Nonlinearity and interconnections are ubiquitous in nature. For nonlinear dynamic networks, the current nonlinear control theory is insufficient to addressing the design challenges and the related engineering application problems. This workshop aims to report the new research results, made by leading researchers, that address some of the new emerging theoretical problems with nonlinear and distributed control, as well as their applications to practical systems.
The small-gain theorem is one of the fundamentally important and systematic tools in modern control theory. Its power in testing robustness of stability and designing robust controllers for interconnected and uncertain systems has clearly been demonstrated in the work of many others, ever since the publication of George Zames's pioneering work in 1966. Taking explicit advantage of Sontag's input-to-state stability (ISS), the first generalized, nonlinear ISS small-gain theorem was proposed by one of the authors in 1994. The nonlinear ISS small-gain theorem distinguishes itself from earlier small-gain theorems by providing a unified framework for internal stability and external stability of interconnected systems. More specifically, in using the ISS concept, the role of the initial conditions is made explicit while, at the same time, the input-output gains are allowed to be nonlinear functions of class K. Applications to a variety of control problems ranging from stabilization and robust adaptive control to decentralized or distributed control and output regulation (asymptotic tracking with disturbance rejection) have generated several novel tools for the design of robust nonlinear controllers. In the past ten years, renewed interest in large-scale nonlinear systems has motivated the further development of small-gain theorems toward a complete, network small-gain theory for network stability and control. This talk provides a survey of some recent developments of the nonlinear small-gain theory and its applications in networked control systems and event-based control subject to communications and computation constraints.
The notion of Input-to-State Stability (ISS), developed by E. D. Sontag in the late 1980s for systems described by ODEs, has enabled the solution of numerous robust nonlinear control problems that were previously inconceivable. No other foundational block in the analysis of forced and interconnected nonlinear systems is of comparable significance. Extensions of ISS have been carried out from ODEs to discrete-time systems, time-delay systems, stochastic nonlinear systems, and hybrid systems. The extension of the ISS to PDEs has remained elusive the longest because of the unbounded nature of the input operators in PDEs with boundary inputs.
One of the major advantages of ISS over other stability notions is the fact that the notion of ISS allowed the development of small-gain analysis. The presentation focuses on the application of the ISS results to systems containing at least one PDE. Many kinds of interconnections are studied: (1) one hyperbolic first-order PDE with ODEs, (2) one parabolic PDE with ODEs, (3) two hyperbolic first-order PDEs, (4) two parabolic PDEs, and (5) one hyperbolic PDE with one parabolic PDE. All kinds of interconnections are studied: interconnections by means of boundary or in-domain terms and interconnections by means of local or non-local terms. Explicit small-gain conditions that guarantee ISS or global exponential stability of the overall system in various spatial Lp norms (with 1 ≤ p < +∞) of the state are given. An emphasis is placed on the spatial sup norm, because it can give spatially pointwise estimates of the solution. Examples and applications that motivate the study of each interconnected system are also provided. These results open the door for a wide range of studies on stability of networks of PDEs.
Event-triggered control aims at improving communication efficiency for networked control systems by sending information only when certain event conditions are satisfied. In this talk, we investigate the consequences of actuator saturation and other nonlinearities on the behavior of the event-triggered control loop in terms of its stability and information exchange. Stability properties are derived showing how the stability of the event-triggered control loop depends on the selection of the event threshold. Is is also shown that a lower bound on the minimum inter-event time exist. As actuator saturation might severely degrade the performance of the event-triggered closed-loop system, the scheme is extended by incorporating an anti-windup mechanism in order to overcome this problem. The results are illustrated by simulations and experiments done with Swedish process control industry.
Significant progress has been made on nonlinear control systems in the past three decades. However, new system analysis and design tools that are capable of addressing more communication and networking issues are still highly desired to handle the emerging theoretical challenges underlying the new engineering problems. As an example, small quantization errors may cause the performance of a ``well-designed'' nonlinear control system to deteriorate. This motivates the recent development of new tools to address the arising theoretical problems. Based on Sontag's input-to-state stability (ISS), refined nonlinear small-gain theorems are able to take advantage of the structural feature of dynamic networks, and provide new solutions to the problems. The results are intended to help solve real-world nonlinear control problems, including quantized control and distributed control aspects.