*3.3. GPI Observer Design*

The voltage tracking accuracy of the DC-DC converter system will be affected by disturbances, such as input voltage fluctuation, parameter uncertainty, load resistance disturbance, etc. An effective method to eliminate these disturbances is to introduce disturbance estimation to compensate accurately. For the Buck converter, two generalized integral observers are designed to estimate the matched disturbance and the unmatched disturbance, respectively, and the disturbance estimation is introduced into the design of the control law to compensate for the influence of these disturbances and uncertainties. The specific design of the two generalized proportional integral observers is as follows. The GPI observer of CPL perturbation based on the Buck converter can be constructed as follows:

$$\begin{cases} \dot{\mathbf{x}}\_{1} = \mathbf{x}\_{2} + \dot{\mathbf{d}}\_{1} + \mathbf{h}\_{11}(\mathbf{x}\_{1} - \hat{\mathbf{x}}\_{1}) \\ \dot{\mathbf{d}}\_{1} = \dot{\hat{\mathbf{d}}}\_{1} + \mathbf{h}\_{12}(\mathbf{x}\_{1} - \hat{\mathbf{x}}\_{1}) \\ \dots \\ \dot{\mathbf{d}}\_{1}(\mathbf{n} - 1) = \dot{\mathbf{d}}\_{1}(\mathbf{n}) + \mathbf{h}\_{1n}(\mathbf{x}\_{1} - \hat{\mathbf{x}}\_{1}) \\ \dot{\mathbf{d}}\_{1}(\mathbf{n}) = \mathbf{h}\_{1(n+1)}(\mathbf{x}\_{1} - \hat{\mathbf{x}}\_{1}) \end{cases} \tag{12}$$

where dˆ <sup>1</sup> (n) is an estimated value of the nth order derivative of d1, and h1i (i = 1, 2, ... , n + 1) represents the parameters to be determined.

To estimate the input voltage disturbance, d2(t), another GPI observer is constructed:

$$\begin{cases} \dot{\mathbf{\hat{x}}}\_{2} = \frac{\mathbf{u}}{\mathbf{LC}} \mathbf{v}\_{\text{in}0} - \frac{\mathbf{x}\_{1} + \mathbf{v}\_{\text{ref}}}{\mathbf{LC}} - \frac{\mathbf{x}\_{2}}{\mathbf{R} \cdot \mathbf{C}} + \mathbf{z}\_{2} + \mathbf{h}\_{21} (\mathbf{x}\_{2} - \mathbf{\hat{x}}\_{2}) \\ \qquad \dot{\mathbf{d}}\_{2} = \dot{\mathbf{d}}\_{2} + \mathbf{h}\_{22} (\mathbf{x}\_{2} - \mathbf{\hat{x}}\_{2}) \\ \qquad \dots \\ \dot{\mathbf{d}}\_{2} (\mathbf{m} - 1) = \dot{\mathbf{d}}\_{2} (\mathbf{m}) + \mathbf{h}\_{2\mathbf{m}} (\mathbf{x}\_{2} - \mathbf{\hat{x}}\_{2}) \\ \qquad \dot{\mathbf{d}}\_{2} (\mathbf{m}) = \mathbf{h}\_{2(\mathbf{m} + 1)} (\mathbf{x}\_{2} - \mathbf{\hat{x}}\_{2}) \end{cases} \tag{13}$$

where dˆ <sup>2</sup> (m) is an estimated value of the nth order derivative of d2, and h2j (j = 1, 2, ... , m + 1) represents the parameters to be determined.

According to Equations (9) and (10), the uncertainties (i = 1, 2) are related to the power of constant power load, so from a practical point of view, their values and derivatives should be bounded.

In a steady state, the power of the constant power load is considered constant. Therefore, the following assumptions can be made:

The uncertain variables, di and . di, of the system (i = 1, 2) meet the following two conditions [47]: .

$$\begin{cases} \mathbf{d}\_{\mathbf{i}}(\mathbf{t}) \in \mathcal{L}\_{\infty} \dot{\mathbf{d}}\_{\mathbf{i}}(\mathbf{t}) \in \mathcal{L}\_{\infty} \\ \lim\_{\mathbf{t} \to \infty} d\_{\mathbf{i}}^{(n)} = 0 \end{cases} \tag{14}$$

According to Equation (8), the uncertain term is defined by the following:

$$\begin{cases} \dot{\mathbf{d}}\_1 = \mathbf{l}\_1(\mathbf{x}\_1 - \mathbf{p}\_1) \\ \dot{\mathbf{p}}\_1 = \mathbf{x}\_2 + \dot{\mathbf{d}}\_1 \end{cases} \tag{15}$$

where p1 is the auxiliary state of the observer, and l1 is a normal number that is expressed as the observer gain. Similarly, the uncertainty term is given by the following:

$$\begin{cases} \dot{\mathbf{d}}\_2 = \mathbf{l}\_2(\mathbf{x}\_2 - \mathbf{p}\_2) \\ \dot{\mathbf{p}}\_2 = \frac{\mathbf{u}}{\mathbf{LC}} \mathbf{v}\_{\mathrm{in}0} - \frac{\mathbf{x}\_1 + \mathbf{v}\_{\mathrm{ref}}}{\mathbf{LC}} - \frac{\mathbf{x}\_2}{\mathbf{R}\mathbf{LC}} + \dot{\mathbf{d}}\_2 \end{cases} \tag{16}$$

where p2 is the observer's auxiliary state, and l2 is a normal number that is expressed as the observer gain.

