Visioli (2001) proposed 3 sets of PID auto-tuning formulas for UFOPDT processes to minimize the ISE, ITSE and ISTE specifications, respectively. The controller settings are computed by genetic algorithms to obtain a global optimal solution. The control system configuration is of one degree of freedom.
Park et al. (1998) proposed an enhanced PID control strategy for UFOPDT and USOPDT processes The control system configuration has double loops to reduce the overshoots and yield reasonable settling time. The unstable process is stabilized by the inner proportional controller for a optimal gain margin. The main PID is then designed for the stabilized inner closed-loop system.
Majhi and Atherton (2000a) proposed another double-loop PI-PD scheme for UFOPTD processes, which is similar to Method B. The inner PD controller is for stabilization and outer PI controller is designed to minimize the ISTE criterion.
Wang and Cai (2002) used gain and phase margin specifications for unstable process control, and consider the same double-loop structure as that of Method B in design. They implement the double loop configuration into an equivalent single-loop PID feedback system with a setpoint filter. The controller setting is obtained by assigning gain margin of 3 and phase margin of 60 degree. The second order Taylor series expansion is employed to approximate the time delay.
Lee et al. (2000) proposed IMC-based PID auto-tuning formulas for FOPDT and SOPDT unstable processes. The control system is in the same 2DOF structure as in Method D.
IMC is not applicable to unstable systems. But an equivalent feedback controller for IMC controller q can be derived as follows: This controller G c can be approximated by a PID controller with the first three terms of its Maclaurin series expansion in s ,i.e.,
Yang et al. (2002) developed another IMC-based method to design PID and high order feedback controllers for unstable processes. In this design methodology, model reduction is employed to approximate the equivalent IMC feedback controller G c by a standard PID controller G c,PID . The non-negative least square method is used to obtain the optimal PID settings to minimize the criterion E , on the desired closed-loop bandwidth. The desired degree of PID approximation to the IMC controller is usually set as 5%. The control system structure is also of 2DOF with the setpoint filter.
Majhi and Atherton (2000b) proposed this modified SP controller for UFOPDT processes, in which the denominator of closed-loop setpoint transfer function is delay-free. Therefore the design of controller G c is facilitated for setpoint tracking. On the inner loops, G c1 is to stabilize to delay-free part of the unstable process, while G c2 is for stabilization and disturbance rejection as well.
Simulation & Comparison 1. Small Normalized Dead-time: 0<L/T<0.693 The plant’s normalized dead-time is 0.5. A unity step setpoint is given at t = 0, and a disturbance of -0.1 is injected at t = 75.
Ranking ( Small Normalized Dead-time ) Setpoint Response Best: Method G Excellent: Method E and F Good: Method C and A Fair: Method B Poor: Method D Disturbance Rejection Excellent: Method E and F Good: Method C and A Fair: Method D and G Poor: Method B
Simulation & Comparison 2. Medium Normalized Dead-time: 0.693<L/T<1 The plant’s normalized dead-time is 0.8, Method B is no longer applicable. Again, the unity step setpoint is given at t = 0, and a disturbance of -0.1 is injected at t = 75.
Ranking ( Median Normalized Dead-time ) Setpoint Response Excellent: Method G Good: Method E and F Fair: Method C and A Poor: Method D Disturbance Rejection Excellent: Method A Good: Method E, F and C Fair: Method D Poor: Method G
Simulation & Comparison 3. Large Normalized Dead-time: 1<L/T<2 The plant’s normalized dead-time is 1.5. Only Method E and F are workable in this scenario. Again, the unity step setpoint is given at t = 0, and a disturbance of -0.1 is injected at t = 75.
Performance Specifications Ranking ( Large Normalized Dead-time ) Setpoint Response Method E is slightly better Disturbance Rejection Method F is slightly better