Robust H∞ estimation of stationary discrete-time linear processes with stochastic uncertainties

The problem of H_∞ filtering of stationary discrete-time linear systems with stochastic uncertainties in the state space matrices is addressed, where the uncertainties are modeled as white noise. The relevant cost function is the expected value, with respect to the uncertain parameters, of the standard H_∞ performance. A previously developed stochastic bounded real lemma is applied that results in a modified Riccati inequality. This inequality is expressed in a linear matrix inequality form whose solution provides the filter parameters. The method proposed is applied also to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H₂/H_∞ filtering for the above system is also treated. The theory developed is demonstrated by a simple tracking example.     

Finite-time H∞ filtering for non-linear stochastic systems

This paper describes the robust H_∞ filtering analysis and the synthesis of general non-linear stochastic systems with finite settling time. We assume that the system dynamic is modelled by Itô-type stochastic differential equations of which the state and the measurement are corrupted by state-dependent noises and exogenous disturbances. A sufficient condition for non-linear stochastic systems to have the finite-time H_∞ performance with gain less than or equal to a prescribed positive number is established in terms of a certain Hamilton–Jacobi inequality. Based on this result, the existence of a finite-time H_∞ filter is given for the general non-linear stochastic system by a second-order non-linear partial differential inequality, and the filter can be obtained by solving this inequality. The effectiveness of the obtained result is illustrated by a numerical example.     

Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> filtering for a class of high-speed sampling uncertain systems

The problem of mixed H2/H∞ filtering for polytopic Delta operator systems is investigated. The aim is to design a linear asymptotically stable filter which guarantees that the filtering error system has different performances in different filtering channels. Based on a parameter-dependent Lyapunov function, a new mixed H2/H∞ performance criterion is presented. Upon this performance criterion, a sufficient condition for the full-order mixed H2/H∞ filter is derived in terms of linear matrix inequalities. The filter can be obtained from the solution of a convex optimization problem. The proposed filter design procedure is less conservative than the strategy based on the quadratic stability notion. A numerical example is given to illustrate the feasibility of the proposed approach.     

H ∞ filtering for discrete-time singular networked systems with communication delays and data missing

In this article, the problem of H _∞ filter design is investigated for discrete-time singular networked systems with both multiple stochastic time-varying communication delays and probabilistic missing measurements. Two kinds of stochastic time-varying communication delays, namely stochastic discrete delays and stochastic distributed delays, are simultaneously considered. The purpose of the addressed filtering problem is to design a filter such that, for the admissible random measurement missing and communication delays, the filtering error dynamics is asymptotically stable in the mean square with a prescribed H _∞ performance index. In terms of linear matrix inequality (LMI) method, a sufficient condition is established that ensures the asymptotical stability in the mean square with a prescribed H _∞ performance index of the filtering error dynamics and then the filter parameters are characterised by the solution to an LMI. A numerical example is introduced to demonstrate the effectiveness of the proposed design procedures.     

H∞ control design in bilinear systems: a tensor formal series approach

In this paper, the H_∞ disturbance attenuation problem of bilinear system is discussed. Dynamic game theory is used to solve this bilinear minimax problem. The solvability of H_∞ disturbance attenuation in bilinear system is also discussed. The techniques of tensor products and formal power series are employed to solve the nonlinear Bellman-Isaac differential equation. Furthermore, the convergence for the tensor formal series approach of this H_∞ control problem is discussed, and the radius of convergence for this H_∞ control to be well defined is also obtained.     

Reduced-order filtering for linear systems with Markovian jump parameters

This paper addresses the reduced-order H_∞ filtering problem for continuous-time Makovian jump linear systems, where the jump parameters are modelled by a discrete-time Markov process. Sufficient conditions for the existence of the reduced-order H_∞ filter are proposed in terms of linear matrix inequalities (LMIs) and a coupling non-convex matrix rank constraint. In particular, the sufficient conditions for the existence of the zero-order H_∞ filter can be expressed in terms of a set of strict LMIs. The explicit parameterization of the desired filter is also given. Finally, a numerical example is given to illustrate the proposed approach.     

