Experimental Evaluation of the Robust Controllers Applied on a Single Inductor Multiple Output DC-DC Buck Converter to Minimize Cross Regulation Considering Parametric Uncertainties and CPL Power Variations

Abstract

This paper presents two control design strategies for voltage regulation in a single inductor dual-output DC-DC buck converter system. Based on a nominal multiple input multiple output plant model and performance requirements, both a Linear Quadratic Regulator and a Decoupled PI control law are designed to control the power converter system under parametric uncertainties such as voltage source variation, constant power load variation, and load resistance variation. Therefore, the converter can cascade operation, which may cause the undesired effects of voltage oscillation and reduce the system’s stability margin. The control performance was assessed under both the uncertainties of resistive loading and the power variation in a constant power load. A single inductor dual-output DC-DC buck converter board was developed for experimental tests. The experimental results show that the Linear Quadratic Regulator strategy, compared to the PI strategy, presented a robust performance in the presence of parametric uncertainties in the resistive load-bearing; however, it was sensitive to uncertainties due to the presence of constant power loads.

1. Introduction

Continued advances in control technology and power electronics have expanded the applications of power conversion systems in spacecraft, aircraft, ships, telecommunication networks, and electric vehicles, aiming to reduce size, weight, and cost, and to improve reliability, power quality, efficiency, and flexibility [1]. Some modern industries, whose processes require high dynamic performance, have used various types of power converters for applications, such as variable speed drivers [2], renewable energy systems [3,4], transportation systems [5,6], hybrid energy storage systems [7,8], and communication systems [9].

In several of these applications, switching power converters are controlled by pulse-width modulation (PWM) techniques to transfer power from a source to loads that typically have constant power characteristics. Because of the switching, the converters have some inherent nonlinear behaviors, such as high switching frequency, increasing harmonics in the system, current and voltage distortion, and instabilities that can occur due to these effects [10,11]. Therefore, it is a challenging task to ensure the stability, transient performance, and higher efficiency of such converters.

In order to reduce the weight as well as to allow different regulated levels of DC voltage at the converter outputs, several studies [12,13,14] have recently proposed new converter topologies by sharing a single inductive element. One of the prominent topologies is the family of non-isolated DC-DC converters, i.e., single inductor multiple output (SIMO) DC-DC converters [15,16]. Some application scenarios for such converters are areas that require high-quality DC voltage regulation, such as the industry sector [17], telecommunications [18], and renewable energy source control [19]. SIMO converters offer a good compromise in terms of the cost, efficiency, and diversity of potential applications. However, to operate effectively, a SIMO converter requires a high-performance control strategy, which leads to a number of challenging engineering problems, such as cross-regulation, high switching frequency, and severe nonlinearities. The main control challenge arises from the need (to reduce weight and cost) to share a single inductive element in a circuit. In short, a SIMO converter behaves similarly to a multivariable system with highly coupled loops, resulting in high dependencies between the main variables: the output voltages, coil current, and DC voltage gains [20].

Therefore, a complex automatic control system is required to maintain good performance regulation at different output voltage levels while minimizing the negative effects of the strong coupling between control loops. To this end, several studies have proposed solutions to mitigate coupling effects in DC-DC SIMO converters [12,15,21,22,23,24,25,26,27].

In other cases, a cascade of power electronic converters is a common feature of almost all converter-domain power systems to help provide the necessary load point regulation. However, tightly regulated electronic switching power converters act as constant power loads (CPLs) and tend to destroy the stability of power systems and upstream converters [28,29], which means that in cascade systems, the dynamic response of the load converter is faster than the dynamic response of the source converter and its switching operating frequency is faster than the source converter. If the source converter is faster than the load converter then it will compensate for disturbances and will regulate its output before the feedback load converter loop responds to disturbances [30,31,32,33]; therefore, CPL can affect the power quality of the power systems and cause its instability and may eventually lead to system failure [34,35].

Despite the many advantages of DC power systems, ensuring their robust stability and performance remains an important technical challenge. This is mainly due to the nonlinearities introduced by the CPL operation of the switching power converters and the nonlinear characteristics of the converter itself. The problem is further aggravated by the interaction of different subsystems [32]. Therefore, the overall stability of the system cannot be guaranteed even if the individual subsystems are stable. It guarantees the analysis of the steady state and dynamic stability of the aggregated system under different loading curves.

Furthermore, most of the papers published so far have focused on mitigating the destabilizing effects of CPL without considering the uncertainty of the system parameters [36]. Therefore, there is still a lack of studies in the literature reporting robust parametric approaches for DC-DC converters driving CPLs to mitigate the oscillation effects caused by CPL. The main contributions of this work are briefly summarized as follows:

• Two control methodologies are experimentally evaluated and compared applied to a multivariable dual-output DC-DC converter. The first control methodology is a decentralized robust control design methodology based on the LQR technique. The other approach is a decentralized control design method based on the decoupled PID technique [37].
• The experimental evaluation uses active control methodologies to solve the cross-regulation problem in the SIDO converter caused by the strong coupled system and nonlinearities. To this end, extensive experimental tests demonstrate the effects of different controllers, showing that the proposed LQR technique has better performance indices than other methods when the system is subjected to parametric variations (input DC voltage and resistance loads) as well as when a CPL is connected to each output once a time. Performance indices (ISE and ISU) are calculated to analyze the performance of the control method.
• For this purpose, a tested system (single inductor dual-output DC-DC converter) was designed to implement the controllers and perform various tests to collect experimental data. The device allows the variation in the physical parameters of the converter (input DC voltage and load resistances) and uses a programmable electronic load to emulate the behavior of the CPL.

The remainder of this article is organized as follows. Section 2 introduces the system description and problem formulation to solve the nonlinear analysis of the system; Section 3 briefly discusses the robust control methods proposed in this work. Section 4 introduces the experimental environment that describes the experiments to be performed in this paper; Section 5 discusses the result. Finally, the main conclusions are presented in Section 6.

2. System Description and Problem Formulation

Mathematical Model for a SIMO Converter

To represent the dynamic behavior of a single inductor multiple output DC-DC converter system, the small-signal approximation model is adopted as an effective mathematical model [38]. Figure 1 shows a single inductor multiple output (SIMO) DC-DC converter with two outputs V₁ and V₂ and the topology used for the control system. The main feature of this converter is that the single inductor current I_L is split into two currents I₁ and I₂ at the converter outputs 1 and 2, respectively. The uncertainties of the system are represented by arrows, i.e., loads and input voltage changes. These uncertainties often occur in real converters and affect performance. Therefore, control strategies must address this issue.

