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This paper presents an overview of the recent developments in the gain-scheduled control and filtering problems for the parameter-varying systems. First of all, we recall several important algorithms suitable for gain-scheduling method including gain-scheduled proportional-integral derivative (PID) control,

During the past decades, the gain-scheduling control and filtering approach has received a lot of attention from the control community. This stems from the fact that the gain-scheduling method is more effective than the traditional ones to cope with the unavoidable nonlinearities and time-varying dynamics of the practical plant. Control and filtering problems are two fundamental issues in control theory and have been intensively discussed by a great number of researchers. The control problems can be classified into two types: state feedback control and output feedback control according to the controller structure. The state feedback controller is based on the state information of the systems, while the purpose of output feedback is to design a controller in terms of measurement output rather than the state of a given system. On the other hand, the general idea of filtering problems is to form a kind of “best estimate” for the true state of some certain system by some potentially noisy observations.

In fact, gain-scheduling is a broad notion that gives rise to many different design ideas, for example, precompensating a nonlinear gain with the inverse gain function, switching gain values according to operating conditions or even according to preset times, controller switching and controller blending, and so on. Therefore, the gain-scheduling idea has been extensively applied to design controllers and filters for many kinds of systems, such as nonlinear stochastic systems with time-varying parameters, T-S fuzzy systems, linear parameter-varying systems. and Markov jumping systems. For these kinds of systems, the algorithms and performance indices have benefited from the gain-scheduling ideas, such as the gain-scheduling proportional-integral-derivative (PID) method, the fuzzy gain-scheduling approach. and the

The randomly occurring incomplete information in system models has been extensively studied by a lot of researchers and many important results have been published; see for example, [

In this paper, we mainly focus on the gain-scheduled control and filtering problems for parameter-varying systems and aim to give a survey on some recent advances in this area. We introduce some important gain-scheduling algorithms, such as gain-scheduling PID control algorithms, fuzzy gain-scheduling methods, and

The remainder of this paper is outlined as follows. In Section

In control area, PID control strategy offers a simple yet efficient solution to many real-world control problems and has become the most widely used control method [

Control performance criteria are key elements in control theory. The most fundamental control objectives are quite naturally and effectively expressed as the norm of certain signals in the control loops. The

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Under guaranteed

Many mathematical models for real-world phenomena are inherently nonlinear, and the stability analysis and synthesis problems for nonlinear systems are generally difficult. To facilitate the mathematical analysis, in the literature, some stringent assumptions have been imposed on the nonlinearities, such as smoothness and Lipschitz continuity, which have inevitably led to considerable conservatism. As an alternative approach, in the past few decades, the fuzzy logic theory has been demonstrated to be effective in dealing with a variety of complex nonlinear systems, which has therefore received a great deal of attention in the literature; see for example, [

The fuzzy controller can be described as

In [

This section gives a systematic overview of recent advances on several common systems, for which the gain-scheduling technique is suitable to design controllers and filters. In general, these systems can be categorized as linear parameter-varying systems, stochastic nonlinear systems, networked control systems, and Markovian jump systems. In the following, we will take a deep investigation of these systems one by one in order to inspire more research interest.

Linear parameter-varying (LPV) systems are a very special type of systems, whose state-space system matrices are functions dependent on unknown but measurable time-varying parameters, and the measurements of these parameters provide real-time information according to the variations of the plant’s characteristics. In the past few years, the research on LPV systems has become a promising work from both theoretical and engineering viewpoints. For example, in [

It should be noticed that, in order to design an appropriate controller/filter for the LPV systems, the gain-scheduling approach has been proven to be an effective one in this process. The idea of gain-scheduling approach is to design controller/filter gains as functions of the scheduling parameters, which are supposed to be available in real time and can be utilized to adjust the controller/filter with hope to get the best performance of the system. Therefore, gain-scheduled control and filtering problems for LPV systems have stirred a great deal of interest in these years; see, for example, [

On the other hand, for the purpose of designing a controller/filter with less conservatism for the LPV systems, it is natural to construct novel Lyapunov functions with scheduling parameters, which are usually called parameter-dependent Lyapunov functions. Very recently, the parameter-dependent Lyapunov function approach has been applied in the gain-scheduled control/filtering problems so as to achieve better control/filtering performance requirements and some results have been reported in the literature [

Owing to pervasive existence of stochastic perturbations in reality, stochastic models have been successfully utilized to describe many practical systems such as mechanical systems, economic systems, and biological systems. Over the past few decades, the study of stabilization, control, and filtering problems for stochastic systems has been paid much attention by many researchers and a large number of results have been obtained in the literature; see, for example, [

Besides, since nonlinearity is inevitable in most real-world systems, it is not surprising that analysis and synthesis of stochastic nonlinear systems have attracted increasing research attention, and some latest results have been published; see, for example, [

Recently, the gain-scheduled control and filtering problems for stochastic nonlinear systems have attracted increasing attention from a variety of engineering areas. For instance, in [

In recent years, with the various applications of networks in the complex dynamical processes such as advanced aircraft, spacecraft, and automotive and manufacturing processes, the networked control systems (NCSs) have attracted much attention owing to low cost, high reliability, reduced weight and power requirements, simple installation and maintenance, and decreasing the hard wiring and implementation difficulties. NCSs are typically made up of sensors, actuators, and controllers, which communicate with a shared network. Review papers about NCSs can be found in [

