Discrete-time Nonlinear Stochastic Systems Research Articles

A Scalable Algorithm for Event-Triggered State Estimation With Unknown Parameters and Switching Topologies Over Sensor Networks

An event-triggered distributed state estimation problem is investigated for a class of discrete-time nonlinear stochastic systems with unknown parameters over sensor networks (SNs) subject to switched topologies. An event-triggered communication strategy is employed to govern the information broadcast and reduce the unnecessary resource consumption. Based on the adopted communication strategy, a distributed state estimator is designed to estimate the plant states and also identify the unknown parameters. In the framework of input-to-state stability, sufficient conditions with an average dwell time are established to ensure the boundedness of estimation errors in mean-square sense. In addition, the gains of the designed estimators are dependent on the solution of a set of matrix inequalities whose dimensions are unrelated to the scale of underlying SNs, thereby fulfill the scalability requirement. Finally, an illustrative simulation is utilized to verify the feasibility of the proposed design scheme.

Compositional construction of infinite abstractions for networks of stochastic control systems

This paper is concerned with a compositional approach for constructing infinite abstractions of interconnected discrete-time stochastic control systems. The proposed approach uses the interconnection matrix and joint dissipativity-type properties of subsystems and their abstractions described by new notions of so-called stochastic storage functions. The interconnected abstraction framework is based on new notions of so-called stochastic simulation functions, constructed compositionally using stochastic storage functions of components. Using stochastic simulation functions, one can quantify the distance between original interconnected stochastic systems and interconnected abstractions in the probabilistic setting. Accordingly, one can leverage the proposed results to perform analysis and synthesis over abstract interconnected systems, and then carry the results back over concrete ones. In the first part of the paper, we derive dissipativity-type compositional reasoning for the quantification of the distance in probability between the interconnection of stochastic control subsystems and that of their abstractions. Moreover, we focus on a class of discrete-time nonlinear stochastic control systems with independent noises in the abstract and concrete subsystems, and propose a computational scheme to construct abstractions together with their corresponding stochastic storage functions. In the second part of the paper, we consider specifications expressed as syntactically co-safe linear temporal logic formulae and show how a synthesized policy for the abstract system can be refined to a policy for the original system while providing a guarantee on the probability of satisfaction. We demonstrate the effectiveness of the proposed results by constructing an abstraction (totally 3 dimensions) of the interconnection of three discrete-time nonlinear stochastic control subsystems (together 222 dimensions) in a compositional fashion such that the compositionality condition does not require any constraint on the number or gains of the subsystems. We also employ the constructed abstraction as a substitute to synthesize a controller enforcing a syntactically co-safe linear temporal logic specification.

New Approach to General Nonlinear Discrete-Time Stochastic $H_\infty$ Control

In this paper, a new approach based on convex analysis is introduced to solve the $H_\infty$ problem for discrete-time nonlinear stochastic systems. A stochastic version of bounded real lemma is proved and the state feedback $H_\infty$ control is studied. Two examples are presented to show the effectiveness of our developed theory.

Event-based Distributed Filtering Approach to Nonlinear Stochastic Systems over Sensor Networks

In this paper, an event-triggered communication strategy and a distributed filtering scheme are designed for discrete-time nonlinear stochastic systems over wireless sensor networks (WSNs). The underlying system is represented by the Takagi-Sugeno (T-S) fuzzy model, and in addition by the description of the WSN under consideration. The structure of the WSN is established on a deterministic one. Based on an event-triggering condition tailored for each sensor, distributed fuzzy filters are established using the triggered measurements of the smart sensors. As a result, an augmented stochastic system is presented for the distributed filtering design. A robust mean-square asymptotic stability criterion is explored using the Lyapunov stability theory and the Disk stability constraint is applied to improve the performance of the distributed filters. An optimization solution to obtaining the parameters of the distributed filters is developed. Subsequently, a computer-simulated example helps to illustrate the validity of the proposed new filtering design techniques.

Forward–backward particle smoother for non‐linear systems with one‐step random measurement delay

This study is concerned with the state smoothing problem for a class of non-linear discrete-time stochastic systems with one-step random measurement delay (ORMD). The main contribution is that the forward–backward particle smoothing scheme is successfully extended to the systems with ORMD. First, the particle filter, specially designed to deal with ORMD, is implemented to obtain the filtering distribution and the joint distribution of state history. Then, by marginalising the joint distribution, the one-step fixed-lag smoothing distribution can be obtained. Finally, based on the forward–backward smoothing scheme, the particle approximation of the fixed-interval smoothing distribution can be obtained by re-weighting the particles which have been used in the foregoing one-step fixed-lag smoothing distribution. Simulation results demonstrate the effectiveness of the proposed smoother.

