Marco Baglietto

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.

Receding-horizon estimation for discrete-time linear systems

The problem of estimating the state of a discrete-time linear system can be addressed by minimizing an estimation cost function dependent on a batch of recent measure and input vectors. This problem has been solved by introducing a receding-horizon objective function that includes also a weighted penalty term related to the prediction of the state. For such an estimator, convergence results and unbiasedness properties have been proved. The issues concerning the design of this filter are discussed in terms of the choice of the free parameters in the cost function. The performance of the proposed receding-horizon filter is evaluated and compared with other techniques by means of a numerical example.

A neural state estimator with bounded errors for nonlinear systems

A neural state estimator is described, acting on discrete-time nonlinear systems with noisy measurement channels. A sliding-window quadratic estimation cost function is considered and the measurement noise is assumed to be additive. No probabilistic assumptions are made on the measurement noise nor on the initial state. Novel theoretical convergence results are developed for the error bounds of both the optimal and the neural approximate estimators. To ensure the convergence properties of the neural estimator, a minimax tuning technique is used. The approximate estimator can be designed offline in such a way as to enable it to process on line any possible measure pattern almost instantly.

On estimation error bounds for receding-horizon filters using quadratic boundedness

Quadratic boundedness is used to deal with stability and design of receding-horizon estimators. Upper bounds on the norm of the estimation error have been found by means of invariant sets that can be constructed by using quadratic boundedness. Moreover, these bounds are expressed in terms of linear matrix inequalities and are well-suited to being minimized for the purpose of design.

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.

Distributed-information neural control: the case of dynamic routing in traffic networks

Large-scale traffic networks can be modeled as graphs in which a set of nodes are connected through a set of links that cannot be loaded above their traffic capacities. Traffic flows may vary over time. Then the nodes may be requested to modify the traffic flows to be sent to their neighboring nodes. In this case, a dynamic routing problem arises. The decision makers are realistically assumed 1) to generate their routing decisions on the basis of local information and possibly of some data received from other nodes, typically, the neighboring ones and 2) to cooperate on the accomplishment of a common goal, that is, the minimization of the total traffic cost. Therefore, they can be regarded as the cooperating members of informationally distributed organizations, which, in control engineering and economics, are called team organizations. Team optimal control problems cannot be solved analytically unless special assumptions on the team model are verified. In general, this is not the case with traffic networks. An approximate resolutive method is then proposed, in which each decision maker is assigned a fixed-structure routing function where some parameters have to be optimized. Among the various possible fixed-structure functions, feedforward neural networks have been chosen for their powerful approximation capabilities. The routing functions can also be computed (or adapted) locally at each node. Concerning traffic networks, we focus attention on store-and-forward packet switching networks, which exhibit the essential peculiarities and difficulties of other traffic networks. Simulations performed on complex communication networks point out the effectiveness of the proposed method.

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.

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.

Robust receding-horizon state estimation for uncertain discrete-time linear systems

An approach to robust receding-horizon state estimation for discrete-time linear systems is presented. Estimates of the state variables can be obtained by minimizing a worst-case quadratic cost function according to a sliding-window strategy. This leads to state the estimation problem in the form of a regularized least-squares one with uncertain data. The optimal solution (involving on-line scalar minimization) together with a suitable closed-form approximation are given. The stability properties of the estimation error for both the optimal filter and the approximate one have been studied and conditions to select the design parameters are proposed. Simulation results are reported to show the effectiveness of the proposed approach.