*3.2. Parameter Estimation Algorithm*

In the parameter estimation algorithm, the augmented state vector **x**ˆ is defined as

$$\hat{\mathbf{x}}' = \underbrace{\begin{bmatrix} s & s & p\_p & p\_1 & p\_2 \\ \hline & & & \end{bmatrix}}\_{\hat{\mathbf{x}}} \underbrace{k\_p \quad k\_0 \quad \mathbf{C\_v}}\_{\hat{\boldsymbol{\theta}}} \Big|^T \tag{37}$$

where *s* is the actuator position, *s*˙ is the actuator velocity, *pp*, *p*1, and *p*<sup>2</sup> are the pressures, *kp* is the pressure flow coefficient, *k*<sup>0</sup> is the flow gain, and **Cv** = **Cv***<sup>a</sup>* **Cv***<sup>b</sup>* **Cv***<sup>c</sup>* **Cv***<sup>d</sup>* are the semi-empiric flow rate coefficients at the corresponding ports of the directional control valve. In Equation (37), **x**ˆ and **y**ˆ present the states and the parameters of the hydraulically driven four-bar mechanism, respectively. Equations (20)–(28) are implemented to estimate the augmented state vector **x**ˆ and the characteristic curves of the directional control valve. In this application, the third-order B-spline interpolation method is combined with the ADEKF. For the case example, three, four, five, and six control points are used in the parameter estimation algorithm to compute Equations (20) and (24). As mentioned earlier, the first, third, and fourth state variables are measured. Therefore, the sensor measurement function **h**(**x**ˆ− *<sup>k</sup>* ) and its Jacobian **hx** can be written as

$$\begin{aligned} \mathbf{h}(\mathbf{\dot{x}}\_{k}^{\prime -}) &= \begin{bmatrix} \mathbf{\dot{x}}\_{k,1}^{\prime -} & \mathbf{\dot{x}}\_{k,3}^{\prime -} & \mathbf{\dot{x}}\_{k,4}^{\prime -} \end{bmatrix}^{T} \\ & \mathbf{h}\_{\mathbf{x}^{\prime}} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\partial \mathbf{h}}{\partial \mathbf{C}\_{\mathbf{v}\_{d}}} & \frac{\partial \mathbf{h}}{\partial \mathbf{C}\_{\mathbf{v}\_{b}}} & \frac{\partial \mathbf{h}}{\partial \mathbf{C}\_{\mathbf{v}\_{c}}} & \frac{\partial \mathbf{h}}{\partial \mathbf{C}\_{\mathbf{v}\_{d}}} \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{\partial \mathbf{h}}{\partial \mathbf{C}\_{\mathbf{v}\_{d}}} & \frac{\partial \mathbf{h}}{\partial \mathbf{C}\_{\mathbf{v}\_{b}}} & \frac{\partial \mathbf{h}}{\partial \mathbf{C}\_{\mathbf{v}\_{c}}} & \frac{\partial \mathbf{h}}{\partial \mathbf{C}\_{\mathbf{v}\_{d}}} \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{\partial \mathbf{h}}{\partial \mathbf{C}\_{\mathbf{v}\_{d}}} & \frac{\partial \mathbf{h}}{\partial \mathbf{C}\_{\mathbf{v}\_{b}}} & \frac{\partial \mathbf{h}}{\partial \mathbf{C}\_{\mathbf{v}\_{c}}} & \frac{\partial \mathbf{h}}{\partial \mathbf{C}\_{\mathbf{v}\_{d}}} \end{bmatrix}, \end{aligned} \tag{38}$$

where **Cv** = **Cv***<sup>a</sup>* **Cv***<sup>b</sup>* **Cv***<sup>c</sup>* **Cv***<sup>d</sup>* . **h**(**x**ˆ− *<sup>k</sup>* ) and **hx** are used in Equations (26) and (27), respectively, in the parameter estimation algorithm.
