The objective of the present paper is finite-dimensional observer-based control of the 1-D linear heat equation with constructive and feasible design conditions. We propose a method which is applicable to boundary or non-local sensing together with non-local actuation, or to Dirichlet actuation with non-local sensing. We use a modal decomposition approach. The dimension of the controller, $N_0$, is equal to the number of modes which decay slower than a given decay rate $\delta >0$. The observer may have a larger dimension $N\ge N_0$. The observer and controller gains are found separately by solving $N_0\times N_0$-dimensional Lyapunov inequalities. We suggest a direct Lyapunov approach to the full-order closed-loop system and provide linear matrix inequalities (LMIs) for finding $N$ and the exponential decay rate of the closed-loop system. We prove that the LMIs are always feasible for large enough $N$. The proposed method is different from existing qualitative methods that do not give easily verifiable and efficient bounds on the observer-based controller dimension and the resulting closed-loop performance. Numerical examples demonstrate that our LMI conditions lead to non-conservative bounds on $N$ and the resulting decay rate.

本文的目的是在具有建设性和可行的设计条件的情况下，基于有限维观测器的一维线性热方程控制。我们提出一种方法，该方法适用于边界或非局部感测以及非局部致动，或者适用于具有非局部感测的狄利克雷致动。我们使用模态分解方法。控制器的尺寸，${ñ}_0$等于衰减比给定衰减率慢的模数 $\delta >0$。观察者的尺寸可能更大$ñ\ge {ñ}_0$。观察者和控制器的增益可通过求解分别找到${ñ}_0\times {ñ}_0$Lyapunov不等式。我们建议对全阶闭环系统采用直接Lyapunov方法，并提供线性矩阵不等式（LMI）$ñ$以及闭环系统的指数衰减率。我们证明LMI足够大总是可行的$ñ$。所提出的方法与现有的定性方法不同，现有的定性方法没有对基于观察者的控制器尺寸和所产生的闭环性能给出容易验证和有效的界限。数值算例表明，我们的LMI条件导致非保守边界$ñ$ 以及由此产生的衰减率。