Voltage Regulation of a DC–DC Boost Converter Using a Vertex-Based Convex PI Controller

Abstract

The regulation of output voltage in power converters often demands nonlinear control techniques; however, their implementation is challenging when deployed on low-cost hardware with limited computational resources. To address this difficulty, the modeling via the sector nonlinearity technique is adopted to represent the converter dynamics as a convex combination of linear vertex models. Building on this representation, this article proposes a vertex-based convex PI controller that significantly reduces the required online computations compared to conventional convex controllers relying on full-state feedback. In the proposed scheme, the inductor current is used solely to evaluate the weighting functions, avoiding the need to compute control gains associated with this state. The effectiveness of the method is demonstrated through offline simulations and validated using hardware-in-the-loop experiments.

1. Introduction

DC–DC boost converters are widely used in a broad range of applications such as renewable energy systems, battery-powered devices, and electric vehicles, where voltage step-up and tight regulation are required [1,2,3,4]. These converters are attractive due to their simplicity, efficiency, and compact structure. However, the inherent nonlinearity, nonminimum phase, and operating point dependency of their dynamics make controller design a challenging task, particularly under wide variations of load and input voltage.

Numerous advanced control techniques have been proposed to improve the dynamic performance and robustness of these converters. Linear controllers such as Proportional–Integral (PI) or Proportional–Integral–Derivative (PID) are simple and easily implementable, but their effectiveness is limited to small-signal conditions around a fixed operating point [5,6]. Sliding-Mode Control (SMC) and Model Predictive Control (MPC) offer improved robustness and fast transient response; however, SMC often induces high-frequency chattering that accelerates device wear [7], and MPC requires an online solution of optimization problems, imposing a significant computational burden that precludes low-cost microcontrollers or small-footprint FPGAs [8,9].

Some recent studies have highlighted additional limitations. The work in [10] presents an interconnection and damping assignment passivity-based control method, which can be effective but may require precise parameter tuning and face challenges with complex system dynamics. The approach in [11] proposes an input–output feedback linearization approach for a Boost converter supplying a constant power load. While this technique stabilizes an otherwise nonminimum phase system, its implementation relies on accurate system inversion and the knowledge of load characteristics, which may limit robustness under parameter variations. Fuzzy-logic and neural-network methods capture nonlinear behavior over wide ranges but typically demand extensive offline training, large rule bases or network parameters, and substantial code memory and real-time inference resources [12,13,14]. Consequently, deploying these sophisticated controllers in cost- and power-constrained embedded platforms remains challenging [15].

In contrast, gain-scheduling strategies based on exact or approximate convex modeling have attracted attention as a practical compromise for handling nonlinear dynamics. These methods recast the nonlinear behavior as a convex combination of linear vertex models, enabling the use of Linear Matrix Inequalities (LMIs) to certify global stability via a single Lyapunov function. The sector nonlinearity approach shows that any static nonlinearity confined within a sector bound can be expressed exactly as a weighted sum of its boundary gains [16]. A comprehensive LMI-based framework for control design has been developed [17], and subsequent work addressed model construction, rule reduction, and robust compensation in the generalized setting [18]. Fuzzy models implement similar convex interpolations via IF–THEN rules [19,20,21,22,23], but the sector formulation provides an exact convex description whenever the nonlinearity lies within the specified sector. This polytopic representation has been successfully applied to a variety of power converters, offering a favorable trade-off between model accuracy and control simplicity [24,25,26,27,28,29,30].

However, existing convex approaches typically require the following:

• Full-state feedback (inductor current, output voltage and integral error).
• Tuning of multiple vertex-specific gains (proportional voltage, proportional current, and integral terms).
• Significant online computation for weight updates.
• Complex implementation on resource-constrained hardware.

This paper presents the following:

• A simplified convex PI controller that
  • Eliminates current-error feedback gains entirely.
  • Reduces per-vertex tuning parameters from three to two (proportional voltage and integral gains only).
  • Maintains current measurement solely for convex weight computation.
  • Integrates a disturbance compensator that estimates input voltage via duty cycle and output voltage measurement.
• Genetic optimization of vertex gains sharing a common matrix P, ensuring stability via LMIs.
• Hardware-efficient implementation through
  • Reduced flash memory requirements for gain storage.
  • Elimination of current-control loop computations.

The remainder of the paper is organized as follows. Section 2 introduces the nonlinear control formulation based on the two-sector nonlinearity method and presents the mathematical model along with its convex representation. Section 3 describes the design of the convex PI controller with its LMI based stability condition and details the integration of the disturbance compensator. Section 4 presents the simulation results and the Hardware-in-the-Loop (HIL) validation of the proposed control strategy. Finally, Section 5 concludes the paper and outlines future research directions.

2. Convex Modeling via Sector Nonlinearity

2.1. Nonlinear Control Based on the Sector Nonlinearity Method

The sector nonlinearity approach has become a widely adopted framework for the exact convex representation of nonlinear systems. This methodology has been extensively applied in the literature for modeling and control, providing a systematic way to rewrite a nonlinear model into a polytopic form without approximation errors [31,32,33]. The main advantage of this formulation lies in its ability to embed nonlinear terms within bounded sectors and express them as convex combinations of their limits, thus enabling the application of LMIs for stability and control synthesis.

