Mixed-Sensitivity H∞ Voltage Regulation of a Boost DC–DC Converter with Integral Augmentation: Design and LQR Benchmarking

The increasing penetration of photovoltaic (PV) generation and other DC-based energy systems, such as DC microgrids, battery energy storage interfaces, and grid-tied power-electronic converters, has intensified the demand for high-efficiency DC-DC conversion with fast and reliable voltage regulation [1, 2]. In such systems, the DC-DC stage acts as a dynamic actuator responsible for maintaining DC-bus voltage stability under load transients, source intermittency, and parameter variations. The Boost converter is widely used to interface low-voltage sources with higher-voltage DC buses; however, in continuous conduction mode (CCM), its duty-to-output dynamics include a right-half-plane (RHP) zero and exhibit pronounced operating-point dependence. These characteristics constrain the achievable bandwidth and complicate robust regulation [3, 4].

Industrial practice often relies on fixed-structure proportional-integral/proportional-integral-derivative (PI/PID) compensators and lead-lag designs because of their simplicity, but their performance may deteriorate when the operating point varies, leading to increased overshoot, longer settling times, and reduced robustness under line and load disturbances [5, 6]. In contrast, state-space averaging and small-signal linearization provide a control-oriented basis for systematic converter design [4, 7, 8]. Within this framework, optimal state-feedback methods such as Linear Quadratic Regulator (LQR) can improve nominal transient performance, whereas mixed-sensitivity H_∞synthesis provides a more explicit frequency-domain framework for balancing tracking accuracy, disturbance attenuation, and control effort under non-minimum-phase constraints [9-13].

Recent literature confirms continuing interest in robust H_∞-based control for Boost and Boost-derived converter topologies, including high-gain Boost converters, interleaved Boost converters, and recent two-phase interleaved Boost applications in battery energy storage systems [14-17]. More broadly, recent robust H_∞regulation results in related power-electronic applications also confirm the continuing relevance of frequency-domain robust design in converter control [18, 19]. These studies demonstrate the value of robust control for converter regulation; however, they do not always incorporate the tracking-error integral state directly into the synthesis model, and they rarely provide a transparent benchmark against an optimal controller designed from the same augmented plant and assessed under matched operating scenarios. Consequently, the source of the observed performance differences often remains difficult to isolate.

The contribution of this paper is not the isolated use of mixed-sensitivity H_∞ control, integral augmentation, or LQR benchmarking, each of which is already established in the literature. Rather, the novelty lies in a controlled and methodologically consistent Boost-converter study that combines these elements within a common evaluation framework. First, the proposed controller is synthesized from an integral-augmented nominal small-signal model so that zero steady-state tracking error is enforced directly within the robust-design stage. Second, the LQR benchmark is constructed from the same augmented nominal plant, allowing differences in closed-loop behavior to be attributed to the control methodology rather than to differences in model structure. Third, the two controllers are assessed under identical reference, load, and input-voltage scenarios in a nonlinear PWM switching model using unified transient, disturbance, and actuator-related metrics. In this way, the paper aims to provide a fair and practically meaningful assessment of mixed-sensitivity H_∞design for Boost-converter voltage regulation.

The remainder of this paper is organized as follows. Section II derives the averaged small-signal model and its integral augmentation. Section III presents the mixed-sensitivity H_∞ synthesis, the frequency-domain analysis, and the LQR benchmark. Section IV reports the comparative nonlinear simulation results, and Section V concludes the paper.

This section presents the control framework adopted for output-voltage regulation of the Boost converter. The objective is to regulate the converter output around its nominal reference by manipulating the duty-ratio perturbation while preserving internal closed-loop stability under reference changes, input-voltage disturbances, and output-side loading perturbations. The proposed design is based on a mixed-sensitivity H_∞controller synthesized from the nominal integral-augmented small-signal model, while an LQR controller designed from the same augmented model is used as a benchmark. To improve clarity, the H_∞ design is first introduced in its generalized-plant form, then verified in the frequency domain, and finally compared with the LQR benchmark.

3.1 Mixed-sensitivity $H_{\infty}$ synthesis

The proposed $H_{\infty}$ controller is synthesized from the nominal integral-augmented small-signal model derived in Section 2. Let $\mathrm{x}_a=\left[\begin{array}{ll}\tilde{\mathrm{x}}^T & \xi\end{array}\right]^T$ denote the augmented state vector, where $\tilde{\mathrm{x}}$ contains the converter state perturbations and $\xi$ is the integral state associated with the output-voltage tracking error. For control synthesis, the nominal model is extended with the exogenous input vector:

$\mathrm{w}=\left[\begin{array}{lll}r & d_{v g} & d_{\text {load }}\end{array}\right]^T$     (13)

where, $r$ is the reference input, $d_{v g}$ is an input-voltage disturbance, and $d_{\text {load}}$ is an equivalent load-disturbance input used to represent abrupt output-side load changes. The resulting synthesis model is written as:

$\begin{aligned} \dot{\mathrm{x}}_a & =\widehat{\mathrm{F}} \mathrm{x}_a+\widehat{\mathrm{G}} u+\mathrm{B}_w \mathrm{w}, \\ y & =\mathrm{C}_y \mathrm{x}_a+D_{y u} u\end{aligned}$           (14)

where, $u=\tilde{d}$ is the duty-ratio perturbation and y denotes the output-voltage perturbation.

