This chapter discusses the design of unknown input observers for a class of time-delay systems, where a constant time delay, \(\tau \) , now also occurs in the output vector, i.e., \(y(t)=Cx(t)+C_dx(t-\tau )\) . To solve this problem, we apply our matrix decomposition technique as well as we impose a constraint on matrix \(C_d\) . Consequently, the decomposition results in a disturbance-free time-delay system with multiple time delays in both the state and output vectors. The unknown input is expressed in its entirety that contains a generalized functional which has multiple state-delayed terms. Thus, the unknown input can be estimated by designing a generalized functional observer to estimate the generalized functional.