For a general nonlinear system described by

$$\begin{gathered} \dot{x}=f\left(x\right)+g\left(x\right)u, \\ y=h\left(x\right), \end{gathered}$$

(1)

where $x\in R^n$ is the state vector, and $\mathrm{u}\in {\mathrm{R}}^{\mathrm{m}}$ is the control input, the nonlinear terms embedded in $f\left(x\right)$ and $g\left(x\right)$ can be bounded using premise variables $z_l\in \left[{\underset{\_}{z}}_l,{\overline{z}}_l\right]$, where the extreme bounds ${\underset{\_}{z}}_l,{\overline{z}}_l$ are induced by the compact $\mathsf{\Omega }$. Each nonlinear term $z_l\left(x\right)$ is expressed as a convex combination of its bounds using weighting functions $w_i^l$:

$$z_l\left(x\right)={\underset{\_}{z}}_l{w}_0^l+{\overline{z}}_l{w}_1^l,$$

(2)

where $l$ denotes the index of the premise variables, and the weights are computed as

$$w_0^l={\displaystyle \frac{{\overline{z}}_l-z_l\left(x\right)}{{\overline{z}}_l-{\underset{\_}{z}}_l}},w_1^l=1-w_0^l,l=1,2,\dots ,p.$$

(3)

These weighting functions are normalized, as follows:

$$w_0^l,w_1^l\ge 0,w_0^l+w_1^l=1.$$

(4)

Remark 1. 

The set $\Omega \subset R^n$ denotes the compact region of the state space over which the nonlinear system (1) is analyzed. This compactness guarantees that all premise variables $z_l\left(x\right)$ remain bounded, making the convex combination (2) well-defined. In practice, $\Omega$ represents the operational range of the system’s states, where the controller is designed to satisfy the desired performance and stability criteria.

For a system with $p$ premise variables, the number of vertex systems is $v=2^p$. The global weighting function for each vertex system is constructed as

$${\mathsf{\rho }}_i=w_{i_1}^1{w}_{i_2}^2\cdots w_{i_p}^p,\hspace{1em}i\in \{1,2,\dots ,v\},\hspace{1em}{i}_k\in \{0,1\}.$$

(5)

The nonlinear system is exactly represented through a convex combination of linear vertex systems:

$$\begin{gathered} \dot{x}=\sum _{i=1}^v{\mathsf{\rho }}_i\left(A_{i}x+B_{i}u\right), \\ y=\sum _{i=1}^v{\mathsf{\rho }}_i{C}_{i}x, \end{gathered}$$

(6)

where $A_i\in R^{n\times n}$, $B_i\in R^{n\times m}$, and $C_i\in R^{q\times n}$ represent, respectively, the state, input, and output matrices for each vertex system. The weighting functions ${\mathsf{\rho }}_i$ satisfy the convexity conditions ${\mathsf{\rho }}_i\ge 0$ and ${\sum }_{i=1}^v{\mathsf{\rho }}_i=1$, ensuring proper interpolation between vertex systems. The resulting convex combination permits stability analysis and controller synthesis while maintaining the exact representation of the original nonlinear dynamics.

A parallel distributed compensation (PDC) is adopted using full state feedback and vertex-specific gains:

$$u=-\sum _{j=1}^v{\mathsf{\rho }}_j{K}_{j}x,$$

(7)

where $K_j\in R^{m\times n}$ are the gain matrices for each linear vertex system. Substituting (7) into (6) yields the interconnected system:

$$\dot{x}=\sum _{i=1}^v\sum _{j=1}^v{\mathsf{\rho }}_i{\mathsf{\rho }}_j\left(A_i-B_i{K}_j\right)x.$$

(8)

The stability of system (8) is analyzed using a quadratic Lyapunov candidate function (9), ensuring that all vertex models satisfy the Hurwitz stability condition.

$$V\left(x\right)=x^{T}Px,\hspace{1em}P>0.$$

(9)

The expression (9) has the following time derivative:

$$\dot{V}\left(x\right)=\sum _{i,j=1}^v{\mathsf{\rho }}_i{\mathsf{\rho }}_j{x}^T\left[{\left(A_i-B_i{F}_j\right)}^{T}P+P\left(A_i-B_i{F}_j\right)\right]x.$$

(10)

Thus, due to the convexity of functions ${\mathsf{\rho }}_i$ and ${\mathsf{\rho }}_j$, $\dot{V}\left(x\right)<0$ is guaranteed for $\left(i,j\right)\in {\left\{1,2,\dots ,v\right\}}^2$:

$${\left(A_i-B_i{F}_j\right)}^{T}P+P\left(A_i-B_i{F}_j\right)<0.$$

(11)

The stability condition (11) contains a bilinear term $P{B}_i{F}_j$, which prevents direct LMI formulation. This is solved through the following variable transformation $S=P^{-1},M_j=F_{j}S$, obtaining

$$S{A}_i^T+A_{i}S-M_j^T{B}_i^T-B_i{M}_j<0.$$

(12)

The solution provides matrices $P$ and $F_j$, ensuring that the outermost Lyapunov level $\left\{x:V\left(x\right)\le c\right\}\subset \mathsf{\Omega }$ serves as a region of attraction where all trajectories converge to the origin, demonstrating the desired closed-loop system properties.