The controller is formulated as a dynamic output-feedback law driven by the measured signal vector.

$\mathrm{y}_m=\left[\begin{array}{lll}\tilde{l}_L & y & \xi\end{array}\right]^T, \quad u=\mathrm{K}(s) \mathrm{y}_m$         (15)

where, $\tilde{l}_L$ is the inductor-current perturbation. This structure allows the controller to exploit both current and voltage information while preserving the integral action required for zero steady-state tracking error.

To encode the design objectives in a unified frequency-domain framework, the tracking error is defined as:

$e=r-y$        (16)

and the weighted performance output is chosen as:

$\mathrm{z}=\left[\begin{array}{l}W_s e \\ W_u u \\ W_t y\end{array}\right]$       (17)

The mixed-sensitivity H_∞ problem then consists in determining a stabilizing controller K(s) that minimizes the closed-loop H_∞ norm from the exogenous input w to the regulated output $Z$ [12], namely:

$\min _{\mathrm{K}(s)}\left\|T_{z w}(s)\right\|_{\infty}$           (18)

The three weighting functions retain their standard interpretations. The weight W_s is used to improve low-frequency tracking accuracy and disturbance rejection, W_u penalizes excessive duty-ratio activity, and W_t enforces high-frequency roll-off of the regulated output in order to limit amplification of unmodeled fast dynamics. In this work, W_s and W_t are selected as low-order stable transfer functions, whereas W_u is chosen as a constant [11-13].

In the present study, the weighting functions are parameterized in the following first-order forms:

$\mathrm{W}_s(\mathrm{~s})=\frac{\mathrm{s} / \mathrm{M}_{\mathrm{s}}+\omega_b}{\mathrm{~s}+\omega_b \mathrm{~A}_{\mathrm{s}}}$    (19)

$\mathrm{W}_u(\mathrm{~s})=1 / u_{\max }$        (20)

$\mathrm{W}_t(\mathrm{~s})=\frac{\mathrm{s}+\omega_t / \mathrm{M}_{\mathrm{t}}}{\mathrm{A}_{\mathrm{t}} \mathrm{s}+\omega_{\mathrm{t}}}$               (21)

where, A_s and M_s determine the low-frequency sensitivity level and the allowable sensitivity peak, respectively, $u_{\max }$ specifies the admissible control activity, and A_t, M_t, and ω_T govern the complementary-sensitivity shaping at high frequency. The frequency ω_b defines the desired sensitivity-shaping bandwidth [21-23].

Because the Boost converter is non-minimum phase, the weighting selection is carried out conservatively so that the implied closed-loop bandwidth remains compatible with the RHP zero of the duty-to-output dynamics. The resulting controller therefore provides a compromise among tracking performance, disturbance attenuation, and control moderation within a unified robust-control framework.

The frequency-domain behavior of the resulting H_∞ controller is examined next in order to verify that the achieved loop shaping is consistent with the intended regulation objectives and the non-minimum-phase limitation of the Boost converter.

3.2 Frequency-domain analysis of the mixed-sensitivity H_∞ design

The frequency-domain behavior of the proposed controller is now examined in order to verify that the achieved loop shaping is consistent with the intended regulation objectives and with the non-minimum-phase nature of the Boost converter. The following results correspond to the reults correspond to the final mixed-sensitivity H_∞ controller whose weighting parameters were optimized by the genetic-algorithm-based tuning procedure described later in Section 3.4.

Figure 2. Mixed-sensitivity H_∞ weighting functions used for controller synthesis

Figure 3. Frequency-domain verification of the mixed-sensitivity H_∞ design for the reference channel

Figure 2 shows the weighting functions used in the synthesis. The sensitivity weight $W_s$ is shaped to enforce strong lowfrequency regulation and disturbance attenuation, the constant weight $W_u$ penalizes excessive duty-ratio activity, and the complementary-sensitivity weight $W_t$ imposes high-frequency roll-off in order to reduce amplification of unmodeled fast dynamics. The corresponding shaping frequencies are $\omega_b= 315.56 \mathrm{rad} / \mathrm{s}$ and $\omega_T=853.78 \mathrm{rad} / \mathrm{s}$, while the nominal RHP zero of the duty-to-output dynamics is located at $\omega_{z, \mathrm{RHP}}=3367.97 \mathrm{rad} / \mathrm{s}$. Therefore, the effective shaping region remains well below the non-minimum-phase limitation of the Boost converter.

