Nonlinear Set Membership Research Articles

Nonlinear set membership filter with state estimation constraints via consensus-ADMM

This paper considers the state estimation problem for nonlinear dynamic systems with unknown but bounded noises. Set membership filter (SMF) is a popular algorithm to solve this problem. In the set membership setting, we investigate the filter problem where the state estimation requires to be constrained by a linear or nonlinear equality. We propose a consensus alternating direction method of multipliers (ADMM) based SMF algorithm for nonlinear dynamic systems. To deal with the difficulty of nonlinearity, instead of linearizing the nonlinear system, a semi-infinite programming (SIP) approach is used to transform the nonlinear system into a linear one, which allows us to obtain a more accurate estimation ellipsoid. For the solution of the SIP, an ADMM algorithm is proposed to handle the state estimation constraints, and each iteration of the algorithm can be solved efficiently. Finally, the proposed filter is applied to typical numerical examples to demonstrate its effectiveness.

Componentwise Hölder Inference for Robust Learning-Based MPC

This article presents a novel learning method based on componentwise Hölder continuity, which allows one to consider independently the contribution of each input to each output of the function to be learned. The method provides a bounded prediction error, and its learning property is proven. It can be used to obtain a predictor for a nonlinear robust learning-based predictive controller for constrained systems. The resulting controller achieves better closed loop performance and larger domains of attraction than learning methods that only consider nonlinear set membership, as illustrated by a case study.

Lazily Adapted Constant Kinky Inference for nonparametric regression and model-reference adaptive control

Techniques known as Nonlinear Set Membership prediction or Lipschitz Interpolation are approaches to supervised machine learning that utilise presupposed Lipschitz properties to perform inference over unobserved function values. Provided a bound on the true best Lipschitz constant of the target function is known a priori, they offer convergence guarantees, as well as bounds around the predictions. Considering a more general setting that builds on Lipschitz continuity, we propose an online method for estimating the Lipschitz constant online from function value observations that are possibly corrupted by bounded noise. Utilising this as a data-dependent hyper-parameter gives rise to a nonparametric machine learning method, for which we establish strong universal approximation guarantees. That is, we show that our prediction rule can learn any continuous function on compact support in the limit of increasingly dense data, up to a worst-case error that can be bounded by the level of observational error. We also consider applications of our nonparametric regression method to learning-based control. For a class of discrete-time settings, we establish convergence guarantees on the closed-loop tracking error of our online learning-based controllers. To provide evidence that our method can be beneficial not only in theory but also in practice, we apply it in the context of nonparametric model-reference adaptive control (MRAC). Across a range of simulated aircraft roll-dynamics and performance metrics our approach outperforms recently proposed alternatives that were based on Gaussian processes and RBF-neural networks.

NSM Converges to a k-NN Regressor Under Loose Lipschitz Estimates

Although it is known that having accurate Lipschitz estimates is essential for certain models to deliver good predictive performance, refining this constant in practice can be a difficult task especially when the input dimension is high. In this letter, we shed light on the consequences of employing loose Lipschitz bounds in the Nonlinear Set Membership (NSM) framework, showing that the model converges to a nearest neighbor regressor (k-NN with $k=1$ ). This convergence process is moreover not uniform, and is monotonic in the univariate case. An intuitive geometrical interpretation of the result is then given and its practical implications are discussed.

Learning Non-Parametric Models with Guarantees: A Smooth Lipschitz Regression Approach

We propose a non-parametric regression methodology that enforces the regressor to be fully consistent with the sample set and the ground-truth regularity assumptions. As opposed to the Nonlinear Set Membership technique, this constraint guarantees the attainment of everywhere differentiable surrogate models, which are more suitable to optimization-based controllers that heavily rely on gradient computations. The presented approach is named Smooth Lipschitz Regression (SLR) and provides error bounds on the prediction error at unseen points in the space. A numerical example is given to show the effectiveness of this method when compared to the other alternatives in a Model Predictive Control setting.