2.2. The Boost Converter Convex Model

The DC–DC boost converter can be modeled as a nonlinear dynamical system describing the time evolution of the inductor current $i_L$ and the output voltage $V_o$ [34]. Assuming a duty cycle $u\in \left[0,1\right]$, the time-domain model is expressed as

$$\begin{gathered} {\displaystyle \frac{d{i}_L}{dt}}={\displaystyle \frac{1}{L_f}}\left(-r{i}_L-\left(1-u\right)V_o+V_s+d_s\right), \\ {\displaystyle \frac{d{V}_o}{dt}}={\displaystyle \frac{1}{C_f}}\left(\left(1-u\right)i_L-{\displaystyle \frac{V_o}{R}}+d_L\right), \end{gathered}$$

(13)

where $u$ is the control input, $L_f$ denotes the inductance, $C_f$ is the output capacitance, $r$ represents the parasitic inductor resistance, $V_s$ is the voltage source, and $R$ is the load resistance. Moreover, the term $d_s$ represents the lumped disturbances caused by the model uncertainty and voltage source variation. On the other hand, $d_L$ represents the parameter variations and external disturbances. A schematic of the boost converter is shown in Figure 1.

The nonlinearities are expressed in terms of premise variables $z_1=i_L$ and $z_2=V_o$, bounded within their operating ranges:

$$z_1=i_L\in \left[\underset{\_}{i_L},\overline{i_L}\right],\hspace{1em}{z}_2=V_0\in \left[\underset{\_}{V_0},\overline{V_0}\right].$$

(14)

The weighting functions $w_k^l$ are constructed to interpolate between the bounds of each premise variable:

$$\begin{gathered} w_0^1={\displaystyle \frac{\overline{i_L}-i_L}{\overline{i_L}-\underset{\_}{i_L}}},\hspace{1em}{w}_1^1=1-w_0^1, \\ {w}_0^2={\displaystyle \frac{\overline{V_0}-V_0}{\overline{V_0}-\underset{\_}{V_0}}},\hspace{1em}{w}_1^2=1-w_0^2. \end{gathered}$$

(15)

These weights satisfy the convexity conditions $w_{i_k}^l\ge 0$ and ${\sum }_{i_k}{w}_{i_k}^l=1$. The global weighting functions combine as

$$\begin{gathered} {\mathsf{\rho }}_1=w_0^1{w}_0^2, \\ {\mathsf{\rho }}_2=w_0^1{w}_1^2, \\ {\mathsf{\rho }}_3=w_1^1{w}_0^2, \\ {\mathsf{\rho }}_4=w_1^1{w}_1^2. \end{gathered}$$

(16)

The nonlinear boost converter model in (13) can be reformulated as a convex representation, where the dynamics are expressed as a convex combination of vertex systems, as follows:

$$\begin{gathered} \dot{x}=Ax+\sum _{i=1}^4{\mathsf{\rho }}_{i}B{z}^{i}u+d_w, \\ y=Cx, \end{gathered}$$

(17)

where $x={\left[\begin{array}{cc}{i}_L& V_o\end{array}\right]}^T$, the disturbance term is given by $d_w={\left[\begin{array}{cc}\left(V_s+d_s\right)/L_f& d_L/C_f\end{array}\right]}^{\top }$, the state, input, and output matrices are defined respectively as

$$A=\left[\begin{array}{cc}-{\displaystyle \frac{r}{L_f}}& -{\displaystyle \frac{1}{L_f}}\\ {\displaystyle \frac{1}{C_f}}& -{\displaystyle \frac{1}{R{C}_f}}\end{array}\right],B=\left[\begin{array}{cc}{\displaystyle \frac{1}{L_f}}& 0\\ 0& -{\displaystyle \frac{1}{C_f}}\end{array}\right],$$

and $C=\left[0 1\right]$.