SIMO converters can be divided into three categories depending on the relationship between the duty cycle inputs d₁ and d₂, which control the MOSFET switches Q1 and Q2 (see Figure 1). If d₁ is greater than d₂, the operation is called Class-A. If d₁ is equal to d₂, it is called Class-B, and if d₁ is less than d₂, it is called Class-C [39]. In this study, a DC-DC SIMO converter operating according to Class-C (i.e., d₁ < d₂) is used. In this case, the nonlinear average model of the converter is a third-order system given by the set of Equation (1), whose state variables are the inductor current i_L and the voltages of the capacitors V₁ and V₂, and whose control inputs are the duty cycle of switches Q₁ and Q₂.

$$\left\{\begin{array}{c}\mathrm{L}\frac{\mathrm{d}{\mathrm{i}}_{\mathrm{L}}}{\mathrm{d}\mathrm{t}}={\mathrm{V}}_{\mathrm{i}\mathrm{n}}{\mathrm{d}}_1-{\mathrm{V}}_1{\mathrm{d}}_2-{\mathrm{V}}_2(1-{\mathrm{d}}_2)\\ {\mathrm{C}}_1\frac{\mathrm{d}{\mathrm{V}}_1}{\mathrm{d}\mathrm{t}}={\mathrm{i}}_{\mathrm{L}}{\mathrm{d}}_2-\frac{{\mathrm{V}}_1}{{\mathrm{R}}_1}\\ {\mathrm{C}}_2\frac{\mathrm{d}{\mathrm{V}}_2}{\mathrm{d}\mathrm{t}}={\mathrm{i}}_{\mathrm{L}}(1-{\mathrm{d}}_2)-\frac{{\mathrm{V}}_2}{{\mathrm{R}}_2}\end{array}\right.$$

(1)

where V_in is the input unregulated voltage source value.

In the SIMO converter of Figure 1, the steady-state value $D_1^0$ of the duty cycle d₁ (switch Q₁) regulates the total input power and thus the coil current of the SIMO converter, while the steady-state value of the duty cycle d₂ (of switch Q₂), $D_2^0$, determines how the coil current is divided between the converter output loads R₁ and R₂. Assuming that the electronic switches and diodes are ideal, the components of the open-loop control-to-output 2 × 2 transfer function matrix at the operational point, $V_1^0, V_2^0,D_1^0,D_2^0,I_L^0$, and $V_{in}^0,$ of the linearized model of the SIMO converter are given by the following formulas:

$${\mathrm{g}}_{11}\left(\mathrm{s}\right)=\frac{{\mathrm{V}}_1\left(\mathrm{s}\right)}{{\mathrm{d}}_1\left(\mathrm{s}\right)}=\frac{{\mathrm{V}}_{\mathrm{i}\mathrm{n}}^0{\mathrm{D}}_2^0{\mathrm{R}}_{\mathrm{e}\mathrm{q}1}}{{\left({\mathrm{D}}_2^0\right)}^2{\mathrm{R}}_{\mathrm{e}\mathrm{q}1}+{\left(1-{\mathrm{D}}_2^0\right)}^2{\mathrm{R}}_{\mathrm{e}\mathrm{q}2}+\mathrm{s}\mathrm{L}}$$

(2)

$${\mathrm{g}}_{12}\left(\mathrm{s}\right)=\frac{{\mathrm{V}}_2\left(\mathrm{s}\right)}{{\mathrm{d}}_1\left(\mathrm{s}\right)}=\frac{{\mathrm{V}}_{\mathrm{i}\mathrm{n}}^0(1-{\mathrm{D}}_2^0){\mathrm{R}}_{\mathrm{e}\mathrm{q}2}}{{\left({\mathrm{D}}_2^0\right)}^2{\mathrm{R}}_{\mathrm{e}\mathrm{q}1}+{\left(1-{\mathrm{D}}_2\right)}^2{\mathrm{R}}_{\mathrm{e}\mathrm{q}2}+\mathrm{s}\mathrm{L}}$$

(3)

$${\mathrm{g}}_{21}\left(\mathrm{s}\right)=\frac{{\mathrm{V}}_1\left(\mathrm{s}\right)}{{\mathrm{d}}_2\left(\mathrm{s}\right)}=\frac{{\mathrm{I}}_{\mathrm{L}}^0{\mathrm{R}}_{\mathrm{e}\mathrm{q}1}\left[\left(1-{\mathrm{D}}_2^0\right){\mathrm{R}}_{\mathrm{e}\mathrm{q}2}+\mathrm{s}\mathrm{L}\right]+{\mathrm{D}}_2^0{\mathrm{R}}_{\mathrm{e}\mathrm{q}1}\left({\mathrm{V}}_2^0-{\mathrm{V}}_1^0\right)}{{\left({\mathrm{D}}_2^0\right)}^2{\mathrm{R}}_{\mathrm{e}\mathrm{q}1}+{\left(1-{\mathrm{D}}_2^0\right)}^2{\mathrm{R}}_{\mathrm{e}\mathrm{q}2}+\mathrm{s}\mathrm{L}}$$

(4)

$${\mathrm{g}}_{22}\left(\mathrm{s}\right)=\frac{{\mathrm{V}}_2\left(\mathrm{s}\right)}{{\mathrm{d}}_2\left(\mathrm{s}\right)}=\frac{-\left[{\mathrm{I}}_{\mathrm{L}}^0{\mathrm{R}}_{\mathrm{e}\mathrm{q}2}\left({\mathrm{D}}_2^0{\mathrm{R}}_{\mathrm{e}\mathrm{q}1}+\mathrm{s}\mathrm{L}\right)+\left(1-{\mathrm{D}}_2^0\right){\mathrm{R}}_{\mathrm{e}\mathrm{q}2}\left({\mathrm{V}}_2^0-{\mathrm{V}}_1^0\right)\right]}{{\left({\mathrm{D}}_2^0\right)}^2{\mathrm{R}}_{\mathrm{e}\mathrm{q}1}+{\left(1-{\mathrm{D}}_2\right)}^2{\mathrm{R}}_{\mathrm{e}\mathrm{q}2}+\mathrm{s}\mathrm{L}}$$

(5)

where

$${\mathrm{R}}_{\mathrm{e}\mathrm{q}1}=\frac{{\mathrm{R}}_1}{1+\mathrm{s}{\mathrm{C}}_1{\mathrm{R}}_1}$$

(6)

$${\mathrm{R}}_{\mathrm{e}\mathrm{q}2}=\frac{{\mathrm{R}}_2}{1+\mathrm{s}{\mathrm{C}}_2{\mathrm{R}}_2}$$

(7)

$${\mathrm{I}}_{\mathrm{L}}^0=\frac{{\mathrm{V}}_1^0}{{\mathrm{R}}_1}+\frac{{\mathrm{V}}_2^0}{{\mathrm{R}}_2}$$

(8)

For the study presented in this paper, the nominal values and assumed variation ranges for the SIMO converter circuit components are given in

Table 1. DC—DC SIMO converter board test system parameters.

Parameter	Unit	Nom. Value	Variation	Description
Vin	V	7	(6–7)	Source input voltage
R1	Ω	10	(5–10)	Loading at output 1
R2	Ω	10	(5–10)	Loading at output 2
CPL1	W	0.5	(0.6–0.5)	Output power of CPL1
CPL2	W	0.25	(0.35–0.25)	Output power of CPL2
C1	µF	2200	-	Capacitor at output 1
C2	µF	1000	-	Capacitor at output 2
L	µH	330	-	Inductor
D10	%	40	-	Operational point for the duty cycle of output 1
D20	%	60	-	Operational point for the duty cycle of output 2
V10	V	3.11	-	Operational point for output voltage 1
V20	V	1.74	-	Operational point for output voltage 2
I10	A	0.311	-	Operational point for current of the output 1
I20	A	0.174	-	Operational point for current of the output 2
fsw	kHz	7.8	-	Switching frequency
fs	kHz	2.0	-	Sampling frequency

.

