In this paper we derive minimum time convergent observers for $n+m$ ${}^1$ systems of linear coupled first-order 1-D hyperbolic PDEs, that use either unilateral (single boundary measured), bilateral (both boundaries measured) or pointwise in-domain sensing. First, a Volterra integral transformation is combined with a Fredholm integral transformation to derive a minimum time unilateral observer for $n+m$ systems. Then, it is shown that an $n+m$ system with bilateral sensing can be transformed to an $(n+m)+(n+m)$ system with unilateral sensing via an invertible coordinate transformation. The $n+m$ bilateral observer is subsequently obtained from the $(n+m)+(n+m)$ minimum time unilateral observer, and it is shown that it converges in a theoretical minimum time for bilateral sensing. In a similar fashion, the observer using pointwise in-domain measurement is derived using the same techniques. The performances of the $n+m$ bilateral and unilateral observers are demonstrated in simulations.

在本文中，我们推导出最小时间收敛观测器 $n+米$ ${}^1$线性耦合一阶一维双曲偏微分方程系统，使用单边（单边界测量）、双边（两个边界测量）或逐点域内传感。首先，将 Volterra 积分变换与 Fredholm 积分变换相结合，推导出最小时间单边观测器$n+米$系统。然后，表明一个$n+米$ 具有双边传感的系统可以转换为 $(n+米)+(n+米)$通过可逆坐标变换实现单边传感的系统。这$n+米$ 双边观察员随后从 $(n+米)+(n+米)$最小时间单边观察者，并且表明它收敛于双边感知的理论最小时间。以类似的方式，使用逐点域内测量的观察者是使用相同的技术导出的。的表演$n+米$ 在模拟中展示了双边和单边观察者。