Riccati-like Difference Equations Research Articles

Recursive state estimation for discrete-time nonlinear systems with event-triggered data transmission, norm-bounded uncertainties and multiple missing measurements

Summary In this paper, we consider the recursive state estimation problem for a class of discrete-time nonlinear systems with event-triggered data transmission, norm-bounded uncertainties, and multiple missing measurements. The phenomenon of event-triggered communication mechanism occurs only when the specified event-triggering condition is violated, which leads to a reduction in the number of excessive signal transmissions in a network. A sequence of independent Bernoulli random variables is employed to model the multiple measurements missing in the transmission. The norm-bounded uncertainties that could be considered as external disturbances which lie in a bounded set. The purpose of the addressed filtering problem is to obtain an optimal robust recursive filter in the minimum-variance sense such that with the simultaneous presence of event-triggered data transmission, norm-bounded uncertainties, and multiple missing measurements; the filtering error is minimized at each sampling time. By solving two Riccati-like difference equations, the filter gain is calculated recursively. Based on the stochastic analysis theory, it is proved that the estimation error is bounded under certain conditions. Finally, two numerical examples are presented to demonstrate the effectiveness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

On co-design of filter and fault estimator against randomly occurring nonlinearities and randomly occurring deception attacks

In this paper, the co-design problem of filter and fault estimator is studied for a class of time-varying non-linear stochastic systems subject to randomly occurring nonlinearities and randomly occurring deception attacks. Two mutually independent random variables obeying the Bernoulli distribution are employed to characterize the phenomena of the randomly occurring nonlinearities and randomly occurring deception attacks, respectively. By using the augmentation approach, the co-design problem of the robust filter and fault estimator is converted into the recursive filter design problem. A new compensation scheme is proposed such that, for both randomly occurring nonlinearities and randomly occurring deception attacks, an upper bound of the filtering error covariance is obtained and such an upper bound is minimized by properly designing the filter gain at each sampling instant. Moreover, the explicit form of the filter gain is given based on the solution to two Riccati-like difference equations. It is shown that the proposed co-design algorithm is of a recursive form that is suitable for online computation. Finally, a simulation example is given to illustrate the usefulness of the developed filtering approach.

A variance-constrained approach to recursive state estimation for time-varying complex networks with missing measurements

In this paper, the recursive state estimation problem is investigated for an array of discrete time-varying coupled stochastic complex networks with missing measurements. A set of random variables satisfying certain probabilistic distributions is introduced to characterize the phenomenon of the missing measurements, where each sensor can have individual missing probability. The Taylor series expansion is employed to deal with the nonlinearities and the high-order terms of the linearization errors are estimated. The purpose of the addressed state estimation problem is to design a time-varying state estimator such that, in the presence of the missing measurements and the random disturbances, an upper bound of the estimation error covariance can be guaranteed and the explicit expression of the estimator parameters is given. By using the Riccati-like difference equations approach, the estimator parameter is characterized by the solutions to two Riccati-like difference equations. It is shown that the obtained upper bound is minimized by the designed estimator parameters and the proposed state estimation algorithm is of a recursive form suitable for online computation. Finally, an illustrative example is provided to demonstrate the feasibility and effectiveness of the developed state estimation scheme.

Event-Based Recursive Distributed Filtering Over Wireless Sensor Networks

In this technical note, the distributed filtering problem is investigated for a class of discrete time-varying systems with an event-based communication mechanism. Each intelligent sensor node transmits the data to its neighbors only when the local innovation violates a predetermined Send-on-Delta (SoD) data transmission condition. The aim of the proposed problem is to construct a distributed filter for each sensor node subject to sporadic communications over wireless networks. In terms of an event indicator variable, the triggering information is utilized so as to reduce the conservatism in the filter analysis. An upper bound for the filtering error covariance is obtained in form of Riccati-like difference equations by utilizing the inductive method. Subsequently, such an upper bound is minimized by appropriately designing the filter parameters iteratively, where a novel matrix simplification technique is developed to handle the challenges resulting from the sparseness of the sensor network topology and filter structure preserving issues. The effectiveness of the proposed strategy is illustrated by a numerical simulation.

A recursive approach to non-fragile filtering for networked systems with stochastic uncertainties and incomplete measurements

In this paper, the non-fragile recursive filtering problem is investigated for a class of networked time-varying nonlinear systems with stochastic uncertainties and incomplete measurements. By employing a stochastic Kronecker delta function, the phenomena of the incomplete measurements are characterized in a unified framework which contain the randomly occurring signal quantization and the missing measurements. Based on the available probability information of the incomplete measurements, a new filtering compensation scheme is proposed to ensure that, for all stochastic uncertainties, incomplete measurements and stochastic perturbations of the filter gain, an upper bound of the filtering error covariance is guaranteed and such an upper bound is minimized by properly designing the filter gain at each sampling instant. It is shown that the desired filter gain can be obtained by solving two Riccati-like difference equations, and the proposed filtering algorithm is of a recursive form which is suitable for online applications. Finally, an illustrative example is provided to demonstrate the feasibility of the developed filtering approach.