3. Proposals of Robust Controller Strategies for the SIMO Power Converter

3.1. Pairing Analysis for the SIMO Converter

The first step in the control design process is to perform a pairwise analysis to determine which output is most affected by a given plant input. There are several ways to perform pairwise analysis on this topic. Nevertheless, the effective relative gain array (ERGA) is easy to implement and understand, and is more efficient than other arrays found in the literature, e.g., relative gain array (RGA) or dynamic relative gain array (DRGA) [20].

The ERGA method is commonly used to determine the sensitivity of each output to each input. This procedure is provided by [37,39]. The main steps of the ERGA-based pairwise analysis are as follows:

(1)

Determine the DC gain array using the DC gain for each element of the transfer function matrix.

(2)

Obtain the cutoff frequency for each element of the transfer function matrix.

(3)

Compute the energy transmission ratio array based on the DC gain and cutoff frequencies.

(4)

Compute the effective gain matrix that indicates the sensitivity between each I/O pair.

(5)

Compute the ERGA number and check for diagonal head dominance.

The effective gain matrix shown in (9) as a result of the ERGA pair analysis shows the optimal pairing, i.e., the first (V₁) and the second output (V₂) are, respectively, more sensitive to the first (d₁) and the second input (d₂).

$$\mathrm{\Phi }=\left[\begin{array}{cc}0.762& 0.238\\ 0.238& 0.762\end{array}\right]$$

(9)

Therefore, a simulation evaluation is performed to check the coupling between the system outputs. So, first a pulse is applied to the duty cycle of the switch Q1 and the system is run in open loop (without any controller) mode, and then the same process is repeated for the switch Q2. Figure 2 shows the resulting coupling for the SIMO converter test system. The results show that V₂ is very sensitive to changes in duty cycle d₂, while V₁ is very sensitive to changes in duty cycle d₁. However, V₁ and V₂ are greatly affected by changes in d₁, although changes in d₂ do not have much relevance in the outputs of the system. This finding reflects the coupling between the loops of the system, which shows the efficient pairing of the ERGA analysis.

3.2. Linear Quadratic Regulator Design

The LQR control is a well-known optimal-based approach to state feedback controller that can solve some problems in systems like instability or low stability margins, manipulating the system inputs and avoiding some typical problems on the stabilization problem like saturation control signal and fast actuator degradation [40]. This controller design is based on the choice of the weight matrices of the states and control signal, Q and R, respectively, which is used in the minimizing cost function.

$$\mathrm{J}=\frac{1}{2}{\sum }_{\mathrm{k}}^{\mathrm{\infty }}(\mathrm{x}{(\mathrm{k})}^{\mathrm{T}}\mathrm{Q}\mathrm{x}(\mathrm{k})+\mathrm{u}{(\mathrm{k})}^{\mathrm{T}}\mathrm{R}\mathrm{u}(\mathrm{k}))$$

(10)

The minimization of the cost function results in a feedback control law.

$$\mathrm{u}(\mathrm{k})=-{\mathrm{R}}^{-1}{\mathrm{B}}^{\mathrm{T}}\mathrm{S}\mathrm{x}(\mathrm{k})\to \mathrm{u}(\mathrm{k})=-\mathrm{K}\mathrm{x}(\mathrm{k})$$

(11)

Positive-definite matrix “S” is a solution of the discrete-time algebraic Ricatti equation.

$$\mathrm{S}=\mathrm{Q}+{\mathrm{A}}_{\mathrm{d}}^{\mathrm{T}}\mathrm{S}{\mathrm{A}}_{\mathrm{d}}-{\mathrm{A}}_{\mathrm{d}}^{\mathrm{T}}\mathrm{S}{\mathrm{B}}_{\mathrm{d}}{\left({\mathrm{B}}_{\mathrm{d}}^{\mathrm{T}}\mathrm{S}{\mathrm{B}}_{\mathrm{d}}+\mathrm{R}\right)}^{-1}{\mathrm{B}}_{\mathrm{d}}^{\mathrm{T}}\mathrm{S}{\mathrm{A}}_{\mathrm{d}}$$

(12)

In this structure, the LQR control on the closed loop has not guaranteed perturbation rejection in low frequency; however, this problem can be solved by an integrator addition on system input, creating a different model called “velocity model” [40]. The discrete integrator $∆=1-z^{-1}$ is inserted on the system plant, by this way creating a new state variable ∆x(k).

$$∆\mathrm{x}\left(\mathrm{k}+1\right)={\mathrm{A}}_{\mathrm{d}}∆\mathrm{x}\left(\mathrm{k}\right)+{\mathrm{B}}_{\mathrm{d}}∆\mathrm{u}\left(\mathrm{k}\right)$$

(13)

$$∆\mathrm{y}\left(\mathrm{k}\right)={\mathrm{C}}_{\mathrm{d}}∆\mathrm{x}(\mathrm{k}+1)$$

(14)

Calling the new state vector ${\mathrm{x}}_{\mathrm{a}}(\mathrm{k})={[\mathrm{y}(\mathrm{k})∆\mathrm{x}(\mathrm{k})]}^{\mathrm{T}}$ and applying it in the incremental model, the velocity model is presented.

$$\left[\begin{array}{c}\mathrm{y}(\mathrm{k}+1)\\ ∆\mathrm{x}(\mathrm{k}+1)\end{array}\right]=\left[\begin{array}{cc}1& {\mathrm{C}}_{\mathrm{d}}{\mathrm{A}}_{\mathrm{d}}\\ 0& {\mathrm{A}}_{\mathrm{d}}\end{array}\right]\left[\begin{array}{c}\mathrm{y}(\mathrm{k}+1)\\ ∆\mathrm{x}(\mathrm{k}+1)\end{array}\right]+\left[\begin{array}{c}{\mathrm{C}}_{\mathrm{d}}{\mathrm{B}}_{\mathrm{d}}\\ {\mathrm{B}}_{\mathrm{d}}\end{array}\right]∆\mathrm{u}(\mathrm{k})$$

(15)

Inserting the augmented space state in the Ricatti transforms the feedback control law structure on a rejection perturbation problem.

$$\mathrm{u}\left(\mathrm{k}\right)=\mathrm{u}\left(\mathrm{k}-1\right)+{\mathrm{K}}_{\mathrm{y}}{\mathrm{y}}_{\mathrm{r}}\left(\mathrm{k}\right)-{\mathrm{K}}_{∆\mathrm{x}}∆\mathrm{x}\left(\mathrm{k}\right)$$

(16)

Then, the control law of the digital LQR for stabilization and perturbation rejection problem is presented.

$$\mathrm{u}\left(\mathrm{k}\right)=\mathrm{u}\left(\mathrm{k}-1\right)+{\mathrm{K}}_{\mathrm{y}}{\mathrm{y}}_{\mathrm{r}}\left(\mathrm{k}\right)-{\mathrm{K}}_{∆\mathrm{x}}∆\mathrm{x}\left(\mathrm{k}\right)$$

