Fixed-time Observer Design for LTI Systems by Time-varying Coordinate Transformation Gilberto Pin , Guitao Yang , Andrea Serrani , and Thomas Parisini Abstract— In this work, we present a novel fixed-time state observer for LTI systems based on a time-varying coordinate transformation yielding the cancellation of the effect of the unknown initial conditions from the state estimates. This coordinate transformation allows one to map the state of the original system into that of an auxiliary system that evolves from initial conditions that are known by construction. After a stable observer is designed in the transformed coordinates, an estimate for the state of the original system can be obtained by inverting the above-mentioned map. The invertibility of the map is guaranteed for any time strictly greater than zero, so that the convergence time can be made arbitrarily small in nominal conditions. The robustness of the observer with respect to bounded measurement disturbances is characterized in terms of both transient and norm bounds on the asymptotic state- estimation error. Compared to existing finite and fixed-time approaches, the proposed method does not require high-gain output-error injection, state augmentation, delay operators, or moving-windows. The dimensionality of the observer matches that of the observed system, and its realization takes the form of an LTV system. I. I NTRODUCTION In many applications such as fault isolation, change-point detection, or in the case of processes that can be observed in finite time-windows, estimates of hidden states are often required to converge with fast dynamics, independently from unknown initial conditions. This task can – in principle – be accomplished by “finite-time” and “fixed-time” observers. In the case of finite-time algorithms, the convergence time depends on the initial estimation error, while in the case of fixed-time observers the convergence time can be chosen by design, being independent from the initial conditions. Since the convergence time of fixed-time observers can be chosen a priori by the designer, the convergence is also termed “prescribed-time” or “free-will” in the literature. A. A Glimpse on the State of the Art While for linear discrete-time systems the design of finite- time (also known as “deadbeat”) observers is trivial, the derivation of a continuous-time counterpart poses technical challenges that have been tackled from different directions. G. Pin is with the Dept. of Information Engineering, University of Padua, Italy (gilberto.pin@unipd.it); G. Yang is with Impe- rial College London, UK (guitao.yang16@imperial.ac.uk); A. Serrani is with the Dept. of Electrical and Computer Eng., The Ohio State University, OH, USA (serrani.1@osu.edu); T. Parisini is with Imperial College London, UK and also with University of Trieste, Italy (t.parisini@gmail.com). Corresponding author: G. Pin, gilberto.pin@unipd.it This work has been partially supported by the Italian Ministry for Education, University, and Research (MIUR) under the initiative ”Departments of Excellence” (Law 232/2016) and in the framework of the 2017 Program for Research Projects of National Interest (PRIN), Grant no. 2017YKXYXJ, and by European Union’s Horizon 2020 research and innovation programme under grant agreement No 739551 (KIOS CoE). The first solution to this problem has been proposed by Kalman and his co-workers in the book [1], where it has been shown that the initial state of a system could be “observed” (according to the lexicon used there) by processing the input/output measurements of a system by causal integra- tion; the same source also offers the first fixed-time state “reconstruction” formula, that permits to estimate exactly the current state of a linear system with unknown initial conditions, again by processing input/output measurements by causal integration. However, said formula is not well suited for implementation due to the presence of nested time- integrals. After the seminal work [1], most of the theory on finite-time state reconstruction was rooted into the use of SM (sliding-mode) observers and homogeneity tools [2]–[10]. Conventional SM observers, however, can only guarantee semi-global stabilizability of the estimation-error dynamics, that is, the convergence can be proven for initial states restricted to a given bounded region. Higher-order SM update laws can be shown to achieve global convergence, at the cost of increased complexity in the implementation. Notably, the SM methodology achieves finite-time convergence by dis- continuous high-gain output injection, such that measurement noise may prevent its applicability. The convergence time of SM observers can be shortened at the cost of boosting the observer gains. The higher-order SM formulation offers a solution to the chattering issue, but still requires high-gain output injection. Among methodologies alternative to sliding mode, one finds the moving-horizon observer described in [11] and the delay-based filters proposed by [12] and [13]. Practical implementation of the aforementioned observers is, how- ever, computationally demanding. Indeed, moving-horizon observers require time discretization and the need to solve re- peated dynamic optimization problems on-line, whose com- plexity depends on the system dimension and on the length of the observation window. Moreover, the implementation of the delay operators required by [12] and [13] poses additional challenges in the continuous-time framework. Further alter- natives to sliding mode observers are provided by Fliess’s algebraic state reconstruction method (see [14]–[16]), or the integral-based modulating functions (see [17]–[20]), that exploit integral operators over compact domains able to an- nihilate the effect of initial conditions. Modulation function methods also found application to signal differentiation [21] and state estimation for fractional order systems [22]. To avoid error accumulation, the implementation of modulating function observers calls for the use of sliding windows (integration over compact domains) or require periodic re- scaling as discussed in [17]. A different finite-time convergent observer, based on im- pulsive innovation updates, has been proposed in [23], [24].