1                      ! "   # $! %&# "" ’ ()*)+,-.*# $!# /" 0’1 2)( 3+* (4*56))7 ,1  89"!""# !9!"" :" Abstract: In this paper, a method is proposed to compute the maximum allowable time delay for first order plus dead time and secondorder plus dead time systems, in order to maintain stability. Designing first order controllers for such systems to preserve stability for a longer time delay is the main aim of this paper. The procedure uses the properties of the phase diagram of the openloop transfer functions of such systems. The effects of the variation of the controller coefficients on the maximum allowable time delay of the systems are investigated and the stability domains in the space of the uncertain delay and the controller coefficients are computed. The results can be used to design firstorder robust controllers for first and second order processes containing uncertain delays. Keywords: Maximum allowable time delay, Nyquist plot, phase diagram, uncertain delay. *" / Recently, uncertain delays appearing in control systems have attracted the attention of control researchers, due to their destabilizing effects and the limitations they introduce in achieving desirable control performance [1]. Computing the allowable ranges of variation of time delays to maintain stability has been the subject of recent research works [2, 3]. However, coupling the delay uncertainty and the variations in the coefficients of the controller, leads to a set of nonlinear equations, which makes the analytical solution of the stability margin problem a difficult task. Some analytical solutions are presented in [411] to stabilize loworder plants containing fixed delays, with loworder controllers. Many industrial delay processes can be modelled as typical loworder plants such as firstorder plus deadtime (FOPDT) or secondorder plus deadtime (SOPDT) systems. In [12], a FOPDT transfer function is used to model a set of n equal cylinder atmospheric tanks and a SOPDT model is used for the heated tanks with long pipes. Similar plants can be found in [13], [14] and [15]. The parameter space method [16] and the Ddecomposition method [17], both graphically present the stability boundaries in the space of uncertain parameters. The Ddecomposition method is applied in [4] to determine the stability domain in two dimensional plane of the FOPDT system parameter and the controller coefficient by sweeping the remaining parameters. In that work, the fact that on the imaginary axis, two symmetric points  satisfy the characteristic equation is used. Then, the delay effect on the stability is investigated and sufficient conditions for stability are presented. The D decomposition method is also used in [5] for computing the stable controller coefficients for a FOPDT system with an integrator. A widelyused approach to determine the stability regions in the controller coefficients space is to use the HermiteBiehler theorem [18]. The theorem has origins in the interlacing property of roots of the real and imaginary parts of a stable polynomial. By employing a version of this theorem applicable to quasipolynomials, the set of stabilizing PI and PID controllers are extracted for FOPDT systems [6,7]. For the case of dependency of controller coefficients to realvalue parameters, [8] computes the PID controller set for FOPDT systems. An extension of Hermite Biehler theorem is applied to SOPDT models in [9]. Both D decomposition and HermiteBiehler methods provide implicit characterization of the stability boundaries in the controller coefficient space, while the latter is mathematically more involved. In [11], the Nyquist plot is used to compute the stable gain intervals for time delay systems with fixed values of delay. The stabilizing ranges of controller coefficients and the upper limit of time delay are derived for SOPDT systems using the Nyquist criterion in [19]. However, this approach does not provide a stability boundary plot for the process parameters or controller coefficients. A frequency domain approach is also employed in [10] to design PI controllers for SOPDT systems, applying the boundary crossing principle, where the transfer function coefficients are fixed and the time delay value is swept. In most previous works [411], determination of stabilizing controllers is investigated for time delay systems with fixed values of delay. In practice, the exact evaluation of delays is difficult and they are usually evaluated by rough estimations and are assumed fixed to simplify the control design and analysis. However, in many cases, the time delay is uncertain and varies in a range. In [10], the time delay is swept to plot the stability boundaries for different values of the delay. Some other works have computed stable ranges for time delay and controller coefficients [19]. Similarly, the LMI methods provide stability margins for time delays [20,21]. However, the major disadvantage of LMI methods is their conservative results. Furthermore, no results are available to compute the set of all stabilizing time delays and controller