Error Recursion Reduction Computational Technique Based Control System Design for a Multivariable Process

Proportional (P), Proportional plus Integral (I), Proportional plus Derivative (PD) and Proportional plus Integral plus Derivative (PID) are generic three-term controllers, widely used in feedback industrial control systems. PID controller can be implemented as a single or acombination of three controls. Proper tuning of these parameters is essential and important to get a stable control system. Over past half-centuries, several sets of PIDs formulas have been discussed. Among all methods, Ziegler Nichols is the basic tuning method, 1942 [1].

Prolong research on PIDs control, new tuning techniques are emerged out such as Cohen and Coon technique, 1953 [2] and Astrom and Hagguland technique, 1984 [3]. Many researchers appreciated these techniques due to maximum efficiency attained by minimum efforts. In recent 20 years, researchers are aiming to implement expert systems and process model parameters [4-6] to obtain PID controller parameters. The outcome is a model-based PID formula. The most well-known control method is the Internal Model Control Technique by Morari and Zafiriou [7-9]. Apart from these techniques, several research works are carried out on the human logical thinking ability and human nerve structure [10-12] based control. Nowadays, to meet the system demands, research communities deal with numerical optimization algorithms [13, 14]. Generally, optimization algorithms are classified as gradient and non-gradient algorithm [15-21]. Genetic algorithms (GA), grid searchers, stochastic, nonlinear simplex are families of non-gradient type algorithms [22]. Non-gradient algorithms are different from intelligent controllers [23-25] and Non-gradient algorithms are used to find optimum solutions, based on objective function evaluations. In the case of a gradient, it requires the presence of constant first subsidiaries of the target work and potentially higher subordinates. It requires the least number of configuration cycles to merge to an ideal contrasted with non-gradient based techniques [26-28].

Zheng et al. [29] designed an adaptive dynamic output-feedback control for a Chemical Continuous Stirred Tank Reactor System [30-32] under varying time delays with Nonlinear Uncertainties and highlighted the betterment of the adaptive control technique then others. Sowmya et al.  proposed Genetic Algorithm Tuned Fuzzy Controller for a Nonlinear Process and this control algorithm different from numerical algorithms [33-35]. A Hybrid controller design was proposed [36] with State Feedback Controller Design for Two various Dynamic nonlinear systems.

For process analyze and purpose of design controllers, researchers are following common practices. They find a model of a physical system represented in terms of mathematical equations and to be used for further analysis. Such a process model is represented as the first or second-order process plus dead time.

This paper addresses the two tuning techniques [21] such as an internal model control (IMC) and PSO based PID tuning techniques [14] and respective PID [19] gain values are computed manually. In this section, tuning formulas, and implementation procedures are explained in the following subsections.

3.1 Conventional tuning

Keeping in mind the end goal to arrive a PID equivalent form for a process with a period delay, we should estimate the dead time by utilizing first order Pade approximation technique is shown in Eq. (7).

$G(s)=\frac{\mathrm{K}_{\mathrm{p}} e^{-t_{d} s}}{\tau s+1}$     (6)

$e^{-t_{d} s}=\frac{-0.5 \mathrm{t}_{\mathrm{d}} s+1}{0.5 \mathrm{t}_{\mathrm{d}} s+1}$    (7)

By solving the above two equations we can get PID controller parameters are follows,

$\mathrm{K}_{\mathrm{p}}=\frac{\tau+0.5 \mathrm{t}_{\mathrm{d}}}{\mathrm{K}_{\mathrm{p}}\left(\tau_{c}+0.5 \tau_{d}\right)}$    (8)

$\mathrm{T}_{\mathrm{i}}=\tau+0.5 \tau_{d}$    (9)

$\mathrm{T}_{\mathrm{d}}=\frac{\tau \tau_{d}}{2 \tau+\tau}$     (10)

3.2 Particle swarm optimization

PSO is a hearty stochastic enhancement procedure in view of the development and collaboration of swarms. PSO is a population-based swarm algorithm, developed by Eberhart et al. The use of PSO algorithm is put ahead by a few specialists who created computational recreations of the development of creatures, for example, schools of fish and rushes of flying creatures. Such reenactments are vigorously in view of controlling the divisions between individuals, i.e., the synchrony of the lead of the swarm was viewed as a push to keep an ideal separation between them.

