This work deals with the problem of finite-time and fixed-time observation of linear multiple input multiple output (MIMO) control systems. The proposed nonlinear dynamic observers guarantee convergence of the observer states to the original system state in a finite and in a fixed (defined a priori) time. Algorithms for the observers parameters tuning are also provided and a robustness analysis against input disturbances and measurement noises is carried out. The theoretical results are illustrated by numerical examples that consider both noisy and noise-free measurements and a comparison with a high-gain observer is included.

A solution to the problem of global fixed-time output stabilization and estimation of a chain of integrators is proposed. A nonlinear scheme comprising a state feedback controller and a dynamic observer are designed in order to guarantee both fixed-time estimation and fixed-time control. Robustness with respect to exogenous disturbances and measurement noises is established and a parameter optimization algorithm is provided. The performance of the obtained control and estimation algorithms are illustrated by numeric experiments.

This work presents Lyapunov analysis conditions for fixed-time stability, a property where all the system's trajectories converge exactly to zero in a finite amount of time that is independent of the system's initial condition. Necessary and sufficient conditions for fixed-time stability without taking into account the regularity of the settling-time function are presented first. Next, a characterization for fixed-time stability with continuous settling-time function is introduced. A particular form of the characterizing functions follows, it allows to establish more constructive conditions and in order to obtain a converse result, the concept of complete fixed-time stability is introduced. A set of academic examples and an example of allocation of mobile agents illustrate the given concepts. Finally, a sufficient condition for fixed-time stabilization of nonlinear affine systems is obtained.