State and parameter estimation of nonlinear systems: a multi-observer approach Michelle S. Chong, Dragan Neˇ si´ c, Romain Postoyan and Levin Kuhlmann Abstract—We present a multi-observer approach for the parameter and state estimation of continuous-time nonlinear systems. For nominal parameter values in the known parameter set, state observers are designed with a robustness property. At any time instant, one observer is selected by a given criterion to provide its state estimate and its corresponding nominal parameter value. Provided that a persistency of excitation condition holds, we guarantee the convergence of state and parameter estimates up to a given margin of error which can be reduced by increasing the number of observers. The potential computational burden of the scheme is eased by introducing a dynamic parameter re-sampling technique, where the nominal parameter values are iteratively updated using a zoom-in procedure on the parameter set. We illustrate the efficacy of the algorithm on a model of neural dynamics. I. I NTRODUCTION Parameter and state estimation of nonlinear systems is a long standing quest in control theory. Adaptive methods that provide joint parameter and state estimates often rely on the structure of the system and hence, no design exists for general nonlinear systems. Another approach is to treat the parameters as states and augmenting the the state vector, thus turning the problem into state estimation only. However, doing so can turn even a linear system to be highly nonlinear and techniques may not exist for dealing with such systems. In this paper, we approach the problem of parameter and state estimation of deterministic nonlinear continuous- time systems by adopting the supervisory framework [22, Chapter 6]. We first sample the parameter set, which is assumed to be known and compact, to form a finite set of nominal parameter values. For each chosen parameter value, a state-observer is designed to satisfy a given robustness property. These observers then form a bank of observers called the multi-observer unit. The chosen state estimate and its corresponding parameter value is decided by the supervisory unit. Using this setup, we are able to guarantee the convergence of the estimates up to a given margin which can be arbitrarily reduced by increasing the number of observers. This static way of sampling the parameter set has the potential to be computationally intensive. To overcome M.S. Chong is with the Center for Control, Dynamical-systems and Computation (CCDC), University of California, Santa Barbara, CA 93106- 9560 USA. mstchong@gmail.com D. Neˇ si´ c and L. Kuhlmann are with the Department of Electri- cal and Electronic Engineering, the University of Melbourne, Australia. {dnesic,levink}@unimelb.edu.au R. Postoyan is with the Universit´ e de Lorraine, CRAN, UMR 7039 and the CNRS, CRAN, UMR 7039, France. He is financially sup- ported by the ANR under the grant SEPICOT (ANR 12 JS03 004 01). romain.postoyan@univ-lorraine.fr this obstacle, we drew inspiration from a quantization scheme for control in [21] and propose a dynamic sampling policy. At discrete time instants, the sampling of the parameter set is updated such that we zoom-in on a region of the parameter set. We show that the dynamic sampling policy ensures the same guarantees as the static scheme, with typically fewer observers. Traditionally, the supervisory framework has been used in the context of stabilization, see [6], [14], [16], [23], [30]. While these works consider systems that depend on some unknown parameters, these do not seek to provide guarantees on the convergence of parameters as the objective is to stabilize the plant. Hence, these results are not applicable in the context of this paper. For the purpose of estimation, a similar approach is pursued for linear systems in works such as [2], [3], [12], [4, Section 8.5], [20]. In this study, we consider nonlinear systems and hence, we envision a different methodology. We believe that the advantages garnered by employing hybrid techniques translate well to the estimation problem for continuous-time systems. While hybrid tools have been used in the control context (see [10], [25], [27] for example), few works have taken this angle for estimation, with the exception of [1], [13] (for parameter estimation) and [19] (for state estimation). This paper investigates both state and parameter estimation from this perspective. Furthermore, the benefits of supervisory control as mentioned in [15] are also present in the estimation context. Firstly, there is no need for the design of adaptive observers, which is challenging for nonlinear systems, e.g. [7], [29], [31]. Secondly, the modu- larity of the supervisory framework allows each component to be designed separately to satisfy the required properties to meet our objective. Hence, we can use readily available state observers in the supervisory framework for the additional purpose of parameter estimation. The paper is organized as follows. We state the problem and describe the supervisory architecture for estimation in Section II. We present the static sampling policy and we give the corresponding convergence guarantees in Section III. The dynamic sampling policy is then proposed in Section IV. We show how the results can be applied a neural mass model in Section V. We conclude the paper with Section VI. The proofs have been omitted due to space constraints, they can be found in [8]. Notation. Let R = (−∞, ∞), R ≥0 = [0, ∞), R >0 = (0, ∞), N = {0, 1, 2,... } and N ≥1 = {1, 2,... }. The notation (u, v) stands for [u T v T ] T , where u ∈ R m and