We present a numerical method to analyse the relative stability of closed-loop single-input–single-output (SISO) dead-time systems on a given left complex half-plane for all positive delays. The well-known boundary crossing method for the imaginary axis is extended to a given vertical line stability boundary in the complex plane for these types of systems. The method allows us to compute the characteristic roots crossing the relative stability boundary and their corresponding delays up to a maximum predefined delay. Based on this method, we analyse the relative stability of the closed-loop system for all positive delays. Both numerical methods are effective for high-order SISO dead-time systems.
BourlèsH (1987) α-stability and robustness of large-scale interconnected systems. International Journal of Control36: 2221–2232.
2.
BredaDMasetSVermiglioR (2005) Pseudospectral differencing methods for characteristic roots of delay differential equations. SIAM Journal on Scientific Computing27: 482–495.
3.
ChenJGuGNettCA (1995) A new method for computing delay margins for stability of linear delay systems. Systems and Control Letters26: 107–117.
4.
CurtainRMorrisK (2009) Transfer functions of distributed parameter systems: A tutorial. Automatica45: 1101–1116.
5.
EngelborghsKLuzyaninaTRooseD (2000) Numerical bifurcation analysis of delay differential equations. Journal of Computational and Applied Mathematics125: 265–275.
6.
EngelborghsKRooseD (2002) On stability of LMS methods and characteristic roots of delay differential equations. SIAM Journal on Numerical Analysis40: 629–650.
7.
FazeliniaHSipahiROlgacN (2007) Stability robustness analysis of multiple time-delayed systems using “building block” concept. IEEE Transactions on Automatic Control52: 799–810.
8.
FilipovicDOlgacN (2002) Delayed resonator with speed feedback—design and performance analysis. Mechatronics12: 393–413.
9.
FuPChenJNiculescuSI (2007) High-order analysis of critical stability properties of linear time-delay systems. Proceedings of the American Control Conference, 11–13 July, New York City, NY, 4921–4926.
10.
FuPNiculescuSIChenJ (2006) Stability of linear neutral time-delay systems: exact conditions via matrix pencil solutions. IEEE Transactions on Automatic Control51: 1063–1069.
11.
GuKKharitonovVChenJ (2003) Stability of Time-delay Systems. Boston, MA: Birkhäuser.
12.
GuKNiculescuSIChenJ (2005) On stability crossing curves for general systems with two delays. Journal of Mathematical Analysis and Applications311: 231–253.
13.
HanQLYuXGuK (2004) On computing the maximum time-delay bound for stability of linear neutral systems. IEEE Transactions on Automatic Control49: 2281–2286.
14.
HuGDLiuM (2007) Stability criteria for linear neutral systems with multiple delays. IEEE Transactions on Automatic Control52: 720–724.
15.
HuangCChouCWangJ (1996) Tuning of PID controllers based on the second-order model by calculation. Journal of the Chineese Institute of Chemical Engineers27: 107–120.
16.
JarlebringE (2009) Critical delays and polynomial eigenvalue problems. Journal of Computational and Applied Mathematics224: 296–306.
17.
JarlebringEMichielsW (2009) Invariance properties in the root sensitivity of time-delay systems with double imaginary roots. In: Proceedings of the 8th IFAC Workshop on Time-Delay Systems, 1–3 September, Sinaia, Romania.
18.
KharitonovV (1998) Robust stability analysis of time delay systems: A survey. Systems Structure and Control1-2: 1–12.
19.
LouisellJ (2001) A matrix method for determining the imaginary axis eigenvalues of a delay system. IEEE Transactions on Automatic Control46: 2008–2012.
20.
MaoWJChuJ (2006) D-stability for linear continuous-time systems with multiple time delays. Automatica42: 1589–1592.
21.
MichielsWNiculescuSI (2007) Stability and Stabilization of Time-delay Systems. In: An Eigenvalue Based Approach (Advances in Design and Control, Vol. 12, Philadelphia, PA: SIAM.
22.
NiculescuSI (2001) Delay Effects on Stability: A Robust Control Approach. Lecture Notes in Control and Information Sciences, Vol. 269, London: Springer-Verlag.
23.
NiculescuSIDionJMDugardL (1994) α-stability criteria for linear systems with delayed state. Internal Note L.A.G, 94–150.
24.
OgataK (1997) Modern Control Engineering, 3rd edn. Englewood Cliffs, NJ: Prentice Hall.
25.
OlgacNSipahiR (2002) An exact method for the stability analysis of time delayed LTI systems. IEEE Transactions on Automatic Control47: 793–797.
26.
OlgacNSipahiR (2004) A practical method for analyzing the stability of neutral type LTI-time delayed systems. Automatica40: 847–853.
27.
RichardJ (2003) Time-delay systems: An overview of some recent advances and open problems. Automatica39: 1667–1694.
28.
SipahiRFazeliniaHOlgacN (2009) Stability analysis of LTI systems with three independent delays—a computationally efficient procedure. Journal of Dynamic Systems, Measurement and Control131: 051013.
29.
SipahiROlgacN (2006) Stability robustness of retarded LTI systems with single delay and exhaustive determination of their imaginary spectra. SIAM Journal on Control and Optimization45: 1680–1696.
30.
ThowsenA (1981) An analytic stability test for a class of time-delay systems. IEEE Transactions on Automatic Control26: 735–736.
