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State/Signal Linear Time-Invariant Systems Theory, Part IV: Affine and Generalized Transfer Functions of Discrete Time Systems

机译:状态/信号线性时不变系统理论,第四部分:离散时间系统的仿射和广义传递函数

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In the present article we continue to develop the linear discrete time-invariant state/signal systems theory that we have introduced and studied in three earlier articles (Parts I—III). The trajectories of a state/signal system ∑ with a Hilbert state space χ and a Hilbert or Kreĭn signal space ${mathcal{W}}$ consists of a pair of sequences (x(·),w(·)) which after an admissible input/output decomposition ${mathcal{W}} = {mathcal{Y}},dot{+},{mathcal{U}}$ of the signal space can be obtained from the set of trajectories (x(·), u(·), y(·)) of a standard input/state/output system by taking w(·) = y(·) + u(·). In Part I we studied the families of all admissible decompositions of W for a given state/signal system ∑ and the corresponding input/state/output representations and their transfer functions. Here we extend that theory to the the non-admissible case, and obtain generalized input/output transfer functions, as well as right and left affine transfer functions. As opposed to a standard transfer function, a generalized transfer function not need to be holomorphic at the origin. This makes it possible to realize a much larger class of transfer functions than what is possible by using the standard input/state/output theory. For example, every rational matrixvalued function (including those that have a pole at the origin) can be realized as the generalized transfer function of a state/signal system whose state space has a finite dimension equal to the McMillan degree of the given function. Likewise, every meromorphic J-contractive matrix-valued function can be realized as the generalized transfer function of a simple conservative state/signal system, or of a minimal passive state/signal system. Similar operator-valued results are also presented. As is well-known, a rational matrix-valued function with a pole at the origin can also be realized as the transfer function of a descriptor system, but this descriptor realization has the drawback that the dimension of its state space is bigger than the dimension of the state space of our s/s realization (i.e., bigger than the McMillan degree of the function). Our s/s realizations also differ significantly from other known realizations which depend on the choice of an auxiliary point in the complex plane where the given function is holomorphic.
机译:在本文中,我们将继续发展线性离散时不变状态/信号系统理论,该理论已在三篇较早的文章(I-III部分)中进行了介绍和研究。具有希尔伯特状态空间χ和希尔伯特或Kreĭn信号空间$ {mathcal {W}} $的状态/信号系统∑的轨迹由一对序列(x(·),w(·))组成,信号空间的可允许输入/输出分解$ {mathcal {W}} = {mathcal {Y}},dot {+},{mathcal {U}} $可以从轨迹集(x(·),通过取w(·)= y(·)+ u(·)来计算标准输入/状态/输出系统的u(·),y(·)。在第一部分中,我们研究了给定状态/信号系统∑的W的所有可允许分解的族,以及相应的输入/状态/输出表示及其传递函数。在这里,我们将该理论扩展到不可受理的情况,并获得广义的输入/输出传递函数,以及左右仿射传递函数。与标准传递函数相反,广义传递函数不必在原点处是全纯的。与使用标准输入/状态/输出理论相比,这可以实现更大范围的传递函数。例如,每个有理矩阵值函数(包括那些在原点处具有极点的值)都可以实现为状态/信号系统的广义传递函数,其状态空间的有限维等于给定函数的McMillan度。同样,每个亚纯J压缩矩阵值函数都可以实现为简单的保守状态/信号系统或最小被动状态/信号系统的广义传递函数。还提供了类似的运算符值结果。众所周知,以极点为原点的有理矩阵值函数也可以实现为描述符系统的传递函数,但是这种描述符实现的缺点是其状态空间的维数大于维数。 s / s实现状态空间的大小(即,大于函数的McMillan度)。我们的s / s实现也与其他已知实现有很大不同,其他已知实现依赖于给定函数为全纯的复杂平面中辅助点的选择。

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