State estimation and reconstruction of unknown inputs with arbitrary relative degree via a predefined-time algebraic solver

A sufficient condition to design unknown input observers for nonlinear systems with arbitrary relative degree

“…As a matter of fact, the use of high-order differentiators in UIOs was already proposed, for instance, to linear time-invariant systems, 29,34 linear parameter-varying systems 19 and nonlinear systems (using sliding-mode UIO). 28 Probably, one of the earliest papers that proposed the use of Levant's differentiator for unknown input estimation is Reference 34. In Reference 19, the proposed nonlinear UIO for LPV systems uses the derivatives of the output, which can be obtained by a sliding-mode differentiator, to avoid the mismatch condition and to allow the UI decoupling even for systems with an arbitrary relative degree greater than one.…”

“…where 𝜇 = 2. Note that this system can be written in the form (28) with 𝜓(x) = −x 1 + 𝜇(1 − x 2 1 )x 2 and 𝛾(x) = 1. In this example, as a constant observer gain is necessary to ensure the global exponential stability of the error dynamics equilibrium, only the LMI condition in ( 14 For the UI given by d = 5 sin(𝜋t) + 5 sin(3𝜋t + 𝜋 2 ) and the initial conditions x(0) = [2 2] ⊤ , x(0) = [0 0] ⊤ , the simulation results with the system's states, the UI, and their respective estimations are depicted in Figure 4A and the related estimation errors are depicted in Figure 4B.…”

“…It is worth noting that it is common to use sliding‐mode differentiators for control and estimation problems. As a matter of fact, the use of high‐order differentiators in UIOs was already proposed, for instance, to linear time‐invariant systems, 29,34 linear parameter‐varying systems 19 and nonlinear systems (using sliding‐mode UIO) 28 . Probably, one of the earliest papers that proposed the use of Levant's differentiator for unknown input estimation is Reference 34.…”

“…Moreover, the design of those observers must consider the domain of validity of local models, as discussed in the control design context by Klug et al, 24 Silva et al, 25 Coutinho et al, 26 and Nguyen et al 27 . An alternative to overcome the first‐order matching condition for nonlinear systems is the use of sliding‐mode UIOs with high‐order differentiators, as proposed in References 28 and 29. In Reference 30, a descriptor Takagi–Sugeno (T‐S) fuzzy representations for discrete‐time nonlinear systems with arbitrary relative degree is used to estimate the states and the UIs with a guaranteed $$$ {\mathcal{L}}_{\infty } $$$ performance.…”