Fast generation of stability charts for time-delay systems using continuation of characteristic roots

Many dynamic processes involve time delays, thus their dynamics are governed by delay differential equations (DDEs). Studying the stability of dynamic systems is critical, but analyzing the stability of time-delay systems is challenging because DDEs are infinite-dimensional. We propose a new approach to quickly generate stability charts for DDEs using continuation of characteristic roots (CCR). In our CCR method, the roots of the characteristic equation of a DDE are written as implicit functions of the parameters of interest, and the continuation equations are derived in the form of ordinary differential equations (ODEs). Numerical continuation is then employed to determine the characteristic roots at all points in a parametric space; the stability of the original DDE can then be easily determined. A key advantage of the proposed method is that a system of linearly independent ODEs is solved rather than the typical strategy of solving a large eigenvalue problem at each grid point in the domain. Thus, the CCR method significantly reduces the computational effort required to determine the stability of DDEs. As we demonstrate with several examples, the CCR method generates highly accurate stability charts, and does so up to 10 times faster than the Galerkin approximation method.Comment: 12 pages, 6 figure

Polynomial approximation of quasipolynomials based on digital filter design principles

This contribution is aimed at a possible procedure approximating quasipolynomials by polynomials. Quasipolynomials appear in linear time-delay systems description as a natural consequence of the use of the Laplace transform. Due to their infinite root spectra, control system analysis and synthesis based on such quasipolynomial models are usually mathematically heavy. In the light of this fact, there is a natural research endeavor to design a sufficiently accurate yet simple engineeringly acceptable method that approximates them by polynomials preserving basic spectral information. In this paper, such a procedure is presented based on some ideas of discrete-time (digital) filters designing without excessive math. Namely, the particular quasipolynomial is subjected to iterative discretization by means of the bilinear transformation first; consequently, linear and quadratic interpolations are applied to obtain integer powers of the approximating polynomial. Since dominant roots play a decisive role in the spectrum, interpolations are made in their very neighborhood. A simulation example proofs the algorithm efficiency. © Springer International Publishing Switzerland 2016

Meromorphic observer-based pole assignment in time delay systems

summary:The paper deals with a novel method of control system design which applies meromorphic transfer functions as models for retarded linear time delay systems. After introducing an auxiliary state model a finite-spectrum observer is designed to close a stabilizing state feedback. The observer finite spectrum is the key to implement a state feedback stabilization scheme and to apply the affine parametrization in controller design. On the basis of the so- called RQ-meromorphic functions an algebraic solution to the problem of time- delay system stabilization and control is presented that practically provides a finite spectrum assignment of the control loop

PREDICTION OF MULTI-DIMENSIONAL MILLING BEHAVIOR

Milling process is a multi-dimensional cutting process accompanied by chatter vibrations because of material removal discontinuity. Also, chatter phenomenon has a great impact on final milled products quality. In this paper, an adequate criterion derived from mathematical literature is used to predict stability of multi-dimensional chatter milling. For that, firstly, the chatter milling system is represented by a system of three Degrees Of Freedom (DOF) and its dynamics is modeled by a system of Retarded Differential Equations (RDEs). Afterward, stability in Lyapunov sense is computed on the basis of the quasi-polynomial characteristic function in frequency domain. Finally, this method is used to predict stability of chatter milling process under different machining conditions and the output related to each case is verified by resolving the system of RDEs and visualizing the displacements in time domain using Matlab software

Quasipolynomial Approach to Simultaneous Robust Control of Time-Delay Systems

A control law for retarded time-delay systems is considered, concerning infinite closed-loop spectrum assignment. An algebraic method for spectrum assignment is presented with a unique optimization algorithm for minimization of spectral abscissa and effective shaping of the chains of infinitely many closed-loop poles. Uncertainty of plant delays of a certain structure is considered in a sense of a robust simultaneous stabilization. Robust performance is achieved using mixed sensitivity design, which is incorporated into the addressed control law

Delay equations and characteristic roots: stability and more from a single curve

Delays appear always more frequently in applications, ranging, e.g., from population dynamics to automatic control, where the study of steady states is undoubt- edly of major concern. As many other dynamical systems, those generated by nonlinear delay equations usually obey the celebrated principle of linearized stability. Therefore, hyperbolic equilibria inherit the stability properties of the corresponding linearizations, the study of which relies on associated characteristic equations. The transcendence of the latter, due to the presence of the delay, leads to infinitely-many roots in the com- plex plane. Simple algebraic manipulations show, first, that all such roots belong to the intersection of two curves. Second, only one of these curves is crucial for stability, and relevant sufficient and/or necessary criteria can be easily derived from its analysis. Other aspects can be investigated under this framework and a link to the theory of modulus semigroups and monotone semiflows is also discussed