(17)

The correct selection of Q and R weight matrices is important because it is responsible for the penalization of the dynamic states and input signal, respectively, in a way that Q and R inform the priority minimization order in the cost function, giving a trade-off balance between conflicting control objectives, such as minimizing control effort and regulation [41,42]. In this work, this selection is based on the loop-shaping of process singular values for improving robustness in high and low frequencies where stability Phase Margin, PM, and Gain Margin, GM, are used in the robustness design, and bandwidth frequency is used in the converter velocity. This method is an interesting solution to the controller’s synthesis for multiple input multiple output process because of the difficulty in the frequency analysis in each loop [43]. For the analysis of minimal stability margins (Phase Margin and Gain Margin) in the frequency singular values, the following constraints formulation, proposed in [44], is used:

$$\mathrm{G}{\mathrm{M}}_{\mathrm{d}\mathrm{B}}\ge 20 {\log }_{10}\mathrm{m}\mathrm{i}\mathrm{n}\left(\frac{{\mathrm{m}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}-1},1+\frac{1}{{\mathrm{m}}_{\mathrm{t}}}\right)$$

(18)

$$\mathrm{P}{\mathrm{M}}_{\mathrm{d}\mathrm{e}\mathrm{g}}\ge \frac{180}{\mathrm{\pi }}\times \min \left(2 {\sin }^{-1}\frac{1}{{2\mathrm{m}}_{\mathrm{s}}},2{\sin }^{-1}\frac{1}{{2\mathrm{m}}_{\mathrm{t}}}\right)$$

(19)

where ${\mathrm{m}}_{\mathrm{s}}={\mathrm{m}\mathrm{a}\mathrm{x}}_{\mathrm{\omega }}\mathrm{\sigma }\left(\mathrm{S}\left(\mathrm{j}\mathrm{\omega }\right)\right)$ and ${\mathrm{m}}_{\mathrm{t}}={\mathrm{m}\mathrm{a}\mathrm{x}}_{\mathrm{\omega }}\mathrm{\sigma }\left(\mathrm{T}\left(\mathrm{j}\mathrm{\omega }\right)\right)$, with ${\mathrm{m}\mathrm{a}\mathrm{x}}_{\mathrm{\omega }}\mathrm{\sigma }\left(\mathrm{T}\left(\mathrm{j}\mathrm{\omega }\right)\right)$ meaning the maximum singular value of a given transfer function matrix.

For LQR design purposes, assuming that the SIMO converter operates in a steady state around the operation point: ${\mathrm{V}}_1^0=3.11 \mathrm{V},{\mathrm{V}}_2^0=1.74 \mathrm{V},{\mathrm{D}}_1^0=0.4,{\mathrm{D}}_2^0=0.6,{\mathrm{I}}_{\mathrm{L}}^0=$ 0.485 A and ${\mathrm{V}}_{\mathrm{i}\mathrm{n}}^0$ = 7 V. Then, the corresponding matrices A, B, C, and D of the linearized state-space model are as follows:

$$\mathrm{A}=\left[\begin{array}{ccc}-45.46& 0& 272.17\\ 0& -100.00& 401.22\\ -1814.50& -1215.80& 0\end{array}\right], \mathrm{B}=\left[\begin{array}{cc}0& 172.09\\ 0& -378.60\\ 21212.00& -2266.70\end{array}\right]$$

(20)

$$\mathrm{C}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right], \mathrm{D}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$$

By using the A, B, C, and D matrix values given by (20), the corresponding open-loop transfer function matrix $\mathrm{G}\left(\mathrm{s}\right)={\mathrm{C}(\mathrm{s}\mathrm{I}-\mathrm{A})}^{-1}\mathrm{B}+\mathrm{D}$ has been computed. For the discrete system model with sample time (ts = 0.5 ms), the resulting maximum singular values for the sensitivity and complementary matrices transfer functions, S(z⁻¹) and T(z⁻¹), for the uncompensated closed-loop system (i.e., with an initial controller matrix K₀ = I_2×2) are presented in Figure 3.

$$\mathrm{S}({\mathrm{z}}^{-1})={({\mathrm{I}}_{2\times 2}+\mathrm{G}({\mathrm{z}}^{-1}))}^{-1}, \mathrm{T}({\mathrm{z}}^{-1})={\mathrm{I}}_{2\times 2}-\mathrm{S}({\mathrm{z}}^{-1})$$

(21)

As can be seen, the maximum singular values for the sensitivity and complementary sensitivity transfer function matrices show very large peaks in high frequency, meaning that the uncompensated converter plant is very sensitive to high-frequency noise and could become unstable due low attenuation property in high-frequency band. Another problem in this uncompensated plant is its low capacity in tracking reference setpoint due to the reduced gain presented by T(s) at the low-frequency band. For this uncompensated system, the computed margin index shows low stability margins $\mathrm{G}{\mathrm{M}}_{\mathrm{d}\mathrm{B}}\ge 1.81$ and $\mathrm{P}{\mathrm{M}}_{\mathrm{d}\mathrm{e}\mathrm{g}}\ge 13.30°$, which confirms the singular value analysis.

For this case, low stability margins could be a problem, because uncertainty in circuit parameters (e.g., capacitors, inductors, and resistors) and in the input voltage are very common. Other problems are associated with noisy measurement (high frequency) and perturbations in low frequency. Aiming at improving the stability margins, the initial specified design requirements were as follows: $\mathrm{G}{\mathrm{M}}_{\mathrm{d}\mathrm{B}}\ge 5$, $\mathrm{P}{\mathrm{M}}_{\mathrm{d}\mathrm{e}\mathrm{g}}\ge 45°$ and ${\mathrm{\omega }}_{\mathrm{B}\mathrm{T}}\approx 200 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$, where ${\mathrm{\omega }}_{\mathrm{B}\mathrm{T}}$ is the angular frequency such that the modulus of T(s = jω) is −3 dB. By using the augmented state-space matrices given by (15), along with choosing weighting states and control matrices, $\mathrm{Q}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(10,10,1000,1000,10\right)$ and $\mathrm{R}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(10,10)$, in LQR cost function (10), the computed value for the control vector gain was as follows:

$$\mathrm{K}=\left[\begin{array}{ccccc}0.0169& \mathrm{-0.0055}& 0.0803& \mathrm{-0.1003}& 0.0199\\ 0.1059& \mathrm{-0.0844}& 1.0188& \mathrm{-0.7250}& \mathrm{-0.0435}\end{array}\right]$$

(22)

The resulting maximum singular values for the sensitivity and complementary matrix transfer functions S(z⁻¹) and T(z⁻¹), for the compensated closed-loop system, are presented in Figure 4.

As can be seen, in Figure 4, the computed controller is able to decrease the sensibility peaks which improves the process robustness. Furthermore, the perturbation rejection is also improved due to decreasing sensibility gain values in the low-frequency band. For this compensated system, the computed minimal stability margins and bandwidth frequency of the system are $\mathrm{G}{\mathrm{M}}_{\mathrm{d}\mathrm{B}}\ge 6.02$, $\mathrm{P}{\mathrm{M}}_{\mathrm{d}\mathrm{e}\mathrm{g}}\ge 56.62$, and ${\mathrm{\omega }}_{\mathrm{B}\mathrm{T}}\approx 200 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$, which satisfy the design specifications.

