## 3 Process Identification
### Three Input-Three Output Model  
The model presented in Section 4.1 was extended about one more input, as described by transfer function \(T_{3}(s)\) (16):  
\[T_{3}(s)=\frac{Y(s)}{U_{3}(s)}=\frac{1}{(10s+1)(30s+1)(60s+1)} \tag{16}\]  
As in the two input-two output case, each sub-model was discretized with a sample time of 2 s. and the step response was simulated over an 1800 s time horizon, which again is 900 time-steps. The resulting step response from each input is presented in Figure 6.  
As in the previous example, the models were reduced to a common denominator and added as shown in (17) in order to meet the requirements for the GPC internal model form shown in (3).  
\[T(s)=T_{1}(s)+T_{2}(s)+T_{3}(s) \tag{17}\]  
This step results in a 9 th order model, which yields numerical errors during the closed-loop simulation. Simulation of transfer function \(T(s)\) for 1800 s (900 steps) is presented in Figure 7.  
Figure 5: GPC with the 6 th order internal model—closed loop simulation of 300 time-steps, trend for MVs signals.  
Figure 6: Step response for T\({}_{1}\), T\({}_{2}\), and T\({}_{3}\) transfer functions simulated for 900 time-steps.  
This example shows that even a correctly executed identification experiment that results in mathematically correct SISO models may eventually lead to a situation in which the internal model is incorrect due to the effects of the high order model. The model's irregularities result from numerical problems and are visible only during simulation of the model in the GPC internal model format. This unfavorable property must be correctly identified at the stage of synthesis of the control algorithm. In the event of this unfavorable situation, appropriate remedial measures must be taken. Otherwise, the overall control performance will be negatively impacted, and the closed loop response of the resulting predictive controller may be unstable. This situation is showed in the next simulation.  
The resulting GPC controller, with the 9 th order internal model, was simulated in closed loop and the three individual modes, before the reduction to common denominator, were used as the plant. The results of the simulation are shown in Figures 8 and 9. As can be easily seen, the control signals are not smooth, resulting from numerical issues. The same experiment was repeated with larger values in the MV penalty matrix \(R\) with similar results. The poor performance is caused by numerical issues associated with the simulation using a high order internal model of the GPC controller. Furthermore, the addition of another input further increases the transfer function order and leads to complete destabilization of the algorithm in closed loop.  
Figure 8: GPC with the 9 th order internal model—closed loop simulation of 300 time-steps; dashed line is a set point, solid is a CV, and control parameters as: _control horizon_\(N_{u}=20\), _prediction horizon_\(N=400\), and the \(R\) matrix = _identity matrix_.  
Figure 7: Step response of the 9 th order model after reduction to common denominator simulated for 900 time-steps.  
The problem outlined above can be solved in several ways. The methods of model order reduction may be considered as the first solution. Linear systems have seen a wide variety of reduction techniques such as modal/eigenvalue truncation [37], Lyapunov balanced truncation [38], frequency weighted model reduction [39], normalized co-prime factorization [40], moment matching [41], optimal Hankel approximations [42], Karhunen-Loeve (proper orthogonal decomposition (POD)) [43], approximation by a lower order time-delay model [44], etc. However, these methods require specialized knowledge in detailed mathematics. In this paper, we strive to find an effective engineering method that can be utilized by engineers and practitioners in the industry.