Christian Ebenbauer

Stability Analysis of Time-Delay Systems With Incommensurate Delays Using Positive Polynomials

We present a new stability condition for linear time-delay systems (TDS) with multiple incommensurate delays based on the Rekasius substitution and positive polynomials. The condition is checked by testing the positivity of multivariate polynomials in certain domains. For this purpose, we propose two alternative algorithms based on linear programming and sum of squares methods, respectively. The efficiency and accuracy of the algorithms is compared in an example to alternative stability conditions taken from the literature.

Model predictive control of linear continuous time singular systems subject to input constraints

In this note we consider the stabilization of linear continuous time singular systems with respect to input constraints. Specifically we propose the use of a sampled-data model predictive control scheme. Stability of the closed-loop is achieved in a similar manner as for nonsingular systems, utilizing a suitable terminal penalty term and a terminal region constraint. One problem that must be overcome is the avoidance of impulsive solutions. This is achieved by enforcing the input to be sufficiently often differentiable. As shown, the resulting sampled-data model predictive control scheme does lead to stability in the sense of convergence and to an impulse free closed-loop.

Refining LaSalle's invariance principle

This work deals with the problem of locating the omega-limit set of a bounded solution of a given autonomous vector field on a Riemannian manifold. The derived results extend LaSalles invariance principle in such a way that the newly obtained conditions provide in certain situations a more refined statement about the location of the omega-limit set when the invariance principle is inconclusive. The derived conditions are for example useful for gradient-type vector fields and cascaded systems.

Certainty-Equivalence Feedback Design With Polynomial-Type Feedbacks Which Guarantee ISS

The purpose of this note is to establish a certainty-equivalence feedback design for inverse optimally controlled affine systems. In particular, it is shown that a class of polynomial-type state feedbacks in conjunction with a globally asymptotically convergent observer leads to a globally asymptotically stable closed-loop. A key step in the proposed certainty-equivalence feedback design procedure is the identification of a new class of polynomial-type inverse optimal feedbacks which guarantees input-to-state stability (ISS) with respect to measurement errors. As a consequence, the proposed certainty-equivalence feedback design has the important feature that the state feedback is allowed to contain polynomial nonlinearities of arbitrarily high degree in the unmeasured states. This feature is illustrated on an example

Stability Analysis for Time-Delay Systems using Rekasius's Substitution and Sum of Squares

In this paper, a new delay-dependent stability analysis for time-delay linear time-invariant (TDLTI) systems is derived. In contrast to many recent approaches, which often utilize Lyapunov-Krasovskii functionals and linear matrix inequalities, an alternative approach is proposed in this paper. The proposed stability analysis is formulated in the frequency domain and investigates the characteristic equation by using the so-called Rekasius substitution and recently established sum of squares techniques from computational semialgebraic geometry. The advantages of the proposed approach are that the stability analysis is often less conservative than many approaches based on Lyapunov-Krasovskii functionals, as demonstrated on a well-known benchmark example, and that the stability analysis is very flexible with respect to additional analysis objectives

A Dissipation Inequality for the Minimum Phase Property

The minimum phase property is an important notion in systems and control theory. In this paper, a characterization of the minimum phase property of nonlinear control systems in terms of a dissipation inequality is derived. It is shown that this dissipation inequality is equivalent to the classical definition of the minimum phase property in the sense of Byrnes and Isidori, if the control system is affine in the input and the so-called input-output normal form exists.

Ensuring Task-Independent Safety for Multi-Agent Systems by Feedback

This paper presents a new safety feedback design for multi-agent systems. In contrast to existing methods, the proposed approach follows the idea to decouple the design of low-level safety features from abstract high-level tasks of the agents. This leads to a modular design that preserves the flexibility of multi-agent systems while increasing their usability in safety critical applications by giving systems theoretic safety guarantees. The approach is illustrated using collision avoidance of two vehicles as an example.

A dynamical system that computes eigenvalues and diagonalizes matrices with a real spectrum

The present paper deals with the problem of diagonalizing matrices using a control system of the form A = [U, A], where [U, A] = UA - AU and A, U are real matrices. It is shown that the feedback U = [N, A + AT] + p[AT, A], N diagonal, rho > 0 allows to solve the diagonalization problem under the assumption that the to be diagonalized matrix has real spectrum. Moreover, in the case of a complex spectrum, the feedback allows to check if a matrix is stable or to compute all eigenvalues of a matrix or roots of a polynomial.

Stability of Networked Systems with Multiple Delays Using Linear Programming

In this paper, we present a new sufficient stability condition for linear time-invariant multiple time-delay systems (MTDS) based on the Rekasius substitution and linear programming. The main advantage of the new stability condition is that it is applicable to the general case of multiple, incommensurate delays yet numerically tractable. In particular, using efficient linear programming algorithms, a numerical stability test is derived to determine a maximum delay tau macr macr such that the system is stable for all delays tauk with tauk les tau macr.

Detecting oscillatory behavior using Lyapunov functions

Conditions are derived which guarantee the existence of oscillatory behavior for general nonlinear, high-dimensional systems. The first result is motivated by the energy transfer which occurs for example in an oscillatory LC circuit. A simple generalization of this energy transfer mechanism leads to conditions which guarantee the existence of oscillations and which can be expressed in terms of Lyapunov-like functions. The second result in this paper are new computationally tractable conditions which allow to use numerical methods, like sum of squares techniques, to verify oscillatory behavior for polynomial systems. Moreover, a simple condition for the nonexistence of oscillatory behavior is pointed out. The applicability of the results is demonstrated by several examples.