Models for Optimal Online Tuning Based on Computational Intelligence of PID Controllers Applied to Operational Processes of Bulk Reclaimers

Abstract

This paper considers the development of models for the optimal online tuning of PID controllers based on computational intelligence approaches that are applied to the operational processes of bulk reclaimers. In the first instance, the optimal gains of a PID controller are determined using a structured artificial neural network (ANN), and in the second instance, a fuzzy system is used to carry out online adjustment through a real-time gain scheduling scheme. The bulk resumption process consists of picking up the stored material in stacks and transporting it using conveyor belts for shipment. For the control system, a model based on data pertaining to the electric current in a bucket wheel motor (a device that picks up material) is estimated and compared with the load measured using a scale. The difference between the load estimated by the model and that measured by the scale is the error, and the proposed control system is designed to minimize it. The results of simulations show that the controller models performed better using structured ANNs and fuzzy logic than PID controllers tuned by the second Ziegler–Nichols method, and the PID–fuzzy controller proposed by Zhao and Tomizuka.

Appendices

Appendix A: PID Controller Model

The PID controller model $C^\mathrm{pid}(s)$ is given by

$$\begin{aligned} \ C^\mathrm{pid}(s)=K^\mathrm{pid} \left( s \right) , \ \end{aligned}$$

(28)

where .

The open-loop transfer function of the plant is given by

$$\begin{aligned} \ G_\mathrm{BR}^{OL}(s) = \frac{{K^\mathrm{pid} \left( s \right) \left( {0.1812s + 0.0871} \right) }}{{s\left( {s^3 + 0.3553s^2 + 0.1117s + 0.01567} \right) }}. \ \end{aligned}$$

(29)

According to Table 2, second Ziegler–Nichols method can be used by tuning through the locus roots to find the critical gain $K_\mathrm{cr}$ and the frequency of oscillations $w_\mathrm{cr}$ (where ). These values can be found from the points of intersection of the roots with axis jw, according to Fig. 12 (Katsuhiko 2009) (if the roots do not cross the jw-axis, this method does not apply).

Table 2 Ziegler–Nichols tuning rule based on critical gain $K_\mathrm{cr}$ and critical period $P_\mathrm{cr}$ (second method)

Fig. 12

Root locus

The gain $K^\mathrm{pid}(s)$ using the second Ziegler–Nichols method is given by

$$\begin{aligned}&K_\mathrm{P} = 0.6K_\mathrm{cr} = 0.6 \times 1.47 = \fbox {0.8820}\\&P_\mathrm{cr} = \frac{{2\pi }}{w_\mathrm{cr}} = \frac{{2 \times 3.1416}}{{0.614}} = 10.2332s\\&T_\mathrm{I} = 0.5 \times 10.2332 = 5.1166s\\&K_\mathrm{I} = \frac{{0.6}}{{5.1166}} = \fbox {0.1175}\\&T_\mathrm{D} = 0.125 \times 1.47 = 0.1837s \\&K_\mathrm{D} = 0.8820 \times 0.1837 = \fbox {0.1620} \end{aligned}$$

$K_\mathrm{cr} = 1.47$ and $w_\mathrm{cr} = 0.614rad/s$ are identified in Fig. 12.

Appendix B: PID–Fuzzy Model

The control system uses a fuzzy rules-based system to tune the PID controller by scheduling gains. The fuzzy system assumes that $K_\mathrm{P}$ and $K_\mathrm{D}$ are at prescribed intervals in $\left[ {K_{p_{\min } },\,K_{p_{\max } } } \right]$ and $\left[ {K_{D_{\min } },\,K_{D_{\max } } } \right]$, respectively. Inputs to the PID–fuzzy controller are the error (e) and its derivative ($\dot{e}$), and the outputs are $K'_\mathrm{P}$, $K'_\mathrm{D}$, and $\alpha$, where $K'_\mathrm{P}$ and $K'_\mathrm{D}$ are the normalized gains in a range of zero to one by means of a linear transformation, and $\alpha$ is a parameter used to calculate the gain $K_\mathrm{I}$.

$$\begin{aligned}&K'_\mathrm{P} = \frac{{K_\mathrm{P} - K_{P_{\min } } }}{{K_{P_{\max } } - K_{P_{\min } } }} \ \end{aligned}$$

(30)

$$\begin{aligned}&K'_\mathrm{D} = \frac{{K_\mathrm{D} - K_{D_{\min } } }}{{K_{D_{\max } } - K_{D_{\min } } }}. \ \end{aligned}$$

(31)

In the design of the PID–fuzzy system, the parameters of the PID are determined based on current error e(k) and the difference in error $\Delta e(k)$. The integral time constant is determined with reference to the derivation time constant.

The parameters $K'_\mathrm{P}$, $K'_\mathrm{D}$, and $K'_\mathrm{I}$ are determined from this set of fuzzy rules:

• IF e(k) is $A_i$ and $\Delta e(k)$ is $B_i$, THEN $K'_\mathrm{P}$ is $C_i$, $K'_\mathrm{D}$ is $D_i$, and $\alpha = \alpha _i$, $i = 1,\,2,\, \ldots ,\,m$.

The set of rules used to adjust gains $K'_\mathrm{P}$ and $K'_\mathrm{D}$ is presented in Table 3, and that used to determine the gain $K'_\mathrm{I}$ is displayed in Table 4, where B (big) and S (small) are the membership functions of the big and small sets, respectively. NB is negative big, NM is negative medium, NS is negative small, ZO is zero, PS is positive small, PM is positive medium, and PB is positive big.

Table 3 Tuning rules for the outputs $K'_\mathrm{P}$ and $K'_\mathrm{D}$ from the inputs e(k) and $\dot{e}(k)$

Table 4 Tuning rules for $\alpha$ from the inputs e(k) and $\dot{e}(k)$