Sampled-data H∞ filtering for Markovian jump singularly perturbed systems with time-varying delay and missing measurements

In this paper, sampled-data H_∞ filtering problem is considered for Markovian jump singularly perturbed systems with time-varying delay and missing measurements. The sampled-data system is represented by a time-delay system, and the missing measurement phenomenon is described by an independent Bernoulli random process. By constructing an ?-dependent stochastic Lyapunov–Krasovskii functional, delay-dependent sufficient conditions are derived such that the filter error system satisfies the prescribed H_∞ performance for all possible missing measurements. Then, an H_∞ filter design method is proposed in terms of linear matrix inequalities. Finally, numerical examples are given to illustrate the feasibility and advantages of the obtained results.     

An LMI approach to periodic discrete-time unbiased filtering

The problem of unbiased filtering for a discrete-time linear periodic system is faced by means of linear matrix inequality techniques. As leading case, we derive the synthesis conditions to obtain an unbiased filter and an unbiased fixed-lag smoother enforcing a bound on the H_∞ performance on the error dynamics.     

Fault detection for interval type-2 fuzzy stochastic systems with D stability constraint

This paper investigates the fault detection problem for interval type-2 (IT2) fuzzy stochastic systems with D stability constraint. In the design process, the constructed IT2 fuzzy stochastic system and fault detection filter use different membership functions and the number of fuzzy rules. The parameter uncertainties in the IT2 membership functions are captured through upper and lower membership functions. For relaxing the stability analysis and deriving the existence conditions of the fault detection filter that guarantee the mean-square asymptotically stable and H_∞ performance of the inferred IT2 fault detection system, the approach of dividing the state space and the values of upper and lower membership functions are exploited. Finally, simulation results are given to show the effectiveness of the presented results.     

Design of robust H ∞ filters for a class of uncertain nonlinear neutral stochastic systems with time delays

This article investigates the problem of robust H _∞ filtering for a class of nonlinear neutral stochastic time-delay systems with norm-bounded parameter uncertainties. The nonlinearities are assumed to satisfy the global Lipschitz conditions. By solving a set of certain linear matrix inequalities, an H _∞ filter is designed, which ensures both the robust stochastic stability and a prescribed H _∞ performance of the filtering error system for all admissible uncertainties. A numerical example is given to show the effectiveness of the design method proposed in this article.     

Linear set-membership state estimation with unknown but bounded disturbances

In this article, we consider the problem of discrete-time linear state estimation when at every discrete instant Δ the Euclidean norm of the discrete-time disturbance ‖w(Δ)‖₂ is bounded within some known value. Specifically, given a hypersphere that contains the uncertain disturbance signal w(Δ) and an ellipsoid containing the uncertain system state x(Δ) at time step Δ, a sub-optimal approach to computing a linear minimax filter which constructs a minimal ellipsoid to contain x(Δ?+?1) is derived. A distinct feature of our approach when compared to earlier solutions is that both the filter and the performance bound can be pre-computed off-line.     

H ∞ filtering for stochastic systems with time-varying delay

This article considers the problem of H _∞ filter design for stochastic systems with time-varying delay. The time delay is assumed to be of interval type. Attention is focused on the design of delay-dependent filters that guarantee the asymptotic stability in mean square and a prescribed noise attenuation level in an H _∞ sense for the filtering error dynamics. The delay-dependent H _∞ filter design scheme is proposed in terms of a linear matrix inequality. A numerical example is used to illustrate the effectiveness of the proposed approach.     

Mode-dependent quantised H∞ filtering for Markovian jump singular system

This paper considers the H_∞ filtering problem for Markovian jump singular systems. In addition, when taking into account the effect of the quantisation, the designed filter not only guarantee admissibility but also satisfy a prescribed H_∞ filtering performance for the filter error system. By using Lyapunov method, a new bounded real lemma is proposed in terms of linear matrix inequalities (LMIs) to ensure that the filtering error system has the mentioned properties. Furthermore, a sufficient condition for the existence of filter is obtained in terms of LMIs. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed approach.     

