Model Predictive Control of Hybrid Dynamical Systems

Organisers

  • Berk Altın
  • Ricardo Sanfelice
  • Francesco Ferrante
  • Mohamed Adlene Maghenem

Speakers

  • Berk Altın, University of California, Santa Cruz, USA
  • Francesco Ferrante, GIPSA Lab, Université Grenoble Alpes
  • Ricardo G. Sanfelice, University of California, Santa Cruz, USA
  • Mohamed A. Maghenem, University of California, Santa Cruz, USA

 

Summary


Hybrid systems model the behavior of dynamical systems where the states can evolve continuously as well as instantaneously. Such systems arise when control algorithms that involve digital devices are applied to continuous-time systems, or due to the intrinsic dynamics (e.g. mechanical systems with impacts, switching electrical circuits). Hybrid control may be used for improved performance and robustness properties compared to conventional control, and hybrid dynamics may be unavoidable due to the interplay between digital and analog components of a system.
This workshop is a complete course on the analysis and design of model predictive control (MPC) schemes for hybrid systems. It presents recently developed results on asymptotically stabilizing MPC for hybrid systems based on control Lyapunov functions. The workshop provides a detailed overview of the state of the art on hybrid MPC, and a short tutorial on a powerful hybrid systems framework (hybrid inclusions) that can model hybrid dynamics described in other frameworks (e.g. switched systems, hybrid automata, impulsive systems). Key analysis tools in this setting are demonstrated, along with several advantages over other frameworks. This background is then used to lay the theoretical foundations of a general MPC framework for hybrid systems, with guaranteed stability and feasibility. The ideas are illustrated in several applications.
The workshop targets a broad audience in academia and industry, including graduate students, looking for an introduction to an active area of research and some modern mathematical analysis tools; control practitioners interested in novel design techniques; researchers in dynamical systems in pursuit of relevant applications; and researchers in industry and labs applying hybrid predictive control methods to engineering systems. The required background is basic familiarity with continuous- and discrete-time nonlinear systems. The lectures are closely related to each other and not meant to be independent research presentations.

Programme

Welcome Remarks, Berk Altın and Ricardo G. Sanfelice


Introduction to Hybrid Dynamical Systems: Modeling, Examples, Asymptotic Stability, Francesco Ferrante

A short tutorial on a powerful hybrid systems framework called hybrid inclusions is provided. The generality of this framework is demonstrated by showing that several classes of hybrid systems often described in other frameworks (e.g. switched systems, hybrid automata, impulsive systems) can be reformulated as hybrid inclusions. The notion of solutions are presented, along with conditions guaranteeing existence and uniqueness. Asymptotic stability of such systems are formalized, sufficient Lyapunov conditions for asymptotic stability are demonstrated.


Background on Hybrid Model Predictive Control: Models, Methods, and Open Questions, Ricardo G. Sanfelice

This module discusses several MPC strategies in the literature that have a hybrid flavor, which, due to the diverse use of the term hybrid, span a wide range of control settings. This module provides a unified presentation of these strategies with the purpose of serving as a self contained summary of the state of the art in hybrid MPC, and as a motivator for research on MPC of hybrid inclusions.


Overview of Model Predictive Control for Hybrid Dynamical Systems, Berk Altın

An MPC algorithm that asymptotically stabilizes a closed set of interest for a given hybrid dynamical system is introduced. An overview of the salient features of the algorithm and the underlying optimal control problem is presented in comparison to continuous- and discrete-time MPC.


Coffee Break


Model Predictive Control for Hybrid Dynamical Systems: Feasibility, Value Function Properties, Lyapunov Stability Analysis, Berk Altın

Feasibility and stability properties of the aforementioned MPC algorithm under appropriate conditions are presented, along with the relevant properties of the optimal control problem. The key stabilizing ingredient of the algorithm, which comes from the consideration of the terminal cost as a control Lyapunov function over the terminal constraint set, is revealed. It is shown that when trajectories of the system persistently flows or jumps, the cost functional defining the underlying optimal control problem simplifies considerably.
 

Lunch


Control Lyapunov Functions for Hybrid MPC, Ricardo G. Sanfelice

The notion of control Lyapunov functions for hybrid inclusions is formalized in the context of MPC and terminal cost functions. Sufficient conditions for the existence of control Lyapunov functions are provided. Methods to construct continuous feedback controllers from such functions are presented.
 

Forward Invariance Tools for Hybrid MPC, Mohamed Maghenem

Notions of forward invariance for hybrid inclusions are presented. Various analysis and synthesis tools to ensure forward invariance, including barrier functions, are demonstrated in the context of MPC and the terminal constraint set.


Evaluating the Cost of Hybrid MPC without Computing, Francesco Ferrante

The problem of evaluation the cost of the optimal control problem arising in the MPC algorithm is considered. It is shown that the value function can be estimated with Lyapunov-like inequalities. In addition, it is shown that when the terminal cost of the problem satisfies the sufficient Lyapunov conditions for asymptotic stability, it is an upper bound on the value function of the corresponding infinite horizon optimal control problem. The results are interpreted in the context of MPC for hybrid inclusions with nonunique solutions.


Coffee Break


Discretizing Hybrid Dynamical Systems to Solve the Hybrid MPC Problem, Berk Altın

A computationally tractable counterpart of the MPC algorithm for hybrid inclusions is introduced, which relies on the time-discretization of the underlying continuous-time dynamics. Feasibility and stability properties of the nondiscretized hybrid MPC algorithm presented before are extended to the discretized case. Computational methods to solve the optimal control problem, such as mixed integer nonlinear programming, are presented. As the discretization of hybrid dynamical systems is a nontrivial problem, appropriate discretization techniques are demonstrated.


Hybrid MPC Applications, Ricardo G. Sanfelice, Mohamed Maghenem, and Francesco Ferrante

A general methodology for the design of the cost functions required for the MPC algorithm is presented. The presented methodology shows a systematic way of satisfying the sufficient conditions for asymptotic stability. The methodology is demonstrated via three different applications with inherently hybrid dynamics.


Open Problems and Concluding Remarks, Ricardo G. Sanfelice and Berk Altın