Port-Hamiltonian系统的能量整形镇定设计
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摘要
机械系统、机电系统和电磁系统等非线性系统是构成工业应用的基本系统,一般可表示为Port-Hamiltonian (PH)系统形式。如何利用PH系统的特点,研究系统的镇定控制器设计方法,为实际应用提供有效手段,具有重要的理论意义和应用前景。论文针对PH系统的镇定控制器设计问题开展研究,取得主要研究成果如下。
(1)提出了基于期望动能函数的动能整形方法。
针对时不变PH系统,利用Hamiltonian函数的特点,提出了一种动能整形方法,利用系统动能函数构造一个正定的期望动能函数,与时不变PH系统建立相应的匹配方程,通过求解匹配方程获得系统的镇定控制律,实现了局部能量对整体能量的匹配,有效避免了传统方法求解期望Hamiltonian函数的复杂过程和求解偏微分方程的困难。
同时,针对时变PH系统,利用期望动能整形方法构造一种时不变期望动能函数,与系统的时变Hamiltonian函数建立匹配方程,通过求解匹配方程获得系统的镇定控制律,实现了状态变量匹配和时间变量匹配的分离,有效避免了传统方法要求时间微分非增性条件的依赖,更具普遍性。
(2)分析和证明了期望阻尼矩阵在时不变PH系统镇定设计的作用。
在时不变PH系统具有相同期望能量函数的情况下,证明了期望阻尼矩阵在能量整形镇定设计的作用,即期望阻尼矩阵越大,被控系统到达期望平衡点的镇定速度越快。同时,通过三相电机系统进行仿真实验,验证了所得结论的有效性。
(3)提出了带输入饱和的时不变PH系统的能量整形镇定设计方法。
针对带输入饱和的时不变PH系统,提出了一种无期望阻尼注入的能量整形方法,利用系统Hamiltonian函数和期望Hamiltonian函数直接建立一种能量平衡模式匹配方程,通过求解匹配方程获得系统的镇定控制律,克服了带输入饱和的系统在能量整形过程中无法进行有效期望阻尼注入的局限性。
(4)建立了带时滞的Hamiltonian控制系统模型,提出了一种基于二对一匹配原则的系统镇定设计方法。
针对仿射非线性控制系统,利用向量场分解方法建立了带时滞的Hamiltonian控制系统模型,提出了一种二对一匹配原则,将系统的两个Hamiltonian函数与期望能量函数进行匹配来实现能量整形,获得系统的镇定控制律,有效避免了传统方法对于Lyapunov-Krasovskii泛函的依赖。
In the industrial applications, the mechanical systems, the electromechanical systems, and the electromagnetic systems are the basic roles, which can be represented as the Port-Hamiltonian (PH) systems. To research the stable method for those systems via the features of PH system, it is with the academic meanings and the pratical backgrounds. For the PH systems, their stable designing methods have been deeply researched in this paper. The main results and innovations are listed as follows.
(1) A kinetic energy-shaping has been proposed via the desired kinetic function.
For the time-invariant PH systems, a kinetic energy-shaping method is proposed via the features of the Hamiltonian function. Depending on it, a matching equation is established such that the original Hamiltonian function is matched with a positive desired kinetic energy function. Solving the equation obtains the control law. For the part energy matched with the whole energy, the proposed methos effectively avoid the complex procedure of desired Hamiltonian function in the tranditional method and the difficult of solving the partial differential equation.
Meanwhile, for the time varying PH system, a time-invariant desired kinetic energy function via the kinetic energy-shaping method is matched with the time varying original Hamiltonian function. Solving this matching equation obtains the control law. Depending on this method, the state variables matching and the time variables matching are separated, thus, it effectively avoids the non-increasing condition of time variable in the tranditional method and is with more application flieds
(2) The effectiveness of desired damping matrix in the procedure of energy-shaping has been analyzed and proven
As the PH systems match with the same desired Hamiltonian functions in the matching equations, the effectiveness of desired damping matrix has been analyzed and proven:when the desired damping matrix is bigger, the stable speed of Hamiltonian system is faster. In the end, examples and their simulations prove the effectiveness of the proposed methods.
(3) A stabilization of PH systems subject to actuator saturation (AS) has been proposed.
For the PH systems subject to AS, an energy-shaping method without the desired damping injection has presented. Due to this method, an energy-balance matching equation is established, which matches the original Hamiltonian function with the desired one. Solving this equation obtains the control law, which effectively avoids the obstacle that energy-shaping with the desired damping injection is employed.
(4) Time-invariant Hamiltonian control systems with a time delay and time varying Hamiltonian control systems with a time delay have been estabilished. In the meantime, the stabilizations of them are also proposed.