3.3. Decoupled Multiloop PI Controller

The presentation is limited to two-input, two-output systems. Our approach was to explore the standard PI multivariable (multi-loop) tuning [45] and observe what can be achieved by adding simple interactions between the feedback loops. The proposed scheme is based on a simple decoupling, which means that it can be easily implemented at the loop level [46]. The advantage is that it improves performance in frequency ranges that modeling predictive control typically cannot handle.

The designable controller is a static decoupler combined with a decentralized PI controller with setpoint weighting. The control law can be written as follows:

$$\left(\begin{array}{c}{\mathrm{U}}_1\left(\mathrm{s}\right)\\ {\mathrm{U}}_2\left(\mathrm{s}\right)\end{array}\right)=\left(\begin{array}{cc}{\mathrm{d}}_{11}& {\mathrm{d}}_{12}\\ {\mathrm{d}}_{21}& {\mathrm{d}}_{22}\end{array}\right)\left(\begin{array}{c}\overline{{\mathrm{c}}_1}\left(\mathrm{s}\right){\mathrm{Y}}_{\mathrm{r}1}\left(\mathrm{s}\right)-{\mathrm{c}}_1\left(\mathrm{s}\right){\mathrm{Y}}_1\left(\mathrm{s}\right)\\ \overline{{\mathrm{c}}_2}\left(\mathrm{s}\right){\mathrm{Y}}_{\mathrm{r}2}\left(\mathrm{s}\right)-{\mathrm{c}}_2\left(\mathrm{s}\right){\mathrm{Y}}_2\left(\mathrm{s}\right)\end{array}\right)$$

(23)

where U is the control signal, Y the process output, and Y_r the reference. The decoupler:

$$\mathrm{D}=\left(\begin{array}{cc}{\mathrm{d}}_{11}& {\mathrm{d}}_{12}\\ {\mathrm{d}}_{21}& {\mathrm{d}}_{22}\end{array}\right)$$

(24)

is a constant matrix. The PI controller $\overline{{\mathrm{c}}_{\mathrm{i}}}$ is different from ${\mathrm{c}}_{\mathrm{i}}$ to allow for setpoint weighting [43]. The controllers are of the form:

$${\mathrm{c}}_{\mathrm{i}}={\mathrm{k}}_{\mathrm{P}\mathrm{i}}+\frac{{\mathrm{k}}_{\mathrm{I}\mathrm{i}}}{\mathrm{s}}$$

(25)

$$\overline{{\mathrm{c}}_{\mathrm{i}}}=\frac{{\mathrm{k}}_{\mathrm{I}\mathrm{i}}}{\mathrm{s}}$$

(26)

The static decoupler is given by:

$$\mathrm{D}={\mathrm{G}}^{-1}\left(0\right)=\frac{1}{\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{G}\left(0\right)}\left(\begin{array}{cc}{\mathrm{g}}_{22}\left(0\right)& {-\mathrm{g}}_{12}\left(0\right)\\ {-\mathrm{g}}_{21}\left(0\right)& {\mathrm{g}}_{11}\left(0\right)\end{array}\right)$$

(27)

where G(0) is non-singular. The transfer function of the decoupled system is given by:

$${\mathrm{q}}_{11}\left(\mathrm{s}\right)=\frac{{\mathrm{g}}_{11}\left(\mathrm{s}\right){\mathrm{g}}_{22}\left(0\right)-{\mathrm{g}}_{12}\left(\mathrm{s}\right){\mathrm{g}}_{21}\left(0\right)}{\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{G}\left(0\right)}$$

(28)

$${\mathrm{q}}_{12}\left(\mathrm{s}\right)=\frac{{\mathrm{g}}_{12}\left(\mathrm{s}\right){\mathrm{g}}_{11}\left(0\right)-{\mathrm{g}}_{12}\left(\mathrm{s}\right){\mathrm{g}}_{11}\left(0\right)}{\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{G}\left(0\right)}$$

(29)

$${\mathrm{q}}_{21}\left(\mathrm{s}\right)=\frac{{\mathrm{g}}_{12}\left(\mathrm{s}\right){\mathrm{g}}_{22}\left(0\right)-{\mathrm{g}}_{21}\left(\mathrm{s}\right){\mathrm{g}}_{22}\left(0\right)}{\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{G}\left(0\right)}$$

(30)

$${\mathrm{q}}_{22}\left(\mathrm{s}\right)=\frac{{\mathrm{g}}_{22}\left(\mathrm{s}\right){\mathrm{g}}_{11}\left(0\right)-{\mathrm{g}}_{21}\left(\mathrm{s}\right){\mathrm{g}}_{12}\left(0\right)}{\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{G}\left(0\right)}$$

(31)

A Taylor series expansion of the transfer function Q(s), for a small s, gives the following:

$$\mathrm{Q}(\mathrm{s})\approx \left(\begin{array}{cc}1& {\mathrm{k}}_{12}\mathrm{s}\\ {\mathrm{k}}_{21}\mathrm{s}& 1\end{array}\right)$$

(32)

for some constants k₁₂ e k₂₁. Then, it is possible to introduce the interaction indices:

$${\mathrm{k}}_1=\left|{\mathrm{k}}_{12}{\mathrm{k}}_{\mathrm{I}2}\right|{\mathrm{M}}_{\mathrm{s}1}{\mathrm{M}}_{\mathrm{s}2}$$

(33)

$${\mathrm{k}}_2=\left|{\mathrm{k}}_{21}{\mathrm{k}}_{\mathrm{I}1}\right|{\mathrm{M}}_{\mathrm{s}1}{\mathrm{M}}_{\mathrm{s}2}$$

(34)

where M_s1 and M_s2 are the maximum sensitivities of each loop. The indices k₁ and k₂ describe the interaction between the loops. The indices are the result of two terms: one is system-dependent and the other is directly the integral gain of the PI controller. Therefore, the interaction can be reduced by decreasing the controller gains [41]. To find the decentralized PI controllers, we need to consider the diagonal terms of Q(s). Hence, standard methods can be used to design the PI controllers for each transfer function q_kk(s).

