Input-to-state stability and stabilization of distributed parameter systems

• Applicants: Professor Dr. Sergey Dashkovskiy; Professorin Dr. Birgit Jacob; Professor Dr. Fabian Wirth
• Subject Area: Mathematics
• Term: from 2015 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 281417092

• Final Report Year: 2019

Final Report Abstract

Modern applications of control theory to chemical reactors, traffic networks, multibody systems (e.g., robotic arms, flexible elements in aircraft), fluid-structure interactions, etc. require methods for robust stabilization of coupled systems, described by partial differential equations (PDEs). How to control such complex systems, if we can physically apply controls only to the boundary of their spatial domain? How to ensure that our control design is efficient in spite of actuator and observation errors, hidden dynamics of a system, and external disturbances? Under which conditions a coupled large-scale infinitedimensional system is stable if all its components are stable? In this project, we have developed the tools and methods which help to tackle these challenging and fascinating problems. Generally speaking, the stability of a system means that small changes in parameters do not change the qualitative behavior of the system. For example, if we change a bit a position of a robot, we would like to know that its future trajectory does not change much, even in the long run. This is called internal stability. A complementary concept is an external stability, which means that the exogenous disturbances do not drastically change the system. This property is also called robustness. To design reliable vehicles, aircraft, robots, biochemical systems, etc. we need to be sure that the system is both internally and externally stable. The concept of input-to-state stability unifies these two properties and constitutes an ideal foundation for robust control of nonlinear systems. In this project, we extended the already classical ISS theory to the infinite-dimensional setting. Our main achievements include: • We have shown that the ISS concept is central for stability theory of infinite-dimensional systems, by relating it to other important concepts, widely used within the systems theory. • For linear systems, we derived powerful criteria for ISS and related concepts, based on the methods of functional analysis which make it possible to use the linear structure of systems under investigation in the most efficient way. • For nonlinear systems, we developed an energy-based approach, or more broadly the method of Lyapunov functions. The idea of this method is to find a suitable energy-like function and investigate its behavior. From this knowledge, we can understand the behavior of the trajectory of the system itself. In particular, we have developed a significant extension of a classical Lyapunov method to allow for so-called non-coercive Lyapunov functions, which are considerably less restrictive than the classical ones. • Finally, for complex networks, we have developed the small-gain theorems which give criteria for stability of complex nonlinear networks consisting of an arbitrary number of components, based on the information on the stability of the subsystems and provided the interconnection structure is so that the so-called small-gain condition holds. Potential applications for these methods include stabilization of multicomponent systems, stability analysis and regulation of boundary control systems, analysis of large-scale multi-agent systems. These problems naturally appear in chemical engineering, robotics, aerospace engineering, additive manufacturing as well as climate modeling.

Publications

• (2020) Stability conditions for infinite networks of nonlinear systems and their application for stabilization. Automatica 112 108643
  Dashkovskiy, Sergey; Pavlichkov, Svyatoslav
  
• Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances
  J. Schmid, H. Zwart
  
• Characterizations of input-to-state stability for infinite-dimensional systems. IEEE Transactions on Automatic Control, 63(6):1602–1617, 2018
  A. Mironchenko, F. Wirth
  
• Infinite-dimensional input-to-state stability and Orlicz spaces. SIAM J. Control Optim. 56(2) (2018), 868-889
  B. Jacob, R. Nabiullin, J.R. Partington, F.L. Schwenninger
  
• Non-coercive Lyapunov functions for infinite-dimensional systems. Journal of Differential Equations, 266(11):7038–7072, 2019
  A. Mironchenko, F. Wirth
  
• On continuity of solutions for parabolic control systems and input-to-state stability, Journal of Differential Equations, 266(10):6284-6306, 2019
  B Jacob, F.L. Schwenninger, H. Zwart