For the nonlinear afflne control systems with a time delay, the time-invariant Hamiltonian control systems with a time delay and time varying Hamiltonian control systems with a time delay are established via the vector decomposing methods. To stabilize the two mentioned systems, energy-shaping and two-to-one matching principle are applied to establish some new matching equations. Solving them yields the control laws stabilizing the systems, as a result, the proposed methods avoid the Lyapunov-Krasovskii function of the tranditional method.
(1)提出了基于期望动能函数的动能整形方法。
针对时不变PH系统,利用Hamiltonian函数的特点,提出了一种动能整形方法,利用系统动能函数构造一个正定的期望动能函数,与时不变PH系统建立相应的匹配方程,通过求解匹配方程获得系统的镇定控制律,实现了局部能量对整体能量的匹配,有效避免了传统方法求解期望Hamiltonian函数的复杂过程和求解偏微分方程的困难。
同时,针对时变PH系统,利用期望动能整形方法构造一种时不变期望动能函数,与系统的时变Hamiltonian函数建立匹配方程,通过求解匹配方程获得系统的镇定控制律,实现了状态变量匹配和时间变量匹配的分离,有效避免了传统方法要求时间微分非增性条件的依赖,更具普遍性。
(2)分析和证明了期望阻尼矩阵在时不变PH系统镇定设计的作用。
在时不变PH系统具有相同期望能量函数的情况下,证明了期望阻尼矩阵在能量整形镇定设计的作用,即期望阻尼矩阵越大,被控系统到达期望平衡点的镇定速度越快。同时,通过三相电机系统进行仿真实验,验证了所得结论的有效性。
(3)提出了带输入饱和的时不变PH系统的能量整形镇定设计方法。
针对带输入饱和的时不变PH系统,提出了一种无期望阻尼注入的能量整形方法,利用系统Hamiltonian函数和期望Hamiltonian函数直接建立一种能量平衡模式匹配方程,通过求解匹配方程获得系统的镇定控制律,克服了带输入饱和的系统在能量整形过程中无法进行有效期望阻尼注入的局限性。
(4)建立了带时滞的Hamiltonian控制系统模型,提出了一种基于二对一匹配原则的系统镇定设计方法。
针对仿射非线性控制系统,利用向量场分解方法建立了带时滞的Hamiltonian控制系统模型,提出了一种二对一匹配原则,将系统的两个Hamiltonian函数与期望能量函数进行匹配来实现能量整形,获得系统的镇定控制律,有效避免了传统方法对于Lyapunov-Krasovskii泛函的依赖。
In the industrial applications, the mechanical systems, the electromechanical systems, and the electromagnetic systems are the basic roles, which can be represented as the Port-Hamiltonian (PH) systems. To research the stable method for those systems via the features of PH system, it is with the academic meanings and the pratical backgrounds. For the PH systems, their stable designing methods have been deeply researched in this paper. The main results and innovations are listed as follows.
(1) A kinetic energy-shaping has been proposed via the desired kinetic function.
For the time-invariant PH systems, a kinetic energy-shaping method is proposed via the features of the Hamiltonian function. Depending on it, a matching equation is established such that the original Hamiltonian function is matched with a positive desired kinetic energy function. Solving the equation obtains the control law. For the part energy matched with the whole energy, the proposed methos effectively avoid the complex procedure of desired Hamiltonian function in the tranditional method and the difficult of solving the partial differential equation.
Meanwhile, for the time varying PH system, a time-invariant desired kinetic energy function via the kinetic energy-shaping method is matched with the time varying original Hamiltonian function. Solving this matching equation obtains the control law. Depending on this method, the state variables matching and the time variables matching are separated, thus, it effectively avoids the non-increasing condition of time variable in the tranditional method and is with more application flieds
(2) The effectiveness of desired damping matrix in the procedure of energy-shaping has been analyzed and proven
As the PH systems match with the same desired Hamiltonian functions in the matching equations, the effectiveness of desired damping matrix has been analyzed and proven:when the desired damping matrix is bigger, the stable speed of Hamiltonian system is faster. In the end, examples and their simulations prove the effectiveness of the proposed methods.
(3) A stabilization of PH systems subject to actuator saturation (AS) has been proposed.
For the PH systems subject to AS, an energy-shaping method without the desired damping injection has presented. Due to this method, an energy-balance matching equation is established, which matches the original Hamiltonian function with the desired one. Solving this equation obtains the control law, which effectively avoids the obstacle that energy-shaping with the desired damping injection is employed.
(4) Time-invariant Hamiltonian control systems with a time delay and time varying Hamiltonian control systems with a time delay have been estabilished. In the meantime, the stabilizations of them are also proposed.
For the nonlinear afflne control systems with a time delay, the time-invariant Hamiltonian control systems with a time delay and time varying Hamiltonian control systems with a time delay are established via the vector decomposing methods. To stabilize the two mentioned systems, energy-shaping and two-to-one matching principle are applied to establish some new matching equations. Solving them yields the control laws stabilizing the systems, as a result, the proposed methods avoid the Lyapunov-Krasovskii function of the tranditional method.
引文
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