The operating space of the boost converter is characterized by vertex vectors constructed from the extreme values of the premise variables:

$$\begin{gathered} z^i={\left[z_2^i\hspace{1em}{z}_1^i\right]}^{\top }\in \mathcal{Z},\hspace{1em}i=1,\dots ,4. \\ \mathcal{Z}=\left[\begin{array}{c}{\underset{\_}{z}}_2\\ {\underset{\_}{z}}_1\end{array}\right],\left[\begin{array}{c}{\overline{z}}_2\\ {\underset{\_}{z}}_1\end{array}\right],\left[\begin{array}{c}{\underset{\_}{z}}_2\\ {\overline{z}}_1\end{array}\right],\left[\begin{array}{c}{\overline{z}}_2\\ {\overline{z}}_1\end{array}\right]. \end{gathered}$$

(18)

3. Proposed Control Strategy

3.1. Convex PI Controller Structure

The proposed control strategy adopts a Parallel Distributed Proportional–Integral (PD-PI) structure formulated in the state–space domain. The integral action is introduced as an additional state $\mathsf{\xi }$, governed by

$$\dot{\mathsf{\xi }}=r_d-V_o,$$

(19)

where $r_d$ is the reference signal. The control input is computed as a convex combination of vertex-dependent PI controllers, given by

$$u=\sum _{j=1}^4{\mathsf{\rho }}_j\left(K_p^j\left(r_d-V_o\right)+K_s^j\mathsf{\xi }\right),$$

(20)

with $K_p^j$ and $K_s^j$ denoting the proportional and integral gains at the $j$-th vertex, respectively, and ${\mathsf{\rho }}_j$ being convex weighting functions satisfying ${\mathsf{\rho }}_j\ge 0$ and ${\sum }_{j=1}^4{\mathsf{\rho }}_j=1$. Substituting (20) into (17), the closed-loop dynamics reduce to

$$\dot{x}=Ax+\sum _{i=1}^4\sum _{j=1}^4{\mathsf{\rho }}_i{\mathsf{\rho }}_{j}B{z}^i\left(K_p^j\left(r_d-Cx\right)+K_s^j\mathsf{\xi }\right)+d_w,$$

(21)

where $A$, $B$, and $C$ are the system matrices defined in Section 2. In this formulation, the output voltage $V_o$ appearing in the PI control law is expressed as $V_o=Cx$, linking the control input directly to the measured system state. This structure explicitly captures the interaction between the vertex-dependent plant model and the vertex-dependent PI controller, with the double summation representing the coupling between the system vertices and the controller vertices.

Rearranging terms, the system can be represented in an augmented state–space form by defining the extended state vector $x_a={\left[\begin{array}{cc}{x}^T& \mathsf{\xi }\end{array}\right]}^T$, leading to

$${\dot{x}}_a=A_{cl}{x}_a+B_{cl}{r}_d+d_w,$$

(22)

where the augmented matrices are

$$A_{cl}={\sum }_{i=1}^4{\sum }_{j=1}^4{\mathsf{\rho }}_i{\mathsf{\rho }}_j\left[\begin{array}{cc}A-K_p^{j}B{z}^{i}C& K_s^{j}B{z}^i\\ -C& 0_{1\times 1}\end{array}\right],$$

and

$$B_{cl}={\sum }_{i=1}^4{\sum }_{j=1}^4{\mathsf{\rho }}_i{\mathsf{\rho }}_j\left[\begin{array}{c}{K}_p^{j}B{z}^i\\ I_{1\times 1}\end{array}\right].$$

3.2. Stability Analysis

Global weighting products satisfy the convexity conditions $0\le {\mathsf{\rho }}_i{\mathsf{\rho }}_j\le 1$, and ${\sum }_{i=1}^4{\sum }_{j=1}^4{\mathsf{\rho }}_i{\mathsf{\rho }}_j=1$. As a result, the system trajectories remain within the convex hull defined by the vertex systems, ensuring that the overall model evolves inside the compact set $\mathsf{\Omega }$. Moreover, the weighting functions ${\mathsf{\rho }}_i$ and ${\mathsf{\rho }}_j$ are linear and convex; thus, the vertex stability can be assumed when $i=j$ and ${\mathsf{\rho }}_i{\mathsf{\rho }}_j=1$. The closed-loop matrix can therefore be expressed compactly as

$$A_{cl}^j=\left[\begin{array}{cc}A-K_p^{j}B{z}^{j}C& K_s^{j}B{z}^j\\ -C& 0_{1\times 1}\end{array}\right].$$

(23)

The Hurwitz property of the vertices constitutes a necessary but not sufficient condition:

$$\mathcal{R}e\left\{{\mathsf{\lambda }}_m\left(A_{cl}^j\right)\right\}<0,\hspace{1em}\forall m\in \left\{1,2,3\right\}.$$

(24)

The disturbance term $d_w$, which incorporates the input voltage-to-inductance ratio $V_s/L_f$, is inherently compensated by the integral action of the controller. At steady-state equilibrium $\dot{x_a}=0$, the integrator state $\mathsf{\xi }$ converges to a value that precisely cancels the effect of $d_w$. This behavior emerges from the equilibrium condition:

$$0=A_{cl}^j{x}_{eq}+B_{cl}^j{r}_d\left(t\right)+d_w,$$

(25)

where the equilibrium point $x_{eq}$ is uniquely determined as

$$x_{eq}=-{\left(A_{cl}^j\right)}^{-1}\left(B_{cl}^j{r}_d\left(t\right)+d_w\right).$$

(26)

The invertibility of $A_{cl}^j$ is assured when the vertex dynamics are stable. To analyze the stability, consider the error dynamics ${\tilde{x}}_a=x_a-x_{eq}$. Substituting $x_a={\tilde{x}}_a+x_{eq}$ into the original system yields the simplified error dynamics:

$${\dot{\tilde{x}}}_a=A_{cl}^j{\tilde{x}}_a,$$

(27)

where the equilibrium terms cancel exactly due to the steady-state definition. In the modeling phase, the inductor’s parasitic series resistance $r$ is explicitly included to improve the accuracy of the control plant. However, other parasitic elements, such as the capacitor’s Equivalent Series Resistance (ESR) and the internal resistances of the switching devices, are neglected. This simplification is justified, because the integral action of the proposed controller effectively compensates for the small voltage drops and damping effects introduced by these omitted components, ensuring zero steady-state error.