Data-Driven Filtering for Nonlinear Systems With Bounded Noises and Quantized Measurements

This paper focuses on the data-driven filtering problem on nonlinear systems with bounded noises and quantized measurements which are subject to logarithmic quantization. Because the accurate system model is unknown, an almost-optimal data-driven filter is designed from an available data set within the nonlinear set membership framework. The set is composed of system inputs and quantized measurements, which can be more easily obtained than the mathematical model. The logarithmic quantization is dependent on the fact that its quantization error serves as a multiplicative noise which widely exists in many applications. For a further improvement of estimation performance, a preliminary approximation is obtained by neural network technique. Two numerical examples are given to illustrate the effectiveness of the proposed algorithm.

Estimation problems for uncertain nonlinear dynamical systems with ellipsoidal state constraints

The paper deals with the estimation problems for uncertain nonlinear dynamical systems in the case when a probabilistic description of noise and errors is not available, but only bounds on them known. Such models are found in many applied areas ranged from engineering problems from physics to economics as well as to biological and ecological modeling when it occurs that a stochastic nature of the errors is questionable because of limited data or because of complexity and nonlinearity of the model. As alternative to a stochastic characterization, a so-called set-membership approach has been proposed and intensively developed in the last decades. The solution of many control and estimation problems under uncertainty involves constructing reachable sets and their analogs. For models with linear dynamics under such set-membership uncertainty, there are several constructive approaches, which allow finding effective estimates of reachable sets. Certainly, concrete problems are mostly nonlinear in their parameters and the set of feasible system states is usually non-convex or even non-connected. The key issue in nonlinear set-membership estimation is to find suitable techniques, which produce related bounds for the set of unknown system states without being too computationally demanding. Here the problem of estimating reachable sets of nonlinear dynamical control systems with combined bilinear and quadratic nonlinearity and with uncertainty in initial states is studied. Applying results of the theory of trajectory tubes of control systems and related techniques of differential inclusions theory we present approaches that allow finding the upper ellipsoidal estimates of reachable sets. The main new result consists in obtaining ellipsoidal estimates for reachable sets of nonlinear dynamical system with additional state constraint of ellipsoidal type. Related numerical algorithms, examples and results of computer simulations are included.

A New Nonlinear Set Membership Filter Based on Guaranteed Bounding Ellipsoid Algorithm

A New Nonlinear Set Membership Filter Based on Guaranteed Bounding Ellipsoid Algorithm

A New Nonlinear Set Membership Filter Based on Guaranteed Bounding Ellipsoid Algorithm

The framework of set membership filter (SMF) with unknown-but-bounded noise assumption provides an attractive alternative for probabilistic filters. However, the potential computational burden and conservation consideration may seriously limit the usage of this filter in practical applications. In this paper, based on guaranteed bounding ellipsoid approximation, a new enhanced set membership filter with better real-time property and reduced conservation is proposed for state estimation problem of nonlinear systems. The nonlinear model is firstly linearized and the DC programming method is used to outer-bound the linearization error, which is incorporated to the model noise with ellipsoidal approximations. A classical two-step prediction-correction procedure consisting vector sum computation between ellipsoids and an iterative outer-bounding ellipsoid algorithm to intersect ellipsoid with strip is presented to compute the ellipsoidal feasible set of the estimated states. Simulation results with comparisons to the nonlinear extended set membership filter are given to demonstrate the effectiveness and improved performances of our proposed algorithm.

Sparse Identification of Nonlinear Functions and Parametric Set Membership Optimality Analysis

Sparse identification can be relevant in the automatic control field to solve several problems for nonlinear systems such as identification, control, filtering, fault detection. However, identifying a maximally sparse approximation of a nonlinear function is in general an NP-hard problem. The common approach is to use relaxed or greedy algorithms that, under certain conditions, can provide sparsest solutions. In this technical note, a combined <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -relaxed-greedy algorithm is proposed and conditions are given, under which the approximation derived by the algorithm is a sparsest one. Differently from other conditions available in the literature, the ones provided here can be actually verified for any choice of the basis functions defining the sparse approximation. A Set Membership analysis is also carried out, assuming that the function to approximate is a linear combination of unknown basis functions belonging to a known set of functions. It is shown that the algorithm is able to exactly select the basis functions which define the unknown function and to provide an optimal estimate of their coefficients.

