Adaptive Observer-Based Robust Control of Mismatched Buck DC–DC Converters for Renewable Energy Applications ^†

Abstract

This paper presents a new robust control strategy for buck DC–DC converters that achieve fast and robust voltage regulation in the presence of load disturbances and model uncertainties. First, an adaptive state observer is designed to estimate the inductor current and capacitor voltage by utilizing the output measurement. The observer gains are tuned online via a Lyapunov-based adaptation law, ensuring that the estimation error remains uniformly bounded, even when the disturbances act on the system. Based on the state estimates, an integral sliding-mode controller is designed in order to eliminate the steady state error and ensure the finite time sliding. The detailed stability proofs for both the observer and the sliding-mode controller are derived showing the finite-time reaching of the sliding surface and exponential convergence of the voltage error. Simulation results under varying load profiles confirm that the proposed scheme outperforms traditional sliding-mode designs in terms of disturbance rejection and settling time, while avoiding excessive chattering.

1. Introduction

Power converters have vast applications in the modern power-electronics systems, from portable electronics to renewable-energy systems. Amongst these power converters, the buck DC–DC converter is widely utilized for stepping down voltage with high efficiency and fast dynamic response [1,2,3]. However, the practical converters must operate under fluctuating loads and parameter uncertainties, which result in the degradation of the regulation performance, and they can even make the system unstable [4].

Traditional control strategies, like PID controllers, are widely utilized in the industry [5,6]. However, such controllers struggle with system uncertainties and disturbances. While the robust control schemes, e.g., sliding-mode control (SMC), have gained traction for their robustness against the disturbances and model inaccuracies, they also come with the trade-offs [7,8]. Conventional SMC applied to buck converters suffer from chattering and steady-state error due to the mismatched disturbances, where the disturbance does not directly affect the control input, and also due to the dependence on accurate measurements of all the system states, i.e., the inductor current and capacitor voltage [9,10]. Obtaining these state measurements usually requires additional, costly, or noisy sensors.

The observer-based SMC controllers are a popular choice because of their ability to estimate the unmeasured states using available outputs [11,12,13]. However, standard fixed-gain observers have limited robustness in case the converter’s parameters are not perfectly known [14]. Similarly, in the presence of large, unpredictable disturbances, their estimated values can become inaccurate and hence degrade the overall control performance [15]. Adaptive observers provide the solution by adjusting their gains online, but ensuring their stability and bounded estimation error under all the operating conditions requires careful design.

To overcome these limitations, this paper proposes a new adaptive observer-based integral sliding-mode control (AO-ISMC) scheme specifically designed for buck converters having mismatched disturbances and model uncertainties. Instead of having the fixed gains, an observer is proposed, whose gains continuously adapt online using a mathematically extensive Lyapunov-based adaptation law. This guarantees that the estimates of the inductor current and capacitor voltage should remain accurate and that the estimation error stays bounded, even when the significant disturbances are acting on the system or its parameters are not exactly known. By utilizing the state estimates, an integral action is included in order to eliminate the steady state error and ensure robustness.

In the following sections, the complete analytical modeling of the system is performed along with the controller design. The results section highlights the usefulness of the proposed controller in terms of disturbance rejection and perfect voltage regulation.

2. Description of Buck DC–DC Converter

Figure 1 shows the circuit diagram of the buck dc–dc converter. The continuous-conduction-mode buck converter is modeled by its inductor current $i_L,$ and the capacitor voltage is given by $v_C$. Let $u\in [0, 1]$ be the duty ratio, $V_{in}$ the input voltage, $R_L$ the inductor resistance, $L$ the inductance, $C$ the capacitance, and $R$ the load resistance, then the inductor dynamics are given as follows.

$$L{\displaystyle \frac{d{i}_L}{dt}}=u{V}_{in}-R_L{i}_L-v_C$$

(1)

The capacitor dynamics are given as follows.

$$C{\displaystyle \frac{d{v}_C}{dt}}=i_L-{\displaystyle \frac{v_C}{R}}$$

(2)

The state vector is defined as $x={[i_L v_C]}^T$, the input to the system is $u$, the system disturbance is $d$, and the measured output is $y=v_C$. For compact analysis and control design, the state space model is then given as follows.

$$\begin{gathered} \dot{x}=Ax+Bu+Dd \\ y=Cx \end{gathered}$$

(3)

where

• $A=\left[\begin{array}{cc}-{\displaystyle \frac{R_L}{L}}& -{\displaystyle \frac{1}{L}}\\ {\displaystyle \frac{1}{C}}& -{\displaystyle \frac{1}{RC}}\end{array}\right]$, $B= \left[\begin{array}{c}{\displaystyle \frac{V_{in}}{L}}\\ 0\end{array}\right]$, $D= \left[\begin{array}{c}{\displaystyle \frac{1}{L}}\\ 0\end{array}\right]$, $C=\left[\begin{array}{cc}0& 1\end{array}\right]$.

Equations (1)–(3) represent the mismatched nature of the buck dc–dc converter system.

3. Control Design

The proposed controller design consists of two phases. In the first step, an adaptive observer is designed in order to estimate the unmeasured state and disturbance, and in the next step, an integral sliding-mode controller is designed that uses these estimates in order to achieve robust voltage regulation.

