 Timedelay systems have been intensively studied for decades. Stability is one of the most important system dynamics properties and the task of stabilization is the main step of controller design. Closed loop characteristic equations of systems with input output or internal delays contain quasipolynomials rather then polynomials. System poles determined by the solution of such equation have (in most cases) as the same meaning as for delayfree systems, thus they decide about system stability. The aim of this paper is to stabilize a selected system with internal delay by a proportional controller. The task can be equivalently formulated as a stabilization of a system with inputoutput delay. The analysis and derivations are based on the argument principle, i.e. on the Mikhaylov criterion, and on the required shape of the Mikhaylov plot. The analogy with the notions of the Nyquist criterion is also presented. Stability bounds for the controller parameter are found analytically through proven lemmas, propositions and theorems. Simulation examples clarify the obtained results. Stabilization, timedelay system, characteristic quasipolynomial, argument principle, Mikhaylov plot, loop shaping. I. INTRODUCTION YSTEMS with aftereffect or dead time, also called hereditary or timedelay systems (TDS), belonging to the class of infinite dimensional systems have been largely studied during last decades due to their interesting and important theoretical and practical features. A number of hypothetic or reallife processes, e.g. in a wide spectrum of natural sciences [1][5] or in pure informatics [6], is affected by delays which can have various forms. Linear timeinvariant dynamic systems with distributed or lumped delays can be represented by the Laplace transfer function as a ratio of socalled quasipolynomials [7] in one complex variable [8][10], instead of polynomials which are usual in system and control theory. Quasipolynomials are formed as linear combinations of product of spowers and exponential terms. Delay can significantly deteriorate the quality of feedback control performance, namely stability and periodicity. The problem of conventional inputoutput delay in the closed loop has been an interesting topic in control theory since its nascence – indeed, the well known Smith predictor has been known for longer than five decades [11]. Since this pioneering work, many control approaches has been investigated and developed, e.g. [12][14]. Linear time delay systems in The authors kindly appreciate the financial support which was provided by the Ministry of Education, Youth and Sports of the Czech Republic, in the grant No. MSM 708 835 2102. technological and other processes have been usually assumed to contain delay elements in inputoutput relations only, which results in shifted arguments on the righthand side of differential equations. However, this conception is somewhat restrictive in effort to fit the real plant dynamics since in many cases; quantities and variables in inner feedbacks are of distributed or delayed nature, which yield delay elements on the lefthand side of a differential equation. Internal delays also appear in the feedback system when control plants with inputoutput transport delays, the dynamics of which is characterized using the Laplace transform by the characteristic quasipolynomial. This quasipolynomial decides (except some special cases) about the control system asymptotic stability because of the fact that its zeros are system poles with the same meaning as for polynomials; however, the number of poles is infinite. A large number of conference and journal papers were dedicated to stability analysis of systems with delay elements on the lefthand side of a differential equation, e.g. in [7], [8], [11], [12], and to control design for those systems. see e.g. [13][15]. In this paper, we address the stabilization of a selected TDS by a proportional controller, which yields a problem of the stability analysis of the characteristic quasipolynomial. In contrast to some other papers, the presented contribution investigates the stability with respect to the single nondelay coefficient and not with respect to the delay. Presented derivations and calculations are based on the fact that the argument principle (i.e. the Mikhaylov criterion) holds for a class of quasipolynomials represented by the studied one as well [7][9]. The information about the admissible interval of the selectable real parameter can serve engineers to decide quickly about closedloop system stability or to set a proportional controller parameter which appears in the characteristic quasipolynomial of a closed loop. Notice that the investigated quasipolynomial was analyzed already e.g. in [16][17]; however, these authors utilized different approaches. The paper is organized as follows: In Chapter II, the solved problem is introduced. Argument principle, the cornerstone of the presented stability analysis, is described in Chapter III. Chapter IV represents the main part of the paper where quasipolynomial stability properties are derived and proven. Coherency with the Nyquist criterion together with simulation examples are demonstrated in Chapter V. Finally, conclusions follow. Stabilization of a Delayed System by a Proportional Controller Libor Pekař and Roman Prokop S INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Issue 4, Volume 4, 2010 282