A MIT-Based Nonlinear Adaptive Set-Membership Filter for the Ellipsoidal Estimation of Mobile Robots' States

The adaptive extended set-membership filter (AESMF) for nonlinear ellipsoidal estimation suffers a mismatch between real process noise and its set boundaries, which may result in unstable estimation. In this paper, a MIT method-based adaptive set-membership filter, for the optimization of the set boundaries of process noise, is developed and applied to the nonlinear joint estimation of both time-varying states and parameters. As a result of using the proposed MIT-AESMF, the estimation effectiveness and boundary accuracy of traditional AESMF are substantially improved. Simulation results have shown the efficiency and robustness of the proposed method.

Design of Robust Predictive Control Laws Using Set Membership Identified Models

AbstractThis paper investigates the robust design of nonlinear model predictive control (NMPC) laws that employ approximated models, derived directly from process input‐output data. In particular, a nonlinear set membership (NSM) identification technique is used to obtain a system model and a bound of the related uncertainty. The latter is used to carry out a robust control design, via a min‐max formulation of the optimal control problem underlying the NMPC methodology. A numerical example with a nonlinear oscillator shows the effectiveness of the proposed approach.

A non-linear set-membership approach for the control of Discrete Event Systems

A variety of problems in non-linear time-evolution systems such as manufacturing plants, operations research, computer networks, etc., can be modelled as min-max-plus systems in which operations of min, max and addition appear simultaneously. It is well know that systems with only maximum (or minimum) constraints can be modelled as max-plus system and handled by max-plus algebra which changes the original non-linear system into linear system in this framework. Several authors have developed methods, in max-plus algebra, to compute control for max-plus systems. In general, these methods use the residuation theory to design a just-in-time control such that the output of the controlled system is, on the one hand, less than the desired reference signal but as close as possible to the given reference and, on the other hand, the control is delayed as much as possible. In this paper, we consider min-max-plus systems which are extensions of max-plus systems and non-linear even in the max-plus algebra view. We proposes a new approach to solve the just-in-time control problem for a non-linear min-max-plus system. This problem is cast into the more general framework of constraint satisfaction problems. This makes it possible to propose a new algorithm that contracts the feasible domains for each uncertain variable optimally (i.e., no smaller domain could be obtained) and efficiently.

A model predictive control approach to vehicle yaw control using identified models

A vehicle equipped with a front steer-by-wire device is considered and the related control design problem dealt with by using a yaw rate feedback structure. In order to effectively handle both the system non-linearities and the input constraints, a Non-linear Model Predictive Control (NMPC) technique is adopted. A novelty of the present paper is that the vehicle model employed by the NMPC algorithm is obtained from previously collected input/output data, using a Non-linear Set Membership (NSM) identification methodology. Since the NSM approach is able to provide a model with minimal worst-case identification error, improved robustness of the closed-loop system is obtained with respect to that of an NMPC law based on a physical vehicle model. Furthermore, the measure of the model uncertainty provided by the NSM approach allows one to perform a theoretical robust stability analysis of the closed-loop system. The effectiveness of the proposed technique is shown through numerical simulations of manoeuvres using a detailed vehicle model.

Robust design of predictive controllers using Set Membership identified models

AbstractThis paper investigates the robust design of Nonlinear Model Predictive Control (NMPC) laws that employ approximated models, derived directly from process input-output data. In particular, a Nonlinear Set Membership (NSM) identification technique is used to derive a system model and a bound of the related uncertainty. The latter is used to carry out a robust control design via a min-max formulation of the optimal control problem underlying the NMPC methodology. A numerical example with a nonlinear oscillator shows the effectiveness of the proposed approach.

