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Moving horizon estimation for networked systems with packet dropouts
Abstract
The moving horizon estimation (MHE) problem is investigated in this paper for a class of networked systems with packet dropouts. The packet dropout is described by a binary switching random sequence. The main purpose of this paper is to design a estimator such that, for all possible packet dropouts, the state estimation error sequence is convergent. By choosing a stochastic cost function, the optimal solution of the MHE optimization problem with packet dropouts is given. Moreover, the convergence properties of the estimator are studied, and the maximum packet dropout probability is given to ensure the convergence of the state estimation error. Finally, the performance of the proposed estimator is evaluated and an example is given to demonstrate the effectiveness of the proposed method.
... At present, Shengxiang Zhang uses rolling time domain method to solve the trajectory planning problem [1]. Andong Liu Design and modeling of rolling time domain controller for time delay, packet loss and quantization problems in NCSs [2]. Zhaowang Fu used rolling time domain control method to model the decision-making of air combat maneuver [3]. ...
In order to generate the command and guidance tactics path based on the current battlefield situation information in real time, a method of online generation of command and guidance strategy based on rolling time domain control is proposed. Firstly, the optimal control model of command and guidance strategy is established. Then the rolling time domain control method is applied to the command and guidance strategy generation model, and conducted a solution study. Finally, a simulation study was conducted on a PC, The simulation results show that the on – line command and guidance strategy generation method based on rolling time domain control is reasonable and effective, which is in line with the actual situation of air combat.
This article is concerned with the problem of event-triggered centralized moving-horizon state estimation for a class of nonlinear dynamical complex networks. An event-triggered scheme is employed to reduce unnecessary data transmissions between sensors and estimators, where the signal is transmitted only when certain condition is violated. By treating sector-bounded nonlinearities as certain sector-bounded uncertainties, the addressed centralized moving-horizon estimation problem is transformed into a regularized robust least-squares problem that can be effectively solved via existing convex optimization algorithms. Moreover, a sufficient condition is derived to guarantee the exponentially ultimate boundedness of the estimation error, and an upper bound of the estimation error is also presented. Finally, a numerical example is provided to demonstrate the feasibility and efficiency of the proposed estimator design method.
This paper deals with state estimation of a spatially distributed system given noisy measurements from pointwise‐in‐time‐and‐space threshold sensors spread over the spatial domain of interest. A maximum a posteriori probability (MAP) approach is undertaken and a moving horizon (MH) approximation of the MAP cost function is adopted. It is proved that, under system linearity and log‐concavity of the noise probability density functions, the proposed MH‐MAP state estimator amounts to the solution, at each sampling interval, of a convex optimization problem. Moreover, a suitable centralized solution for large‐scale systems is proposed with a substantial decrease of the computational complexity. The latter algorithm is shown to be feasible for the state estimation of spatially dependent dynamic fields described by partial differential equations via the use of the finite element spatial discretization method. A simulation case study concerning estimation of a diffusion field is presented in order to demonstrate the effectiveness of the proposed approach. Quite remarkably, the numerical tests exhibit a noise‐assisted behavior of the proposed approach in that the estimation accuracy results optimal in the presence of measurement noise with non‐null variance.
In this note, a moving horizon estimation (MHE) strategy for detectable linear systems is proposed. Like the idea of “pre-stabilizing” model-predictive control, the states are estimated by a forward simulation with a pre-estimating observer in the MHE formulation. Compared with standard linear MHE approaches, it has more degrees of freedom to optimize the noise filtering. Tuning parameters are chosen to minimize the effects of measurement noise and model errors, which is implemented by finding tightest estimation error bounds. The performance of the proposed observer is demonstrated on one linear discrete-time example.
In this paper we consider a nonlinear constrained system observed by a sensor network and propose a distributed state estimation scheme based on moving horizon estimation (MHE). In order to embrace the case where the whole system state cannot be reconstructed from data available to individual sensors, we resort to the notion of MHE detectability for nonlinear systems, and add to the MHE problems solved by each sensor a consensus term for propagating information about estimates through the network. We characterize the error dynamics and provide conditions on the local exchanges of information in order to guarantee convergence to zero and stability of the state estimation error provided by any sensor. Copyright (C) 2010 John Wiley & Sons, Ltd.