Based on Equations (12) and (13), the standard model and observer estimate of the load power can be provided according to the sliding-mode control design of the proposed composite controller. Them we take the switching function of the system as follows:

$$\begin{cases} \mathbf{s} = \mathbf{k}\_1 \mathbf{x}\_1 + \mathbf{x}\_2 + \hat{\mathbf{d}}\_1\\ \dot{\mathbf{s}} = \mathbf{k}\_1 (\mathbf{x}\_2 + \mathbf{d}\_1) + \dot{\mathbf{x}}\_2 + \dot{\mathbf{d}}\_1 \end{cases} \tag{17}$$

where k1 > 0 is a parameter to be selected.

As a high-order sliding-mode algorithm, the realization of the high-order slidingmode algorithm usually requires the derivative of sliding-mode variables, while the superdistortion algorithm is a second-order sliding-mode algorithm in nature, so its realization does not require the derivative of sliding-mode variables, thus simplifying the controller structure. Through the design of the control rate, the sliding-mode variable structure rapidly converges within a limited time [30].

The general form of the super-twisting algorithm is as follows:

$$\begin{cases} \frac{d\mathbf{a}\_1}{dt} = -\lambda |\mathbf{a}\_1|^{\frac{1}{2}} \text{sign}(\mathbf{a}\_1) + \mathbf{a}\_2 + \rho\_1\\ \frac{d\mathbf{a}\_2}{dt} = -\text{csign}(\mathbf{a}\_1) + \rho\_2 \end{cases} \tag{18}$$

In Equation (16), a1 and a2 are the state variables; λ and c are the positive constants; and ρ<sup>1</sup> and ρ<sup>2</sup> are the disturbance quantities.

In order to weaken chattering, saturation function is often used to replace the sign function [48]. The form of saturation function is as follows:

$$\text{sat}(\mathbf{s}, \boldsymbol{\delta}) = \begin{cases} \frac{\mathbf{s}}{\mathfrak{S}} & |\mathbf{s}| \le \boldsymbol{\delta} \\ \text{sign}(\mathbf{s}) & |\mathbf{s}| > \boldsymbol{\delta}' \end{cases} \tag{19}$$

By combining Equations (18) and (19), the form of the super-twisting algorithm becomes the following:

$$\begin{cases} \mathbf{u}\_{\rm sta} = -\alpha |\mathbf{s}|^{\frac{1}{2}} \text{sat}(\mathbf{s}) + \boldsymbol{\omega} \\ \frac{d\boldsymbol{\omega}}{\rm dt} = -\beta \text{sat}(\mathbf{s}) \end{cases} \text{,} \tag{20}$$

When the super-twisting algorithm is used to design the sliding-mode control function, let ω = a2 + ρ1, and α,β are the parameters to adjust the dynamic velocity and set the steady-state error, respectively. Among them, the parameters α,β affect the convergence rate of the sliding-mode surface. In general, the system can reach the sliding-mode surface faster by taking a larger value and a smaller value.

When the first derivative of the sliding-mode surface is zero, the switching signal, u, is equivalent to a continuous value, ueq.

$$\mathbf{u\_{eq}} = -\frac{\mathbf{L}\mathbf{C}}{\mathbf{v\_{in0}}} \left[ \mathbf{k\_{1}} \left( \mathbf{x\_{2}} + \hat{\mathbf{d}}\_{1} \right) - \frac{\mathbf{x\_{1}} + \mathbf{v\_{ref}}}{\mathbf{L}\mathbf{C}} - \frac{\mathbf{x\_{2}}}{\mathbf{R}\_{\mathbf{L}}\mathbf{C}} + \hat{\mathbf{d}}\_{2} + \hat{\mathbf{d}}\_{1} + \eta \mathbf{sat}(\mathbf{s}) \right],\tag{21}$$

where the control parameter, k1 > 0, and the switching gain, η > 0, are the parameters to be designed. Then, the total switch signal, u, is as follows:

$$\mathfrak{u} = \mathfrak{u}\_{\mathsf{eq}} + \mathfrak{u}\_{\mathsf{sta}\prime} \tag{22}$$

where ueq is used to ensure that the trajectory of the system phase is maintained on the sliding-mode surface, and usta is used to overcome the disturbance effect and to ensure the robustness of the system. The following proves the existence and accessibility of the switching surface, and the Lyapunov function is used to analyze the switching function, so as to enable the stability of the controller's control voltage and the state curve to quickly converge to the sliding surface.
