Passive and Conservative Continuous-Time Impedance and Scattering Systems. Part I: Well-Posed Systems
Passive and Conservative Continuous-Time Impedance and Scattering Systems. Part I: Well-Posed Systems
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无源和保守的连续时间阻抗和散射系统。
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发表时间：
2002
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通讯作者：
O. Staffans
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作者：
O. Staffans
抽象的。 设 U 为希尔伯特空间。开右半平面上的 ℒ (U) 值正解析函数是指满足条件 的解析函数。该函数不需要是适当的，即，它不需要限制在任何右半平面上。我们研究在什么条件下可以实现这样的函数作为阻抗无源系统的传递函数的问题。我们指的是一个连续时间状态空间系统，其控制和观察算子并不比系统的（主）半群生成元更无界，此外，还存在与吸收能量和内能相关的一定能量不等式。如果该能量不等式是等式，则该系统是（阻抗）能量守恒的；如果该系统及其对偶都是能量守恒的，则该系统是保守的。阻抗保守系统的典型示例是具有并置传感器和执行器的双曲线型系统。我们给出了几组等效条件，这些条件表征了系统何时处于阻抗无源、能量守恒或保守状态。我们证明，当且仅当阻抗无源系统是适当的时，它才是适定的。我们还表明，所谓的对角变换（可以被视为稍微修改的反馈变换）将适当的阻抗无源（或能量守恒或保守）系统映射到（适定的）散射无源（或能量守恒或守恒）系统。这意味着，就像在有限维情况下一样，如果我们将负输出反馈应用于适当的阻抗无源系统，则所得系统是（能量）稳定的。最后，我们证明右半平面上的每个适当的正解析函数都具有（本质上唯一的）适定阻抗保守实现，并且它还具有最小阻抗无源实现。
Abstract. Let U be a Hilbert space. By an ℒ (U)-valued positive analytic function on the open right half-plane we mean an analytic function which satisfies the condition . This function need not be proper, i.e., it need not be bounded on any right half-plane. We study the question under what conditions such a function can be realized as the transfer function of an impedance passive system. By this we mean a continuous-time state space system whose control and observation operators are not more unbounded than the (main) semigroup generator of the system, and, in addition, there is a certain energy inequality relating the absorbed energy and the internal energy. The system is (impedance) energy preserving if this energy inequality is an equality, and it is conservative if both the system and its dual are energy preserving. A typical example of an impedance conservative system is a system of hyperbolic type with collocated sensors and actuators. We give several equivalent sets of conditions which characterize when a system is impedance passive, energy preserving, or conservative. We prove that a impedance passive system is well-posed if and only if it is proper. We furthermore show that the so-called diagonal transform (which may be regarded as a slightly modified feedback transform) maps a proper impedance passive (or energy preserving or conservative) system into a (well-posed) scattering passive (or energy preserving or conservative) system. This implies that, just as in the finite-dimensional case, if we apply negative output feedback to a proper impedance passive system, then the resulting system is (energy) stable. Finally, we show that every proper positive analytic function on the right half-plane has a (essentially unique) well-posed impedance conservative realization, and it also has a minimal impedance passive realization.