Nonlinear control systems - A brief overview of historical and recent advances <s> Historical Perspective <s> We show that the well-known Lyapunov sufficient condition for "input-to-state stability" (ISS) is also necessary, settling positively an open question raised by several authors during the past few years. Additional characterizations of the ISS property, including one in terms of nonlinear stability margins, are also provided. <s> BIB001 </s> Nonlinear control systems - A brief overview of historical and recent advances <s> Historical Perspective <s> This article presents a review of some of the early developments in nonlinear control engineering. It begins by briefly examining the status prior to World War II and then concentrates on the developments in approximately the following two decades. A significant amount of the work will be seen to have as its foundation the need to obtain methods to solve wartime problems related to servomechanisms, fire control, and missile control. The major analytical techniques of the phase plane, the describing function and Tsypkin's method for relay systems, are discussed at length. Two important aspects in obtaining control engineering solutions to problems are the existence of simulation facilities and the capabilities of the available hardware to implement specific solutions at an allowable cost. A brief review of some of the early papers on nonlinear control is also given, and finally some comments are provided on the continuing relevance of these early methods and some still unsolved problems. <s> BIB002 </s> Nonlinear control systems - A brief overview of historical and recent advances <s> Historical Perspective <s> In the early days of nonlinear control theory most of the stability, optimality and uncertainty concepts were descriptive rather than constructive. This survey describes their 'activation' into design tools and constructive procedures. Structural properties of nonlinear systems, such as relative degree and zero dynamics, are connected to passivity, while dissipativity, as a finite L"2-gain property, also appears in the disturbance attenuation problem, a nonlinear counterpart of robust linear control. Passivation-based designs exploit the connections between passivity and inverse optimality, and between Lyapunov functions and optimal value functions. Recursive design procedures, such as backstepping and forwarding, achieve certain optimal properties for important classes of nonlinear systems. The survey concludes with four representative applications. The selection of the topics and their interpretations are greatly influenced by the experience and personal views of the senior author. <s> BIB003 </s> Nonlinear control systems - A brief overview of historical and recent advances <s> Historical Perspective <s> 1State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou 310027, China 2Department of Aeronautics, Imperial College London, London SW7 2AZ, UK 3Frick Chemistry Laboratory, Princeton University, Princeton, NJ 08544, USA 4Department of Electrical Engineering and Electronics, The University of Liverpool, Liverpool L69 3GJ, UK 5Department of Electrical Engineering, School of Electrical and Information Engineering, Jiangsu University, Zhenjiang 212013, China <s> BIB004
History of nonlinear control dates back to 18 th century when centrifugal "flyball" governor was invented to control steam engine by regulating the admission of steam into the cylinder. However, the governor was made to work without concrete analytical concepts BIB002 . In 1892, A.M. Lyapunov, a Russian mathematician, presented two methods in order to determine the stability of dynamic systems described by Ordinary Differential Equations (ODE). The second method called as direct method of Lyapunov, can determine stability without actually solving the ODE and thus finds potential in stability analysis of nonlinear control systems  . Lyapunov showed that if the linear approximation of a system is stable near an equilibrium point, then the truly nonlinear system will be stable for some neighborhood of that point. Lyapunov's work was translated into French in 1907 and recovered in control context later by Kalman and Bertram. Other noticeable works which benefited nonlinear control include Van der Pol's research on electronic oscillations and Duffing's findings on nonlinear vibrations. Based on the proposed nonlinear second order equations, approaches were developed to predict various phenomena of nonlinear systems, which include subharmonic oscillations, limit cycles, jump phenomena and frequency entrainment. As a subject, control engineering was in its infancy till late 1930s when scientific community started to face the problem of servomechanisms control  . Such mechanisms, approximated by second order systems, were dealt with the phase plane method, introduced by Poincaré. The event of Second World War boosted research in nonlinear control of servomechanisms due to the functional requirements imposed by fire-control systems and control of guided vehicles. During 1940-1960, three main analytical approaches used for analyzing nonlinear systems include; the describing function, the phase plane method and various methods involving relay systems. In the classical era, most problems involved single input, single output, linear, finite dimensional and time-invariant systems. Year 1960 is considered as start of modern era for nonlinear control BIB002 . This is the same time when the first International Federation of Automatic Control (IFAC) congress was held in Moscow, which brought theory of mechanics and Nyquist-Bode feedback theory closer to create to-day's control theory. The two key application drivers during this time were defense and space race. Other industrial avenues where nonlinear control were applied include automobiles, ships, steel, paper, minerals etc. The nonlinear, time varying, highly dimensional, poorly modelled and multivariable nature of the encountered real systems were outside the bounds of classical control theory. The digital computer was first introduced as a design tool and later as a component of a control system. In early 1970s, scientists investigated that the notions of energy and dissipation are linked with Lyapunov theory. So, dynamic systems can be viewed as energy transformation mechanisms. Based on this concept, Willems Jan proposed a theory for dissipative systems  . In 1990s, Sontag and Wang proposed theory of input-to-state stability for nonlinear control systems BIB001 , which can analyze stability of complex structures based on behavior of elementary subsystems and has been successfully applied to biological and chemical processes BIB004 . In 1995, Isidori presented geometric control theory by introducing the concept of zero dynamics  . With the ability to analyze controllability and observability, differential geometry finds enormous potential in the domain of nonlinear control systems. A critical review of early history of nonlinear control shows that concepts related with optimality, stability and uncertainty were descriptive rather than constructive. 1990s is considered as decade of 'activation process' when the descriptive research was being transformed into formal design methods through constructive procedures BIB003 . Table 1 summarizes prominent historical advances which directly or indirectly enriched the domain of nonlinear control systems.