## 3 Trajectory Tracking Control of a Nonlinear Reactor: A Case Study
### A Preliminary Simulation Analysis of the I/O FBL Control Design
In order to employ the state compensator provided in Equation (44), measurements regarding the system variable states are required. Particularly, the concentration needs to be known for the I/O FBL control law (44) to prevail. Since the possibility of concentration measurements has already been removed, this issue can be fixed by adopting a model-based control strategy. The CSTR model is employed as an observer in order to feed the control input with concentration value. It needs to be noted that the disturbance \(D_{2}\) remains unmeasured. In the beginning, this disturbance was taken into account for the process but not considered in the CSTR model. Moreover, since the output is already recognized, this information is employed for the model. The extraction of the manipulated variable to the plant (\(U\)) is established throughout the equivalent I/O FBL model to feed identical input to both Process and Model. Thus, the system and the model are impacted by bounds over the input naturally. The I/O FBL approach had satisfactory performance when all the disturbances were measured and states are obtainable, provided that modeling errors and parameter uncertainty are not there. In order to offer a comparative analysis versus the case of a disturbed system, Figure 2 shows the system's closed-loop stabilization in the neighborhood of the focused equilibrium point while employing the feedback control design (44). Should the measurable disturbance \(D_{1}\) exist, modification is done only to the control law's magnitude, while the output remains the same because of decoupling. During startup, a predetermined trajectory needs to be adopted by the temperature towards the steady-state. This trajectory can generally be represented as:  
\[y_{d}(t)=X_{2s}[1+k_{1}\exp(-k_{2}t)] \tag{45}\]  
Here, \(k_{2}\) and \(k_{1}\) rely on real plant limitations and \(X_{2s}\) represents the anticipated steady-state features. Figure 3 shows the CSTR state variables dynamics obtained for several values of \(k_{2}\), when there are no disturbances. Now, a 10% step disturbance is taken into account for the feed concentration \(D_{2}\). At first, the system would tend to stray away from the equilibrium, but later, when the disturbance is gone, it proceeds to the steady state caused by the action of the controller. [38] contest there is no need to measure this disturbance, at most when all the CSTR states variables have become measurable. Nevertheless, referring to this case study, the info on the disturbance \(D_{2}\) is not entirely extant in the model variables since the concentration measurement has not been done. Subsequently, it is not possible to decouple this unmeasured disturbance without properly estimating the value of the concentration. This later condition becomes more complex when \(D_{2}\) remains for a long time, which commits the system steady state to move towards another one (Figure 4). Figure 3: Tracking during startup. Figure 2: Stabilization and tracking by I/O linearization. To better interpret the problem, let a random disturbance be injected into the CSTR model. This later disturbance was considered for the process simulation, nonetheless, ignored in the CSTR control model calculation. The result of this study is illustrated in Figures 5 and 6 by the following figures:  
It is obvious that the disturbance hardly affects the state variables and the control input dynamics. Equally important when implementing the SMC, the CSTR variables dynamics are largely improved. However, the chattering appears and affects the temperature dynamics, which is not accepted in terms of chemical process constraints. The investigated control problem in this work is similar to the case of the uncertainty modeling problem. As stated in Section 2, plant/model inconsistency is a substantial restriction in the use of I/O FBL control methods since the controller design is attained based on the plant modeling outcomes [43; 44].