Networked control systems (NCSs) are spatially distributed systems for which the communication between sensors, actuators, and controllers is supported by a shared communication network. We review several recent results on estimation, analysis, and controller synthesis for NCSs. The results surveyed address channel limitations in terms of packet-rates, sampling, network delay, and packet dropouts. The results are presented in a tutorial fashion, comparing alternative methodologies
Moving Horizon Estimation (MHE) is an efficient state estimation method used for nonlinear systems. Since MHE is optimization-based it provides a good framework to handle bounds and constraints when they are required to obtain good state and parameter estimates. Recent research in this area has been directed to develop computationally efficient algorithms for on-line application. However, an open issue in MHE is related to the approximation of the so-called arrival cost and of the parameters associated with it. The arrival cost is very important since it provides a means to incorporate information from the previous measurements to the current state estimate. It is difficult to calculate the true value of the arrival cost; therefore approximation techniques are commonly applied. The conventional method is to use the Extended Kalman Filter (EKF) to approximate the covariance matrix at the beginning of the prediction horizon. This approximation method assumes that the state estimation error is Gaussian. However, when state estimates are bounded or the system is nonlinear, the distribution of the estimation error becomes non-Gaussian. This introduces errors in the arrival cost term which can be mitigated by using longer horizon lengths. This measure, however, significantly increases the size of the nonlinear optimization problem that needs to be solved on-line at each sampling time. Recently, particle filters and related methods have become popular filtering methods that are based on Monte-Carlo simulations. In this way they implement an optimal recursive Bayesian Filter that takes advantage of particle statistics to determine the probability density properties of the states. In the present work, we exploit the features of these sampling-based methods to approximate the arrival cost parameters in the MHE formulation. Also, we show a way to construct an estimate of the log-likelihood of the conditional density of the states using a Particle Filter (PF), which can be used as an approximation of the arrival cost. In both cases, because particles are being propagated through the nonlinear system, the assumption of Gaussianity of the state estimation error can be dropped. Here we developed and tested EKF and eight different types of sample based filters for updating the arrival cost parameters in the weighted 2-norm approach (see Table 1 for the full list). We compare the use of constrained and unconstrained filters, and note that when bounds are required the constrained particle filters give a better approximation of the arrival cost parameters that improve the performance of MHE. Moreover, we also used PF concepts to directly approximate the negative of the log-likelihood of the conditional density using unconstrained and constrained particle filters to update the importance distribution. Also, we show that a benefit of having a better approximation of the arrival cost is that the horizon length required for the MHE can be significantly smaller than when using the conventional MHE approach. This is illustrated by simulation studies done on benchmark problems proposed in the state estimation literature.
In situ adaptive tabulation or ISAT based moving horizon estimation (MHE) is suggested as a fast and robust approach for state estimation. Computational issues with a moving horizon constrained state estimation technique like MHE are discussed. Implementation of storage and retrieval approach of ISAT for state estimation is proposed to maintain the accuracy and robustness of MHE, while generating the estimates at a reduced computational cost (∼300 times faster). Comparison with the widely used nonlinear state estimation technique of extended Kalman filtering (EKF) shows better performance using ISAT–MHE. Case studies with nonlinear discrete-time and continuous-time systems are carried out using ISAT, which is tailored for solving the optimal estimation problem.
This paper is concerned with the robust H∞ finite-horizon filtering problem for discrete time-varying stochastic systems with multiple randomly occurred sector-nonlinearities (MROSNs) and successive packet dropouts. MROSNs are proposed to model a class of sector-like nonlinearities that occur according to the multiple Bernoulli distributed white sequences with a known conditional probability. Different from traditional approaches, in this paper, a time-varying filter is designed directly for the addressed system without resorting to the augmentation of system states and measurement, which helps reduce the filter order. A new H∞ filtering technique is developed by means of a set of recursive linear matrix inequalities that depend on not only the current available state estimate but also the previous measurement, therefore ensuring a better accuracy. Finally, two illustrative examples are used to demonstrate the effectiveness and applicability of the proposed filter design scheme. Copyright © 2010 John Wiley & Sons, Ltd.
a b s t r a c t Moving horizon estimation (MHE) is an efficient optimization-based strategy for state estimation. Despite the attractiveness of this method, its application in industrial settings has been rather limited. This has been mainly due to the difficulty to solve, in real-time, the associated dynamic optimization problems. In this work, a fast MHE algorithm able to overcome this bottleneck is proposed. The strategy exploits recent advances in nonlinear programming algorithms and sensitivity concepts. A detailed anal-ysis of the optimality conditions of MHE problems is presented. As a result, strategies for fast covariance information extraction from general nonlinear programming algorithms are derived. It is shown that highly accurate state estimates can be obtained in large-scale MHE applications with negligible on-line computational costs.
