Angelo Alessandri

Moving-horizon state estimation for nonlinear discrete-time systems: New stability results and approximation schemes

A moving-horizon state estimation problem is addressed for a class of nonlinear discrete-time systems with bounded noises acting on the system and measurement equations. As the statistics of such disturbances and of the initial state are assumed to be unknown, we use a generalized least-squares approach that consists in minimizing a quadratic estimation cost function defined on a recent batch of inputs and outputs according to a sliding-window strategy. For the resulting estimator, the existence of bounding sequences on the estimation error is proved. In the absence of noises, exponential convergence to zero is obtained. Moreover, suboptimal solutions are sought for which a certain error is admitted with respect to the optimal cost value. The approximate solution can be determined either on-line by directly minimizing the cost function or off-line by using a nonlinear parameterized function. Simulation results are presented to show the effectiveness of the proposed approach in comparison with the extended Kalman filter.

Technical communique: Design of state estimators for uncertain linear systems using quadratic boundedness

The notion of quadratic boundedness, which allows one to address the stability of a dynamic system in the presence of bounded disturbances, is applied to the design of state estimators for discrete-time linear systems with polytopic uncertainties. Necessary and sufficient stability conditions are stated and upper bounding sequences on the estimation error are derived. For the purpose of design, such conditions can be expressed in terms of linear matrix inequalities (LMIs), thus guaranteeing the numerical tractability. Simulation results are reported to show the effectiveness of the approach.

Modeling and Feedback Control for Resource Allocation and Performance Analysis in Container Terminals

A dynamic discrete-time model of container flows in maritime terminals is proposed as a system of queues. Such queues are controlled via input variables that account for the use of the available resources given by the capacities of the handling machines used to move containers inside a terminal. Two feedback control strategies for the allocation of such resources are described. The first consists of a resource assignment that is proportional to the corresponding queue lengths; in the second, the assignment is obtained by the one-step-ahead optimization of a performance cost function according to a myopic approach. Simulation results are reported to compare such methodologies for the purpose of sensitivity and scenario analyses in the management of a maritime terminal.

Luenberger observers for switching discrete-time linear systems

State estimation using Luenberger-like observers is considered for a class of switching discrete-time linear systems. The switching is assumed to be unknown among the various system modes described by known matrices. The convergence of the error dynamics is ensured, even in the presence of bounded noises, by conditions that can be expressed by means of Linear Matrix Inequalities (LMIs). The design of such observer may be accomplished by minimizing an upper bound on a quadratic cost function of the estimation error using LMI-based optimization techniques. Moreover, an improvement to the estimator is presented that is based on a projection technique.

Design of Asymptotic Estimators: An Approach Based on Neural Networks and Nonlinear Programming

A methodology to design state estimators for a class of nonlinear continuous-time dynamic systems that is based on neural networks and nonlinear programming is proposed. The estimator has the structure of a Luenberger observer with a linear gain and a parameterized (in general, nonlinear) function, whose argument is an innovation term representing the difference between the current measurement and its prediction. The problem of the estimator design consists in finding the values of the gain and of the parameters that guarantee the asymptotic stability of the estimation error. Toward this end, if a neural network is used to take on this function, the parameters (i.e., the neural weights) are chosen, together with the gain, by constraining the derivative of a quadratic Lyapunov function for the estimation error to be negative definite on a given compact set. It is proved that it is sufficient to impose the negative definiteness of such a derivative only on a suitably dense grid of sampling points. The gain is determined by solving a Lyapunov equation. The neural weights are searched for via nonlinear programming by minimizing a cost penalizing grid-point constraints that are not satisfied. Techniques based on low-discrepancy sequences are applied to deal with a small number of sampling points, and, hence, to reduce the computational burden required to optimize the parameters. Numerical results are reported and comparisons with those obtained by the extended Kalman filter are made