On the Robustness of Receding Horizon Control using Nonlinear Approximated Models

This paper investigates the robustness of Nonlinear Model Predictive Control (NMPC) laws that employ approximated models, derived with Nonlinear Set Membership (NSM) techniques, with bounded model uncertainty. Although theoretical results already exist in the literature for the robustness analysis and for the robust design of NMPC in the face of a given description and related bound of the model uncertainty, the issue of how to practically estimate such an uncertainty bound is still an open one. This work shows that, with the proposed NSM approach, measured process input/output data can be employed to derive directly an input/output model and a bound of the related model uncertainty. The obtained uncertainty bound can be then employed to evaluate the effects of model uncertainty on the closed loop system performance. Moreover, since the NSM approach is able to provide an input/output model with minimal uncertainty bound, it is shown that improved robustness of the closed loop system can be obtained, with respect to that of a NMPC law based on a physical model. The effectiveness of the proposed approach is shown in a vehicle lateral stability control problem, through numerical simulations of harsh maneuvers using a detailed nonlinear vehicle model.

NSM constrained approximation of Lipschitz functions from data

The problem of approximating a function from a set of its values in the presence of constraints is considered. Solving this problem is useful in system identification, state estimation, and control. The typical approach to this problem consists in (1) deriving an approximation without considering the constraints, and (2) using the constraints to clip the approximation. However, using this approach may give poor results and, in any case, no optimality properties can be guaranteed. In this paper, a Nonlinear Set Membership method is developed, allowing optimal constrained approximation. The effectiveness of the proposed method is shown through its application to the problem of model predictive control approximation.

The filter design from data (FD2) problem: Nonlinear Set Membership approach

In the paper, we consider the problem of designing a filter that, operating on noisy measurements of input u and output y of a dynamical system, gives estimates (possibly optimal in some sense) of some other variable of interest z . A large body of literature exists, which investigates this problem assuming that the system equations relating u , y and z are known. However, in most practical situations, the system equations are not (completely) known, but a data set composed of noisy measurements of u , y and z is available. In such situations, a two-step procedure is typically adopted: a model is identified from the set of measured data, and the filter is designed on the basis of the identified model. In this paper, we propose an alternative solution, which uses the available data set of measured u , y and z not for the identification of system dynamics, but for the direct design of the filter. Such a direct design is investigated within the Nonlinear Set Membership framework. In the case of full observability, an almost optimal filter is derived, where optimality refers to minimizing a worst-case estimation error. In the case of partial observability, conditions are given for which the direct design is guaranteed to give bounded estimation error. Three examples are presented, related to the Lorenz chaotic system the first two, and to an automotive application the third one.

A Nonlinear Set Membership Approach for the Localization and Map Building of Underwater Robots

This paper proposes a set membership method based on interval analysis to solve the simultaneous localization and map building (SLAM) problem. The principle of the approach is to cast the SLAM problem into a constraint satisfaction problem for which interval propagation algorithms are particularly powerful. The resulting propagation method is illustrated on the localization and map building of an actual underwater robot.

Experimental modeling of controlled suspension vehicles from onboard sensors

In this article, identification of vertical dynamics of vehicles with controlled suspensions is considered. Identification is performed from experimental data measured on a four-poster bench test of a segment C car, equipped with a CDC-Skyhook dampers control system. The measurements are obtained from the onboard accelerometers needed by the control system. A nonlinear model in regression form is identified, having the road profile and damper control currents as inputs and chassis accelerations as outputs. The model is identified by means of a set membership structured identification method, which takes advantage of physical information on the structure of the system, decomposing the system into three subsystems: one represents the chassis and engine and the other two represent the overall behavior of front and rear suspensions, wheels and tires. This decomposition allows us to avoid the complexity accuracy problems derived from the high dimension of required regression space. Indeed, the overall high-dimensional identification problem is reduced to the identification of lower dimensional subsystems and to the estimation of their interactions. An iterative scheme is used for solving the decomposed identification problem. As the chassis pitch is small for the usual road profiles, the chassis-engine block is considered linear and standard linear methods are used for its identification. The other two subsystems are the main sources of nonlinearities in the system, mainly due to the significant nonlinearities of controlled dampers and of tires. Owing to the complexity/accuracy problems of a physical modeling of these subsystems, an input–output approach is taken. In particular, a nonlinear set membership method that does not require the search of the functional form of involved nonlinearities is used for the identification of these subsystems. The iterative algorithm converged in two iterations to a model providing a quite satisfactory simulation accuracy for all the considered road profiles and CDC-Skyhook settings.