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Now, applying the Ziegler–Nichols tuning rules (see Table 1.2), Cohen–Coon tuning rules (see Table 1.3) and Wang–Cluett tuning rules (see Table 1.6), we obtain the PI controller parameters for the fired heater process shown in Table 1.9. The PI controller parameters obtained are drastically different. The PI controllers using Ziegler–Nichols and Wang–Cluett tuning rules produce stable closed-loop system for the fired heater process, however the PI controller using Cohen–Coon tuning rules does not lead to a stable closed-loop system, which was verified using closed-loop simulation. To evaluate the closed-loop control performance, a unit step input signal is used as a reference and a step input disturbance with magnitude of is added to the closed-loop simulation at half of the simulation time. Figure 1.22(a) shows the control signals generated by the PI controllers and Figure 1.22(b) shows the output responses to the reference change and the disturbance signal. Both closed-loop systems have oscillations, but in comparison, the controller using Wang–Cluett tuning rules leads to a slightly better closed-loop performance with less oscillations.
Figure 1.21 Unit step response of the fired heater process.
Table 1.9 PI controller parameters with reaction curve.
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| Ziegler–Nichols | 0.4239 | 28.6200 | ||
| Cohen–Coon | 0.4517 | 13.2353 | ||
| Wang–Cluett | 0.2835 | 15.7610 |
Figure 1.22 Comparison of closed-loop responses using Ziegler–Nichols and Wang–Cluett tuning rules (Example 1.9). (a) Control signal. (b) Output. Key: line (1) Ziegler–Nichols tuning rule; line (2) Wang–Cluett tuning rule.
1.6 Summary
This chapter has introduced the basics in PID control systems. It is important for us to understand the roles of proportional control, integral control and derivative control. The simple modification with the implementation of PID controller by putting the proportional control on output only reduces the overshoot in the step reference response. For some applications, avoiding the overshoot is important because the requirement is associated with the system's operational constraints while for other applications the overshoot is preferred if the PID controller is used in an inner-loop system (see Chapter 7). Additionally the derivative control should be implemented with a derivative filter to avoid amplification of measurement noise and the derivative control should be implemented on the output only to avoid the scenario of derivative ‘kick’.
Several sets of tuning rules have been introduced in this chapter. These tuning rules are very simple and easy to use if the system can be approximated by a first-order-plus delay model. However, they offer no guarantees on the closed-loop performance as demonstrated by the simulation examples. Some of the examples used in this chapter will be analyzed using the Nyquist stability criterion and sensitivity functions in Chapter 2.
1.7 Further Reading
1 Text books in control engineering include, Franklin et al. (1998), Franklin et al. (1991), Ogata (2002), Golnaraghi and Kuo (2010), Goodwin et al. (2000) and Astrom and Murray (2008).
2 Process control books include Marlin (1995), Ogunnaike and Ray (1994), Seborg et al. (2010).
3 There are many books published on PID control, including Astrom and Hagglund (1995), Astrom and Hagglund (2006), Yu (2006), Johnson and Moradi (2005), Visioli (2006), Tan et al. (2012). PID control of multivariable systems is discussed in Wang et al. (2008).
4 Tuning rules are compiled as a book (O'Dwyer (2009)). PID controller tuning rules with performance specifications derived from frequency response analysis are introduced in Wang and Cluett (2000).
5 The survey and tutorial papers on PID control include Åström and Hägglund (2001), Ang et al. (2005), Li et al. (2006), Knospe (2006), Cominos and Munro (2002) and Visioli (2012), Blevins (2012).
6 A web-based laboratory for teaching PID control is introduced in Ko et al. (2001), Yeung and Huang (2003).
7 The issues associated with derivative filters in PID controller are addressed in Luyben (2001), Hägglund (2012), Hägglund (2013), Isaksson and Graebe (2002), Larsson and Hägglund (2011).
8 Improving reference tracking and reducing overshoot is discussed for an existing controller in an industrial environment (Visioli and Piazzi (2003)).
9 Ziegler-Nichols tuning formula is refined in Hang et al. (1991) and for unstable systems with time delay in De Paor and O'Malley (1989). The ZieglerNichols step response method is revisited from the point of view of robust loop shaping (Åström and Hägglund (2004)). There is a set of tuning rules for integrating plus delay model in Tyreus and Luyben (1992), which was extended to PID controllers in Luyben (1996).
Problems
1 1.1 The following system is given to practice the Ziegler-Nichols tuning rules based on oscillation testing data to determine the PID controller parameters:Build a Simulink simulator using closed-loop proportional control with a controller for the system. The sampling interval is chosen to be 0.1 sec.Find the PI and PID controller parameters using Table 1.1.Implement the PI and PID controllers in Simulink with a step reference signal by putting both proportional and derivative control on the output only.What are your observations with respect to the closed-loop performance of the PI and PID control system?
2 1.2 In the majority of the tuning rules, the key step is to find a first order plus delay approximate model for step response testing data. Because the step response of a first order system () to a unit step input signal can be expressed aswe can determine the time constant using 63.2 percent of the rising time in the step response.Why does the 6.32 percent of the rising time correspond to the time constant ?Construct a graphic method to find a first order plus time delay model for the following system:Compare this first order plus delay model with that found in Example 1.7.Find the PID controller parameters using Table