We address the peak covariance stability of time-varying Kalman filter with possible packet losses in transmitting measurement outputs to the filter via a packet-based network. The packet losses are assumed to be bounded and driven by a finite-state Markov process. It is shown that if the observability index of the discrete-time linear time-invariant (LTI) system under investigation is one, the Kalman filter is peak covariance stable under no additional condition. For discrete LTI systems with observability index greater than one, a sufficient condition for peak covariance stability is obtained in terms of the system dynamics and the probability transition matrix of the Markov chain. Finally, the validity of these results is demonstrated by numerical simulations. Copyright © 2008 John Wiley & Sons, Ltd.
Moving-horizon estimation (MHE) is an optimization-based strategy for process monitoring and state estimation. One may view MHE as an extension for Kalman filtering for constrained and nonlinear processes. MHE, therefore, subsumes both Kalman and extended Kalman filtering. In addition, MHE allows one to include constraints in the estimation problem. One can significantly improve the quality of state estimates for certain problems by incorporating prior knowledge in the form of inequality constraints. Inequality constraints provide a flexible tool for complementing process knowledge. One also may use inequality constraints as a strategy for model simplification. The ability to include constraints and nonlinear dynamics is what distinguishes MHE from other estimation strategies. Both the practical and theoretical issues related to MHE are discussed. Using a series of example monitoring problems, the practical advantages of MHE are illustrated by demonstrating how the addition of constraints can improve and simplify the process monitoring problem.
In this paper, the robust filtering problem is studied for stochastic uncertain discrete time-delay systems with missing measurements. The missing measurements are described by a binary switching sequence satisfying a conditional probability distribution. We aim to design filters such that, for all possible missing observations and all admissible parameter uncertainties, the filtering error system is exponentially mean-square stable, and the prescribed performance constraint is met. In terms of certain linear matrix inequalities (LMIs), sufficient conditions for the solvability of the addressed problem are obtained. When these LMIs are feasible, an explicit expression of a desired robust filter is also given. An optimization problem is subsequently formulated by optimizing the filtering performances. Finally, a numerical example is provided to demonstrate the effectiveness and applicability of the proposed design approach.
This paper studies the H∞ filtering problem for networked discrete-time systems with random packet losses. The general multiple-input–multiple-output (MIMO) filtering system is considered. The multiple measurements are transmitted to the remote filter via distinct communication channels, and each measurement loss process is described by a two-state Markov chain. Both the mode-independent and the mode-dependent filters are considered, and the resulting filtering error system is modelled as a discrete-time Markovian system with multiple modes. A necessary and sufficient condition is derived for the filtering error system to be mean-square exponentially stable and achieve a prescribed H∞ noise attenuation performance. The obtained condition implicitly establishes a relation between the packet loss probability and two parameters, namely, the exponential decay rate of the filtering error system and the H∞ noise attenuation level. A convex optimization problem is formulated to design the desired filters with minimized H∞ noise attenuation level bound. Finally, an illustrative example is given to show the effectiveness of the proposed results.
This paper is concerned with the optimal linear estimation problem for linear discrete-time stochastic systems with multiple packet dropouts. Based on a packet dropout model, the optimal linear estimators including filter, predictor and smoother are developed via an innovation analysis approach. The estimators are computed recursively in terms of the solution of a Riccati difference equation of dimension equal to the order of the system state plus that of the measurement output. The steady-state estimators are also investigated. A sufficient condition for the convergence of the optimal linear estimators is given. Simulation results show the effectiveness of the proposed optimal linear estimators.
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.
This article considers moving horizon strategies for constrained linear state estimation. Additional information for estimating state variables from output measurements is often available in the form of inequality constraints on states, noise, and other variables. Formulating a linear state estimation problem with inequality constraints, however, prevents recursive solutions such as Kalman filtering, and, consequently, the estimation problem grows with time as more measurements become available. To bound the problem size, we explore moving horizon strategies for constrained linear state estimation. In this work we discuss some practical and theoretical properties of moving horizon estimation. We derive sufficient conditions for the stability of moving horizon state estimation with linear models subject to constraints on the estimate. We also discuss smoothing strategies for moving horizon estimation. Our framework is solely deterministic.
This paper studies the problem of H∞ filtering in networked control systems (NCSs) with multiple packet dropouts. A new formulation enables us to assign separate dropout rates from the sensors to the controller and from the controller to the actuators. By employing the new formulation, random dropout rates are transformed into stochastic parameters in the system’s representation. A generalized H∞-norm for systems with stochastic parameters and both stochastic and deterministic inputs is derived. The stochastic H∞-norm of the filtering error is used as a criterion for filter design in the NCS framework. A set of linear matrix inequalities (LMIs) is given to solve the corresponding filter design problem. A simulation example supports the theory.
