**4. Controller Stability Analysis**

**Theorem 1.** *Consider a DC-DC converter system with both CPL and supply voltage perturbations and combine Equation (14). Under the proposed control law (21), the effect of the time-varying perturbation is removed from the output voltage channel, provided that the switching gain, η* > (*k*1*e*<sup>1</sup> <sup>∗</sup> + *e*<sup>2</sup> <sup>∗</sup> + *e*<sup>3</sup> ∗*), and the observer parameters in selected Equations (12) and (13) are appropriate, such that (25) is the Hurwitz matrix.*

**Proof of Theorem 1.** For the GPI observer, the estimated error is defined as e1 <sup>=</sup> d1 <sup>−</sup> dˆ 1, e2 <sup>=</sup> d2 <sup>−</sup> dˆ 2, and e3 <sup>=</sup> . dˆ <sup>1</sup> <sup>−</sup> <sup>ˆ</sup> . d1. The upper bound of the estimation error is defined as ei ∗, (i = 1, 2, 3). Then, the estimated error of the observer can be expressed as follows:

$$\begin{cases} \begin{aligned} \mathbf{e}\_{11} &= \mathbf{x}\_{1} - \hat{\mathbf{x}}\_{1} \\ \mathbf{e}\_{12} &= \mathbf{d}\_{1} - \hat{\mathbf{d}}\_{1} \\ &\cdots \\ \mathbf{e}\_{1n} &= \mathbf{d}\_{1} \left(^{n-1}\right) - \hat{\mathbf{d}}\_{1} \left(^{n-1}\right)' \\ \mathbf{e}\_{1(n+1)} &= \mathbf{d}\_{1} \left(^{n}\right) - \hat{\mathbf{d}}\_{1} \left(^{n}\right) \end{aligned} \end{cases} \tag{23}$$
 
$$\begin{cases} \begin{aligned} \mathbf{e}\_{21} &= \mathbf{x}\_{2} - \hat{\mathbf{x}}\_{2} \\ \mathbf{e}\_{22} &= \mathbf{d}\_{2} - \hat{\mathbf{d}}\_{2} \\ &\cdots \\ \mathbf{e}\_{2m} &= \mathbf{d}\_{2} \left(^{m-1}\right) - \hat{\mathbf{d}}\_{1} \left(^{m-1}\right)' \\ \mathbf{e}\_{2(m+1)} &= \mathbf{d}\_{2} \left(^{m}\right) - \hat{\mathbf{d}}\_{1} \left(^{m}\right) \end{aligned} \tag{24}$$

where e = [e11 e12 ...e1(n+1) e21 e22 ...e2(m+1)] <sup>T</sup> takes the derivative of the estimate error. Then, the observer error can be dynamically expressed as follows:

$$
\dot{\mathbf{e}} = \mathbf{H}\_{\mathbf{e}} \mathbf{e} + \dot{\mathbf{d}}\_{\prime} \tag{25}
$$

where  $\mathbf{d} = \begin{bmatrix} 0 \, 0 \dots \mathbf{d}\_1 \, ^{(i)} 0 \, 0 \dots \mathbf{d}\_2 \, ^{(j)} \end{bmatrix}^\mathrm{T}$  
$$\mathrm{H}\_\mathrm{e} = \begin{bmatrix} \mathrm{H}\_\mathrm{e1} & 0 \\ 0 & \mathrm{H}\_\mathrm{e2} \end{bmatrix}^\mathrm{'} $$
 
$$\mathrm{H}\_\mathrm{e1} = \begin{bmatrix} -\mathrm{h}\_{11} & 1 & 0 & \dots & 0 \\ -\mathrm{h}\_{12} & 0 & 1 & \dots & 0 \\ \dots & \\ -\mathrm{h}\_{1n} & 0 & 0 & \dots & 1 \\ -\mathrm{h}\_{1(n+1)} & 0 & 0 & 0 & 0 \end{bmatrix}^\prime $$
 
$$\mathrm{H}\_\mathrm{e2} = \begin{bmatrix} -\mathrm{h}\_{21} & 1 & 0 & \dots & 0 \\ -\mathrm{h}\_{22} & 0 & 1 & \dots & 0 \\ \dots & \\ -\mathrm{h}\_{2m} & 0 & 0 & \dots & 1 \\ -\mathrm{h}\_{2(m+1)} & 0 & 0 & 0 & 0 \end{bmatrix}^\prime $$