Moving-Horizon State Estimation for Nonlinear Systems Using Neural Networks

In recent results, a moving-horizon state estimation problem has been addressed for a class of nonlinear discrete-time systems with bounded noises acting on the system and measurement equations. For the resulting estimator, suboptimal solutions can be addressed for which a certain error is allowed in the minimization of the cost function. Building on such results, in this paper the use of nonlinear parameterized functions is studied to obtain suitable state estimators with guaranteed performance. Thanks to the off-line optimization of the parameters, the estimates can be generated on line almost instantly. A new technique based on the approximation of the cost value (and not of its argument) is proposed and the properties of such a scheme are studied. Simulation results are presented to show the effectiveness of the proposed approach in comparison with the extended Kalman filter.

Advances in moving horizon estimation for nonlinear systems

In the last decade, moving horizon estimation (MHE) has emerged as a powerful technique for tackling the problem of estimating the state of a dynamic system in the presence of nonlinearities and disturbances. MHE is based on the idea of minimizing an estimation cost function defined on a sliding window composed of a finite number of time stages. The cost function is usually made up of two contributions: a prediction error computed on a recent batch of inputs and outputs; an arrival cost that serves the purpose of summarizing the past data. However, the diffusion of such techniques has been hampered by: i) the difficulty in choosing the arrival cost so as to ensure stability of the overall estimation scheme; ii) the request of an adequate computational effort on line. In this paper, both problems are addressed and possible solutions are proposed. First, by means of a novel stability analysis, it is constructively shown that under very general observability conditions a quadratic arrival cost is sufficient to ensure the stability of the estimation error provided that the weight matrix is adequately chosen. Second, a novel approximate MHE algorithm is proposed that is based on nonlinear programming sensitivity calculations. The approximate MHE algorithm has the same stability properties of the optimal one which make the overall approach suitable to be applied in real settings. Preliminary simulation results confirm the effectiveness of proposed method.

Feedback Optimal Control of Distributed Parameter Systems by Using Finite-Dimensional Approximation Schemes

Optimal control for systems described by partial differential equations is investigated by proposing a methodology to design feedback controllers in approximate form. The approximation stems from constraining the control law to take on a fixed structure, where a finite number of free parameters can be suitably chosen. The original infinite-dimensional optimization problem is then reduced to a mathematical programming one of finite dimension that consists in optimizing the parameters. The solution of such a problem is performed by using sequential quadratic programming. Linear combinations of fixed and parameterized basis functions are used as the structure for the control law, thus giving rise to two different finite-dimensional approximation schemes. The proposed paradigm is general since it allows one to treat problems with distributed and boundary controls within the same approximation framework. It can be applied to systems described by either linear or nonlinear elliptic, parabolic, and hyperbolic equations in arbitrary multidimensional domains. Simulation results obtained in two case studies show the potentials of the proposed approach as compared with dynamic programming.

A recursive algorithm for nonlinear least-squares problems

Abstract The solution of nonlinear least-squares problems is investigated. The asymptotic behavior is studied and conditions for convergence are derived. To deal with such problems in a recursive and efficient way, it is proposed an algorithm that is based on a modified extended Kalman filter (MEKF). The error of the MEKF algorithm is proved to be exponentially bounded. Batch and iterated versions of the algorithm are given, too. As an application, the algorithm is used to optimize the parameters in certain nonlinear input–output mappings. Simulation results on interpolation of real data and prediction of chaotic time series are shown.

Increasing-gain observers for nonlinear systems

For the purpose to estimate the state of nonlinear continuous-time systems, we focus on the increasing-gain observer regarded as generalization of the well-known high-gain observer. As compared with the previous results on increasing-gain observers, the assumptions under which the global asymptotic stability of the estimation error is proved are relaxed. The stability analysis is drawn by using a more general type of Lyapunov functional, where design parameters can be tuned to set the time-varying gain. A new design method based on such a Lyapunov functional is proposed to construct the observer. Simulation results affirm the effectiveness of the increasing-gain approach as compared with the high-gain observer.