本文从判定离散系统混沌存在性和强弱的Lyapunov指数判据出发,在Chen-Lai算法等的基础上和启发下,提出并解决了一系列离散系统反馈混沌化与控制的基本问题。本文的几个结果可以用于达到不同的控制目标。本文所做的研究工作和所取得的结果主要有以下几方面:
1、对Chen-Lai算法和Wang-chen算法进行了一种扩展,用于离散系统混沌化:通过引入指定形式的反馈,仅配置一个Lyapunov指数为大于一个给定的正常数c,而其他Lyapunov指数均不为零。这个结果与Lyapunov指数混沌判据完全吻合。该算法可以使得一个非混沌的离散系统混沌化,也可以加强某离散系统中的混沌。
2、对Chen-Lai算法和Wang-chen算法进行了另一种扩展,用于离散系统超混沌化:通过引入指定形式的反馈,配置两个Lyapunov指数为大于一个给定的正常数c,而其他Lyapunov指数均不为零。这个结果与Lyapunov指数超混沌判据完全吻合。将离散系统反馈混沌化问题统一为一个问题:配置若干个Lyapunov指数为正。所得到的结果还可以自然地演变成Chen-Lai算法和Wang-chen算法。
3、提出配置离散系统的若干个最大的Lyapunov指数为大于一个给定的正常数c的一种算法。这个结果与Lyapunov指数混沌判据和超混沌判据完全吻合。算法所引入的反馈增益更小。通过引入指定形式的反馈,将受控离散系统的若干个最大的Lyapunov指数配置为大于一个给定的正常数c,包括仅配置一个、两个最大Lyapunov指数为正。在这个结果的基础上,分别将Chen-Lai算法和Wang-chen算法做了3种不同程度的改进。
4、提出增强和抑制离散系统中的混沌的一种算法。研究离散系统的Lyapunov指数与不动点处的Jacobi矩阵元素的关系。建立一些特征值条件,用于估计离散系统中Lyapunov指数的符号和数值。基于新建立的特征值条件,通过引入指定的非对角形式的反馈,调整受控系统雅可比矩阵的元素,使Lyapunov指数增大或者减小。这个结果可以用于增强和抑制离散系统中的混沌。
5、提出可以精确配置离散系统所有Lyapunov指数的一种算法。该算法可以灵活配置Lyapunov指数的符号和数值。因此该算法也可以用于加强和减弱离散系统中的混沌。
所提出的几种算法均给出了相应的数学证明和仿真例子。仿真结果表明了这些算法都是有效的。
The problem about control of chaos as well as anticontrol of chaos via feed-back in discrete dynamical systems is studied in this dissertation. Research on control and anticontrol of chaos is a very novel issue. It is important and valuable to chaotify a discrete dynamical system via feed-back control, and so is to control a chaotic discrete system.
There are several different goals to chaotify a discrete system via feed-back control. Some problems should be solved as following: to chaotify a discrete system, to make a discrete system hyper-chaotic, to strengthen chaos in discrete system, to weaken or eliminate chaos in discrete system. One way to get these goals is to place the sign and value of Lyapunov exponents in a discrete system. Chen-Lai’s algorithm are representative among those very few algorithms to chaotify a discrete system. The main result of Chen-Lai’s algorithm and Wang-Chen’s algorithm is very important which can place all Lyapunov exponents greater than a given positive constant, c. But it is a regret that this result does not stand just in line with the criterion to judge a discrete system chaotic or not.
On the basis of these two algorithms some important problems about how to chaotify a discrete system are put forward and solved in this dissertation. A series of different algorithms to place Lyapunov exponents is proposed. The correspond research as well as some main results consist of several respects as followed:
1.An algorithm to chaotify a discrete system: to place only one Lyapunov exponent greater than a given positive constant, c, via some special feed-back while other Lyapunov exponents are none-zero. This result agree with the criterion very well. The algorithm can be used to chaotify a discrete system effectively.
