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The multiobjective control and filtering problems for nonlinear stochastic systems with variance constraints are surveyed. First, the concepts of nonlinear stochastic systems are recalled along with the introduction of some recent advances. Then, the covariance control theory, which serves as a practical method for multi-objective control design as well as a foundation for linear system theory, is reviewed comprehensively. The multiple design requirements frequently applied in engineering practice for the use of evaluating system performances are introduced, including robustness, reliability, and dissipativity. Several design techniques suitable for the multi-objective variance-constrained control and filtering problems for nonlinear stochastic systems are discussed. In particular, as a special case for the multi-objective design problems, the mixed

It is widely recognized that, in almost all engineering applications, nonlinearities are inevitable and could not be eliminated thoroughly. Hence, the nonlinear systems have gained more and more research attention, and lots of results have been published. On the other hand, due to the wide appearance of the stochastic phenomena in almost every aspect of our daily life, stochastic systems which have found successful applications in many branches of science and engineering practice have stirred quite a lot of research interests during the past few decades. Therefore, the control and filtering problems for nonlinear stochastic systems have been studied extensively so as to meet ever-increasing demand toward systems with both nonlinearities and stochasticity.

In many engineering control/filtering problems, the performance requirements are naturally expressed by the upper bounds on the steady state covariance which is usually applied to scale the control/estimation precision, one of the most important performance indices of stochastic design problems. As a result, a large number of control and filtering methodologies have been developed to seek a convenient way to solve the variance-constrained design problems, among which the LQG control and Kalman filtering are two representative minimum variance design algorithms.

On the other hand, in addition to the variance constraints, real-world engineering practice also desires the simultaneous satisfaction of many other frequently seen performance requirements including stability, robustness, reliability, and energy constraints, to name but a few key ones. It gives the rise to the so-called multiobjective design problems, in which multiple cost functions or performance requirements are simultaneously considered with constraints being imposed on the system. An example of multiobjective control design would be to minimize the system steady-state variance indicating the performance of control precision, subject to a prespecified external disturbance attenuation level evaluating system robustness. Obviously, multiobjective design methods have the ability to provide more flexibility in dealing with the tradeoffs and constraints in a much more explicit manner on the prespecified performance requirements than those conventional optimization methodologies like LQG control scheme or

When coping with the multiobjective design problem with variance constraints for stochastic systems, the well-known covariance control theory provides us with a useful tool for the system analysis and synthesis. For linear stochastic systems, it has been shown that multiobjective control/filtering problems can be formulated using linear matrix inequalities (LMIs), due to their ability to include desirable performance objectives such as variance constraints,

In this paper, we focus mainly on the multiobjective control and filtering problems for nonlinear systems with variance constraints and aim to give a comprehensive survey on some recent advances in this area. The design objects (nonlinear stochastic system), design requirements (multiple performance specifications including variance constraints), several design techniques, and a special case of the addressed problem, mixed

Architecture of surveyed contents.

The rest of the paper is organized as follows. In Section

For several decades, nonlinear stochastic systems have been attracting increasing attention in the system and control community due to their extensive applications in a variety of areas ranging from communication and transportation to manufacturing, building automation, computing, automotive, and chemical industry, to mention just a few key areas. In this section, the analysis and synthesis problems for nonlinear systems and stochastic systems are recalled, respectively, and some recent advances in these areas are also given.

It is well recognized that in almost all engineering applications, nonlinearities are inevitable and could not be eliminated thoroughly. Hence, the nonlinear systems have gained more and more research attention, and lots of results have been reported; see, for example, [

On the other hand, in real-world applications, one of the most inevitable and physically important features of some sensors and actuators is that they are always corrupted by different kinds of nonlinearities, either from within the devices themselves or from the external disturbances. Such nonlinearities are generally resulting from equipment limitations as well as the harsh environments such as uncontrollable elements (e.g., variations in flow rates, temperature) and aggressive conditions (e.g., corrosion, erosion, and fouling) [

Recently, the system with randomly occurring nonlinearities (RONs) has started to stir quite a lot of research interests as it reveals an appealing fact that, instead of occurring in a deterministic way, a large quantity of nonlinearities in real-world systems would probably take place in a random way. Some of the representative publications can be discussed as follows. The problem of randomly occurring nonlinearities was raised in [

It should be emphasized that, for nonlinearities, there are many different constraints conditions for certain aim, such as Lipschitz conditions, among which the kind of stochastic nonlinearities described by statistical means has drawn particular research focus since it covers several well-studied nonlinearities in stochastic systems; see [

As is well known, in the past few decades, there have been extensive study and application of stochastic systems because the stochastic phenomenon is inevitable and cannot be avoided in the real-world systems. When modeling such kinds of systems, the way neglecting the stochastic disturbances, which is a conventional technique in traditional control theory for deterministic systems, is not suitable anymore. Having realized the necessity of introducing more realistic models, nowadays, a great number of real-world systems such as physical systems, financial systems, and ecological systems, as well as social systems, are more suitable to be modeled by stochastic systems, and therefore the stochastic control problem which deals with dynamical systems, described by difference or differential equations, and subject to disturbances characterized as stochastic processes has drawn much research attention; see [

Parallel to the control problems, the filtering and prediction theory for stochastic systems which aims to extract a signal from observations of signals and disturbances has been well studied and found widely applied in many engineering fields. It also plays a very important role in the solution of the stochastic optimal control problem. The research on filtering problem was originated in [

In this section, we first review the covariance control theory which provides us with a powerful tool in variance-constrained design problems with multiple requirements specified by the engineering practice. Then, we discuss several important performance specifications including robustness, reliability, and dissipativity. Two common techniques for solving the addressed problems for nonlinear stochastic systems are introduced. The mixed

As we have stated in the previous section, engineering control problems always require upper bounds on the steady state covariances [

On the other hand, it is always the case in real-world applications such as the tracking of a maneuvering target, that the filtering precision is characterized by the error variance of estimation [

It should be pointed out that most of the available literature regarding covariance control theory has been concerned with

When it comes to nonlinear stochastic systems, limited work has been done in the covariance-constrained analysis and design problems, just as what we have anticipated. A multiobjective filter has been designed in [

In the following, several performance indices originated from the engineering practice and frequently applied in multiobjective design problems are introduced.

