Tutorial 17: Control theory, Part 3

The aim of this contribution is to revise and extend results about stability and stabilization of a retarded quasipolynomial and systems obtained using the Mikhaylov criterion in our papers earlier. Not only retarded linear time-invariant time-delay systems (LTI-TDS) are considered in this paper; neutral as well as distributed-delay systems are the matter of the research. A LTI-TDS system of retarded type is said to be asymptotically stable if all its poles rest in the open left half plane. Asymptotic stability of neutral systems described by its spectrum is not sufficient to express the notion of stability at whole since neutral LTI-TDS are sensitive to infinitesimal delay changes. This yields the concept of so called strong stability involving this fact. Moreover, stability can not be studied using the characteristic quasipolynomial when distributed delays in either input-output or internal relation appear in a model. The contribution transforms the formulation of the Mikhaylov cr...

This paper extends results about stability and stabilization of a retarded quasipolynomial obtained using the Mikhaylov criterion earlier. Retarded quasipolynomials appear as numerators and denominators of linear time-invariant time-delay systems (LTI-TDS). A LTI-TDS system of retarded type (destitute of distributed delays) is said to be stable if all roots of its characteristic quasipolynomial are located in the open left-half complex plane. The contribution transforms the formulation of spectrum assignment of a characteristic quasipolynomial into the language of the Nyquist criterion for the open loop of a control system. Again, the argument principle is utilized to derive generalized Nyquist criterion for LTI-TDS. Stability measures related to the criterion are discussed with the specifications for LTI-TDS. An illustrative example is presented to illuminate the results.

Mathematical Problems in Engineering, 2018

Quarterly of Applied Mathematics, 1982

The changes in the stability of a system of linear differential delay equations resulting from the delay are studied by analyzing the associated eigenvalues of the characteristic equation. A specific contour is mapped by the characteristic equation into the complex plane to give an easy test for stability from an application of the argument principle. When the real part of an eigenvalue is positive, the contour gives bounds on the imaginary part which are important in certain applications to nonlinear problems.

2008 47th IEEE Conference on Decision and Control, 2008

In this paper we show how stability of an infinitedimensional linear time-delay system can be assessed by studying the stability of an associated finite-dimensional linear system, constructed after substituting the exponential function in the characteristic equation of the delay-system by a high enough finite power of the bilinear transformation.

2017

We introduce a framework for the description of a large class of delay-differential algebraic systems, in which we study three core problems: first we characterize abstractly the well-posedness of the initial-value problem, then we design a practical test for well-posedness based on a graph-theoretic representation of the system; finally, we provide a general stability criterion. We apply each of these results to a structure that commonly arises in the control of delay systems.

Robust Control - Theoretical Models and Case Studies, 2016

This chapter presents a tool for analysis of robust stability, consisting of a graphical method based on the construction of the value set of the characteristic equation of an interval plant that is obtained when the transfer function of the mathematical model is connected with a feedback controller. The main contribution presented here is the inclusion of the time delay in the mathematical model. The robust stability margin of the closed-loop system is calculated using the zero exclusion principle. This methodology converts the original analytic robust stability problem into a simplified problem consisting on a graphic examination; it is only necessary to observe if the value-set graph on the complex plane does not include the zero. A case of study of an internal combustion engine is treated, considering interval uncertainty and the time delay, which has been neglected in previous publications due to the increase in complexity of the analysis when this late is considered.

1980

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IEEE Control Systems, 2000

Control systems often operate in the presence of delays, primarily due to the time it takes to acquire the information needed for decision-making, to create control decisions, and to execute these decisions, as shown in . Systems with delays arise in engineering, biology, physics, operations research, and economics.

IEEE Access

This paper gives an overview of the stability analysis of systems with delay-dependent coefficients. Such systems are frequently encountered in various scientific and engineering applications. Most such analyses are generalization of those on systems with delay-independent coefficients. Therefore an introduction on systems with delay-independent coefficients is also given, with an emphasis on the τ-decomposition approach. Methods for two key ingredients of this approach are discussed, namely the identification of imaginary characteristic roots with the corresponding delays, and local behavior analysis of these roots as the delay increases through these critical values. For systems with delay-dependent coefficients, we review the methods of analysis for systems with a single delay and commensurate delays, their application to output feedback control and a geometric perspective that establishes a link between systems with and without delay-dependent coefficients. We provide the main ideas of various stability analysis methods and their advantages and limitations. We also present our perspectives on future directions of research on this interesting topic.