Input-to-State Stabilization in Probability for Nonlinear Stochastic Systems Under Quantization Effects and Communication Protocols.

In this paper, the observer-based stabilization problem is investigated for a class of discrete-time nonlinear stochastic networked control systems (NCSs) with exogenous disturbances. The signal transmission from the sensors to the observer is implemented via a shared digital network, in which both uniform quantization effect and stochastic communication protocol (SCP) are taken into account to reflect several network-induced constraints. The notion of input-to-state stability in probability is introduced to describe the dynamical behaviors of the closed-loop stochastic NCS that is effectively characterized by a general nonlinear stochastic difference equation with Markovian jumping parameters. A theoretical framework is first established to felicitate the dynamics analysis of the closed-loop system in virtue of the switched Lyapunov function method and the stochastic analysis techniques. By making full use of the quantized measurement output under the scheduling of the SCP, the existence conditions for an observer-based controller are established under which the closed-loop system is input-to-state stable in probability. Then, the explicit expression of the gain matrices of the desired controller is given by resorting to a set of feasible solutions of certain matrix inequalities. The effectiveness of the theoretical results is demonstrated by a numerical simulation example.

Security Control for Discrete-Time Stochastic Nonlinear Systems Subject to Deception Attacks

This paper is concerned with the security control problem with quadratic cost criterion for a class of discrete-time stochastic nonlinear systems subject to deception attacks. A definition of security in probability is adopted to account for the transient dynamics of controlled systems. The purpose of the problem under consideration is to design a dynamic output feedback controller such that the prescribed security in probability is guaranteed while obtaining an upper bound of the quadratic cost criterion. First of all, some sufficient conditions with the form of matrix inequalities are established in the framework of the input-to-state stability in probability. Then, an easy-solution version on above inequalities is proposed by carrying out the well-known matrix inverse lemma to obtain both the controller gain and the upper bound. Furthermore, the main results are shown to be extendable to the case of discrete-time stochastic linear systems. Finally, two simulation examples are utilized to illustrate the usefulness of the proposed controller design scheme.

Non-fragile – control for discrete-time stochastic nonlinear systems under event-triggered protocols

In this paper, the non-fragile – control problem is investigated for a class of discrete-time stochastic nonlinear systems under event-triggered communication protocols, which determine whether the measurement output should be transmitted to the controller or not. The main purpose of the addressed problem is to design an event-based output feedback controller subject to gain variations guaranteeing the prescribed disturbance attenuation level described by the – performance index. By utilizing the Lyapunov stability theory combined with S-procedure, a sufficient condition is established to guarantee both the exponential mean-square stability and the – performance for the closed-loop system. In addition, with the help of the orthogonal decomposition, the desired controller parameter is obtained in terms of the solution to certain linear matrix inequalities. Finally, a simulation example is exploited to demonstrate the effectiveness of the proposed event-based controller design scheme.

Feedback Controllers as Financial Advisors for Low-Income Individuals

Feedback controllers are introduced to help manage an individual’s or household’s financial life and build savings. The controllers can be viewed as financial advisors for an individual’s resource allocation problem, which is modeled as a nonlinear discrete time stochastic system with income uncertainties, asset losses, and constraints on cash flow and credit. We introduce a model predictive controller (MPC) and a proportional-integral-derivative (PID) controller, and compare them with a benchmark method employed in finance and economics, stochastic dynamic programming (DP). Both MPC and PID produce similar consistency in financial management compared with DP. They also offer the advantage of low computational complexity relative to DP, which allows us to efficiently perform assessments of robustness (reliability) and disturbance rejection (e.g., effects of uncertainties), both of which are of significant practical engineering importance. In addition, this flexibility enables us to uncover the system’s properties, such as the existence of a “poverty trap” caused by constraints in the control space and dynamics. The effectiveness of a PID-based aid intervention for low-skilled and low-endowed agents that lie within the trap is assessed and contrasted with other existing cash transfer programs, with results that support a further implementation analysis. These assessments constitute a novel application of feedback controllers that, besides effectively dealing with scarcity constraints, present practical advantages that are shown to translate into an ability to implement the MPC or PID controller in a variety of ways for low-income individuals via computer-assistive methods (e.g., a cell phone).

