A Review of Some Subtleties of Practical Relevance

This paper reviews some subtleties in time-delay systems of neutral type that are believed to be of particular relevance in practice. Both traditional formulation and the coupled differential-difference equation formulation are used. The discontinuity of the spectrum as a function of delays is discussed. Conditions to guarantee stability under small parameter variations are given. A number of subjects that have been discussed in the literature, often using different methods, are reviewed to illustrate some fundamental concepts. These include systems with small delays, the sensitivity of Smith predictor to small delay mismatch, and the discrete implementation of distributed-delay feedback control. The framework prsented in this paper makes it possible to provide simpler formulation and strengthen, generalize, or provide alternative interpretation of the existing results

Linear stability in networks of pulse-coupled neurons

In a first step towards the comprehension of neural activity, one should focus on the stability of the various dynamical states. Even the characterization of idealized regimes, such as a perfectly periodic spiking activity, reveals unexpected difficulties. In this paper we discuss a general approach to linear stability of pulse-coupled neural networks for generic phase-response curves and post-synaptic response functions. In particular, we present: (i) a mean-field approach developed under the hypothesis of an infinite network and small synaptic conductances; (ii) a "microscopic" approach which applies to finite but large networks. As a result, we find that no matter how large is a neural network, its response to most of the perturbations depends on the system size. There exists, however, also a second class of perturbations, whose evolution typically covers an increasingly wide range of time scales. The analysis of perfectly regular, asynchronous, states reveals that their stability depends crucially on the smoothness of both the phase-response curve and the transmitted post-synaptic pulse. The general validity of this scenarion is confirmed by numerical simulations of systems that are not amenable to a perturbative approach.Comment: 13 pages, 7 figures, submitted to Frontiers in Computational Neuroscienc

On feedback stabilization of linear switched systems via switching signal control

Motivated by recent applications in control theory, we study the feedback stabilizability of switched systems, where one is allowed to chose the switching signal as a function of $x(t)$ in order to stabilize the system. We propose new algorithms and analyze several mathematical features of the problem which were unnoticed up to now, to our knowledge. We prove complexity results, (in-)equivalence between various notions of stabilizability, existence of Lyapunov functions, and provide a case study for a paradigmatic example introduced by Stanford and Urbano.Comment: 19 pages, 3 figure

Approximate Explicit MPC and Closed-loop Stability: Analysis based on PWA Lyapunov Functions

Model Predictive Control (MPC) is the de facto standard in advanced industrial automation systems. There are two main formulations of the MPC algorithm: an implicit one and an explicit MPC one. The first requires an optimization problem to be solved on-line, which is the main limitation when dealing with hard real-time applications. As the implicit MPC algorithm cannot be guaran- teed in terms of execution time, in many applications the explicit MPC solution is preferable. In order to deal with systems integrating mixed logic and dynam- ics, the class of the hybrid and piecewise affine models (PWA) were introduced and tackled by the explicit MPC strategy. However, the resulting controller complexity leads to a requirement on the CPU/memory combination which is as strict as the number of states, inputs and outputs increases. To reduce drasti- cally the complexity of the explicit controller while preserving the controller’s performance, a strategy combining switched MPC with discontinuous simpli- cial PWA models is introduced in this thesis. The latter is proven to be circuit implementable, e.g., in FPGA. To ensure that closed-loop stability properties are guaranteed, a stability analysis tool is proposed which exploits suitable and possibly discontinuous PWA Lyapunov-like functions. The tool requires solving offline a linear programming problem. Moreover, the tool is able to compute an invariant set for the closed-loop system, as well as ultimate boundedness and input-to-state stability properties

Approximation of explicit model predictive control using regular piecewise affine functions: an input-to-state stability approach

Abstract Piecewise affine (PWA) feedback control laws defined on general polytopic partitions, as for instance obtained by explicit MPC, will often be prohibitively complex for fast systems. In this work we study the problem of approximating these high-complexity controllers by low-complexity PWA control laws defined on more regular partitions, facilitating faster on-line evaluation. The approach is based on the concept of input-to-state stability (ISS). In particular, the existence of an ISS Lyapunov function (LF) is exploited to obtain a priori conditions that guarantee asymptotic stability and constraint satisfaction of the approximate low-complexity controller. These conditions can be expressed as local semidefinite programs (SDPs) or linear programs (LPs), in case of 2-norm or 1, ∞-norm based ISS, respectively, and apply to PWA plants. In addition, as ISS is a prerequisite for our approximation method, we provide two tractable computational methods for deriving the necessary ISS inequalities from nominal stability. A numerical example is provided that illustrates the main results. The authors are with the Hybrid an

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