The stability of the convex model set is verified through a common quadratic Lyapunov function. A symmetric positive-definite matrix $P$ must satisfy

$${\tilde{x}}_a^T\left[{\left(A_{cl}^j\right)}^{T}P+P{A}_{cl}^j\right]{\tilde{x}}_a<0,\forall j\in \left\{1,\dots ,4\right\}.$$

(28)

Remark 2. 

The symmetric positive-definite matrix $P$ is common for all vertices and ensures a quadratic Lyapunov function that guarantees the stability of the convex model set.

To enhance numerical tractability, the condition is relaxed with a tolerance ${\mathsf{\delta }}_1>0$, the Lyapunov function is defined as

$$L^j={\left(A_{cl}^j\right)}^{T}P+P{A}_{cl}^j< -{\mathsf{\delta }}_1{I}_{3\times 3},$$

(29)

ensuring stability across the entire operating envelope. This formulation guarantees the convergence of the error dynamics while accounting for practical implementation constraints. Although the proposed control law has a PI structure and relies only on output voltage feedback and the integral of the tracking error, it can be interpreted as a state-feedback controller acting on an augmented system that includes the integral state. This augmented representation explicitly captures the interaction between the vertex-dependent plant model and the vertex-dependent PI gains. As a result, the stability analysis is carried out on the augmented closed-loop system, and the global stability over the entire operating envelope is guaranteed through the common Lyapunov-based LMI condition in (29).

3.3. PI Controller Tuning via Genetic Algorithm

The dependency of the closed-loop matrix $A_{cl}^j$ on the proportional $K_p^j$ and integral gain $K_s^j$ results in a hybrid convex–heuristic optimization problem, which is solved using a genetic algorithm (GA). The use of a GA for controller tuning is motivated by the nonlinear nature of the system and the presence of multiple control parameters. Unlike traditional tuning methods based on linearized models, the GA enables direct optimization of nonlinear performance criteria and provides enhanced robustness to parametric variations and changing operating conditions. A cost function is proposed to guarantee a desired dynamic performance of the closed-loop system, preventing excessive overshoot and avoiding sustained oscillations. The cost function is defined as follows:

$$J=\hspace{0.17em}\mathrm{tr}\left(P\right)+{\mathsf{\eta }}_1\hspace{0.17em}\mathrm{cond}\left(P\right)+{\mathsf{\eta }}_2{\displaystyle \frac{1}{{\mathsf{\lambda }}_{min}\left(P\right)+\mathsf{\epsilon }}},$$

(30)

where the matrix $P$ is computed through the feasibility of the LMIs associated with (29). The first term $tr\left(P\right)$ penalizes the magnitude of the Lyapunov matrix to avoid excessively large solutions, and the second term $cond\left(P\right)$ penalizes the condition number to ensure good numerical sensitivity and is defined as follows:

$$cond\left(P\right)={\displaystyle \frac{{\mathsf{\sigma }}_{max}\left(P\right)}{{\mathsf{\sigma }}_{min}\left(P\right)}},$$

(31)

where ${\mathsf{\sigma }}_{max}$ and ${\mathsf{\sigma }}_{min}$ represent the maximum and minimum singular values of $P$. The third term penalizes the proximity of the minimum eigenvalue ${\mathsf{\lambda }}_{min}\left(P\right)$ zero in order to guarantee positive definiteness. By adding a small parameter $\mathsf{\epsilon }$, this term ensures that $P$ remains strictly positive definite, avoiding numerical issues. The weighting factors ${\mathsf{\eta }}_1$ and ${\mathsf{\eta }}_2$ allow adjusting the relative importance of the conditioning and eigenvalue margin in the optimization process.

In addition to the previous terms, the cost function incorporates a penalty for the maximum violation of the Lyapunov inequality across all vertices of the polytopic system, denoted as $v_{max}$, which enforces closed-loop stability and accelerates the convergence of the genetic algorithm, the maximum violation measure is then obtained as

$$v_{max}\left(P\right)=\underset{j\in \{1,\dots ,4\}}{\max }{\mathsf{\lambda }}_{max}\left(A_{cl}^j\right).$$

(32)

A third weighting factor, denoted as ${\mathsf{\eta }}_3$, emphasizes solutions that better satisfy the stability constraints. To further refine the transient response, an integral squared error (ISE) term is incorporated, promoting faster and smoother dynamics. These contributions yield the extended cost function defined as

$$J_T=J+{\mathsf{\eta }}_3{v}_{max}+{\int }_0^T{\left(r_d-V_o\right)}^{2}dt.$$

(33)

The optimization variables are arranged as the global gain matrix

$$K=\left[\begin{array}{cccc}{K}_p^1& K_p^2& K_p^3& K_p^4\\ K_s^1& K_s^2& K_s^3& K_s^4\end{array}\right],$$

(34)

where the first row collects the proportional gains, and the second row collects the integral gains associated with each operating vertex.

