## 2 RTD-A Control Scheme  
The basic control scheme of the RTD-A controller is discussed in this section. The general closed loop block diagram with the RTD-A controller is shown in Figure 1. There is adequate research going on to find the appropriate alternative algorithm for the PID controller. This alternative RTD-A controller is also known as the regulatory controller [11]. This controller is basically an implementation of an algorithm designed to have the robustness characteristics, tracking ability, disturbance rejection capability and aggressive property. Initially, [PERSON] and [PERSON] have developed a tuning rule that deals with the parametric uncertainty of the controller [13]. In 2011, [PERSON] designed the second tuning method that overcomes the problem of more aggressiveness [14]. The proposed idea concentrates on designing the simulation tool for a RTD-A controller in MATLAB. The control scheme is implemented in three steps [4; 12]:  
1. Predicting the process output;
2. Updating the model prediction;
3. Computing the control action. Most of the processes can be represented in the form of the first order process with a dead time (FOPDT) model. It turns out to be very useful to investigate the system output because it has all the necessary components in the unambiguous form. At the first step, the higher order over-damped system transfer function reduced to the FOPDT model using different methods such as fraction incomplete, two-point and Skogestad in the Tool-I [28]. The FOPDT system output used in this paper is given in (1) [29]:  
\[y(s)=\frac{Ke^{-as}}{Ts+1}u(s), \tag{1}\]  
Figure 1: General closed loop diagram with a robust, tracking, disturbance rejection and aggressiveness (RTD-A) controller. where \(K\) is the process gain, \(\alpha\) is the dead time or transportation lag, \(\tau\) is the time constant, \(u(s)\) is the process input and \(y(s)\) is the process output. The discrete model for (1) is given in (2). \[\overline{y}(k+1)=\overline{ay}(k)+bu(k-m),\ k=0,1,2\ldots, \tag{2}\]  
where,  
\(a=e^{-\Delta t/\tau}\), \(\Delta t\) is the sampling time;  
\(b=K(1-e^{-\Delta t/\tau})\) and \(m=\mathrm{round}(\alpha/\Delta t)\). From (2), the output prediction for the next \(m\) steps can be written as in (3) [11]. \[\overline{y}(k+m+1)=a^{(m+1)}\overline{y}(k)+b\mu(k,m)+bu(k), \tag{3}\]  
where,  
\(\mu(k,m)=\sum_{i=1}^{m}a^{i}u(k-i)\). \(\mu\) is the weighted sum of the past control action taken during the \(m\) period and \(u(k)\) is the current control action. There will be only one control \(u(k)\) allowed from the current time instant and is given in (4). \[u(k+i)=u(k),\ i=1,2,\ldots N. \tag{4}\]  
Equation (5) is obtained by considering the predicted process output over the next \(N\)-steps. \[\overline{y}(k+m+i)=a^{m+i}\overline{y}(k)+a^{i-1}b\mu(k,m)+b\eta_{i}u(k);\ 1 \leq i\leq N, \tag{5}\]  
where,  
\[\eta_{i}=\frac{1-a^{i}}{1-a}. \tag{6}\]  
The model prediction requires a model error \(e(k)=y(k)-\overline{y}(k)\) that is decomposed into two parts such as the effect of model mismatch \(e_{m}(k)\) and unmeasured disturbance \(e_{D}(k)\). Therefore, the total error is (7),  
\[e(k)=e_{m}(k)+e_{D}(k). \tag{7}\]  
The estimate of \(e_{D}(k)\) is given as (8). \[\widetilde{e}_{D}(k)=\theta_{R}\widetilde{e}_{D}(k-1)+(1-\theta_{R})e(k) \tag{8}\]  
where \(\theta_{R}\) lies between 0 and 1. The controller has the robustness ability to handle the plant uncertainties. The future disturbance effect is predicted according to (9). \[y(s)=\frac{Ke^{-as}}{Ts+1}u(s), \tag{1}\]0  
for \(m+1\leq j\leq m+N\), where \(\
abla e_{D}(k)=e_{D}(k)-e_{D}(k-1)\). Here \(\theta_{D}\) is the disturbance rejection controller tuning parameter. Finally, by solving the (5) and (9) the updated \(N\)-step model output prediction is given by (10). \[y(s)=\frac{Ke^{-as}}{Ts+1}u(s), \tag{1}\]1  
A plant with the setpoint trajectory \(y^{*}\) has the control action to minimize the error. At a discrete time interval, the control action will be updated.