Finite time estimation through a continuous‐discrete observer

Abstract

Summary We study two broad classes of nonlinear time‐varying continuous‐time systems with outputs. For the first class, we build an observer in the case where a state dependent disturbance affects the linear approximation. When the disturbances are the zero functions, our observer provides exact values of the state at all times larger than a suitable finite time, and it provides an approximate estimate when there are nonzero disturbances, so our observers are called finite time observers. We use this construction, which is of interest for its own sake, to design a globally exponentially stabilizing dynamic output feedback for a family of nonlinear systems whose outputs are only available on some finite time intervals. Our simulations illustrate the efficacy of our methods.

Reduced Order Finite Time Observers for Time-Varying Nonlinear Systems

“…We will study other extensions. In particular, we hope to combine the main result of the present paper with the result of [8] and to the case where there are a delay and a disturbance in the input and where the outputs are only available on some finite time intervals.…”

Output Feedback Stabilization by Reduced Order Finite Time Observers using a Trajectory Based Approach

“…Since our observers only required computing fundamental matrices for subsystems that have the dimensions of the unknown states, our method can reduce the computational burden relative to existing methods. We hope to combine Theorem 1 with the result of [10] to cover delays and disturbances in the input.…”

“…This paper continues our development and use of finite time observers that can cope with uncertain or intermittent output observations and nonlinearities, while also reducing the dimension of the required observers. While our work [10] provided full order finite time observers (whose dimensions equal the dimension of the original system) that allowed intermittent output observations, and our work in [11] provided reduced order observers that led to continuous output feedback controllers that have distributed terms (meaning, the control is implicitly defined by integrals that contain past control values), the present work provides an alternative to [11] in which the controls contain a mixture of continuous and discrete time dynamics (and therefore are called continuous-discrete) but do not contain any distributed control terms. This can help further reduce the computational burden by eliminating the need for distributed terms; see [2] and [13] for the relevance of distributed terms.…”

Finite time estimation for time-varying systems with delay in the measurements

“…The pioneering work by Engel and Kreisselmeier (2002) on finite time observer design has been built upon by significant results such as Mazenc et al (2015) (which provides finite time observers with guaranteed bounds for solutions, including systems with disturbances), Menard et al (2010), Menold et al (2003), and Sauvage et al (2007). See also Mazenc et al (2018) for finite time continuous-discrete observers for systems with temporary loss of measurements. However, these works do not allow delayed measurements,…”

“…We show how the difference between the value of the state and its estimation is bounded by a function of the past output value, the input, and the disturbances. We illus-trate our observer design using a class of dynamics that includes Mathieu's equation from the study of vibrating membranes, which was studied in Mazenc et al (2018) for the case where there are no measurement delays.…”