State-space formulae for all stabilizing controllers that satisfy an H∞-norm bound and relations to relations to risk sensitivity

Given a linear system, all stabilizing controllers such that a specified closed-loop transfer function has H_∞ norm less than a given scalar, are parametrized. This characterization involves the solution to two algebraic Riccati equations, each with the same order as the system, and further gives gives feasible controllers also with this order. The relationship to the risk-sensitive LQG stochastic control problem is established, giving an equivalence between robust and stochastic control.     

Decentralised H ∞ filtering of interconnected discrete-time fuzzy systems with time delays

This article considers the decentralised H _∞ filtering of interconnected discrete-time fuzzy systems with time delays based on piecewise Lyapunov–Krasovskii functionals. The fuzzy system consists of J interconnected time-delay discrete-time Takagi–Sugeno fuzzy subsystems and the decentralised H _∞ filter is designed for each subsystem. It is shown that the stability with H _∞ performance of overall filtering error system can be established if a piecewise Lyapunov–Kroasovskii functional can be constructed, and moreover, the functional can be obtained by solving a set of linear matrix inequalities that are numerically feasible. A simulation example is given to show the effectiveness of the presented approach.     

Event-triggered network-based ℓ1-gain filtering for positive linear systems

The ?₁-gain filtering problem of positive linear discrete-time systems based on networked communication is investigated in this paper. A filter system model in which the sampled signals are transmitted through the unreliable communication channels is constructed for a positive system. An event-triggered scheme in a linear form, which is different from the prior literatures, is designed to determine whether the signal packet should be transmitted to the filter or not. Network-induced delays are considered while handling the packet transmission. By using the linear Lyapunov function method, a sufficient condition to ensure the existence of the network-based positive filter satisfying ?₁-gain performance is proposed. The desired filter design method for the positive system is presented by using a linear programming approach. A numerical example with practical considerations is given to verify the proposed theoretical results.     

Convergent algorithms for frequency weighted L2 model reduction

This paper is concerned with computing an L₂-optimal reduced-order model for a given stable multivariable linear system in the presence of input and output frequency weightings. By parametrizing a class of reduced-order models in terms of an orthogonal projection and using manifold techniques as tools, both continuous and iterative algorithms are derived and their convergence properties are established. As an application, we show that an L₂ optimal reduced-order filter in the closed-loop sense can be computed using these algorithms.     

Analysis, design, and performance limitations of optimal filtering in the presence of an additional input with known frequency

In this paper, the inputs are considered to be of two types. The first type of input, as in standard H₂ optimal filtering, is a zero mean wide sense stationary white noise, while the second type is a linear combination of sinusoidal signals each of which has an unknown amplitude and phase but known frequency. The generalized H₂ optimal filtering problem seeks to find a linear stable filter that estimates a desired output such that the H₂ norm of the transfer matrix from the white noise input to the estimation error is minimized subject to the constraint that the mean of the error converges to zero for all initial conditions of the given system and filter and for all possible external sinusoidal signals. The analysis, design, and performance limitations of generalized H₂ optimal filters are presented here.     

Reduced-order ℓ2-ℓ∞ filtering for discrete-time T-S fuzzy systems with stochastic perturbation

This paper is concerned with the problem of ?₂-?_∞ filtering for discrete-time Takagi-Sugeno (T-S) fuzzy systems with stochastic perturbation. Firstly, a basis-dependent existence condition of desirable ?₂-?_∞ filters is proposed. Then by means of the convex linearisation technique, the derived condition is transformed into some strict linear matrix inequality (LMI) constraints, by which both full- and reduced-order filters can be designed. Moreover, for the reduced-order ?₂-?_∞ filter design, a novel analysis and design method with the projection lemma is also provided. Finally, a numerical example is provided to illustrate the feasibility and effectiveness of the proposed full- and reduced-order ?₂-?_∞ filter design methods.