By considering the same operating point used for the design of the LQR controller (${\mathrm{V}}_1^0=3.11 \mathrm{V},{\mathrm{V}}_2^0=1.74 \mathrm{V},{\mathrm{D}}_1^0=0.4,{\mathrm{D}}_2^0=0.6,{\mathrm{I}}_{\mathrm{L}}^0=$ 0.485 A, and ${\mathrm{V}}_{\mathrm{i}\mathrm{n}}^0$ = 7 V) the following values were computed for the transfer functions ${\mathrm{g}}_{11}\left(\mathrm{s}\right),{\mathrm{g}}_{12}\left(\mathrm{s}\right),{\mathrm{g}}_{21}\left(\mathrm{s}\right),\mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{g}}_{22}\left(\mathrm{s}\right)$:

$${\mathrm{g}}_{11}\left(\mathrm{s}\right)=\frac{5.773\times {10}^6\mathrm{s}+5.773\times {10}^8}{{\mathrm{s}}^3+145.5{\mathrm{s}}^2+9.862\times {10}^5\mathrm{s}+7.156\times {10}^7}$$

(35)

$${\mathrm{g}}_{12}\left(\mathrm{s}\right)=\frac{8.511\times {10}^6\mathrm{s}+3.868\times {10}^8}{{\mathrm{s}}^3+145.5{\mathrm{s}}^2+9.862\times {10}^5\mathrm{s}+7.156\times {10}^7}$$

(36)

$${\mathrm{g}}_{21}\left(\mathrm{s}\right)=\frac{172.1{\mathrm{s}}^4-5.841\times {10}^5{\mathrm{s}}^3+9.337\times {10}^7{\mathrm{s}}^2+1.217\times {10}^{10}\mathrm{s}+3.048\times {10}^{11}}{{\mathrm{s}}^5+236.4{\mathrm{s}}^4+1.001\times {10}^6{\mathrm{s}}^3+1.615\times {10}^8{\mathrm{s}}^2+8.543\times {10}^9\mathrm{s}+1.478\times {10}^{11}}$$

(37)

$${\mathrm{g}}_{22}\left(\mathrm{s}\right)=\frac{\mathrm{-378.6}{\mathrm{s}}^4-1.002\times {10}^6{\mathrm{s}}^3-5.427\times {10}^8{\mathrm{s}}^2-7.999\times {10}^{10}\mathrm{s}+3.536\times {10}^{12}}{{\mathrm{s}}^5+345.5{\mathrm{s}}^4+1.025\times {10}^6{\mathrm{s}}^3+2.703\times {10}^8{\mathrm{s}}^2+2.417\times {10}^{10}\mathrm{s}+7.156\times {10}^{11}}$$

(38)

Thus, it is possible to determine the matrix static decoupler D by using (27), the computed value obtained of D was as follows:

$$\mathrm{D}=\frac{1}{\mathrm{-51.012}}\left(\begin{array}{cc}\mathrm{-4.941}& \mathrm{-5.406}\\ \mathrm{-2.062}& 8.068\end{array}\right)$$

$$\mathrm{D}=\left(\begin{array}{cc}0.097& 0.106\\ 0.040& \mathrm{-0.156}\end{array}\right)$$

(39)

The transfer function of the decoupled system can be obtained considering Equation (28) to Equation (31); hence, to design the decentralized PI controllers, we have to consider the diagonal terms of Q(s):

$${\mathrm{q}}_{11}\left(\mathrm{s}\right)=\frac{9.032\times {10}^5\mathrm{s}+7.156\times {10}^7}{{\mathrm{s}}^3+145.5{\mathrm{s}}^2+9.862\times {10}^5\mathrm{s}+7.156\times {10}^7}$$

(40)

$${\mathrm{q}}_{22}\left(\mathrm{s}\right)=\frac{78.12{\mathrm{s}}^3+8.655\times {10}^4{\mathrm{s}}^2+7.533\times {10}^7\mathrm{s}+3.252\times {10}^9}{{{\mathrm{s}}^4+190.9\mathrm{s}}^3+9.928\times {10}^5{\mathrm{s}}^2+1.164\times {10}^8\mathrm{s}+3.253\times {10}^9}$$

(41)

Therefore, for each transfer function ${\mathrm{q}}_{11}(\mathrm{s})$ and ${\mathrm{q}}_{22}(\mathrm{s})$, the root-locus design method for the PI controllers was used with the chosen requirements: settling time less than 0.5 s and damping factor greater than 0.2 (these nominal specifications give reasonable performance for DC-DC power converters [47]), and the corresponding parameters were obtained as follows:

$${\mathrm{k}}_{\mathrm{P}1}=0.87,{\mathrm{k}}_{\mathrm{I}1}=30.0,{\mathrm{k}}_{\mathrm{P}2}=14.5,{\mathrm{k}}_{\mathrm{I}2}=500.0$$

(42)

Thus, the corresponding transfer functions of the controllers PI₁ and PI₂ have the structures of Equations (25) and (26). Thus, the corresponding state-space representation is as follows:

$$\left[\begin{array}{c}{\mathrm{U}}_1\left(\mathrm{s}\right)\\ {\mathrm{U}}_2\left(\mathrm{s}\right)\end{array}\right]=\left[\begin{array}{cccc}-{\mathrm{d}}_{11}{\mathrm{k}}_{\mathrm{P}1}& {\mathrm{d}}_{11}{\mathrm{k}}_{\mathrm{I}1}& -{\mathrm{d}}_{12}{\mathrm{k}}_{\mathrm{P}2}& {\mathrm{d}}_{12}{\mathrm{k}}_{\mathrm{I}2}\\ -{\mathrm{d}}_{21}{\mathrm{k}}_{\mathrm{P}1}& {\mathrm{d}}_{21}{\mathrm{k}}_{\mathrm{I}1}& -{\mathrm{d}}_{22}{\mathrm{k}}_{\mathrm{P}2}& {\mathrm{d}}_{22}{\mathrm{k}}_{\mathrm{I}2}\end{array}\right]\left[\begin{array}{c}{\mathrm{Y}}_1\left(\mathrm{s}\right)\\ {\mathrm{X}}_{\mathrm{I}1}\left(\mathrm{s}\right)\\ {\mathrm{Y}}_2\left(\mathrm{s}\right)\\ {\mathrm{X}}_{\mathrm{I}2}\left(\mathrm{s}\right)\end{array}\right]$$

(43)

where

$${\mathrm{X}}_{\mathrm{I}1}\left(\mathrm{s}\right)=\frac{{\mathrm{E}}_1\left(\mathrm{s}\right)}{\mathrm{s}},⟶ {\mathrm{X}}_{\mathrm{I}2}\left(\mathrm{s}\right)=\frac{{\mathrm{E}}_2\left(\mathrm{s}\right)}{\mathrm{s}}, {\mathrm{Y}}_1\left(\mathrm{s}\right)={\mathrm{V}}_1\left(\mathrm{s}\right),{\mathrm{Y}}_2\left(\mathrm{s}\right)={\mathrm{V}}_2\left(\mathrm{s}\right)$$

(44)

Thereby, the control law equation can be represented like

$$\mathrm{U}\left(\mathrm{s}\right)=\mathrm{k}\mathrm{C}\mathrm{X}\left(\mathrm{s}\right)+{\mathrm{k}}_{\mathrm{I}}\frac{\mathrm{E}\left(\mathrm{s}\right)}{\mathrm{s}}$$