By selecting parameters correctly in the GPI observer, we can get the Hurwitz stability matrix; that is, the state matrix of the system is the Hurwitz matrix. Then, the error dynamic is asymptotically stable, which means the following:

$$\lim\_{\mathbf{i}\to\infty} \mathbf{e}\_{\mathbf{i}} = \mathbf{d}\_{\mathbf{i}} - \hat{\mathbf{d}}\_{\mathbf{i}} = \mathbf{0},\\ (\mathbf{i} = 1, 2), \tag{26}$$

Take the Lyapunov function as follows:

$$\mathbf{V} = \frac{1}{2}\mathbf{s}^2,\tag{27}$$

$$
\dot{\mathbf{V}} = \mathbf{s}\dot{\mathbf{s}}\_{\prime} \tag{28}
$$

Substituting (17) and (22) into (28) gives the following:

$$\begin{array}{l}\dot{\mathbf{V}} \\ \leq & -\eta|\mathbf{s}| + (\mathbf{k}\_{1}\mathbf{e}\_{1} + \mathbf{e}\_{2} + \mathbf{e}\_{3})\mathbf{s} + \frac{\mathbf{v}\_{\text{in}0}}{\mathbf{LC}}\mathbf{u}\_{\text{sta}}\mathbf{s} \\ \leq & -[\eta - (\mathbf{k}\_{1}\mathbf{e}\_{1} + \mathbf{e}\_{2} + \mathbf{e}\_{3})]|\mathbf{s}| + \frac{\mathbf{v}\_{\text{in}0}}{\mathbf{LC}}\mathbf{u}\_{\text{sta}}\mathbf{s} \\ \leq & -\sqrt{2}[\eta - (\mathbf{k}\_{1}\mathbf{e}\_{1}^{\*} + \mathbf{e}\_{2}{}^{\*} + \mathbf{e}\_{3}{}^{\*})]\mathbf{V}^{\frac{1}{2}} + \frac{\mathbf{v}\_{\text{in}0}}{\mathbf{LC}}\mathbf{u}\_{\text{sta}}\mathbf{s} \end{array} \tag{29}$$

where the coefficient α > 0,β > 0 of usta can be seen from Equation (17); when s < 0, usta > 0, and when s > 0, usta < 0. Then, when the switching gain η > (k1e1 <sup>∗</sup> + e2 <sup>∗</sup> + e3 ∗) meets the condition, it is satisfied, . V < 0. According to Lyapunov's sliding-mode reachability condition, the system can reach the designed sliding-mode surface in a finite time. The system state will reach the defined sliding surface, s = 0, in a finite time.

By integrating Equations (11) and (17), we obtain the following:

$$
\dot{\mathbf{x}}\_1 = -\mathbf{k}\_1 \mathbf{x}\_1 - \left(\mathbf{d}\_1 - \hat{\mathbf{d}}\_1\right) \tag{30}
$$

According to Reference [18], if the following system exists,

$$
\dot{\mathbf{x}} = \mathbf{f}(\mathbf{t}, \mathbf{x}, \mathbf{u}), \mathbf{x} \in \mathbb{R}, \mathbf{u} \in \mathbb{R}, \tag{31}
$$

when the system input reaches stability, if the input signal meets lim <sup>t</sup>→<sup>∞</sup> <sup>u</sup> <sup>=</sup> 0, then the states satisfy lim <sup>t</sup>→<sup>∞</sup> <sup>x</sup> <sup>=</sup> 0. According to Equations (26) and (17), lim <sup>t</sup>→<sup>∞</sup> e1 <sup>=</sup> 0,k1 <sup>&</sup>gt; 0. It is easy to arrive at the conclusion that the voltage tracking error converges asymptotically to zero along the sliding surface, thus completing the proof. Similarly, the Boost mode is a similar proof. -