2.An algorithm to make a discrete system hyperchaotic: to place only two Lyapunov exponent greater than a given positive constant, c, via some special feed-back while other Lyapunov exponents are none-zero. The problems about how to make a discrete system chaotic or hyperchaotic are united to one: how to place some Lyapunov exponents greater than a given positive constant, c. The algorithm even can be evolved to Chen-Lai’s algorithm and Wang-Chen’s algorithm.
3. An algorithm to place only those greatest Lyapunov exponents of discrete systems greater than a given positive constant, c: It is discussed that smaller feed-back gain than that in Chen-Lai’s algorithm and Wang-Chen’s algorithm can place Lyapunov exponents positive. The problem about placing only one or two greatest Lyapunov exponents is solved and these results agree with the criterion very well. On the basis of these results, Chen-Lai’s algorithm and Wang-Chen’s algorithm are improved on three different level.
4. An algorithm to strengthen or weaken chaos in discrete systems: the relation between Lyapunov exponents of discrete systems and elements of Jacobian matrix is studied. Some new conditions on elements of Jacobian matrix and eigenvalues is proposed to estimate the signs and values of Lyapunov exponents. Some new none-diagonal feed-back are introduced to arrange the elements of Jacobian matrix. The Lyapunov exponents are placed to be different range of value via special feed-back, and as a result chaos in discrete systems is strengthened or weakened.
5. An algorithm to place all Lyapunov exponents of discrete systems precisely. The problem about how to place all Lyapunov exponents of discrete systems precisely is discussed and solved.
Correspond proofs and simulation experiements all algorithms proposed in this dissertation was presented. The simulation results show the effectiveness of these algorithms.
离散系统反馈混沌化与控制有几种不同的目标,包括混沌化一个离散系统,超混沌化一个离散系统,增强混沌,抑制或消除混沌等。Lyapunov指数是混沌系统的重要的特征量,可以用于判定混沌的存在性和强弱。解决离散系统反馈混沌化与控制问题的一种方法是将Lyapunov指数配置为不同的符号和数值。这方面的结果比较少,其中Chen-Lai算法具有代表性。Chen-Lai算法的主要结果是将受控系统的Lyapunov指数全部配置为大于一个给定的正常数c。Chen-Lai算法等在一定的程度上成功地解决了此类问题。然而Chen-Lai算法等的结果与目前学术界所接受的关于利用Lyapunov指数判定混沌存在性的判据并不完全吻合,在实用上也有一定的局限性。
本文从判定离散系统混沌存在性和强弱的Lyapunov指数判据出发,在Chen-Lai算法等的基础上和启发下,提出并解决了一系列离散系统反馈混沌化与控制的基本问题。本文的几个结果可以用于达到不同的控制目标。本文所做的研究工作和所取得的结果主要有以下几方面:
1、对Chen-Lai算法和Wang-chen算法进行了一种扩展,用于离散系统混沌化:通过引入指定形式的反馈,仅配置一个Lyapunov指数为大于一个给定的正常数c,而其他Lyapunov指数均不为零。这个结果与Lyapunov指数混沌判据完全吻合。该算法可以使得一个非混沌的离散系统混沌化,也可以加强某离散系统中的混沌。
2、对Chen-Lai算法和Wang-chen算法进行了另一种扩展,用于离散系统超混沌化:通过引入指定形式的反馈,配置两个Lyapunov指数为大于一个给定的正常数c,而其他Lyapunov指数均不为零。这个结果与Lyapunov指数超混沌判据完全吻合。将离散系统反馈混沌化问题统一为一个问题:配置若干个Lyapunov指数为正。所得到的结果还可以自然地演变成Chen-Lai算法和Wang-chen算法。
3、提出配置离散系统的若干个最大的Lyapunov指数为大于一个给定的正常数c的一种算法。这个结果与Lyapunov指数混沌判据和超混沌判据完全吻合。算法所引入的反馈增益更小。通过引入指定形式的反馈,将受控离散系统的若干个最大的Lyapunov指数配置为大于一个给定的正常数c,包括仅配置一个、两个最大Lyapunov指数为正。在这个结果的基础上,分别将Chen-Lai算法和Wang-chen算法做了3种不同程度的改进。
4、提出增强和抑制离散系统中的混沌的一种算法。研究离散系统的Lyapunov指数与不动点处的Jacobi矩阵元素的关系。建立一些特征值条件,用于估计离散系统中Lyapunov指数的符号和数值。基于新建立的特征值条件,通过引入指定的非对角形式的反馈,调整受控系统雅可比矩阵的元素,使Lyapunov指数增大或者减小。这个结果可以用于增强和抑制离散系统中的混沌。
5、提出可以精确配置离散系统所有Lyapunov指数的一种算法。该算法可以灵活配置Lyapunov指数的符号和数值。因此该算法也可以用于加强和减弱离散系统中的混沌。
所提出的几种算法均给出了相应的数学证明和仿真例子。仿真结果表明了这些算法都是有效的。
The problem about control of chaos as well as anticontrol of chaos via feed-back in discrete dynamical systems is studied in this dissertation. Research on control and anticontrol of chaos is a very novel issue. It is important and valuable to chaotify a discrete dynamical system via feed-back control, and so is to control a chaotic discrete system.