From Equation (3), the plant model is written as below.

$$\dot{x}=Ax+Bu+Dd$$

(4)

A Luenberger-type observer with time-varying gain $L\left(t\right)$ is constructed, having the observer’s state denoted by $\widehat{x}$ and its output by $\widehat{y}$. The observer equations are given as follows.

$$\begin{gathered} \dot{\widehat{x}}=A\widehat{x}+Bu+L(t)\left(y-C\widehat{x}\right) \\ \widehat{y}=C\widehat{x} \end{gathered}$$

(5)

The estimation error is defined as $e=x-\widehat{x}$. Subtracting Equation (5) from the true model, i.e., Equation (4) yields as follows.

$$\dot{e}=\left(A-L_{0}C\right)e-\tilde{l}(t)\left(Ce\right)+Dd$$

(6)

where the gain is split into a fixed nominal part $L_0$ and an adaptive correction $\tilde{l}\left(t\right)$, so that $L\left(t\right)=L_0+\tilde{l}\left(t\right)$ and $\tilde{l}\left(t\right)=\left[\begin{array}{c}{\tilde{l}}_1(t)\\ {\tilde{l}}_2(t)\end{array}\right]$.

For driving both the state error $e$ and the gain error $\tilde{l}$ to manageable levels, we adopt the Lyapunov function as:

$$V(e,\tilde{l})={\displaystyle \frac{1}{2}}{e}^{T}Pe+{\displaystyle \frac{1}{2\gamma }}{\tilde{l}}^T\tilde{l}$$

(7)

where $P=P^T>0$ is the positive-definite matrix solving the below equation.

$${\left(A-L_{0}C\right)}^{T}P+P\left(A-L_{0}C\right)=-Q, Q>0$$

(8)

and $\gamma >0$ is an adaptation rate. Equation (8) guarantees that, if $\tilde{l}\equiv 0$ and $d\equiv 0$, the nominal error dynamics $\dot{e}=(A-L_{0}C)e$ would decay exponentially:

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{1}{2}}{e}^{T}Pe\right)=-{\displaystyle \frac{1}{2}}{e}^{T}Qe<0.$$

Differentiating $V$ along the trajectories of both $e$ and $\tilde{l}$ gives the following result.

$$\dot{V}={\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{1}{2}}{e}^{T}Pe\right)+{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{1}{2\gamma }}{\tilde{l}}^T\tilde{l}\right)$$

$$\dot{V}=e^{T}P\dot{e}+{\displaystyle \frac{1}{\gamma }}{\tilde{l}}^T\dot{\tilde{l}}$$

Substituting the value of $\dot{e}$ from Equation (6) gives the following result.

$$e^{T}P\dot{e}=e^{T}P\left[\left(A-L_{0}C\right)e-\tilde{l}\left(Ce\right)+Dd\right]$$

$$e^{T}P\left(A-L_{0}C\right)e=-{\displaystyle \frac{1}{2}}{e}^{T}Qe$$

$${-e}^{T}P\left[\tilde{l}\left(Ce\right)\right]=-{\tilde{l}}^T\left(CPe\right)$$

Therefore, putting these values in $\dot{V}$ equation gives the following result.

$$\dot{V}=-{\displaystyle \frac{1}{2}}{e}^{T}Qe-{\tilde{l}}^T\left(CPe\right)+e^{T}PDd+{\displaystyle \frac{1}{\gamma }}{\tilde{l}}^T\dot{\tilde{l}}$$

(9)

The adaptive control law is designed as follows.

$$\dot{\tilde{l}}=\gamma CPe$$

(10)

Substituting Equation (10) into Equation (9) leads to the following bound.

$$\dot{V}\le -{\displaystyle \frac{1}{2}}{e}^{T}Qe+‖e‖‖PD‖\left|d\right|$$

Since $Q>0,$

$$e^{T}Qe\ge {\lambda }_{min}\left(Q\right){‖e‖}^2$$

Therefore,

$$\dot{V}\le -{\displaystyle \frac{1}{2}}{\lambda }_{min}\left(Q\right){‖e‖}^2+‖e‖‖PD‖\left|d\right|$$

(11)

which shows that $e\left(t\right)$ remains uniformly ultimately bounded. In practice, increasing the design gains $Q$ or $\gamma$ reduces the estimation error bound.

