Safety Control Synthesis with Input Limits: a Hybrid Approach

We introduce a hybrid (discrete--continuous) safety controller which enforces strict state and input constraints on a system---but only acts when necessary, preserving transparent operation of the original system within some safe region of the state space. We define this space using a Min-Quadratic Barrier function, which we construct along the equilibrium manifold using the Lyapunov functions which result from linear matrix inequality controller synthesis for locally valid uncertain linearizations. We also introduce the concept of a barrier pair, which makes it easy to extend the approach to include trajectory-based augmentations to the safe region, in the style of LQR-Trees. We demonstrate our controller and barrier pair synthesis method in simulation-based examples.Comment: 6 pages, 7 figures. Accepted for publication at the 2018 American Controls Conference. Copyright IEEE 201

Analysis of parametric biological models with non-linear dynamics

In this paper we present recent results on parametric analysis of biological models. The underlying method is based on the algorithms for computing trajectory sets of hybrid systems with polynomial dynamics. The method is then applied to two case studies of biological systems: one is a cardiac cell model for studying the conditions for cardiac abnormalities, and the second is a model of insect nest-site choice.Comment: In Proceedings HSB 2012, arXiv:1208.315

Model predictive control techniques for hybrid systems

This paper describes the main issues encountered when applying model predictive control to hybrid processes. Hybrid model predictive control (HMPC) is a research field non-fully developed with many open challenges. The paper describes some of the techniques proposed by the research community to overcome the main problems encountered. Issues related to the stability and the solution of the optimization problem are also discussed. The paper ends by describing the results of a benchmark exercise in which several HMPC schemes were applied to a solar air conditioning plant.Ministerio de Eduación y Ciencia DPI2007-66718-C04-01Ministerio de Eduación y Ciencia DPI2008-0581

Composite control Lyapunov functions for robust stabilization of constrained uncertain dynamical systems

This work presents innovative scientific results on the robust stabilization of constrained uncertain dynamical systems via Lyapunov-based state feedback control. Given two control Lyapunov functions, a novel class of smooth composite control Lyapunov functions is presented. This class, which is based on the R-functions theory, is universal for the stabilizability of linear differential inclusions and has the following property. Once a desired controlled invariant set is fixed, the shape of the inner level sets can be made arbitrary close to any given ones, in a smooth and non-homothetic way. This procedure is an example of ``merging'' two control Lyapunov functions. In general, a merging function consists in a control Lyapunov function whose gradient is a continuous combination of the gradients of the two parents control Lyapunov functions. The problem of merging two control Lyapunov functions, for instance a global control Lyapunov function with a large controlled domain of attraction and a local one with a guaranteed local performance, is considered important for several control applications. The main reason is that when simultaneously concerning constraints, robustness and optimality, a single Lyapunov function is usually suitable for just one of these goals, but ineffective for the others. For nonlinear control-affine systems, both equations and inclusions, some equivalence properties are shown between the control-sharing property, namely the existence of a single control law which makes simultaneously negative the Lyapunov derivatives of the two given control Lyapunov functions, and the existence of merging control Lyapunov functions. Even for linear systems, the control-sharing property does not always hold, with the remarkable exception of planar systems. For the class of linear differential inclusions, linear programs and linear matrix inequalities conditions are given for the the control-sharing property to hold. The proposed Lyapunov-based control laws are illustrated and simulated on benchmark case studies, with positive numerical results

Algorithmic Verification of Continuous and Hybrid Systems

We provide a tutorial introduction to reachability computation, a class of computational techniques that exports verification technology toward continuous and hybrid systems. For open under-determined systems, this technique can sometimes replace an infinite number of simulations.Comment: In Proceedings INFINITY 2013, arXiv:1402.661

Lyapunov methods for time-invariant delay difference inclusions

Motivated by the fact that delay difference inclusions (DDIs) form a rich modeling class that includes, for example, uncertain time-delay systems and certain types of networked control systems, this paper provides a comprehensive collection of Lyapunov methods for DDIs. First, the Lyapunov–Krasovskii approach, which is an extension of the classical Lyapunov theory to time-delay systems, is considered. It is shown that a DDI is KL-stable if and only if it admits a Lyapunov–Krasovskii function (LKF). Second, the Lyapunov–Razumikhin method, which is a type of small-gain approach for time-delay systems, is studied. It is proved that a DDI is KL-stable if it admits a Lyapunov–Razumikhin function (LRF). Moreover, an example of a linear delay difference equation which is globally exponentially stable but does not admit an LRF is provided. Thus, it is established that the existence of an LRF is not a necessary condition for KL-stability of a DDI. Then, it is shown that the existence of an LRF is a sufficient condition for the existence of an LKF and that only under certain additional assumptions is the converse true. Furthermore, it is shown that an LRF induces a family of sets with certain contraction properties that are particular to time-delay systems. On the other hand, an LKF is shown to induce a type of contractive set similar to those induced by a classical Lyapunov function. The class of quadratic candidate functions is used to illustrate the results derived in this paper in terms of both LKFs and LRFs, respectively. Both stability analysis and stabilizing controller synthesis methods for linear DDIs are proposed