Particle filter for estimating multi-sensor systems using one- or two-step delayed measurements

In this paper, we focus on designing a particle filter for a class of nonlinear discrete-time stochastic systems, where the multi-sensor measurements can be randomly and asynchronously delayed by one- or two- sampling periods. Under the independence assumption of multi-sensor delays, asynchronous delay model is built by using a separate set of Bernoulli random variables to describe the relationship between the ideal measurement and the actual measurement for each sensor. Based on the model, a new weighting scheme for particles is derived with the measurement delay fully considered. By incorporating the weighting scheme into the particle filtering framework, we obtain a new filter for systems with delayed measurements. The performance of the proposed filter is demonstrated by two numerical examples.

Discrete-time controller for stochastic nonlinear polynomial systems with Poisson noises

ABSTRACTThis paper presents a solution to the optimal control problem for discrete-time stochastic nonlinear polynomial systems confused with white Poisson noises subject to a quadratic criterion. The solution is obtained in the following way: a nonlinear optimal controller is first developed for polynomial systems, considering the state vector completely available for control design. Then, based on the solution of the state estimation problem for polynomial systems with white Poisson noises, the state estimate vector is used in the control law to obtain a closed-form solution. Performance of this controller is compared to that of the controller employing the extended Kalman filter and the linear-quadratic regulator and the controller designed for polynomial systems confused with white Gaussian noises.

State estimators for systems with random parameter matrices, stochastic nonlinearities, fading measurements and correlated noises

Using the innovation analysis approach, the optimal linear state estimators, including the filter, predictor and smoother, in the linear minimum variance (LMV) sense are presented for a class of nonlinear discrete-time stochastic uncertain systems with fading measurements and correlated noises. Stochastic uncertainties of parameter matrices are depicted by correlated multiplicative noises. Stochastic nonlinearities are characterized by a known conditional mean and covariance. Different sensor channels have different fading measurement rates. The process and measurement noises are finite-step auto- and/or cross-correlated with each other. Two simulation examples verify the effectiveness of the proposed algorithms.

A novel cubature statistically linearized Kalman filter for fractional-order nonlinear discrete-time stochastic systems

In this paper, the state estimation problem for fractional-order nonlinear discrete-time stochastic systems is considered. A new method for the state estimation of fractional nonlinear systems using the statistically linearized method and cubature transform is presented. The fractional extended Kalman filter suffers from two problems. Firstly, the dynamic and measurement models must be differentiable and, secondly, nonlinearity is approximated by neglecting the higher order terms in the Taylor series expansion; by the proposed method in this paper, these problems can be solved using a statistically linearized algorithm for the linearization of fractional nonlinear dynamics and cubature transform for calculating the expected values of the nonlinear functions. The effectiveness of this proposed method is demonstrated through simulation results and its superiority is shown by comparing our method with some other present methods, such as the fractional extended Kalman filter.

Multi-constrained fuzzy intelligent control for uncertain discrete systems with complex noises: an application to ship steering systems

ABSTRACTIn this paper, a performance constrained fuzzy controller design is investigated for the nonlinear uncertain discrete-time ship steering system with complex noises. The performance constraints considered in this paper include individual state variance constraint and passivity constraint. The nonlinear discrete-time ship steering system considered in this paper is represented by a discrete-time Takagi–Sugeno fuzzy model with perturbations, additional noise and multiplicative noise. According to this complex nonlinear discrete-time stochastic system, a fuzzy controller design approach is investigated based on Lyapunov theory, passivity theory and covariance control theory. Some sufficient conditions are derived to satisfy stability constraint, individual state variance constraint and passivity constraint, simultaneously. These sufficient conditions are constructed as linear matrix inequality forms that can be solved by the convex optimal programming algorithm. Finally, some simulation results are provided to verify the validity and applicability of the proposed fuzzy control approach with multiple performance constraints.

Stabilisation and control of a class of discrete-time nonlinear stochastic output-dependent system with random missing measurements

ABSTRACTControl and stabilisation of a class of discrete-time output-dependent nonlinear stochastic system subject to missing measurements is tackled in this article. First, a Bernoulli-distributed white sequence satisfying a known conditional probability distribution is introduced into the output measurement equation to describe the missing measurement phenomenon. Then, an observer-based controller is designed by employing the backstepping technique and the stability of the resulting closed-loop system is investigated by utilising the Lyapunov stability theory. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed control scheme.

