**Appendix A. The Estimation of the Curve Using Second-Order (Linear) B-Spline Interpolation**

When using second-order (linear) B-spline interpolation, the characteristic curves of the directional control valve are not continuous, as shown in the Figure A1. Therefore, to demonstrate the real-world application, it is recommended to use the third-order B-spline or above.

**Figure A1.** The estimation of the characteristic curve of the directional control valve by using the ADEKF algorithm with second-order (linear) B-spline interpolation. (**a**) Three-point B-spline estimation. (**b**) Four-point B-spline estimation. (**c**) Five-point B-spline estimation. (**d**) Six-point B-spline estimation.

#### **Appendix B. Estimation of the Pressure Flow Coefficient and the Flow Gain**

The estimation of the pressure flow coefficient *kp* and the flow gain *k*<sup>0</sup> in the case example is represented below.

**Figure A2.** Requirements for the accuracy in the system states for parameter estimation. (**a**) Error in *kp* with 95% CI. (**b**) Error in *k*<sup>0</sup> with 95% CI.