The design problem is posed as

$$\begin{gathered} \underset{K}{\mathit{min}} J_T, \\ \mathit{such} \mathit{that} {\mathrm{K}}_{\mathrm{p}}^{j,\mathit{min}}\le K_p^j\le K_p^{j,\mathit{max}}, \\ {K}_s^{j,\mathit{min}}\le K_s^j\le K_s^{j,\mathit{max}}, \\ P\succ {\mathsf{\delta }}_{2}I, \\ {L}^j<-{\mathsf{\delta }}_1{I}_{3\times 3}, \\ \forall j\in \{1,\dots ,4\}, \end{gathered}$$

(35)

where ${\mathsf{\delta }}_2>0$ enforces strict positive definiteness. This formulation ensures that a single Lyapunov matrix $P$ simultaneously guarantees stability across all vertices, while the gain bounds enforce physically realizable controller parameters. For each candidate solution, the procedure was as follows:

• Attempt to solve the LMIs for a common $P$ with margins ${\mathsf{\delta }}_1$ and ${\mathsf{\delta }}_2$;
• If $P$ is feasible, compute the associated metrics $\mathrm{tr}\left(P\right)$, $\mathrm{cond}\left(P\right)$, ${\mathsf{\lambda }}_{min}\left(P\right)$, $v_{max}$, and ISE;
• Form the cost function $J_T$ using the predefined coefficients ${\mathsf{\eta }}_1$, ${\mathsf{\eta }}_2$, and ${\mathsf{\eta }}_3$;
• If no feasible $P$ is found, assign a large penalty to discard the individual.

The proposed hybrid convex–heuristic optimization method is summarized in Algorithm 1.

Algorithm 1. GA-based Gain Tuning Procedure

1: Initialize the population of candidate gains (34)

2: while termination criterion is not satisfied, do

3:  for all individuals in population (in parallel) do

4:   Solve LMIs

5:   if $P\succ {\mathsf{\delta }}_{2}I$ and (29) then

6:    Compute $\mathrm{tr}\left(P\right)$$, \mathrm{cond}\left(P\right)$$, {\mathsf{\lambda }}_{min}\left(P\right),$$v_{max}$

7:    Simulate system response and compute ISE

8:    Evaluate (33)

9:   else

10:     Assign large penalty cost

11:   end if

12:  end for

13:  Apply selection, crossover, and mutation operations

14: end while

15: Output optimal vertex-dependent gains $K_p$, $K_s$

3.4. Disturbance Compensator

The boost converter is inherently susceptible to exogenous disturbances, primarily input voltage source variations, $d_s$, and load changes, $d_L$. To enhance the closed-loop performance without modifying the main operation of the proposed controller, a disturbance compensator is incorporated.

This compensator is explicitly designed to directly mitigate the effects of the input voltage source variations. However, it is also capable of attenuating part of the effects of load changes. A key design requirement is that the compensation must be performed without resorting to complex state observers, relying instead on the direct measurement of the output voltage, $V_o$.

The compensator’s design is derived from the steady-state voltage gain relationship of the boost converter in Continuous Conduction Mode (CCM). This gain expression can be directly obtained by manipulating the state–space averaging model (13), yielding.

$$G_v={\displaystyle \frac{V_o}{V_s}}={\displaystyle \frac{R\left(1-u\right)}{r+R{\left(1-u\right)}^2}}.$$

(36)

In this analysis, $R$ corresponds to the nominal load resistance ($R_{\mathrm{nom}}$) used for the controller design. This nominal value represents the operating point around which the system is optimized. Furthermore, the control input $u$ directly influences the relationship between the power supply and output voltage. For a fixed control input $u$, the converter’s output voltage $V_o$ is positively correlated with the supply voltage. Consequently, a decrease in $V_o$ is attributed to a drop in the supply voltage, while an increase in $V_o$ results from a higher supply voltage.

The disturbance compensator is expressed as

$$u_c=K_c\left(V_{nom}-G_v^{-1}{V}_o\right).$$

(37)

where $V_{nom}$ is the nominal voltage source, $u_c$ is the compensator control input, and $K_c$ is the compensator gain. It follows that the total control input $u_T$ is given by

$$u_T=u+u_c.$$

(38)

The compensator (37) is designed to operate only under transient conditions. Therefore, an adequate selection of $K_c$ is essential to prevent control saturation. Equation (37) is derived under the assumption that external disturbances, mainly load and input voltage, can be modeled as slowly varying or bounded signals, which is a common hypothesis in practical DC–DC converter applications. The parameter $K_c$ acts as the gain of the disturbance compensator and must be selected by balancing the performance and robustness: higher values of $K_c$ improve the disturbance rejection speed but may amplify the noise and excite unmodeled dynamics, whereas excessively low values reduce the effectiveness of the compensator.