Network-induced delay and packet dropout are key features of NCSs. Because of the devices connected to the shared medium, the transmission capacity of the communication network is usually limited, which in turn affects the number of bits or packets per second transported via the network. Consequently, the networked-induced delays and packet losses have become unavoidable and constitute the main causes for degrading the achievable performance of the networked systems. Therefore, in the past decade, the filtering and control problems for NCSs with communication delays and/or missing measurements have been extensively considered by many researchers; see, for example, [

As an important method, gain-scheduling can also be applied in the NCSs. In [

Markovian jump systems are the hybrid systems with two components in the state [

The jump systems have the advantage of modeling the dynamic systems subject to abrupt variation in their structures caused by component failures or repairs, sudden environmental disturbance, changing subsystem interconnections, or operating in different points of a nonlinear plant, which often take place in a lot of dynamics systems [

Gain-scheduling method can also be applied to analyze the Markovian jump systems. In [

The randomly occurring incomplete information has recently raised a great deal of interest within the control community; it refers to these phenomena appearing in a random way based on a certain kind of probabilistic law which mainly caused by some environment reasons, such as random failures and repairs of the components, and intermittently switching in the interconnections of subsystems. These randomly occurring incomplete information phenomena under consideration mainly include missing measurements [

As we all know, the Bernoulli distribution model is perhaps the most effective one to be utilized in different systems (e.g., time-delay systems [

As we have introduced before, the gain-scheduling approach is one of the most popular ways to design a controller or filter, whose gains can be updated by a set of tuning parameters in order to optimize the closed-loop system’s performance in time. On the other hand, the randomly occurring incomplete information often occurs with time-varying probabilities, which also can be considered as a tuning parameter for the controller or filter. Under such considerations, a novel gain-scheduling approach, namely, probability-dependent gains-scheduling approach, has been proposed to deal with systems with randomly occurring incomplete information.

It is noteworthy that by utilizing the probability-dependent gains-scheduling approach, the designed gain-scheduled controller/filter has not only the constant part but also the time-varying part which can be scheduled on-line according to the corresponding time-varying probability parameters; therefore, it will naturally lead to less conservatism than the conventional ones with fixed gains only. Associated with that, the probability-dependent Lyapunov functional has also been constructed in a sense that it can reduce the potential conservatism. With the development of the related research in the past several years, the probability-dependent gains-scheduled controller/filter has turned out to be a very useful tool to cope with system with randomly occurring incomplete information.

Since firstly introduced in [

The probability-dependent gains-scheduling approach has been firstly proposed in [

The study on two-dimensional (2D) systems has recently attracted considerable attention due to their extensive applications in many engineering fields such as thermal process in chemical reactors, multidimensional digital filtering, and electron heating systems, [

The polynomial nonlinear systems are a rather general class of nonlinear systems. It is mainly about the nonlinearity disturbance in the nonlinear systems which can be approximated by polynomials via the Taylor expansion centered on the point we are interested in, and the introduced conservatism that came from the approximation error can be reduced by increasing the degree of the polynomials. Recently, the control and filtering issues for polynomial nonlinear systems have attracted some initial research attention. For instance, a nonlinear gain-scheduling output-feedback control problem has been addressed in [

Recently, in the gain-scheduled paradigm, the scheduling parameters with uncertainties have received scattered attention. For instance, in [

A limitation of the original gain-scheduling approaches is that the closed-loop stability can only be assured when the underlying parameters vary sufficiently slowly. A remedy exists but requires the implementation of possible solution of asymptotic Riccati equations (AREs) for an infinite number of different parameter values and the on-line solution of a Riccati differential equation (RDE) with time-varying coefficient matrices. The method in [

There is a long history of gain-scheduling in applications areas. Recently, in [

In this paper, we have summarized some recent advances on the gain-scheduled control and filtering for several kinds of systems with randomly occurring incomplete information. Several kinds of techniques related to gain-scheduled control and filtering algorithms have been surveyed. Next, the research and development of various system models have been reviewed, such as stochastic nonlinear systems, networked control systems, and linear parameter-varying systems. Furthermore, the probability-dependent gain-scheduled control and filtering problems for systems with randomly occurring incomplete information have been introduced. To conclude this survey paper, based on the literature review, some related topics for the future research work are listed as follows.

In engineering applications, there still exist many more complex yet important randomly occurring issues which, however, have not been considered. Therefore, it would be a promising research topic to discuss these new phenomena of incomplete information and establish a unified measurement model accounting for these issues simultaneously.

In the existing literatures, a lot of results are based on the LMI conditions. While the interior-point LMI solvers are significantly faster than the classical convex optimization algorithms, it should be kept in mind that the complexity of LMI computations remains higher than that of solving, for example, a Riccati equation. For instance, problems with thousand design variables typically take over an hour on today’s workstations. Therefore, another future research direction is to reduce the computation cost while keeping the desired performances.

In case that multiple randomly occurring incomplete information appear simultaneously and influence each other in the same systems, the probability-dependent gain-scheduling analysis and synthesis problem is a challenge, which constitutes one of the future research topics.

When there is randomly occurring incomplete information in two-dimensional system, the probability-dependent gain-scheduled control and filtering problems are potential research topics.

An additional trend for future research is to discuss the probability-dependent gain-scheduled synchronization, control, and filtering problems for nonlinear stochastic complex networks with randomly occurring incomplete information.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported in part by the National Natural Science Foundation of China under Grants 61074016, 61374039, 61304010, and 61329301; the Natural Science Foundation of Jiangsu Province of China under Grant BK20130766; the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning; the Program for New Century Excellent Talents in University under Grant NCET-11-1051, the Leverhulme Trust of the U.K., the Alexander von Humboldt Foundation of Germany.