Chapter 1
Introduction

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 many results have been published. On the other hand, due to the wide appearance of stochastic phenomena in almost every aspect of our daily lives, stochastic systems that have found successful applications in many branches of science and engineering practice have stirred quite a lot of research interest during the past few decades. Therefore, control and filtering problems for nonlinear stochastic systems have been studied extensively in order to meet an 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 linear quadratic Gaussian(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, energy constraints, to name but a few key ones. This gives rise to the so-called multi-objective design problems, in which multiple cost functions or performance requirements are simultaneously considered with constraints being imposed on the system. An example of multi-objective control design would be to minimize the system steady-state variance indicating the performance of control precision, subject to a pre-specified external disturbance attenuation level evaluating system robustness. Obviously, multi-objective design methods have the ability to provide more flexibility in dealing with the tradeoffs and constraints in a much more explicit manner on the pre-specified performance requirements than those conventional optimization methodologies like the LQG control scheme or H8 design technique, which do not seem to have the ability of handling multiple performance specifications.

When coping with the multi-objective design problem with variance constraints for stochastic systems, the well-known covariance control theory provides us with a useful tool for system analysis and synthesis. For linear stochastic systems, it hasbeen shown that multi-objective control/filtering problems can be formulated using linear matrix inequalities (LMIs), due to their ability to include desirable performance objectives such as variance constraints, H2 performance, H8 performance, and pole placement as convex constraints. However, as nonlinear stochastic systems are concerned, the relevant progress so far has been very slow due primarily to the difficulties in dealing with the variance-related problems resulting from the complexity of the nonlinear dynamics. A key issue for the nonlinear covariance control study is the existence of the covariance of nonlinear stochastic systems and its mathematical expression, which is extremely difficult to investigate because of the complex coupling of nonlinearities and stochasticity. Therefore, it is not surprising that the multi-objective control and filtering problems for nonlinear stochastic systems with variance constraints have not been adequately investigated despite the clear engineering insights and good application prospect.

In this chapter, we focus mainly on the multi-objective control and filtering problems for nonlinear systems with variance constraints and aim to give a survey on some recent advances in this area. We shall give a comprehensive discussion from three aspects, i.e., design objects (nonlinear stochastic system), design requirements (multiple performance specifications including variance constraints), and several design techniques. Then, as a special case of the addressed problem, mixed H2/H8 design problems have been discussed in great detail with some recent advances. Subsequently, the outline of this book is given. The contents that are reviewed in this chapter and the architecture are shown in Figure. 1.1.

Figure 1.1 Architecture of surveyed contents.

1.1 Analysis and Synthesis of Nonlinear Stochastic Systems

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 industries, 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.

1.1.1 Nonlinear Systems

It is well recognized that in almost all engineering applications, nonlinearities are inevitable and could not be eliminated thoroughly. Hence, nonlinear systems have gained more and more research attention, and many results have been reported; see, for example, Refs [1-3]. When analyzing and designing nonlinear dynamical systems, there are a wide range of nonlinear analysis tools, among which the most common and widely used is linearization because of the powerful tools we know for linear systems. It should be pointed out that, however, there are two basic limitations of linearization [4]. (1) As is well known, linearization is an approximation in the neighborhood of certain operating points. Thus, the resulting linearized system can only show the local behavior of the nonlinear system in the vicinity of those points. Neither nonlocal behavior of the original nonlinear system far away from those operating points nor global behavior throughout the entire state space can be correctly revealed after linearization. (2) The dynamics of a nonlinear system are much richer than that of a linear system. There are essentially nonlinear phenomena, like finite escape time, multiple isolated equilibria, subharmonic, harmonic or almost periodic oscillations, to name just a few key ones that can take place only in the presence of nonlinearity; hence, they cannot be described by linear models [5-8]. Therefore, as a compromise, during the past few decades, there has been tremendous interest in studying nonlinear systems, with nonlinearities being taken as the exogenous disturbance input to a linear system, since it could better illustrate the dynamics of the original nonlinear system than the linearized one with less sacrifice of the convenience on the application of existing mathematical tools. The nonlinearities emerging in such systems may arise from the linearization process of an originally highly nonlinear plant or may be an external nonlinear input, which would drastically degrade the system performance or even cause instability; see, for example, Refs [9-11].

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 device themselves or from the external disturbances. Such nonlinearities generally result from equipment limitations as well as the harsh environments such as uncontrollable elements (e.g., variations in flow rates, temperature, etc.) and aggressive conditions (e.g., corrosion, erosion, and fouling, etc.) [12]. Since the sensor/actuator nonlinearity cannot be simply ignored and often leads to poor performance of the controlled system, a great deal of effort in investigating the analysis and synthesis problems has been devoted by many researchers to the study of various systems with sensor/actuator nonlinearities; see Refs [13-18].

Recently, the systems with randomly occurring nonlinearities (RONs) have started to stir quite a lot of research interest as it reveals an appealing fact that, instead of occurring in a deterministic way, a large amount 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 Ref. [19], where an iterative filtering algorithm has been proposed for the stochastic nonlinear system in the presence of both RONs and output quantization effects. The filter parameters can be obtained by resorting to solving certain recursive linear matrix inequalities. The obtained results have quickly been extended to the case of multiple randomly occurring nonlinearities [20]. Such a breakthrough on how to deal with nonlinear systems with RONs has been well recognized and quickly followed by other researchers in the area. Using similar techniques, the filtering as well as control problems have been solved for a wide range of systems containing nonlinearities that are occurring randomly, like Markovian jump systems [21, 22], sliding mode control systems [23], discrete-time complex networks [24], sensor networks [25], time-delay systems [26], and other types of nonlinear systems [27-29], which therefore has proven that the method developed in Ref. [19] is quite general and is applicable to the analysis and synthesis of many different kinds of...