## 4 Results and Discussion  
In this paper, the used parameters are based on A123 LiFePO4 battery parameters. In the 1-h of simulation, the controllable input was the charging rate (=0.9 C-constant value). The uncontrollable one, ambient temperature, was assumed as a half-sine wave. Figure 4 presents the system inputs. Figure 3: Voltages sensor model. Figure 2: Current sensor model; (**a**) the Matlab/Simulink block, and (**b**) current sensors with and without noises. For the outputs of the simulation, after 1-h charging, the SOC reached 90%. As the SOC increased, the terminal voltage increased. As time passed, SOH decreased. Surface and core temperatures of the LiFePO4 battery changed as they are affected by the ambient temperature and the current. Figure 5 presents these outputs. With adding white noises to both surface temperature and terminal voltage data with SNR = 60, the generated synthetic data produced surface temperature and terminal voltage measurements. Figure 6 introduces these synthetic measurements. Figure 4: System inputs. (**a**) Temperature input values; (**b**) Current input values. Figure 5: Systems outputs; (**a**) SOC%, (**b**) terminal voltage, (**c**) SOH%, and (**d**) temperature changing. Figure 8: Voltage-measured observer estimation test using wrong parameters; (**a**) terminal voltage, (**b**) service temperature changing, (**c**) core temperature changing, (**d**) SOH, (**e**) SOC. Figure 7: Voltage-measured observer estimation values using wrong initial values; (**a**) terminal voltage, (**b**) service temperature changing, (**c**) core temperature changing, (**d**) SOH, (**e**) SOC. Both terminal voltage and SOC were estimated well for the voltage-measured observer. After 1000 s, the estimated core and surface temperatures converged with their true values. However, that did not happen to the estimated SOH. That result indicated that the SOH was unobservable. The reason for that was that the observability matrix of the voltage-measured observer was not full rank. Span ([0, 0, 0, 1, 0, 0]T, [0, 0, 0, 0, 1, 0]T, [0, 0, 0, 0, 1]T) is the observability matrix null space. That indicates that the \(T_{c}\), \(T_{s}\) states, and SOH were unobservable. Nevertheless, when the model of the observer is accurate, the convergence of the temperatures to their true values would occur, although temperatures are unobservable for the voltage-measured observer. The reason for that is the core and surface temperatures would only reach their equilibrium states after a long time as the thermal system is asymptotically stable. Figures 9 and 10 present temperature-measured observer estimation tests using wrong initial estimates and wrong parameter values, respectively. For that observer, the core and surface temperatures are estimated well. The estimated SOC and SOH did not converge to their true values. The result of that test indicated that SOC and SOH were unobservable for that observer. The observability matrix rank of that observer was 4, which was not a full rank. Span ([1, 0, 0, 0, 0, 0]T, [0, 0, 0, 0, 0, 1]T) is the observability matrix null space, which indicates the unobservability of SOC and SOH for the observer as well. Figures 11 and 12 present the results of the estimation test for the VT-measured observer using wrong initial estimates and wrong parameter values, respectively. For the VT-measured observer, \(V_{T}\), \(T_{C}\), \(T_{S}\), and SOC were well estimated, except for the SOH. The observability matrix rank of that observer was 5. Span ([0, 0, 0, 0, 0, 1]T)  
Figure 10: Temperature-measured observer estimation test using wrong parameters; (**a**) SOC, (**b**) service temperature changing, (**c**) SOH, (**d**) terminal voltage, (**e**) core temperature changing. was the observability matrix null space, which indicated the SOH unobservability as well. For the three observers' performance evaluation, six battery states' observability of the observers are shown in Tables 4-6, which present these three observers' root-mean-squared error (RMSE) and two tests.