3.3 Selection of PSO parameters

To fire up with PSO, certain constraints should be characterized. The choice of these settings chooses, as it were, the capacity of global minimization [21]. Design constants are of the Population size of 50, Number of iterations are 50 [27] and Velocity constants are c1=1.2 and c2=2.

velocity=w *velocity+c1*(R1.*(L_b_position-current_position))+c2*(R2.* (g_b_position-current_position))    (11)

where, c1 and c2 are sure constants address the scholarly and social parameter separately; R1 and R2 are discretionary numbers reliably passed on and w is idleness weight to adjust the worldwide and neighborhood seeks capacity.

3.4 ERRC

Error Recursion - Reduction Computational Technique (ERRC) is to improve the performance of the stabilized system or to stabilize an unstable system, a feedback controller is designed using the controller tuning technique. When the process undergoes changes in the operation or by the action of disturbances, the controller designed with classical tuning methods does not yield acceptable results. Hence, it is necessary to design the controller, which takes action to uncertainties and tackle disturbances.

This technique can be considered as a generalized framework for all types of processes. It can be easily incorporated into and adapt to any kind of controller.

The primary idea behind this technique is to minimize the error in order to get a perfect control action, in an integrated manner with the classical controller. Possible to estimate the error dynamically and can be minimized. This technique can be applied to both linear as well as non-linear systems.

3.4.1 Methodology 

In the proposed technique, an error is optimally predicated or determined as a step-ahead output prediction method. Here, instant error e(k_n) can be calculated from the present process output y(k_n). Based upon the instant error value, the error value E(k_n+1) has been predicted and to be processed to Parameter Identification (PI)space. In PI space, the ultimate output y(k_n+N) should be found based upon the present data analysis. The predicted present ultimate output y(k_n+N) to be forwarded to Parameter Evaluation (PE) space to check the fitness of the predicted value and should be sent to a loop analyzer to get the fitting error value to meet the system requirements. After all this process, error will be minimized and also the controller performance is improved. Figure 6 shows the block diagram of the ERRC technique for the multi-process station and Figure 7 shows the flow chart of the computational progress in the feedback. Where S_P is setpoint, C is a comparator, C_1, …, C_n-1, C_n are controllers for n processes and P_1, …, P_n-1, P_n are representing processes. X_ref is equal to the value of reference input and sensor.

The physical equation is consisting of present process output (Po), the respective error (eo), prediction of output (Pi) and error (ei) i=1,2,3,4….

$\text { Output, } \mathrm{P}_{\mathrm{i}+1}=\left[\left(\mathrm{P}_{\mathrm{i}-1}-\mathrm{t}_{\mathrm{p}}\right)+\mathrm{e}_{\mathrm{i}}\right] $    (12)

1.6.png

Figure 6. Block diagram of ERRC technique for multi process variables

1.7.png

Figure 7. Flow chart of computational progress

The objective of the computational technique is to trim down the error to zero. In general, an error is calculated on the basis of relationship exists between the actual output (y_p) and target output (t_p),

Error, $\mathrm{E}=\left(\mathrm{t}_{\mathrm{p}}-\mathrm{y}_{\mathrm{N}}\right)$    (13)

The parameter estimation equation is formed on the basis of process,

$\mathrm{E}_{0}=\left(\mathrm{t}_{\mathrm{p}}-\mathrm{P}_{0}\right)$    (14)

Which gives present error value e₀.

Based upon this analog consecutive output, error values are predicted (i.e. P_i and e_i are predicted, i.e. “i” varies from one to infinity). The steps to follow in the analysis, determine the best fit of the minimum error value and the output. This technique seems to be an iterative method, iteration continuous on error prediction, till the error converges to e<0.0……1.

y_c is the present process output (C(t)), in both cases y_{c is} added is to the error and depends on error signal polarity, whether the present process output value is added or subtracted to the error value.