There are several different goals to chaotify a discrete system via feed-back control. Some problems should be solved as following: to chaotify a discrete system, to make a discrete system hyper-chaotic, to strengthen chaos in discrete system, to weaken or eliminate chaos in discrete system. One way to get these goals is to place the sign and value of Lyapunov exponents in a discrete system. Chen-Lai’s algorithm are representative among those very few algorithms to chaotify a discrete system. The main result of Chen-Lai’s algorithm and Wang-Chen’s algorithm is very important which can place all Lyapunov exponents greater than a given positive constant, c. But it is a regret that this result does not stand just in line with the criterion to judge a discrete system chaotic or not.
On the basis of these two algorithms some important problems about how to chaotify a discrete system are put forward and solved in this dissertation. A series of different algorithms to place Lyapunov exponents is proposed. The correspond research as well as some main results consist of several respects as followed:
1.An algorithm to chaotify a discrete system: to place only one Lyapunov exponent greater than a given positive constant, c, via some special feed-back while other Lyapunov exponents are none-zero. This result agree with the criterion very well. The algorithm can be used to chaotify a discrete system effectively.
2.An algorithm to make a discrete system hyperchaotic: to place only two Lyapunov exponent greater than a given positive constant, c, via some special feed-back while other Lyapunov exponents are none-zero. The problems about how to make a discrete system chaotic or hyperchaotic are united to one: how to place some Lyapunov exponents greater than a given positive constant, c. The algorithm even can be evolved to Chen-Lai’s algorithm and Wang-Chen’s algorithm.
3. An algorithm to place only those greatest Lyapunov exponents of discrete systems greater than a given positive constant, c: It is discussed that smaller feed-back gain than that in Chen-Lai’s algorithm and Wang-Chen’s algorithm can place Lyapunov exponents positive. The problem about placing only one or two greatest Lyapunov exponents is solved and these results agree with the criterion very well. On the basis of these results, Chen-Lai’s algorithm and Wang-Chen’s algorithm are improved on three different level.
4. An algorithm to strengthen or weaken chaos in discrete systems: the relation between Lyapunov exponents of discrete systems and elements of Jacobian matrix is studied. Some new conditions on elements of Jacobian matrix and eigenvalues is proposed to estimate the signs and values of Lyapunov exponents. Some new none-diagonal feed-back are introduced to arrange the elements of Jacobian matrix. The Lyapunov exponents are placed to be different range of value via special feed-back, and as a result chaos in discrete systems is strengthened or weakened.
5. An algorithm to place all Lyapunov exponents of discrete systems precisely. The problem about how to place all Lyapunov exponents of discrete systems precisely is discussed and solved.
Correspond proofs and simulation experiements all algorithms proposed in this dissertation was presented. The simulation results show the effectiveness of these algorithms.
引文
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