Linear continuous or discrete time-varying systems in which the sum of a quadratic form of the initial state and the integral or sum of quadratic forms of a disturbance on a finite horizon is bounded above by a given value are considered. It is demonstrated that the reachability set of such a continuous- or discrete-time system is an evolving ellipsoid, and its ellipsoid matrix satisfies a linear matrix differential or difference equation, respectively. The optimal ellipsoidal observer and identification algorithm that yield the best ellipsoidal estimates of the system’s state and unknown parameters are designed. In addition, the optimal controllers ensuring that the system’s state will fall into a target set or that the system’s trajectory will stay within the ellipsoidal tube are designed. A connection between the optimal ellipsoidal observer and the Kalman filter is established. Some illustrative examples for the Mathieu equation, which describes the parametric oscillations of a linear oscillator, are given.

考虑了线性连续或离散时变系统，在该系统中，初始状态的二次形式的总和与有限水平线上的扰动的积分形式或二次形式的总和以给定值为界。证明了这种连续或离散时间系统的可达性集是一个演化的椭球，其椭球矩阵分别满足线性矩阵微分或差分方程。设计了产生系统状态和未知参数的最佳椭圆估计的最佳椭圆观测器和识别算法。此外，还设计了确保系统状态落入目标集或系统轨迹将保持在椭圆管内的最佳控制器。在最佳椭圆形观测器和卡尔曼滤波器之间建立了联系。给出了Mathieu方程的一些说明性示例，这些示例描述了线性振荡器的参数振荡。