Robust Stability, Stabilization, andH∞Control of a Class of Nonlinear Discrete Time Stochastic Systems

This paper studies robust stability, stabilization, andH∞control for a class of nonlinear discrete time stochastic systems. Firstly, the easily testing criteria for stochastic stability and stochastic stabilizability are obtained via linear matrix inequalities (LMIs). Then a robustH∞state feedback controller is designed such that the concerned system not only is internally stochastically stabilizable but also satisfies robustH∞performance. Moreover, the previous results of the nonlinearly perturbed discrete stochastic system are generalized to the system with state, control, and external disturbance dependent noise simultaneously. Two numerical examples are given to illustrate the effectiveness of the proposed results.

Robust Stability and Stabilization of a Class of Uncertain Nonlinear Discrete-Time Stochastic Systems with Interval Time-Varying Delays

This paper deals with the problems of the robust stochastic stability and stabilization for a class of uncertain discrete-time stochastic systems with interval time-varying delays and nonlinear disturbances. By utilizing a new Lyapunov-Krasovskii functional and some well-known inequalities, some new delay-dependent criteria are developed to guarantee the robust stochastic stability of a class of uncertain discrete-time stochastic systems in terms of the linear matrix inequality (LMI). Then based on the state feedback controller, the delay-dependent sufficient conditions of robust stochastic stabilization for a class of uncertain discrete-time stochastic systems with interval time-varying delays are established. The controller gain is designed to ensure the robust stochastic stability of the closed-loop system. Finally, illustrative examples are given to demonstrate the effectiveness of the proposed method.

Output-Feedback Control for Nonlinear Stochastic Systems With Successive Packet Dropouts and Uniform Quantization Effects

In this paper, the dynamic output-feedback control problem is studied for a class of discrete-time nonlinear stochastic systems with successive packet dropouts and uniform quantization effects. The stochastic system under investigation involves state-, control-, and disturbance-dependent noises (also called ( ${x}$ , ${u}$ , ${v}$ )-dependent noises) that bring in substantial difficulties in the stability analysis. The phenomenon of successive packet dropouts is governed by a binary switching random sequence. The measurement output is subject to the uniform quantization which results in the norm-bounded disturbances, and the concept of input-to-state stability in probability is introduced to deal with this kind of disturbances. In virtue of intensive stochastic analysis, several sufficient conditions are established to guarantee that the closed-loop system is input-to-state stable in probability under the effects of probabilistic packet dropouts as well as uniform quantizations. As an easy consequence, the design problem with linear output-feedback controllers is discussed for the benefits of practical applications and some simplified conditions are derived. Finally, a numerical example is presented to illustrate the effectiveness of the proposed method.

Event-triggered consensus control for discrete-time stochastic multi-agent systems: The input-to-state stability in probability

This paper is concerned with the event-triggered consensus control problem for a class of discrete-time stochastic multi-agent systems with state-dependent noises. A novel definition of consensus in probability is proposed to better describe the dynamics of the consensus process of the addressed stochastic multi-agent systems. The measurement output available for the controller is not only from the individual agent but also from its neighboring ones according to the given topology. An event-triggered mechanism is adopted with hope to reduce the communication burden, where the control input on each agent is updated only when a certain triggering condition is violated. The purpose of the problem under consideration is to design both the output feedback controller and the threshold of the triggering condition such that the closed-loop system achieves the desired consensus in probability. First of all, a theoretical framework is established for analyzing the so-called input-to-state stability in probability (ISSiP) for general discrete-time nonlinear stochastic systems. Within such a theoretical framework, some sufficient conditions on event-triggered control protocol are derived under which the consensus in probability is reached. Furthermore, both the controller parameter and the triggering threshold are obtained in terms of the solution to certain matrix inequalities involving the topology information and the desired consensus probability. Finally, a simulation example is utilized to illustrate the usefulness of the proposed control protocol.

Mixed sliding mode fuzzy control for discrete‐time non‐linear stochastic systems subject to variance and passivity constraints

In this study, a novel multi-objective sliding mode fuzzy control technique is presented for a class of discrete-time non-linear stochastic systems that are represented by Takagi–Sugeno (T–S) fuzzy models. Subject to variance and passivity constraints, parallel distributed compensation based sliding mode fuzzy control approach is investigated for the discrete-time T–S fuzzy models. Combining Lyapunov theory, passivity theory and covariance theory, sufficient conditions are derived to satisfy stability, passivity and variance constraints. The iterative linear matrix inequality algorithm is employed in this study to solve these sufficient conditions. The contribution of this study is to develop a novel sliding mode fuzzy control method such that the closed-loop discrete-time non-linear stochastic system achieves passivity constraint and individual state variance constraints, simultaneously. At last, the simulations of controlling a discrete-time non-linear ship steering system are given in a numerical example to demonstrate the effectiveness and usefulness of proposed sliding mode fuzzy control methodology.