We give a simple proof of the (perhaps not so) well known fact that exponential polynomials of order with real exponents have at most real zeros. We deduce several results that relate to impulsive controllability and sampled observability of finite-dimensional linear time-invariant dynamical systems. We prove that the initial state of a continuous-time linear time-invariant dynamical system of dimension can be uniquely reconstructed from the sampled output regardless of its sampling time sequence of length if and only if the system is observable and all the eigenvalues of the system matrix are real. This result thus characterizes sampled observability with arbitrary sampling times. Likewise, we prove that the system is controllable by means of impulses regardless of when they occur if and only if the system is controllable and all the eigenvalues of the system matrix are real. We also show that, if the system is observable and all the eigenvalues of the system matrix are real, then there exists an initial state such that the times where the output crosses a prescribed threshold are prescribed times with .

中文翻译：

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脉冲可控性和采样可观测性的强版本


我们给出一个简单的证明（也许不是）众所周知的事实：实指数阶指数多项式最多有实数零。我们推导出与有限维线性时不变动力系统的脉冲可控性和采样可观性相关的几个结果。我们证明，无论其采样时间序列的长度如何，当且仅当系统可观且系统的所有特征值都可以从采样输出唯一地重构维数连续时间线性时不变动力系统的初始状态矩阵是实数。因此，该结果表征了任意采样时间的采样可观察性。同样，我们证明系统是可通过脉冲控制的，无论脉冲何时发生，当且仅当系统是可控的并且系统矩阵的所有特征值都是实数时。我们还表明，如果系统是可观的，并且系统矩阵的所有特征值都是实数，则存在一个初始状态，使得输出跨越规定阈值的次数为规定次数。