仿射依赖LPV状态空间表示的实现理论
Realization Theory for LPV State-Space Representations with Affine Dependence
摘要：
在本文中，我们提出了一个卡尔曼风格的实现理论的线性参数变化的状态空间表示，其矩阵依赖于调度变量的仿射方式（简称为LPV-SSA表示）。我们处理离散时间和连续时间的情况。我们表明，这样的LPV-SSA表示是一个最小的（在这个意义上具有最少数量的状态变量）表示其输入输出函数，当且仅当它是可观察的和跨度可达的。证明了同一输入输出函数的任意两个极小LPV-SSA表示之间存在线性同构关系，且同构关系不依赖于调度变量，证明了输入输出函数可用LPV-SSA表示表示的充要条件是输入输出函数的Hankel矩阵具有有限秩.事实上，Hankel矩阵的秩给出了最小LPV-SSA表示的维数。此外，我们可以制定一个对应的部分实现理论的LPV-SSA表示和证明的正确性的Kalman-Ho算法。这些结果代表了LPV-SSA表示的系统理论基础。
英文摘要：
In this paper we present a Kalman-style realization theory for linear parameter-varying state-space representations whose matrices depend on the scheduling variables in an affine way (abbreviated as LPV-SSA representations). We deal both with the discrete-time and the continuous-time cases. We show that such a LPV-SSA representation is a minimal (in the sense of having the least number of state-variables) representation of its input-output function, if and only if it is observable and span-reachable. We show that any two minimal LPV-SSA representations of the same input-output function are related by a linear isomorphism, and the isomorphism does not depend on the scheduling variable.We show that an input-output function can be represented by a LPV-SSA representation if and only if the Hankel-matrix of the input-output function has a finite rank. In fact, the rank of the Hankel-matrix gives the dimension of a minimal LPV-SSA representation. Moreover, we can formulate a counterpart of partial realization theory for LPV-SSA representation and prove correctness of the Kalman-Ho algorithm. These results thus represent the basis of systems theory for LPV-SSA representation.
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