Design of RCGA-based PID controller for two-input two-output system

2.1 Multi-variable system

In many cases, control systems commonly found in industrial sites have to control more than two variables. Additionally, there can be interference in mutual outputs, as shown in Figure 1. Here, when the pressure (P) is controlled by operating the pressure control valve (V_p), the flow (F) within the pipe changes at the same time. When the flow is controlled by operating the flow control valve (V_f), the flow within the pipe changes along with the pressure. That is, under the condition that the flow is consistent with a target value while the pressure differs from a target value, when is operated to adjust the pressure, the flow—which has the correct value—is inevitably changed. A similar situation occurs for the pressure when is operated to adjust the flow.

Figure 1:

Flow and pressure control system

2.2 TITO System

In this paper, a TITO system, which is one of the most common multivariable systems, is examined. As shown in Figure 2, a TITO system comprises two inputs and two outputs.

Figure 2:

Block diagram of a TITO system

As shown in the figure, the TITO system is divided into the transfer functions G₁₁(s), G₁₂(s), G₂₁(s), and G₂₂(s) and G₁₂(s) and G₂₁(s) induce mutual interference in their outputs.

R(s)∈R² is the reference input, E(s)∈R² is the error, U(s)∈R² is the control input, and Y(s)∈R² is the output vector.

The transfer function matrix G(s)of this TITO system is expressed by Equation (1).

As shown in Figure 2, each control input is determined by the PID controller using each error variable. This can be expressed in a diagonal matrix PID controller format excluding the transfer matrix, as shown in Equation (2).

$$ k_{p1}+\frac{k_{i1}}{s}+k_{d1}s$$ and $$ k_{p2}+\frac{k_{i2}}{s}+k_{d2}s$$ represent the controllers PID1 and PID2, respectively.

To obtain each gain matrix, Equation (2) can be modified into Equation (3).

where K_p∈R^2×2 is the proportional gain matrix, K_i∈R^2×2 is the integral gain matrix, and K_d∈R^2×2 is the derivative gain matrix. They can be expressed as follows.

3.1 PID controller based on Z-N

In the TITO system, the outputs of the two controllers have a mutual effect on each other; thus, the tuning rule of Z-N can not be directly applied to a single-input single-output (SISO) system. Thus, it is necessary to divide a TITO system into two SISO systems. When the control input U(s) in Figure 2 is set as ‘0’ in order to design a PID1 controller, it can be divided into two SISO systems for the input R₁(s), as shown in Figure 3. In this regard, one SISO system becomes a closed-loop system, as shown in Figure 3 (a), and the other becomes an open-loop system, as shown in Figure 3 (b).

Figure 3:

Decoupled TITO system(U2(s)=0)

In Figure 3 (b), the input U₁(s) can be considered as a disturbance or sensor noise added to the output Y₂(s), and this does not affect the error E₁(s) necessary for the controller design. Thus, to design a PID1 controller for the reference input R₁(s), only Figure 3 (a) must be considered.

On the other hand, to design a PID2 controller, the control variable U₁(s) must be removed. Then, the system in Figure 2 can be divided into two SISO systems for the input R₂(s), similarly to Figure 3. Here, one SISO system becomes a closed-loop system, and the other becomes an open-loop system.

The Z-N tuning formulas for the PI and PID controllers are shown in Table 1. We considered only the PID controller.

Table 1:

Ziegler-Nichols tuning rules

3.2 PID controller based on RCGA

3.2.1 Structure and operator of RCGA

In this section, an RCGA used to resolve an optimization problem that occurs during the design of PID controllers is briefly examined [7].

An RCGA is an algorithm based on natural selection and genetics. It starts from an initial group that consists of a number of individuals (chromosomes) and a group that has been newly formed through fitness evaluation. A genetic operator repeatedly performs simulated evolution until an optimal solution is found. Figure 4 shows the operation process of the RCGA employed in this paper.

Figure 4:

Basic structure of RCGA adopted

For the genetic operator used in the RCGA, reproduction similar to the gradient, a modified simple crossover, and dynamic mutation are used. A scaling-window method is used to increase the selection pressure, and an elite strategy is used to guarantee that the strongest individual is passed onto the next generation by preventing the optimal individual extinction of a generation.

4.1 System and controller design

To examine the validity of the proposed method, a linearized TITO system [1] is considered, as shown in Equation (8). This system is not a typical process-control problem, as the plant is not diagonally dominant and also has one large and negative diagonal gain.

When the closed-loop transfer function is obtained by setting the critical proportional gain K_cr for G₁₁(s) , it is expressed by Equation (9).

The K_cr value of the stability limit where an output induces oscillation with a constant amplitude can be obtained by the Routh–Hurwitz stability criterion.

In this case, the value is K_cr=9. When the period of oscillation P_cr is calculated by determining the oscillation frequency using the characteristic equation in Equation (9), it is 0.56 sec. Therefore, the parameters of the PID controller according to the Z-N tuning rule (refer to Table 1) are determined as follows: K_p1 = K_p2 = 5.4, T_p1 = T_p1 = 0.28 and T_d1 = T_d2 = 0.07.

For the control variables of the RCGA, the population size N = 20, reproduction coefficient η = 1.8, crossover probability P_c = 0.9, and mutation probability P_m = 0.2 are used. The upper boundary values of the PID controller parameters $$ K_p^{\mathit{UL}}$$, $$ K_i^{\mathit{UL}}$$, and $$ K_d^{\mathit{UL}}$$ are set to 25, including a sufficient margin based on the PID controller parameters obtained by the Z-N method. By decreasing the search range of RCGA, the computation time is reduced, and a global solution is guaranteed.

The parameters of the PID controller are determined by using the ISE, IAE, and ITAE as objective functions of the RCGA. Figure 5–7 show the determination of the PID controller parameters by using the ISE, IAE, and ITAE.

Figure 5:

Tuning process of PID parameter using ISE

Figure 6:

Tuning process of PID parameter using IAE

Figure 7:

Tuning process of PID parameter using ITAE

where t_f is a time that is sufficiently large that the integral after this time can be ignored.

Table 2 summarizes the PID controller parameters based on Z-N and the RCGA

Table 2:

Parameters of PID controllers by Z-N and RCGA

4.2 Performance comparison

To examine the validity of the proposed method, a stepwise set-point tracking experiment is conducted. The target value r(t) = [1 0]^T is applied to each controller, and the output y(t) = [y₁(t) y₂(t)]^T is examined. Figure 8–9 show the responses of the PID control systems tuned by the Z-N method and the proposed method. As shown, both the responses of the Z-N method and the proposed method reach the set point, but the response of the Z-N method oscillate excessively, with a large overshoot and a long settling time.

Figure 8:

Response of PID control system by Z-N

Figure 9:

Responses of PID control systems by RCGA

Table 3 shows the percentage overshoot (M_p), rising time (t_r), and 2% settling time (t_s) for a quantitative comparison of the performance of each method. Here, t_r = t₉₀ - t₁₀ is used, where t₁₀ and t₉₀ are the times taken for the output to reach 10% and 90%, respectively, of the set point. As summarized in the table, the control performances of the PID control systems based on the RCGA are superior to those for the Z-N method in every aspect.

Table 3:

Comparison of control performances