Nature is nonlinear in general as the responses of physics and engineering systems are nonlinear. The approximate method is a simple and good tool to deal with nonlinear effects sometimes, but it usually damages the original characteristics of nonlinear systems and leads to inaccuracy, misunderstanding, or incorrect conclusions. The designer must be acquainted with the basic techniques available for considering nonlinear systems. He must be able to analyze the effects of unwanted nonlinearities in the system and to synthesize nonlinearities into the system to improve dynamic performance [1].

In the past few years, nonlinear control systems have experienced a growing popularity and these developments are motivated by extensive applications, in particular, to such area as mechanical systems, aircraft flight control systems, and electrical systems. A number of new ideas, approaches, and results have appeared in the field of nonlinear control systems.

For all control systems, stability is the primary requirement. One of the most widely used stability concepts in control theory is Lyapunov stability. Lyapunov stability theory was used extensively in system analysis and design, and many nonlinear system technologies are developed based on it.

The notions of energy and dissipation are related to Lyapunov theory and a theory for dissipative systems is developed in [2]. Energy and dissipation are the fundamental concepts in science and engineering practice, where it is common to view dynamical systems as energy-transformation devices. The control problem can then be recast as finding a dynamical system and an interconnection pattern to make the overall energy function take the desired form. This energy shaping approach is the essence of passivity-based control (PBC), a controller design technique that is very well known in mechanical systems and electrical systems now [3].

The notions of control Lyapunov functions and input-to-state stability for nonlinear control systems were introduced in [4]. The successes of this theory have been due in large part to its ability to analyze complicated structures on the basis of the behavior of elementary subsystems in a suitable input-output sense (stable, passive, etc.), in conjunction with the use of tools such as the small gain theorem to characterize interconnections. The theory has been applied to chemical process and biology.

Differential geometry has also been proved to be an effective means of analysis and design for nonlinear control systems, and the notions of controllability and observability of nonlinear systems were investigated [5]. Isidori introduced the notion of zero dynamics and gave the geometry control theory in [6]. There are many interesting applications of geometrical control theory, for example, aircraft flying at high angles of attack, walking robots, and quantum systems [1].

The estimation for some internal information or states naturally arises in control systems, as one cannot use sensors to measure the signals in some conditions. Then the problem of observer design is also important for nonlinear control systems. Many kinds of observer designs for nonlinear control systems and applications are shown in [7].

Another important research field for nonlinear control systems is to design controllers to deal with uncertainty, mainly due to lack of knowledge of the system parameters, modeling errors, and external disturbances. According to these requirements, adaptive control is developed to estimate the unknown parameters and has grown to be one of the richest fields in terms of algorithms, design techniques, analytical tools, and modifications [8, 9]. However, an adaptive scheme design for a disturbance-free plant model may go unstable in the presence of small disturbances; then robust adaptive control is developed, which can deal with the parameter uncertainties, modeling errors, and external disturbances [10].

Modern heuristic black-box type control approaches for nonlinear control systems, also called intelligent control, such as neural networks, machine learning, and fuzzy logic, have been used widely. It does not necessarily require an analytical model, and they are developed on the basis of data. These methodologies could be applied to big data research, which is very interesting research field in computer science.

Last but not least, a special kind of nonlinear control systems, hybrid dynamical system, exhibits characteristics of both continuous-time and discrete-time dynamical systems [11]. It actually arises in a great variety of applications, such as manufacturing systems, air traffic management, automotive engine control, and chemical process. Hybrid dynamical systems have also a central role in networked embedded control systems which interact with the physical world and human operators, like cyber-physical systems (CPS). So far, many theories and technologies for hybrid dynamical systems have been developed.

The field of nonlinear control systems has a bright future since there are many important and interesting challenges. The applications of nonlinear control systems, such as energy, health care, robots, biology, and big data research, will make the advanced theories and technologies be developed quickly. We hope that the readers of system theory will find their interesting research topics for nonlinear control systems in this special issue.

Zhitao Liu
Deqing Huang
Yifan Xing
Chuanke Zhang
Zhengguang Wu
Xiaofu Ji