The controller structure mainly depends on the general form of the nonlinear model; so, the proposed approach is generalizable to other commonly used DC–DC conversion topologies, such as buck–boost, Ćuk, SEPIC, and flyback converters, among others. The procedure can be applied by replacing the corresponding dynamic equations of each architecture, while preserving the same controller design philosophy and disturbance compensator. The stability analysis and tuning scheme remain valid as long as the system can be represented as an input-affine nonlinear system.

4. Simulation Results and Discussion

4.1. Offline Simulation

The first validation of the proposed controller was carried out through time-domain simulations of the boost converter under different operating conditions. The closed-loop structure is depicted in Figure 2, where the vertex-based convex PI controller is integrated with the weighting functions. The electrical parameters of the converter, summarized in

Table 1. Boost converter parameters.

Parameter	Description	Value
$V_s$	Input voltage	48 V
$L_f$	Inductance	1.5 mH
$C_f$	Capacitance	220 $\mathsf{\mu }$F
$R$	Nominal load resistance	$50 \mathsf{\Omega }$
$r$	Inductor series resistance	$0.25 \mathsf{\Omega }$
$i_L$	Inductor current range	0.42–4.5 A
$V_o$	Output voltage range	48–150 V
$P_o$	Nominal power output	200 W

, define the operating ranges of the inductor current and the output voltage.

Based on these parameters, the vertex models were constructed, and a Genetic Algorithm (GA) was employed to determine the proportional and integral gains at each vertex. The GA configuration included a population size of 300, a maximum of 1000 generations, a function tolerance of ${10}^{-8}$, tournament selection, a crossover fraction of 0.8, and an adaptive feasible mutation rate of 5%. The cost function combined the trace and condition number of the Lyapunov matrix $P$, the minimum eigenvalue margin, the maximum violation of the Lyapunov inequality, and ISE. Specifically, the weighting coefficients were set as ${\mathsf{\eta }}_1=0.1$ for the condition number, ${\mathsf{\eta }}_2=1$ for the eigenvalue margin, and ${\mathsf{\eta }}_3=10$ for the violation penalty.

To guarantee feasibility, the Lyapunov inequalities were formulated with small margins defined by ${\mathsf{\delta }}_1={10}^{-4}$ and ${\mathsf{\delta }}_2={10}^{-3}$. These margins were consistently enforced during the GA optimization and the subsequent feasibility verification. For each individual in the population, the feasibility of the Lyapunov condition in (29) was assessed by solving the corresponding LMIs through the YALMIP toolbox in combination with the SeDuMi solver. Only when a common $P$, satisfying (29), was found, were the candidate gains were considered valid and further evaluated in terms of performance metrics.

The optimization yielded a common positive-definite Lyapunov matrix

$$P=\left[\begin{array}{ccc}0.0097& 0.0002& -0.4672\\ 0.0002& 0.0014& -0.1973\\ -0.4672& -0.1973& 298.6545\end{array}\right],$$

which guarantees stability across the entire polytopic region. The optimal vertex-dependent gains were obtained as: Vertex 1: $K_p=0.0025386$, $K_s=4$; Vertex 2: $K_p=0.00076261$, $K_s=1.2032$; Vertex 3: $K_p=0.0013458$, $K_s=2$; and Vertex 4: $K_p=0.00046931$, $K_s=1.039$. In addition, the disturbance compensator (37) with gain $K_c=0.05$ was integrated into the control law to attenuate the input-voltage perturbations and improve transient recovery.

The closed-loop performance of the proposed control strategy was evaluated through detailed simulations carried out in the software PSIM version 2021.a environment. PSIM was selected due to its robustness and efficiency in modeling power electronic converters. Three representative operating scenarios were simulated to assess the dynamic response, steady-state behavior, and overall stability of the system. In the first case, the output reference was sequentially modified from 80 V to 100 V, then to 120 V, back to 100 V, and finally to 80 V, as shown in Figure 3. The controller exhibited fast tracking with negligible overshoot and steady-state error. The second test considered abrupt load changes between 50% and 100% of the nominal power (200 W). The load was switched from 100 $\mathsf{\Omega }$ to 50 $\mathsf{\Omega }$ at 0.1 s and back to the original value at 0.3 s, as illustrated in Figure 4. During this test, the controller exhibited a response time around 15 ms, a transient overshoot of approximately 5%, and a steady state error near to zero. This behavior is consistent with a control design that prioritizes fast dynamic response and robust stability, while keeping the overshoot within acceptable limits according to industrial standards for DC–DC converters. The output voltage remained well regulated, demonstrating a disturbance rejection capability during load transients. Finally, Figure 5 shows the response under input voltage disturbances, where the source was subjected to an 8 V drop (from 48 V to 40 V) and later restored to its nominal value. The integral action, together with the disturbance compensator, ensured fast recovery of the output voltage despite the supply perturbation.