(45)

where

$$\mathrm{k}=\left[\begin{array}{cc}-{\mathrm{d}}_{11}{\mathrm{k}}_{\mathrm{P}1}& -{\mathrm{d}}_{12}{\mathrm{k}}_{\mathrm{P}2}\\ -{\mathrm{d}}_{21}{\mathrm{k}}_{\mathrm{P}1}& -{\mathrm{d}}_{22}{\mathrm{k}}_{\mathrm{P}2}\end{array}\right],{\mathrm{k}}_{\mathrm{I}}=\left[\begin{array}{cc}{\mathrm{d}}_{11}{\mathrm{k}}_{\mathrm{I}1}& {\mathrm{d}}_{12}{\mathrm{k}}_{\mathrm{I}2}\\ {\mathrm{d}}_{21}{\mathrm{k}}_{\mathrm{I}2}& {\mathrm{d}}_{22}{\mathrm{k}}_{\mathrm{I}2}\end{array}\right]$$

(46)

4. Experimental Methodology and Tests

4.1. Description of the SIMO Converter System Test Board

To allow the experimental performance and robustness evaluation of the proposed control methods, namely LQR and Decoupled PI (see Figure 1), a single inductor dual-output DC-DC buck converter board test system was developed. This test board (together with the measuring equipment) is shown in Figure 5. This test system was designed according to the specifications in Table 1. It allows two different levels of regulated DC voltage and allows for variations in the input voltage and load converter outputs. A microcomputer system that communicates via USB is used to set the desired set point and to monitor and acquire converter-relevant signals (i.e., voltages, inductor current, and control efforts). Note that the special design of the power converter allows the circuit parameter values to be easily changed to emulate parametric uncertainties as well as changing operating conditions.

The DC-DC SIMO converter is represented by its multivariable two-input two-output transfer function G(s), while the controllers are represented by C(z)_LQR and C(z)_PI. The transfer functions H₁(s), H₂(s), and H₃(s) represent the dynamic effects of the measurements and conditioning of the output voltage signals V₁ and V₂ and the inductor current I_L.

The single inductor multiple output (SIMO) test system (Figure 5) implements the control laws, LQR, and Decoupled PI by using vector and matrix structures in a 32-bit ARM core microcontroller AT91SAM3X8E ARDUINO UNO, with the programming performed by a C compiler.

In addition, two other subsystems have been developed for the use of parametric variations. The first allows performing variations in the input voltage value, while the second allows us to change the load conditions of the converter using parallel addition (or deletion) resistances to the converter output, or by adding a CPL with variations in its power consumption. In addition, the microcomputer system uses an application that allows the real-time visualization of data and the recording of data for posterior analysis.

4.2. Description of Experiments

Tree experiments are performed to evaluate the performance and robustness of the real closed-loop SIMO converter system. The specific uncertainty family is defined in Table 1; therefore, to execute the experiments and the analysis of the results, only the worst case of the voltage level uncertainty and load variations were considered, which has a more significant impact on the dynamic response presented by all controllers.

The first experiment is designed to evaluate the controller performance for variations in the value of converter input voltage, V_in. With the system operating in a steady state at the operating point provided in Table 1, a step variation was applied in V_in, from 7 V to 6 V.

Remark 1.

Note that it is not advisable to reduce the input voltage of the SIMO converter to a greater extent as it will reduce the inductor current, which would then change the operational conduction mode of the converter from the continuous to the discontinuous conduction mode.

The second experiment evaluates the performance of the controller for a single variation in the load resistance R₁ and R₂ after the steady state is achieved at the nominal operation point, from R = 10 Ω to R = 5 Ω in each case.

Remark 2.

Note that the rig system’s physical limits are limited by the maximum operation of the voltage source power, while the minimum operation is limited by the current level that maintains the operational conduction mode of the SIMO converter.

The third experiment evaluates the closed-loop performance when the CPLs undergo negative power variations in parallel with R₁ and R₂. When the SIMO converter system reaches its stable state, CPL₁ is connected in parallel with R₁ and after a few seconds a negative power change is given to CPL₁ (CPL₂ is not yet connected). The same procedure is adopted for the output voltage at R₂. The exact values of the CPL variations are described in Section 5.3.

The aim of these experiments is to show that the proposed controllers can compensate for the perturbations when the input voltage and load change, thereby maintaining the desired performance in the uncertainty region and at different operation points without increasing the measurement noise. The integral square error (ISE) and the integral square control (ISU) indices are used to evaluate the performance of the control strategies.

5. Experimental Results and Discussion

5.1. Input Voltage Variation

With the real SIMO converter test system (see Figure 4), operating in a steady state around the operating point ($V_1^0=3.7 \mathrm{V}, V_2^0=2.7 \mathrm{V}, d_1^0=0.7, d_2^0=0.7)$, a negative step variation was applied to the value of the external converter input voltage, V_in, from 7.0 to 6.0 Volts, at the time instant t = 69.0. Figure 6 presents the acquired signals of output voltages, V₁ and V₂, as well as the corresponding control efforts, d₁ and d₂, provided by LQR and Decoupled PI controllers, respectively. As can be observed, both designed controllers, LQR and Decoupled PI, were able to ensure the system’s stability. However, only the LQR strategy has succeeded in compensating for the variation in V_in by adequately regulating both outputs, V₁ and V₂, for their respective pre-disturbance values, without excessive control effort or saturation in both duty cycle signals d₁ and d₂, showing a good performance obtained by the LQR strategy. In contrast, Figure 6 also shows that the experimental results obtained for the Decoupled PI strategy at the same test were not satisfactory. Although the PI strategy was able to provide a stable closed-loop system, it failed to adequately regulate the voltage outputs V₁ and V₂ to their respective pre-disturbance values. The PI multiloop methodology may cause this situation because it is not suitable for the design of controllers that take into account the uncertainty of the parameters, where the operating point may change, so this method cannot ensure the performance of the system, and the system can present a steady state error. In addition, the SIMO system presents several nonlinearities, and the two outputs are highly coupled [20]; this situation can intensify the controller´s difficulty in adequately regulating the outputs when parametric variations are presented. Further research is needed on this particular point.

Table 2. Values of the ISE and ISU indices of the experimental data collected when the system submitted a parametric variation in input voltage.

Test	OUTPUT	ISE		ISU		J = ISE + ISU
		LQR	PI	LQR	PI	LQR	PI
Vin	V1	0.00039	0.0368	0.5549	0.5227	0.5553	0.5595
	V2	0.0028	0.0355	0.4201	0.3211	0.4229	0.3566

shows the comparison of ISE and ISU performance indices for the single inductor dual-output converter test system between the two approaches.

Those evaluate the impact of voltage variation on the controller performance. In the case of the output V₁, it is possible to notice that the ISE index value of the LQR strategy is almost one hundred times smaller in comparison with the Decoupled PI index value. Furthermore, the ISU index values show that the improvement in the voltage regulation happened with less than a 10 percent increase in the LQR ISU index in comparison with the PI ISU index. In the case of the output V₂, it is possible to notice that the ISE index value of the LQR strategy is about ten times smaller in comparison with the Decoupled PI index value. Moreover, the ISU index values show that the improvement in the voltage regulation happened with less than a 30 percent increase in the LQR ISU index in comparison with the PI ISU index. Therefore, the preview analysis supports improving the robustness of the proposed LQR control.

