# Low Temperature Combustion Modeling and Predictive Control of Marine Engines
## 3 Results
### Closed-Loop Control Simulations  
The closed-loop system was simulated in steady state and transient cases for both mid and high engine load. The control signals were the same as in the previous simulations, i.e., \(\Delta\text{BR}\) and \(\Delta E_{\text{fuel}}\) and process outputs were \(\theta_{50}\) and \(\text{IMEP}_{\text{gross}}\). The sampling time was set to 80 ms, corresponding to one engine cycle at the speed of 750 RPM. Figures 10 and 11 present the results with constant and step inputs in nominal operating conditions. In the nominal conditions, the convergence of \(\text{IMEP}_{\text{gross}}\) towards the reference value is slow. However, the steady state error remains small. As shown in Figure 10, the maximum steady state error is 1.1 bar. However, \(\text{IMEP}_{\text{gross}}\) decreases rapidly during the first 50 cycles, after which the error is below 0.5 bar. In Figure 10 the steady state error of \(\text{IMEP}_{\text{gross}}\) remains below 0.5 bar in high load condition, although it never reaches the reference value. In both cases \(\theta_{50}\) reaches the reference values in relatively short time. Figure 11,b present the nominal mid and high load conditions with step inputs. \(\theta_{50}\) is changed by 2 \({}^{\circ}\)CA and \(\text{IMEP}_{\text{gross}}\) by 0.5 bar. In both cases, \(\theta_{50}\) accurately follows the reference value. \(\text{IMEP}_{\text{gross}}\) fully reaches the reference value after the step change is activated. Similar behavior between the inputs and outputs can be observed as previously shown in Section 3.3. At the 50 th cycle, the step change for \(\theta_{50}\) takes place. This can be seen as an  
Figure 11: Simulation results of the system for \(\theta_{50}\) and \(\text{IMEP}_{\text{gross}}\) control using \(\Delta E_{\text{fuel}}\) and \(\Delta\text{BR}\) as control variables at nominal mid and high engine loads. A small step change has been applied on the reference values. (**a**) Step response of the system at mid engine load. (**b**) Step response of the system at high engine load. Figure 10: Simulation results of the system for \(\theta_{50}\) and \(\text{IMEP}_{\text{gross}}\) control using \(\Delta E_{\text{fuel}}\) and \(\Delta\text{BR}\) as control variables at nominal mid and high engine loads. No step change has been applied on the reference values. (**a**) Steady state results at mid engine load. (**b**) Steady state results at high engine load. increase of \(\Delta\)BR, which decreases the \(\text{IMEP}_{\text{gross}}\). Similarly, with \(\Delta\)BR decreasing \(\text{IMEP}_{\text{gross}}\) increases as can be seen at the 100 th cycle. This behavior could be explained by the fact that \(E_{\text{fuel}}\) imposes BR as shown in Equation (18). In the transient case, the nominal condition of the system is changed from mid load to high load, as shown in Figure 12. Here, both output signals showcase a good absolute and trendwise accuracy. The peak of \(\Delta\)BR does not seem to affect the \(\text{IMEP}_{\text{gross}}\) during the step change. This might be due to a high value of \(\Delta E_{\text{fuel}}\), which overcomes the affect of \(\Delta\)BR. Additionally, \(\Delta E_{\text{fuel}}\) achieved its maximum value during the transition. Increasing or decreasing the maximum constraint of \(\Delta E_{\text{fuel}}\) would be seen as overshoot or undershoot of \(\text{IMEP}_{\text{gross}}\), respectively. The nominal operating conditions with step inputs and their transient were performed again with small disturbance added to the process inputs \(T_{\text{IVC}}\) and \(m_{\text{EGR}}\). The disturbances were added as random noise. The mean (\(\mu\)) and standard deviation (\(\sigma\)) for each input disturbance are presented in Table 6.  
Despite the process disturbance, the system remains stable, as shown in Figures 13 and 14.