In real-world engineering practice, various reasons such as variations of the operating point, aging of devices, and identification errors, would lead to the parameter uncertainties which result in the perturbations of the elements of a system matrix when modeling the system in a state-space form. Such a perturbation in system parameters cannot be avoided and would cause degradation (sometimes even instability) to the system performance. Therefore, in the past decade, considerable attention has been devoted to different issues for linear or nonlinear uncertain systems, and a great number of papers have been published; see [

On another research frontier of robust control, the

It is worth mentioning that, in contrast to the

In practical control systems especially networked control systems (NCSs), due to a variety of reasons including the erosion caused by severe circumstance, abrupt changes of working conditions, the intense external disturbance, and the internal physical equipment constraints and aging, the process of signal sampling and transmission has always confronted with different kinds of failures such as measurements missing, signal quantization, and sensor and actuator saturations. Such a phenomenon is always referred to as incomplete information, which would drastically degrade the system performance. In recent years, as requirements increase toward the reliability of engineering systems, the reliable control problem which aims to stabilize the systems accurately and precisely in spite of incomplete information caused by possible failures has therefore attracted considerable attention. In [

It should be pointed out that, for nonlinear stochastic systems, the relevant results of reliable control/filtering with variance constraints are relatively fewer, and some representative results can be summarized as follows. By means of linear matrix inequality approach, a reliable controller has been designed for nonlinear stochastic system in [

In recent years, the theory of dissipative systems, which plays an important role in system and control areas, has been attracting a great deal of research interests, and many results have been reported so far; see [

Although the dissipativity theory provides us a useful tool for the analysis of systems with multiple performance criteria, unfortunately, when it comes to nonlinear stochastic systems, few of the literature has been concerned with the multiobjective design problem for nonlinear stochastic systems, except [

The complexity of nonlinear system dynamics challenges us to come up with systematic design procedures to meet control objectives and design specifications. It is clear that we cannot expect one particular procedure to apply to all nonlinear systems; therefore, quite a lot of tools have been developed to deal with control and filtering problems for nonlinear stochastic systems, including T-S fuzzy model approximation approach, linearization, gain scheduling, sliding mode control, and backstepping, to name but a few key ones. In the sequel, we will investigate two nonlinear design tools that can be well combined with the covariance control theory for the purpose of providing a theoretical framework within which the variance-constrained control and filtering problems can be solved systematically for nonlinear stochastic systems.

The T-S fuzzy model approach occupies an important place in the study of nonlinear systems for its excellent capability in nonlinear system descriptions. Such a model allows one to perfectly approximate a nonlinear system by a set of local linear subsystems with certain fuzzy rules, thereby carrying out the analysis and synthesis work within the linear system framework. Therefore, T-S fuzzy model is extensively applied in both theoretical research and engineering practice of nonlinear systems; see [

In the past few decades, the sliding mode control (also known as variable structure control) problem originated in [

Along with the development of continuous-time sliding mode control theory, in recent years, as most control strategies are implemented in a discrete-time setting (e.g., networked control systems), the sliding mode control problem for discrete-time systems has gained considerable research interests, and a large amount of literature has appeared on this topic. For example, in [

As a special case of multiobjective control problem, the mixed

Parallel to the mixed

As far as nonlinear systems are concerned, the mixed

Very recently, the variance-constrained multiobjective control as well as filtering problem for nonlinear stochastic systems has been intensively studied, and some elegant results have been reported. In this section, we highlight some of the newest work with respect to this topic.

In [

For the stochastic system with nonlinearities of both the matched and unmatched forms, in [

In [

For the same type of nonlinear stochastic systems as mentioned above, in [

When the nonlinear stochastic system is time-varying, [

When it comes to the finite-horizon multiobjective filtering for time-varying nonlinear stochastic systems, [

In [

In this paper, the variance-constrained multiobjective control and filtering problems have been reviewed with some recent advances for nonlinear stochastic systems. Latest results on analysis and synthesis problems for nonlinear stochastic systems with multiple performance constraints have been surveyed. Based on the literature review, some related topics for the future research work are listed as follows.

In practical engineering, there are still some more complicated yet important kinds of nonlinearities that have not been studied. Therefore, the variance-constrained multiobjective control and filtering problems for more general nonlinear systems still remain open and challenging.

Another future research direction is to further investigate new performance indices (e.g., system energy constraints) that can be simultaneously considered with other existing ones. Also, variance-constrained multiobjective modeling, estimation, filtering, and control problems could be considered for more complex systems [

It would be interesting to study the problems of variance-constrained multiobjective analysis and design for large scale nonlinear interconnected systems that are frequently seen in modern industries.

A practical engineering application of the existing theories and methodologies would be the target tracking problem.

This work was supported in part by the National Natural Science Foundation of China under Grants 61304010, 61134009, 61329301, 61374127, and 61374039, the Royal Society of the UK, and the Alexander von Humboldt Foundation of Germany.

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