Contracting Nonlinear Observers: Convex Optimization and Learning from Data

A new approach to design of nonlinear observers (state estimators) is proposed. The main idea is to (i) construct a convex set of dynamical systems which are contracting observers for a particular system, and (ii) optimize over this set for one which minimizes a bound on state-estimation error on a simulated noisy data set. We construct convex sets of continuous-time and discrete-time observers, as well as contracting sampled-data observers for continuous-time systems. Convex bounds for learning are constructed using Lagrangian relaxation. The utility of the proposed methods are verified using numerical simulation.Comment: conference submissio

A review of convex approaches for control, observation and safety of linear parameter varying and Takagi-Sugeno systems

This paper provides a review about the concept of convex systems based on Takagi-Sugeno, linear parameter varying (LPV) and quasi-LPV modeling. These paradigms are capable of hiding the nonlinearities by means of an equivalent description which uses a set of linear models interpolated by appropriately defined weighing functions. Convex systems have become very popular since they allow applying extended linear techniques based on linear matrix inequalities (LMIs) to complex nonlinear systems. This survey aims at providing the reader with a significant overview of the existing LMI-based techniques for convex systems in the fields of control, observation and safety. Firstly, a detailed review of stability, feedback, tracking and model predictive control (MPC) convex controllers is considered. Secondly, the problem of state estimation is addressed through the design of proportional, proportional-integral, unknown input and descriptor observers. Finally, safety of convex systems is discussed by describing popular techniques for fault diagnosis and fault tolerant control (FTC).Peer ReviewedPostprint (published version

A Hybrid Observer for a Distributed Linear System with a Changing Neighbor Graph

A hybrid observer is described for estimating the state of an $m>0$ channel, $n$-dimensional, continuous-time, distributed linear system of the form $\dot{x} = Ax,\;y_i = C_ix,\;i\in\{1,2,\ldots, m\}$. The system's state $x$ is simultaneously estimated by $m$ agents assuming each agent $i$ senses $y_i$ and receives appropriately defined data from each of its current neighbors. Neighbor relations are characterized by a time-varying directed graph $\mathbb{N}(t)$ whose vertices correspond to agents and whose arcs depict neighbor relations. Agent $i$ updates its estimate $x_i$ of $x$ at "event times" $t_1,t_2,\ldots$ using a local observer and a local parameter estimator. The local observer is a continuous time linear system whose input is $y_i$ and whose output $w_i$ is an asymptotically correct estimate of $L_ix$ where $L_i$ a matrix with kernel equaling the unobservable space of $(C_i,A)$. The local parameter estimator is a recursive algorithm designed to estimate, prior to each event time $t_j$, a constant parameter $p_j$ which satisfies the linear equations $w_k(t_j-\tau) = L_kp_j+\mu_k(t_j-\tau),\;k\in\{1,2,\ldots,m\}$, where $\tau$ is a small positive constant and $\mu_k$ is the state estimation error of local observer $k$. Agent $i$ accomplishes this by iterating its parameter estimator state $z_i$, $q$ times within the interval $[t_j-\tau, t_j)$, and by making use of the state of each of its neighbors' parameter estimators at each iteration. The updated value of $x_i$ at event time $t_j$ is then $x_i(t_j) = e^{A\tau}z_i(q)$. Subject to the assumptions that (i) the neighbor graph $\mathbb{N}(t)$ is strongly connected for all time, (ii) the system whose state is to be estimated is jointly observable, (iii) $q$ is sufficiently large, it is shown that each estimate $x_i$ converges to $x$ exponentially fast as $t\rightarrow \infty$ at a rate which can be controlled.Comment: 7 pages, the 56th IEEE Conference on Decision and Contro

LMI-Based Reset Unknown Input Observer for State Estimation of Linear Uncertain Systems

This paper proposes a novel kind of Unknown Input Observer (UIO) called Reset Unknown Input Observer (R-UIO) for state estimation of linear systems in the presence of disturbance using Linear Matrix Inequality (LMI) techniques. In R-UIO, the states of the observer are reset to the after-reset value based on an appropriate reset law in order to decrease the $L_2$ norm and settling time of estimation error. It is shown that the application of the reset theory to the UIOs in the LTI framework can significantly improve the transient response of the observer. Moreover, the devised approach can be applied to both SISO and MIMO systems. Furthermore, the stability and convergence analysis of the devised R-UIO is addressed. Finally, the efficiency of the proposed method is demonstrated by simulation results

A detectability criterion and data assimilation for non-linear differential equations

In this paper we propose a new sequential data assimilation method for non-linear ordinary differential equations with compact state space. The method is designed so that the Lyapunov exponents of the corresponding estimation error dynamics are negative, i.e. the estimation error decays exponentially fast. The latter is shown to be the case for generic regular flow maps if and only if the observation matrix H satisfies detectability conditions: the rank of H must be at least as great as the number of nonnegative Lyapunov exponents of the underlying attractor. Numerical experiments illustrate the exponential convergence of the method and the sharpness of the theory for the case of Lorenz96 and Burgers equations with incomplete and noisy observations