Quantised recursive filtering for a class of nonlinear systems with multiplicative noises and missing measurements

This article is concerned with the recursive finite-horizon filtering problem for a class of nonlinear time-varying systems subject to multiplicative noises, missing measurements and quantisation effects. The missing measurements are modelled by a series of mutually independent random variables obeying Bernoulli distributions with possibly different occurrence probabilities. The quantisation phenomenon is described by using the logarithmic function and the multiplicative noises are considered to account for the stochastic disturbances on the system states. Attention is focused on the design of a recursive filter such that, for all multiplicative noises, missing measurements as well as quantisation effects, an upper bound for the filtering error covariance is guaranteed and such an upper bound is subsequently minimised by properly designing the filter parameters at each sampling instant. The desired filter parameters are obtained by solving two Riccati-like difference equations that are of a recursive form suitable for online applications. Finally, two simulation examples are exploited to demonstrate the effectiveness and applicability of the proposed filter design scheme.

Error variance-constrained ℋ∞ filtering for a class of nonlinear stochastic systems with degraded measurements: the finite horizon case

This article is concerned with the robust ℋ∞ filtering problem for a class of time-varying nonlinear stochastic systems with error variance constraint. The stochastic nonlinearities considered are quite general, which contain several well-studied stochastic nonlinear systems as special cases. The purpose of the filtering problem is to design a filter which is capable of achieving the pre-specified ℋ∞ performance and meanwhile guaranteeing a minimised upper-bounded on the filtering error variance. By means of the adjoint system method, a necessary and sufficient condition for satisfying the ℋ∞ constraint is first given, expressed as a forward Riccati-like difference equation. Then an upper-bound on the variance of filtering error system is given, guaranteeing the error variance is not more than a certain value at each sampling instant. The existence condition for the desired filter is established, in terms of the feasibility of a set of difference Riccati-like equations, which can be solved forward in time, hence is suitable for online computation. A numerical example is presented finally to show the effectiveness and applicability of the proposed method.

Extended Kalman filtering with stochastic nonlinearities and multiple missing measurements

In this paper, the extended Kalman filtering problem is investigated for a class of nonlinear systems with multiple missing measurements over a finite horizon. Both deterministic and stochastic nonlinearities are included in the system model, where the stochastic nonlinearities are described by statistical means that could reflect the multiplicative stochastic disturbances. The phenomenon of measurement missing occurs in a random way and the missing probability for each sensor is governed by an individual random variable satisfying a certain probability distribution over the interval [0,1]. Such a probability distribution is allowed to be any commonly used distribution over the interval [0,1] with known conditional probability. The aim of the addressed filtering problem is to design a filter such that, in the presence of both the stochastic nonlinearities and multiple missing measurements, there exists an upper bound for the filtering error covariance. Subsequently, such an upper bound is minimized by properly designing the filter gain at each sampling instant. It is shown that the desired filter can be obtained in terms of the solutions to two Riccati-like difference equations that are of a form suitable for recursive computation in online applications. An illustrative example is given to demonstrate the effectiveness of the proposed filter design scheme.

LQ optimal control for 2D Roesser models of finite extent

This paper investigates several aspects of linear–quadratic (LQ) optimal control for Roesser models over a two-dimensional (2D) signal-index set of finite extent. First, we consider the characterisation and computation of open-loop control laws when constraints on the system semi-states are imposed at both the south-west and north-east boundaries of the frame (i.e. signal-index set) of interest; by virtue of the quarter-plane causal structure of the Roesser model, the south-west and north-east boundary conditions are analogous to initial conditions and terminal constraints, respectively. A necessary and sufficient characterisation of optimality is obtained and explicitly computable formulae are derived to characterise the corresponding control inputs and performance index under reasonable assumptions on the problem data. In the second part of the paper, the problem of optimal LQ control via semi-state feedback is considered. A 2D Riccati-like difference equation is introduced to characterise, in a sufficient sense, a solution to this problem.

A control scheme for a class of discrete nonlinear stochastic systems

A control scheme is presented for effective stabilization of discrete-time, nonlinear stochastic systems where the nonlinearity involves a zero-mean, independent random sequence. The control is of a constant feedback type and makes use of finite time solutions of a Riccati-like matrix difference equation. Stability results are both in terms of mean square and almost sure stochastic stability. Moreover, a matrix inequality is given to check the existence of weighting matrices which would result in a stable regulator problem.

Constant feedback stabilization of discrete-time systems with random-coefficients†

A method is proposed to effectively stabilize discrete-time systems involving random coefficients. The stability results are both of sample solutions and second moments of system states. The method is handy in that any finite number of steps of a Riccati-like difference equation is sufficient to guarantee the construction of a stabilizing feedback gain. The results are given for cases both with and without the presence of an additive noise term. An easy-to-test stabilizability condition accompanies the method, enhancing its utility.

On discrete-time Riccati-like matrix difference equations with random coefficients

In the present paper it is shown that the optimal solution to the stochastic adaptive control of linear dynamic systems with uncertainties depends on the unconditional means and covariances of the uncertainties. The random Riccati-like systems of equations generated as a result of the presence of multiplicative and additive noise leads to the necessity of constraint on the means and covariances of the uncertainties. Furthermore, the existence and uniqueness of an optimal solution of the given system depends upon the above constraints : steady-state solutions exist only under restrictive statistical conditions of the system parameters