4.2. Controller Hardware in the Loop Simulation

Discrete-time equations of the boost converter were required for the real-time simulation, following the methodology presented in [35]. The difference equations were programmed in fixed-point using LabVIEW. The boost converter model (13) and the proposed PD–PI controller (20) were implemented on two NI cRIO-9067 operating with a 40 MHz clock. In the case of the PD–PI controller, the PWM gate-drive signal was generated through the NI 9401 digital I/O module, while the NI 9291 analog input module was used for current and voltage acquisition. Conversely, the boost converter model required the NI 9262 DAC module to generate the inductor current and capacitor voltage and other NI 9401 digital I/O for the acquisition of the PWM signal. The schematic diagram of the overall system interconnections is shown in Figure 6.

To further validate the proposed control strategy under real-time computational constraints, the same three test scenarios were reproduced using the HIL setup. This configuration allowed the controller to operate on the actual embedded platform while interacting with a real-time model of the power converter.

In the first scenario, the output reference was modified following the same sequence used in the offline simulations. The reference was set initially to 80 V, then increased to 100 V, later to 120 V, subsequently reduced again to 100 V, and finally returned to 80 V. As shown in Figure 7, the controller preserved its tracking capability under real-time execution, achieving transitions without overshoot and maintaining a negligible steady-state error. The second scenario consisted of sudden load variations between 50% and 100% of the nominal load. The resistance was switched from 100 Ω to 50 Ω and back to 100 Ω, as depicted in Figure 8. The voltage regulation remained robust despite the abrupt changes in operating conditions, and the controller exhibits a similar behavior to the simulation results (a response time around 15 ms, an overshoot of 5%, and a steady state error near to zero). The HIL results confirmed that the disturbance rejection mechanism performed reliably when subjected to realistic timing and quantization effects.

Finally, Figure 9 presents the system response under input-voltage disturbances. The source voltage experienced an 8 V drop, decreasing from 48 V to 40 V, and was later restored to its nominal value. The controller recovered the output voltage rapidly, preserving the stability and tracking accuracy. The synergy between the offline and real-time simulations provides a comprehensive verification of the control system’s performance. While the PSIM environment enabled the design and optimization of the proposed controller, the HIL platform confirmed the feasibility of its execution on a real-time digital system. This dual-verification methodology significantly mitigates the risk in subsequent hardware implementation and testing phases, establishing a reliable pathway from the theoretical model to a functional prototype.

To quantitatively support the claim of the reduced online computational burden, a comparison of the arithmetic operations required per sampling cycle is carried out between a conventional full-state feedback LMI/PDC controller (7) and the proposed vertex-based convex PI controller (20). The results are summarized in

Table 2. Online computational complexity comparison per sampling cycle.

Controller	State Used Online	Gains Stored	Matrix–Vector Products	Multiplications	Additions
Full-State Feedback LMI/PDC	$i_L,V_o, \mathsf{\xi }$	12 (3 per vertex)	Yes	16	11
Proposal	$V_o, \mathsf{\xi }$	8 (2 per vertex)	No	12	7

.

The proposed controller eliminates matrix–vector products and reduces both the number of stored gains and the required arithmetic operations per sampling cycle. This reduction directly translates into lower execution time and memory usage, which is critical for real-time implementation on low-cost embedded hardware.

For embedded platforms based on ARM Cortex microcontrollers operating at approximately 84 MHz and without hardware floating-point units, such as those commonly used in low-cost control applications, the matrix–vector products required by conventional full-state feedback LMI/PDC controllers may represent a considerable computational burden. In contrast, the proposed control law relies only on scalar proportional and integral gains per vertex and can be efficiently implemented using single-precision floating-point arithmetic.

On the other hand, the stability guarantees provided by the proposed vertex-based convex PI controller are inherently linked to the operating region defined by the sector nonlinearity approach. The selected vertex models correspond to physically meaningful ranges of the boost converter variables, such as the inductor current and output voltage, around a nominal operating point. As a result, the common Lyapunov-based LMI condition ensures stability only within the convex hull spanned by these vertices. Outside this region, no formal mathematical guarantee can be established.

5. Conclusions

In this paper, a vertex-based convex PI controller for voltage regulation in a boost converter was presented. The proposed method achieves accurate voltage tracking while avoiding the computation of control gains associated with the current measurement, thus reducing the online complexity. The controller also exhibits a fast dynamic response under reference variations. Additionally, a sector-nonlinearity-based dynamical model was introduced to obtain a convex representation of the nonlinear converter, enabling the design of the vertex-based PI gains. A disturbance compensator that estimates the input voltage using only the duty cycle and the output voltage was incorporated, enhancing the rejection of input-voltage disturbances.

The experimental results demonstrate that the proposed scheme regulates the output voltage using a simple control loop. The disturbance rejection is maintained not only under input voltage variations but also under load changes exceeding 50%. These characteristics suggest that the proposed voltage regulation strategy is well suited for renewable-energy conversion systems.