5.2. Load Variation

The load variation test is performed by changing the load at each converter output, keeping the value V_in = 7 V. The load resistance value at each converter output is gradually reduced by adding resistors in parallel with it. As a result, the corresponding output current value changes. Figure 7 shows the closed-loop response of voltages V₁ and V₂ using LQR and Decoupled PI controllers, and the corresponding control efforts, d₁ and d₂. The system starts with a load resistance of R₁ = 10 Ω. At t = 90 s, a 10 Ω resistor is inserted in parallel, reducing the load resistance value to 5 Ω. It can be seen that only the controller method using the LQR strategy can effectively regulate the two outputs V₁ and V₂ to reach their respective pre-disturbance values without excessive control effort or the saturation of the duty cycle signals d₁ and d₂ and has a fast response. Furthermore, the Decoupled PI controller causes steady-state errors mainly at V₁, and it can also be seen that V₁ is more affected because of its sensitivity to loop interactions.

Similar results were obtained for the load variations in output 2 of the LQR strategy and the Decoupled PI control structure, as shown in Figure 8. However, the controller designed with the LQR strategy correctly compensated the oscillations due to load variation in output 2 of the single inductor dual-output converter. The results for outputs 1 and 2 were collected simultaneously for the same load variation, i.e., adding a 10 Ω resistor in parallel with the resistor R₂ = 10 Ω. The LQR strategy is capable of quickly compensating for the load variation, nearly eliminating steady-state error in V₁ and V₂ without saturating the control signals. On the other hand, the Decoupled PI controller method cannot compensate for the load variations at outputs 1 and 2, resulting in significant performance degradation, which is mainly reflected in output 2’s high steady-state error value.

Table 3. Values of the ISE and ISU indices of the experimental data collected when the system submitted a parametric variation in load R₁.

Test	OUTPUT	ISE		ISU		J = ISE + ISU
		LQR	PI	LQR	PI	LQR	PI
R1	V1	0.0036	0.8938	0.7614	0.7779	0.7649	1.672
	V2	0.0011	0.0042	0.8099	0.3336	0.8111	0.3378

shows the ISE and ISU indices for each output due to the load changes at output 1 for the LQR strategy and the Decoupled PI control structure. Compared to the Decoupled PI structure values, the ISE of the proposed LQR strategy is lower for both outputs. However, it can be seen that the ISU index value of the LQR case is much higher than in the Decoupled PI case, especially for output 2. Also,

Table 4. Values of the ISE and ISU indices of the experimental data collected when the system submitted a parametric variation in load R₂.

Test	OUTPUT	ISE		ISU		J = ISE + ISU
		LQR	PI	LQR	PI	LQR	PI
R2	V1	0.0637	0.0638	0.7023	0.7360	0.7660	0.7998
	V2	0.0443	0.1881	0.7758	0.3343	0.8201	0.5223

shows the same index values for variations in output 2 for the LQR strategy and Decoupled PI control structure. The proposed LQR strategy offers lower ISE values compared to the Decoupled PI values. These results show that the proposed LQR strategy provides significant performance improvements and justifies its application in single inductor dual-output converter systems.

5.3. CPL Power Variation

The CPL power variation test is performed by placing the CPL in parallel with the load resistor and varying its power value at each output of the converter. The load resistance value at each converter output is fixed at 10 Ω. By using a programmable DC electronic load in parallel with each resistor, the CPL power value at each output of the converter is gradually increased. As a result, the corresponding output current value is changed. Figure 9 shows the closed-loop response of the V₁ voltage using the LQR strategy and the Decoupled PI controller. The system starts with CPL power value CPL₁ = 0.5 W. At t = 43.6 s, the power value of CPL₁ becomes 0.7 W. It can be observed that when CPL₁ is connected, all the control efforts of loop 1 remain saturated, while the control effort of the Decoupled PI controller of loop 2 drops to a very low value. Although these control methods cannot compensate for the output errors caused by the CPL₁ perturbation (only the Decoupled PI controller can mitigate the steady-state error of V₂), they can ensure system stability and fast response.

Similar results are also obtained for the CPL₂ variation at output 2 for the LQR strategy and Decoupled PI control structure, as shown in Figure 10; nevertheless, the Decoupled PI controller can compensate for the oscillations in output 1 due to the power variation in the CPL₂. The system starts with a CPL power value of CPL₂ = 0.25 W, and then at t = 54.8 s, the power value of the CPL₂ changes to 0.35 W. It can be observed that the control efforts of loop 2 stay saturated when the CPL₂ is connected, and on the other hand, the control effort of the Decoupled PI controller of loop 2 falls to a very low value. Although the control methodologies cannot compensate for the error at the outputs due to the perturbation in CPL₂ (only the Decoupled PI controller can mitigate the steady state error in V₁), they can ensure the stability of the system with a quick response.

Table 5. Values of the ISE and ISU indices of the experimental data collected when the system submitted a parametric variation in the added CPL₁.

Test	OUTPUT	ISE		ISU		J = ISE + ISU
		LQR	PI	LQR	PI	LQR	PI
CPL1	V1	0.0525	0.0803	0.9025	0.8655	0.955	0.9458
	V2	0.0994	0.0553	0.9025	0.0545	1.002	0.1099

shows the ISE indices for each output resulting from the CPL power change at output 1 for the LQR strategy and the Decoupled PI control structure. The ISE of the Decoupled PI controller is low for both outputs. Also,

Table 6. Values of the ISE and ISU indices of the experimental data collected when the system submitted a parametric variation in the added CPL₂.

Test	OUTPUT	ISE		ISU		J = ISE + ISU
		LQR	PI	LQR	PI	LQR	PI
CPL2	V1	0.1246	0.0546	0.9025	0.8796	1.027	0.9342
	V2	0.1500	0.1134	0.9025	0.0331	1.052	0.1465

shows the same indices for power variations in CPL at output 2 using the LQR strategy and the Decoupled PI control structure. The Decoupled PI approach has lower ISE values compared to the LQR strategy. These results show that the proposed Decoupled PI controller has significant performance improvements in this case and demonstrate its application in single inductor dual-output converter systems under such conditions.

6. Conclusions

This paper proposes the use of a robust LQR strategy and a Decoupled PI controller in a single inductor dual-output DC-DC converter system to ensure robust stability and performance over the entire predetermined uncertain region, thereby reducing oscillations effects caused by the connection and variation in a constant power load, in parallel with a resistor of each converter output.

The proposed controllers have been exhaustively evaluated in physical experiments performed on a DC-DC SIMO converter board. The performances of the proposed methods, the LQR strategy and the Decoupled PI structure, are compared. Based on the experimental results, it is concluded that the proposed LQR strategy is able to maintain the required performance and reduce the steady-state errors V₁ and V₂ without excessive control effort or the saturation of the duty cycle signals d₁ and d₂, which indicates good performance. On the other hand, when a CPL is connected to the outputs of the system, and it is subjected to a certain variation in the power of each CPL, both strategies are able to maintain the system stable; however, none of the controllers can mitigate the steady-state errors in V₁ and V₂ under these conditions. Thus, the results indicate that the proposed LQR strategy approach is reasonable and provides adequate performance improvements in the SIMO converter controls, offering robust performance and stability, while more research is needed to handle constant power loads (CPLs). The authors are currently investigating strategies to deal with CPL, and results will be reported in future papers.