By utilizing the observer’s voltage estimate ${\widehat{v}}_C$, the output tracking error $e_v$ is defined as follows.

$$e_v={\widehat{V}}_C-V_{\mathrm{ref}}$$

(12)

where $v_{\mathrm{ref}}$ is the desired DC output. The integral sliding surface is introduced as follows.

$$\sigma \left(\mathrm{t}\right)=e_v\left(t\right)+\lambda {\int }_0^t{e}_v\left(\tau \right)d\tau , \lambda >0.$$

(13)

Differentiating Equation (13) and substituting the observer dynamics from Equation (5) (with constant reference) yields the following result.

$$\dot{\mathsf{\sigma }}=C\dot{\widehat{x}}+\mathsf{\lambda }{e}_v$$

$$\dot{\sigma }=CA\widehat{x}+CBu+CL\left(y-C\widehat{x}\right)+\lambda e_v$$

(14)

The reaching law is designed as follows.

$$\dot{\sigma }=-\eta sign\left(\sigma \right), \eta >0.$$

(15)

Equation (14) is solved for the control input as the sum of an equivalent term and a switching term as follows.

$$u=u_{eq}-u_{sw}$$

(16)

Therefore,

$$u_{\mathrm{eq}}=-{\left(CB\right)}^{-1}\left(CA\widehat{x}+CL\left(y-C\widehat{x}\right)+\lambda e_v\right)$$

(17)

The switching term is given as follows.

$$u_{\mathrm{sw}}={\left(CB\right)}^{-1}\eta sign\left(\sigma \right)$$

(18)

The stability on the sliding surface is established by choosing the following Lyapunov function.

$$V_s(\sigma )={\displaystyle \frac{1}{2}}{\sigma }^2$$

(19)

By utilizing the reaching law given in Equation (15), the following result is obtained.

$$\dot{V_s}=\sigma \dot{\sigma =}-\eta \left|\sigma \right|<0 for \sigma \ne 0,$$

(20)

The above result guarantees finite-time convergence to $\sigma =0$. Once on the manifold, it satisfies the following relation.

$${\dot{e}}_v+\lambda e_v=0,$$

Therefore, the voltage error decays exponentially to zero, which completes the control design.

4. Results

To validate the proposed adaptive integral sliding-mode control, time-domain simulations are conducted in MATLAB/Simulink 2022 under realistic operating scenarios. The simulations were conducted using Simulink, with a fixed-step solver (auto, automatic solver selection) and a sampling time of 0.0001 s. The simulation duration was set to 20 s. These settings were chosen to ensure accurate and reliable results. The nominal values of the parameters for the mismatched buck converter are provided in

Table 1. Buck DC–DC converter’s nominal parameters.

Description	Parameter	Value
Reference Voltage	$V_{ref}$	20→15 V
Input Voltage	$V_{in}$	30 V
Filter Inductor	$L$	1.5 mH
Load Resistance	$R$	10→20$\mathsf{\Omega }$
Filter Capacitor	$C$	2200 $\mathsf{\mu }$F

. Initially, the value of the reference voltage is set at 20 V, and the load resistance value is set to 20 $\mathsf{\Omega }$. The Lyapunov weight vector $Q$ is set to $1\times {10}^6{I}_2$. The adaptation rate $\gamma$ is chosen as $1\times {10}^4$, while the integral weight $\lambda$ is selected as 100. The reaching rate $\eta$ is set to 5, and the discrete update interval for adaptation law $∆t$ is set to 50 $\mathsf{\mu }\mathrm{s}$.

Figure 2 shows the reference tracking performance of the proposed adaptive controller. The output voltage tracks the reference voltage accurately in the presence of mismatched disturbance. The response time is 23 ms, and there is no overshoot present in the output response.

For verifying the performance of the designed controller during the load variation scenario, the reference voltage is kept fixed at 20 V, while the load resistance is varied from 10 $\mathsf{\Omega }$ to 20 $\mathsf{\Omega }$. The resulting voltage regulation process of the proposed controller is shown in Figure 3. The settling time of the resulting response is 43 ms, with a voltage variation of 0.34 V due to the load variation.

Similarly, the load is increased from 20 $\mathsf{\Omega }$ to 10 $\mathsf{\Omega },$ and the resulting response of the converter is given in Figure 4. The settling time is 45 ms, and the observed voltage overshoot is 0.34 V.

In order to verify the robustness properties of the designed controller in the presence of model uncertainties, the inductor and capacitor values are perturbed as $\overline{L}=L+20\%$ and $\overline{C}=C+20\%,$ respectively. The resulting output response is given in Figure 5. Despite the presence of uncertainty, the converter is still able to achieve good performance, indicating the robustness of the design controller.

The reference voltage is changed from 20 V to 12 V while keeping the load resistance constant. The resulting output voltage waveform is given in Figure 6, showing good robustness against the reference voltage variation.

Finally, the performance of the designed adaptive controller is compared against the conventional SMC controller. The resulting output response of the two controllers is provided in Figure 7. The comparison indicates the effectiveness of the proposed control system in terms of settling time and overshoot in the presence of mismatched uncertainties.

5. Conclusions

An integrated adaptive observer and integral sliding-mode control framework was designed for the buck DC–DC converters having mismatched load disturbances and model uncertainties. By designing a real-time Lyapunov-based adaptation law for the observer gains, a fast and accurate estimation of both inductor current and disturbance dynamics was achieved by utilizing only the output voltage measurement. Incorporating an integral term directly into the sliding surface eliminated the steady-state voltage error without sacrificing the finite-time convergence. Simulation results under the reference voltage variation, sudden load changes, and model uncertainties validated the effectiveness of the proposed adaptive controller. For future work, the proposed control framework can be extended to real-time embedded implementation and experimental validation on a hardware prototype, which would then provide further insights into its practical feasibility and performance under the realistic conditions.