This paper presents a new delay system approach to network-based control. This approach is based on a new time-delay model proposed recently, which contains multiple successive delay components in the state. Firstly, new results on stability and H∞ performance are proposed for systems with two successive delay components, by exploiting a new Lyapunov–Krasovskii functional and by making use of novel techniques for time-delay systems. An illustrative example is provided to show the advantage of these results. The second part of this paper utilizes the new model to investigate the problem of network-based control, which has emerged as a topic of significant interest in the control community. A sampled-data networked control system with simultaneous consideration of network induced delays, data packet dropouts and measurement quantization is modeled as a nonlinear time-delay system with two successive delay components in the state and, the problem of network-based H∞ control is solved accordingly. Illustrative examples are provided to show the advantage and applicability of the developed results for network-based controller design.
A moving horizon estimation (MHE) method of the constrained system with uncertain measurement output and norm-bounded parameter uncertainties is proposed. Based on the probability of known error of the output signal, the initial state variables and disturbance are estimated by minimizing performance object at each sampling moment. The estimation values of the state variables are calculated by the optimization theory of model predictive control (MPC). Simulation results show that MHE method has better ability than the method of the robust filter.
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.
We analyze the stability properties of an approximate algorithm for moving horizon estimation (MHE). The strategy provides instantaneous state estimates and is thus suitable for large-scale feedback control. In particular, we study the interplay between numerical approximation errors and the convergence of the estimator error. In addition, we establish connections between the numerical properties of the Hessian of the MHE problem and traditional observability denitions. We demonstrate the developments through a simulation case study.
This paper presents three novel moving-horizon estimation (MHE) methods for discrete-time partitioned linear systems, i.e., systems decomposed into coupled subsystems with non-overlapping states. The MHE approach is used due to its capability of exploiting physical constraints on states and noise in the estimation process. In the proposed algorithms, each subsystem solves reduced-order MHE problems to estimate its own state and different estimators have different computational complexity, accuracy and transmission requirements among subsystems. In all cases, proper tuning of the design parameters, i.e., the penalties on the states at the beginning of the estimation horizon, guarantees convergence of the estimation error to zero. Numerical simulations demonstrate the viability of the approach.
Moving-horizon (MH) state estimation is addressed for nonlinear discrete-time systems affected by bounded noises acting on system and measurement equations by minimizing a sliding-window least-squares cost function. Such a problem is solved by searching for suboptimal solutions for which a certain error is allowed in the minimization of the cost function. Nonlinear parameterized approximating functions such as feedforward neural networks are employed for the purpose of design. Thanks to the offline optimization of the parameters, the resulting MH estimation scheme requires a reduced online computational effort. Simulation results are presented to show the effectiveness of the proposed approach in comparison with other estimation techniques.
This note studies the problem of optimal H<sub>2</sub> filtering in networked control systems (NCSs) with multiple packet dropout. A new formulation is employed to model the multiple packet dropout case, where the random dropout rate is transformed into a stochastic parameter in the system's representation. By generalization of the H<sub>2</sub>-norm definition, new relations for the stochastic -norm of a linear discrete-time stochastic parameter system represented in the state-space form are derived. The stochastic H<sub>2</sub>-norm of the estimation error is used as a criterion for filter design in the NCS framework. A set of linear matrix inequalities (LMIs) is given to solve the corresponding filter design problem. A simulation example supports the theory.
In this paper, we consider estimation with lossy measurements. This problem can arise when measurements are communicated over wireless channels. We model the plant/measurement loss process as a Markovian jump linear system. While the time-varying Kalman estimator (TVKE) is known to be optimal, we introduce a simpler design, termed a jump linear estimator (JLE), to cope with losses. A JLE has predictor/corrector form, but at each time selects a corrector gain from a finite set of precalculated gains. The motivation for the JLE is twofold. The computational burden of the JLE is less than that of the TVKE and the estimation errors expected when using JLE provide an upper bound for those expected when using TVKE. We then introduce a special class of JLE, termed finite loss history estimators (FLHE), which uses a canonical gain selection logic. A notion of optimality for the FLHE is defined and an optimal synthesis method is given. The proposed design method is compared to TVKE in a simulation study.
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.
State estimator design for a nonlinear discrete-time system is a challenging problem, further complicated when additional physical insight is available in the form of inequality constraints on the state variables and disturbances. One strategy for constrained state estimation is to employ online optimization using a moving horizon approximation. We propose a general theory for constrained moving horizon estimation. Sufficient conditions for asymptotic and bounded stability are established. We apply these results to develop a practical algorithm for constrained linear and nonlinear state estimation. Examples are used to illustrate the benefits of constrained state estimation. Our framework is deterministic.