This article considers the problem of determining the complete stabilizing set of proportional–integral controllers, which is applied in a class of high-order processes with time delay in complex-frequency domain. First, we give a necessary condition for a proportional–integral controller to stabilize the process with certain constant delay using Descartes’ rule of signs. Then by employing the generalization of the Hermite–Biehler theorem applicable to quasi-polynomials, a complete set for all stabilizing proportional–integral parameters is derived to stabilize the open-loop stable and unstable systems. In the case of uncertain time delay, we provide a design approach to stabilize the related plant with a robust proportional–integral controller, where the unknown but constant time delay lies inside a known interval. Three examples illustrate the effectiveness of the proposed results.
ShafieiZShentonAT. Frequency-domain design of PID controllers for stable and unstable systems with time delay. Automatica1997; 33(12): 2223–2232.
2.
NiculescuS-I. Delay effects on stability: a robust control approach. Heidelberg: Springer-Verlag, 2001.
3.
GuKKharitonovVLChenJ. Stability of time-delay systems. Boston, MA: Birkhäuser, 2003.
4.
SipahiRNiculescuS-IAbdallahT. Stability and stabilization of systems with time delay. IEEE Contr Syst Mag2011; 31(1): 38–65.
5.
RichardJ-P. Time-delay systems: an overview of some recent advances and open problems. Automatica2003; 39(10): 1667–1694.
6.
ÅströmKHägglundT. Advanced PID control. Research Triangle Park, NC: Instrumentation, Systems, and Automation Society, 2006.
7.
LozanoRValeraAAlbertosP. PD control of robot manipulators with joint flexibility, actuators dynamics and friction. Automatica1999; 35(10): 1697–1700.
8.
WangQGZouBLeeTH. Auto-tuning of multivariable PID controllers from decentralized relay feedback. Automatica1997; 33(3): 319–330.
9.
FungH-WWangQ-GLeeT-H. PI tuning in terms of gain and phase martins. Automatica1998; 34(9): 1145–1149.
10.
SilvaGJDattaABhattacharyyaSP. PID controllers for time-delay systems. Boston, MA: Birkhäuser, 2005.
11.
TaylorCJChotaiAYongPC. Proportional-integral-plus (PIP) control of time delay systems. Proc IMechE, Part I: J Systems and Control Engineering1998; 212(1): 37–48.
12.
KoumboulisFNKouvakasNDParaskevopoulosPN. Linear approximant-based metaheuristic proportional-integral-derivative controller for a neutral time delay central heating system. Proc IMechE, Part I: J Systems and Control Engineering2009; 223(5): 605–618.
13.
Normey-RicoJEBordonsCCamachoEF. Improving the robustness of dead-time compensating PI controllers. Control Eng Pract1997; 5(6): 801–810.
14.
WangBReesDZhongQ-C. Control of integral processes with dead time. Part IV: various issues about PI controllers. IEE P: Contr Theor Ap2006; 153(3): 302–306.
15.
SilvaGJDattaABhattacharyyaSP. PI stabilization of first-order systems with time delay. Automatica2001; 37(12): 2025–2031.
16.
LuoYChenYQ. Stabilizing and robust fractional order PI controller synthesis for first order plus time delay systems. Automatica2012; 48(9): 2159–2167.
17.
PadulaFVisioliA. On the stabilizing PID controllers for integral processes. IEEE T Automat Contr2012; 57(2): 494–499.
18.
SilvaGJDattaABhattacharyyaSP. New results on the synthesis of PID controllers. IEEE T Automat Contr2002; 47(2): 241–252.
19.
OuLZhangWGuD. Sets of stabilising PID controllers for second-order integrating processes with time delay. IEE P: Contr Theor Ap2006; 153(5): 607–614.
WangDJ. Further results on the synthesis of PID controllers. IEEE T Automat Contr2007; 52(6): 1127–1132.
22.
WangDJ. Synthesis of PID controllers for high-order plants with time-delay. J Process Contr2009; 19(10): 1763–1768.
23.
OuLZhangWYuL. Low-order stabilization of LTI systems with time delay. IEEE T Automat Contr2009; 54(4): 774–787.
24.
HohenbichlerN. All stabilizing PID controllers for time delay systems. Automatica2009; 45(11): 2678–2684.
25.
OliveiraVACossiLVTeixeiraMCM. Synthesis of PID controllers for a class of time delay systems. Automatica2009; 45(7): 1778–1782.
26.
BozorgMTermehF. Domains of PID controller coefficients which guarantee stability and performance for LTI time-delay systems. Automatica2011; 47(9): 2122–2125.
27.
CookeKLVan den DriesscheP. On zeroes of some transcendental equations. Funkcialaj Ekuacioj1986; 29: 77–90.
28.
GóreckiHFuksaSGrabowskiP. Analysis and synthesis of time delay systems. Warsaw: Polish Scientific Publishers, 1989.
