We discuss an algebraic design of unknown input observers (UIOs) for linear systems with unknown inputs that do not satisfy the observer matching condition. Such a condition is often required for the existence of a UIO. To circumvent the restriction imposed by the observer matching condition, auxiliary outputs defined as the higher-order derivatives of the output measurements are introduced so that the observer matching condition is satisfied with respect to unknown inputs. The augmented outputs consisting of both the output measurements and the auxiliary outputs are defined, where the auxiliary outputs are estimated using higher-order sliding-mode exact differentiators based on the output measurements. Through suitable coordinate transformations, a system with the augmented outputs can be partitioned into two interconnected subsystems: an unknown input-free subsystem and an unknown input-dependent subsystem. The state vectors of the second subsystem are described in terms of augmented outputs. The unknown input-free subsystem is then used to design a UIO for estimation of the state vector. The performance of the UIO is verified through simulated numerical examples.
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