The achieved closed-loop behavior is verified in Figure 3 through the weighted maps $W_s S, W_u K S$, and $W_t T$, where $S, K S$, and $T$ denote the sensitivity, control-sensitivity, and complementary-sensitivity functions, respectively. For the reference channel, the obtained norms are $\left\|W_s S\right\|_{\infty}=1.092$, $\left\|W_u K S\right\|_{\infty}=0.0127$, and $\left\|W_t T\right\|_{\infty}=0.444$. These values confirm that the proposed controller achieves the intended frequency-domain compromise on the nominal plant: the sensitivity channel remains close to its desired low-frequency target, the control-effort channel is only weakly stressed, and the complementary-sensitivity channel is sufficiently attenuated at high frequency.

A channel-wise examination of the generalized closed-loop map provides additional insight into the achieved H_∞shaping. The reference and input-voltage-disturbance channels exhibit moderate weighted amplification, whereas the equivalent load-disturbance channel remains the most demanding component of the generalized design. This observation is physically consistent with the nonlinear switched simulations reported later, in which abrupt load changes also constitute the most severe operating condition.

Overall, the frequency-domain analysis confirms that the proposed mixed-sensitivity H_∞ controller achieves the desired compromise among low-frequency voltage regulation, moderated control effort, and high-frequency attenuation, while respecting the bandwidth limitation imposed by the RHP zero of the Boost converter.

Having established the frequency-domain behavior of the proposed H_∞ controller, a benchmark design is introduced next in order to assess the extent to which the observed closed-loop improvement is attributable to the control methodology itself rather than to the underlying plant model.

3.3 Linear Quadratic Regulator benchmark

To provide a transparent baseline for comparison, an LQR controller is designed from the same nominal integral-augmented small-signal model used for the H_∞ synthesis. Specifically, the augmented dynamics defined in Eqs. (10)-(12) are used for the LQR design. The corresponding control law is expressed as [16, 22, 23]:

$u=-\mathrm{K}_{\mathrm{LQR}} \mathrm{x}_a$,       (22)

where, $\mathrm{K}_{\mathrm{LQR}}$ is the optimal state-feedback gain obtained by minimizing the quadratic cost function.

$J=\int_0^{\infty}\left(\mathrm{x}_a^T \mathrm{Qx}_a+u^T R u\right) d t$.               (23)

The weighting matrix Q penalizes deviations of the augmented states, while R limits excessive control activity. In particular, a relatively large weight is assigned to the integral state to enforce negligible steady-state tracking error, whereas the remaining state weights are selected to moderate transient overshoot and settling behavior. The scalar R is chosen to avoid overly aggressive duty-ratio variations.

Because the LQR benchmark is constructed from the same integral-augmented nominal model as the H_∞ controller, the subsequent comparison focuses on the effect of the control design methodology rather than on differences in plant representation. For both controllers, the physical duty ratio applied to the PWM stage is reconstructed from the nominal operating duty ratio and the control perturbation, with saturation enforced over the admissible interval.

3.4 Controller-parameter tuning by genetic algorithm

To avoid manual trial-and-error tuning and to ensure a fair comparison, the final parameters of both controller families were obtained using a common Genetic Algorithm (GA)-based outer-loop optimization procedure applied to the nominal integral-augmented small-signal model.

For each candidate parameter set, the corresponding controller was synthesized and evaluated under the same nominal closed-loop scenarios, namely reference tracking, input-voltage disturbance rejection, and equivalent load-disturbance rejection. The optimization objective was defined as the sum of the scenario-wise ITAE indices,

$J_{G A}=\mathrm{ITAE}_{\mathrm{ref}}+\mathrm{ITAE}_{\mathrm{vg}}+\mathrm{ITAE}_{\mathrm{load}}$,             (24)

For the proposed controller, the GA tuned the parameters defining the weighting functions W_s, W_u, and W_t. For the benchmark controller, the GA tuned the diagonal entries of Q and the scalar R. The final retained parameters are summarized in Tables 1 and 2.

After fixing the final controller parameters through the common GA-based tuning procedure, both controllers were implemented in the nonlinear switched MATLAB/Simulink model for comparative validation under realistic operating conditions.

Table 1. Final Genetic Algorithm (GA)-tuned weighting parameters of the mixed-sensitivity H_∞ controller

Table 2. Final Genetic Algorithm (GA)-tuned Linear Quadratic Regulator (LQR) parameters